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Well posedness and asymptotic analysis for the stochastic equations of geophysical fluid dynamics

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Well posedness and asymptotic analysis for the stochastic equations of geophysical fluid dynamics

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Content WELL POSEDNESS AND ASYMPTOTIC ANALYSIS FOR THE STOCHASTIC EQUATIONS OF GEOPHYSICAL FLUID DYNAMICS by Nathan Edward Glatt-Holtz A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (APPLIED MATHEMATICS) August 2008 Copyright 2008 Nathan Edward Glatt-Holtz Dedication To my mother, Diane Glatt, in loving memory. ii Acknowledgments I would like to express my gratitude to the many people who made my years at USC a pleasure. In particular I would like to mention Peter Baxendale, Simon Guest, Igor Kukavica, John Mayberry, Remigijus Mikulevicius, Paul Newton, Gary Rosen, and VladVicol. Iamespeciallyindebtedtomythesisadvisor,MohammedZiane,whohas been a thoughtful and caring mentor as well as a tireless advocate and friend. Finally and most importantly I would like to thank my mother Diane Glatt, my fatherDavidHoltz,andtheloveofmylife(andnowfianc` ee!),PamelaMangan. Their constant and unconditional support has sustained and encourged me through all the ups and downs of the long process of earning a doctorate in mathematics. iii TableofContents Dedication ii Acknowledgments iii Abstract vi Chapter 1: A Model for The Primitive Equations in Two Space Dimensions with Multiplicative Noise 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 A Model for the Stochastic Primitive Equations in Two Space Dimensions . 4 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 The Galerkin Systems and A Priori Estimates . . . . . . . . . . . . . . . . 15 Existence and Uniqueness of Solutions . . . . . . . . . . . . . . . . . . . . 24 Appendix: Physical Background . . . . . . . . . . . . . . . . . . . . . . . 37 Fundamental Equations . . . . . . . . . . . . . . . . . . . . . . . . . 39 The Boussinesq Approximation . . . . . . . . . . . . . . . . . . . . 40 Scale Analysis and The Hydrostatic Approximation . . . . . . . . . . 41 Eddy Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Temperature and Salinity . . . . . . . . . . . . . . . . . . . . . . . . 42 The Equation of State . . . . . . . . . . . . . . . . . . . . . . . . . . 43 Other Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . 44 Chapter 2: Strong Pathwise Solutions of the Stochastic Navier-Stokes System 45 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 The Abstract Functional Analytic Setting . . . . . . . . . . . . . . . . . . . 47 The Galerkin Scheme and Comparison Estimates . . . . . . . . . . . . . . 59 Existence and Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 Existence of Local Strong Solutions . . . . . . . . . . . . . . . . . . 70 Maximal Existence Time and Blow-Up . . . . . . . . . . . . . . . . 77 Global Existence for Dimension Two . . . . . . . . . . . . . . . . . . 82 Abstract Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 A Pairwise Comparison Theorem . . . . . . . . . . . . . . . . . . . . 85 Weak Convergence Lemmas . . . . . . . . . . . . . . . . . . . . . . 90 A Gr¨ onwall Lemma For Stochastic Processes . . . . . . . . . . . . . 92 Chapter 3: Singular Perturbation Systems with Small Stochastic Forcing and the Renormalization Group Method 95 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 iv A Derivation of the Renormalization Group . . . . . . . . . . . . . . . . . 99 Resonance Analysis for the Nonlinear Term . . . . . . . . . . . . . . 102 The Antisymmetric Case: Improved Convergence Rates . . . . . . . . 106 ThePositiveSemidefiniteCase: ImprovedConvergenceRatesandthe Case of Large Noise . . . . . . . . . . . . . . . . . . . . . . 108 Rigorous Approximation Results . . . . . . . . . . . . . . . . . . . . . . . 110 Applications: Some Examples from Fluid Dynamics . . . . . . . . . . . . 122 Large Time Estimates for Slowly Varying Processes . . . . . . . . . . . . . 125 Appendix I: Stochastic Convolution Estimates . . . . . . . . . . . . . . . . 129 Appendix II: Small Noise Asymptotic Results . . . . . . . . . . . . . . . . 133 References 142 v Abstract Thisworkcollectsthreeinterrelatedprojectsthatdeveloprigorousmathematicaltools for the study of the stochastically forced equations of geophysical fluid dynamics and turbulence. Since the presence of persistent random fluctuations are ubiquitous in fluid flow problems, the development of new analytical techniques in this setting are ultimately of paramount practical interest in the pursuit of more accurate models. The first two projects address the question of well posedness. One work consid- ers the Primitive equations of the ocean. These systems are widely used in numerical studies of geophysical scale fluid flow. We consider a two dimensional model forced by multiplicative noise terms. For the case when the initial data and its vertical gra- dient take values in L 2 , we prove the global existence and uniqueness of solutions in a fixed probability space. The proof is based on finite dimensional approximations, anisotropicSobolevestimates,andweakconvergencetechniques. Theworkconcludes with a sketch of the physical derivation of the governing equations. The second project introduces new results for the stochastic Navier-Stokes sys- tem on bounded domains in space dimensions two and three. We establish the local existence and uniqueness of pathwise solutions evolving in H 1 . For the two dimen- sional case global existence is established without employing moment estimates at fixedtimes. Theproofreliesonapairwisecomparisontechniquefacilitatedbydecom- positions into high and low modes to surmount the critical issue of compactness. In the final chapter we introduce a class of singular perturbation systems in the presence of a small white noise. Modifying the renormalization group procedure developed by Chen, Goldenfeld and Oono, we derive an associated reduced system which we use to construct an approximate solution that separates scales. Rigorous vi results demonstrating that these approximate solutions remain valid with high prob- ability on large time intervals are established. As a special case we infer new small noiseasymptoticresultsforaclassofprocessesexhibitingaphysicallymotivatedcan- cellation property in the nonlinear term. We next consider some concrete perturbation systems arising in geophysical fluid dynamics and in the study of turbulence. For each system we exhibit the associated renormalization group which is seen to decou- ple the interactions between the different scales inherent in the original system. The articleconcludeswithageneralresultconcerningslowlyvaryingprocessesthatforms the backbone of the proof of the main theorem. Appendixes collect mostly classical results on the stochastic convolution and further novel small noise asymptotic results. vii Chapter1 AModelforThePrimitiveEquations inTwoSpaceDimensionswith MultiplicativeNoise 1.1 Introduction The Primitive equations are a ubiquitous model in the study of geophysical fluid dynamics. They can be derived from the compressible Navier-Stokes equations by taking advantage of various properties common to geophysical flows. In particular, oneusestheBoussinesqapproximation,thatfluctuationsinthedensityofthefluidare much smaller than the mean density throughout. A scale analysis relying on the rela- tive shallowness of the ocean at geophysical scales is employed to show that the pres- sure and the gravitational forces are the only relevant terms in the vertical momentum equation. This isreferred to as the hydrostatic approximation. For furtherbackground and detailed physical derivations see [14] or [42], for example. ThemathematicalstudyofthePrimitiveequationswasinitiatedintheearly1990’s with the work of Lions, Temam and Wang [32], [31], [30]. In these initial works the existence of global weak solutions and weak attractors were established and numer- ical schemes were developed. The case of local strong solutions was addressed by Guill´ en-Gonz ´ alez, Masmoudi, and Rodr´ ıguez-Bellido [24] and by Hu, Temam and 1 Ziane[25]. AlsoseethesurveyarticleofTemamandZiane[50]. Recentbreakthroughs have yielded global existence of strong solutions in three space dimensions. The case ofNeumannboundaryconditionswasaddressedbyCaoandTiti[8]andlaterindepen- dently by Kobelkov [28]. In subsequent work of Kukavica and Ziane [29] a different proof was discovered which covers the case of physically relevant boundary condi- tions. For the two dimensional deterministic setting we mention the work of Petcu, Temam,andWirosoetisnoin[43]andBresch,KazhikhovandLemoine[6]whereboth the cases of weak and strong solutions are considered. The 2-D primitive equations seem to be more difficult mathematically than the 2-D Navier-Stokes equations. For instance, it is still an open problem whether weak solutions of the Primitive equations in the deterministic setting are unique. This is an easy exercise for the 2-D Navier Stokes equations. The addition of white noise driven terms to the basic governing equations for a physicalsystemisnaturalforbothpracticalandtheoreticalapplications. Forexample, these stochastically forced terms can be used to account for numerical and empiri- cal uncertainties and thus provide a means to study the robustness of a basic model. Specificallyinthecontextoffluids,complexphenomenarelatedtoturbulencemayalso beproducedbystochasticperturbations. Forinstance,intherecentworkofMikulevi- cius and Rozovsky [36] such terms are shown to arise from basic physical principals. A wide body of mathematical literature exists for the stochastic Navier-Stokes equations. This analytic program dates back to the early 1970’s with the work of Bensoussan and Temam [2]. For the study of well-posedness new difficulties related to compactness often arise due to the addition of a probabilistic parameter. For situa- tionswherecontinuousdependenceoninitialdataremainsopen(forexampleind = 3 2 when the initial data merely takes values inL 2 ) it has proven fruitful to consider Mar- tingale solutions. Here one constructs a probabilistic basis as part of the solution. For this context we refer the reader to the works of Cruzeiro [13], Capinski and Gatarek [9], Flandoli and Gatarek [21] and of Mikulevicius and Rozovskii [38]. On the other hand, when working in spaces where continuous dependence on the initial data can be expected, existence of solutions can sometimes be established on a preordained probability space. Such solutions are often referred to as “strong” or “pathwise” solutions. In the two dimensional setting, Da Prato and Zabczyk [16] and later Breckner [5] as well as Menaldi and Sritharan [34] established the existence of pathwisesolutionswhereu∈L ∞ ([0,T],L 2 ),P−a.s.Ontheotherhand,Bensoussan and Frehse [1] have established local solutions in 3-d for the class C β ([0,T];H 2s ) where 3/4 < s < 1 and β < 1−s. In the works of Mikulevicius and Rozovsky [37] and of Brzezniak and Peszat [7] the case of arbitrary space dimensions for local solutions evolving in Sobolev spaces of typeW 1,p forp>d is addressed. It is with this background in mind that we present the following examination of the two dimensional Primitive equations in the presence of multiplicative white noise terms on a preordained probability space. In the first section we introduce the model, providing an overview of the relevant function spaces and establish some anisotropic Sobolev type estimates on the nonlinear terms of the equation. A variational defini- tion for solutions is then presented. We next turn to the Galerkin scheme. Since the best estimates for the nonlinear terms are closer to those currently available for the three dimensional Navier Stokes equations we need to make use of special cancella- tion properties available for thez direction. In this way we are able to infer uniform bounds for ∂ z u (n) in L p (Ω;L 2 (0,T;V)∩L ∞ (0,T;H)) along with those typical for u (n) . To establish existence we apply weak convergence methods to identify a limit 3 system. A comparison technique is then employed taking advantage of the additional regularityestablishedforu (n) . Thisallowsustoidentifycertainpoint-wiselimitsfrom whichweconcludethatthesystemisindeedthedesiredequation. Uniquenessisestab- lished to conclude this section. We conclude the work with an appendix that outlines the physical derivation of the Primitive equations. 1.2 A Model for the Stochastic Primitive Equations in TwoSpaceDimensions The two dimensional Primitive equations can be formally derived from the full three dimensional system under the assumption of invariance with respect to the second horizontal variable y. As such we will assume that the initial data and the external forcing are independent of y. By adding a second external forcing term driven by a white noise, we arrive at the following non-linear stochastic evolution system: ∂ t u−νΔu+u∂ x u+w∂ z u+∂ x p =f +g(u,t) ˙ W(t) (1.1a) ∂ x u+∂ z w = 0. (1.1b) In this formulation we have ignored the coupling with the temperature and salinity equationsinordertofocusourattentiontowardsdifficultiesarisingfromthenonlinear term (see [6]). This omission will be remedied in future work. The unknowns (u,w), p represent the field of the flow and the pressure respectively. The fluid fills a domain M = [0,L]×[0,−h]3 (x,z). Note thatp does not depend on the vertical variablez. We partition the boundary into the top Γ i ={z = 0}, the bottom Γ b ={z =−h} and the sides Γ s = {x = 0}∪{x = L}. Regarding the boundary conditions, we 4 assumetheDirichletconditionu = 0onΓ s ,whileonΓ i ∪Γ b wepositthefreeboundary condition∂ z u = 0,w = 0. We further suppose with no loss of generality that: Z 0 −h fdz = 0, Z 0 −h gdz = 0, Z 0 −h udz = 0. (1.2) Due to (1.1b) we have that 1 : w(x,z) =− Z z −h ∂ x u(x,˜ z)d˜ z. (1.3) The stochastic term can be written in the expansion: g(u,t) ˙ W(t) = X k g k (u,t) ˙ β k (t). (1.4) The ˙ β k are the formal time derivatives of a collection of independent standard Brow- nian motionsβ k relative to some ambient, filtered, right continuous, probability space (Ω,F,(F t ) t≥0 ,P). Theseriesconvergesintheappropriatefunctionspacesandissub- ject to uniform Lipschitz conditions in u. All of this will be formulated rigorously below. 1.3 Definitions We will be working on the Hilbert spaces: H = v∈L 2 (M) : Z 0 −h vdz = 0 (1.5) 1 In the geophysical literaturew is often referred as a diagnostic variable as its value is completely determined byu 5 and V = v∈H 1 (M) : Z 0 −h vdz = 0,v = 0 onΓ s . (1.6) ThesespacesareendowedwiththeL 2 andH 1 normswhichwerespectivelydenoteby |·| andk·k. We shall also need the intermediate space: X ={v∈H :∂ z v∈H} (1.7) with the norm|u| X = (|u| 2 +|∂ z u| 2 ) 1/2 . TakeV 0 to be the dual ofV with the pairing notated byh·,·i. The Leray operatorP H is the orthogonal projection ofL 2 (M) onto H. The action of this operator is given explicitly by: P H v =v− 1 h Z 0 −h vdz. (1.8) It will also be useful to consider: V = v∈C ∞ ( ¯ M) : Z 0 −h vdz = 0,v = 0 onΓ s ,∂ z v = 0 onΓ b ∪Γ i (1.9) which is a dense subset ofH,X andV. We would now like to make (1.1) precise as an equation in V 0 . To this end, we define a Stokes-type operator A as a bounded map fromV toV 0 via: hv,Aui = ((v,u)). 6 AcanbeextendedtoanunboundedoperatorfromH toH accordingtoAu =−P H Δu with the domain: D(A) = v∈H 2 (M) : Z 0 −h vdz = 0,v = 0 onΓ s ,∂ z v = 0 onΓ b ∪Γ i . (1.10) By applying the theory of symmetric compact operators for A −1 , one can prove the existenceofanorthonormalbasis{e k }forH ofeigenfunctionsofA. Heretheassoci- ated eigenvalues{λ k } form an unbounded, increasing sequence. Define: H n = span{e 1 ,...,e n } and takeP n to be the projection fromH onto this space. LetQ n =I−P n . 1.3.1Remark. Foru∈D(A): Z 0 −h ∂ xx udz =∂ xx Z 0 −h udz = 0, Z 0 −h ∂ zz udz =∂ z u] z=0 z=−h = 0. Thus, as in the case of periodic boundary conditions, we have−P H ∂ xx = −∂ xx and −P H ∂ zz =−∂ zz and therefore that−P H Δ =−Δ onD(A). The eigenvalue problem forA reduces to: Δu =λu Z 0 −h udz = 0 u(0,z) = 0 =u(L,z), ∂ z u(x,−h) = 0 =∂ z u(x,0). 7 As such, the eigenfunctions and associated eigenvalues can be explicitly identified: 2 √ hL sin( k 1 πx L )cos( k 2 πz h ) k 1 ,k 2 ≥1 , π 2 k 2 1 L 2 + k 2 2 h 2 k 1 ,k 2 ≥1 . This has the useful consequence that whenj6=l: h∂ zz e j ,e l i = 0 (1.11) and hence: P n (−∂ zz v) =−∂ zz v wheneverv∈H n . (1.12) Next we address the nonlinear term. In accordance with (1.3) we take: W(v)(x,z) :=− Z z −h ∂ x v(x,˜ z)d˜ z v∈V and let: B(u,v) :=P H (u∂ x v +W(u)∂ z v). Below it will sometimes be convenient to denoteB(u) :=B(u,u). One would like to establishthatB isawelldefinedandcontinuousmappingfromV×V →V 0 according to: hB(u,v),φi =b(u,v,φ) where the associated trilinear form is given by: b(u,v,φ) := Z M (u∂ x vφ−W(u)∂ z vφ)dM :=b 1 (u,v,φ)−b 2 (u,v,φ). This and more is contained in the following lemma: 8 1.3.1Lemma. (i) b is a continuous linear form onV ×V ×V and: |b(u,v,φ)|≤C |u| 1/2 kuk 1/2 kvk|φ| 1/2 kφk 1/2 +|∂ x u||∂ z v||φ| 1/2 kφk 1/2 (1.13) for anyu,v,φ∈V (ii) b satisfies the cancellation propertyb(u,v,v) = 0 (iii) b is also a continuous form onD(A)×D(A)×H (iv) Foru∈D(A) we have the additional cancellation property: hB(u),∂ zz ui = 0 (v) Moreover, for any> 0: |hB(u),∂ xx ui|≤C(|∂ x u| 2 k∂ x uk+|∂ x u| 1/2 k∂ x uk 3/2 |∂ z u| 1/2 k∂ z uk 1/2 ) ≤k∂ x uk 2 +C()(|∂ x u| 4 +|∂ x u| 2 |∂ z u| 2 k∂ z uk 2 ) (1.14) (vi) Ifu,v∈V and∂ z v∈V then: |B(u,v)| V 0 ≤C((|u|+|∂ z u|)|∂ x v|+|u|k∂ z vk+|u||∂ z v| 1/2 k∂ z vk 1/2 ) (1.15) Proof. Fixu,v,φ∈V. The first termb 1 admits the classical 2-D estimate: |b 1 (u,v,φ)|≤C|u| 1/2 kuk 1/2 kvk|φ| 1/2 kφk 1/2 . 9 The second term is estimated anisotropically: |b 2 (u,v,φ)| ≤ Z L 0 sup z∈[−h,0] Z z −h ∂ x ud¯ z Z L 0 |∂ z vφ|dz ! dx ≤C Z L 0 Z 0 −h |∂ x u| 2 dz· Z 0 −h |∂ z v| 2 dz· Z 0 −h |φ| 2 dz 1/2 dx ≤C sup x∈[0,L] Z 0 −h |φ| 2 dz 1/2 Z L 0 Z 0 −h |∂ x u| 2 dz· Z 0 −h |∂ z v| 2 dz 1/2 dx ≤C|∂ x u||∂ z v||φ| 1/2 kφk 1/2 . For the final inequality we make use of the boundary conditions: sup x∈[0,L] Z 0 −h |φ| 2 dz = sup x∈[0,L] Z x 0 ∂ x Z 0 −h φ 2 dzdx ≤ 2|φ|kφk. To establish the cancellation property in (ii): b(u,v,v) =− 1 2 Z M ∂ x uv 2 dM− 1 2 Z M Z z −h ∂ x ud˜ z ∂ z (v 2 )dM =− 1 2 Z M ∂ x uv 2 dM+ 1 2 Z M ∂ x uv 2 dM = 0. Property(iii)isadirectapplicationofH¨ older’sInequalityandSobolevembedding inequalities and is omitted. For (iv), noting that−P H ∂ zz =−∂ zz onD(A): hB(u),∂ zz ui = Z M (−∂ x u(∂ z u) 2 −u∂ xz u∂ z u+ 1 2 ∂ x u(∂ z u) 2 )dM = 0. 10 The inequality given in (v) is addressed by estimating: |hB(u),∂ xx ui|≤ 1 2 |∂ x u| 3 L 3 (M) + Z M (W(u)∂ zx u∂ x u+∂ x W(u)∂ z u∂ x u)dM ≤|∂ x u| 3 L 3 (M) + Z M |∂ x W(u)∂ z u∂ x u|dM. For the first term above we use the Sobolev embeddingH 1/3 ⊂ L 3 . For the second term we haveH 1/2 ⊂L 4 which justifies the estimate: Z M |∂ x W(u)∂ z u∂ x u|dM≤C|∂ xx u||∂ z u| L 4|∂ x u| L 4 ≤C|∂ x u| 1/2 k∂ x uk 3/2 |∂ z u| 1/2 k∂ z uk 1/2 . The second inequality is just an application of-Young. For the final item (vi) fixφ∈V. In this case we estimateb 1 anisotropically: b 1 (u,v,φ) = Z L 0 Z 0 −h u∂ x vφdM ≤C Z L 0 Z 0 −h (|∂ z u|+|u|) 2 dz· Z 0 −h |∂ x v| 2 dz· Z 0 −h |φ| 2 dz 1/2 dx ≤C(|∂ z u|+|u|)|∂ x v||φ| 1/2 kφk 1/2 . (1.16) Forb 2 , by integrating by parts in,x we find: b 2 (u,v,φ) = Z L 0 Z 0 −h Z z −h ∂ x ud¯ z ∂ z vφdM =− Z L 0 Z 0 −h Z z −h ud¯ z ∂ zx vφdM − Z L 0 Z 0 −h Z z −h ud¯ z ∂ z v∂ x φdM :=T 1 (u,v,φ)+T 2 (u,v,φ). (1.17) 11 ForT 1 the anisotropic estimates yield: |T 1 (u,v,φ)|≤C Z L 0 Z 0 −h |u| 2 dz· Z 0 −h |∂ zx v| 2 dz· Z 0 −h |φ| 2 dz 1/2 dx ≤|u|k∂ z vk|φ| 1/2 kφk 1/2 . (1.18) TheestimateforT 2 issimilarexceptthatwemaketheL ∞ x estimateonthemiddleterm: |T 2 (u,v,φ)|≤C Z L 0 Z 0 −h |u| 2 dz· Z 0 −h |∂ z v| 2 dz· Z 0 −h |∂ x φ| 2 dz 1/2 dx ≤C|u||∂ z v| 1/2 k∂ z vk 1/2 kφk. (1.19) It remains to examine the stochastically forced termg ={g k } k≥1 in order to make precise the Lipschitz condition alluded to above. For this purpose we introduce some notation. SupposeU is any (separable) Hilbert space. One defines` 2 (U) via the inner product: (h,g) ` 2 (U) = X k (h k ,g k ) U . ForanynormedspaceY,wesaythatg :Y×[0,T]×Ω→` 2 (U)isuniformlyLipschitz with constantK Y if: |g(x,t,ω)−g(y,t,ω)| ` 2 (U) ≤K Y |x−y| Y forx,y∈Y (1.20) and |g(x,t,ω)| ` 2 (U) ≤K Y (1+|x| Y ) (1.21) where K Y is independent of t and ω. We denote the collection of all such map- pings Lip u (Y,` 2 (U)). For the analysis below we will often assume that g ∈ 12 Lip u (H,` 2 (H))∩ Lip u (X,` 2 (X)). It is worth noting at this juncture that the con- dition imposed ong is not overly restrictive by considering some examples where the above conditions are satisfied: 1.3.1Example. • (Independently Forced Modes) Suppose (κ k (t,ω)) is any sequence uniformly bounded inL ∞ ([0,T]×Ω). We force the modes independently defining: g k (v,t,ω) =κ k (t,ω)(v,e k )e k . In this case the Lipschitz constants can be taken to be K H = K X = K V = sup ω,k,t |κ k (t,ω)|. • (Uniform Forcing) Given a uniformly square summable sequence a k (t,ω) we can take: g k (v,t,ω) =a k (t,ω)v withK H =K X =K V = (sup t,ω P k a k (t,ω) 2 ) 1/2 as the Lipschitz constants. • (Additive Noise) We can also include the case when the noise term does not depend on the solution: g k (v,t,ω) =g k (t,ω) HeretheuniformconstantscanbetakentobeK U := sup t,ω ( P k |g k (t,ω)| 2 U ) 1/2 forU =H,X,V as desired. With the above framework in place we now give a variational definition for solu- tionsofthesystem(1.1). Notethatweakreferstothespatial-temporalregularityofthe solutions. Strong refers to the fact that the probabilistic basis is given in advance (see Remark 2.2.1 below). 13 1.3.1 Definition (Weak-Strong Solutions) . Suppose that (Ω,P,F,(F t ) t≥0 ,(β k )) is a fixed stochastic basis,T > 0 andp∈ [2,∞]. For the data assume thatu 0 ∈L p (Ω;H) and isF 0 -measurable. Wesuppose that f andg are respectivelyH and` 2 (H)valued, predictable processes with: f ∈L p (Ω,L 2 (0,T;V 0 )), g∈ Lip u (H,` 2 (H)). (1.22) We say that anF t adapted process u is a weak-strong solution to the stochastically forced primitive equation if: u∈C([0,T];H) a.s. u∈L p (Ω;L ∞ ([0,T];H)∩L 2 ([0,T];V)) (1.23) and satisfies: dhu(t),vi+hνAu+B(u),vidt =hf,vidt+ ∞ X k=1 hg k (u,t),vidβ k hu(0),vi =hu 0 ,vi (1.24) for anyv∈V. Several remarks are in order regarding this definition: 1.3.2Remark. • Note that, as in the theory of the Navier-Stokes equations, the pressure disap- pears in the variational formulation. Suppose that∂ x p in (1.1) is integrable and does not depend on the vertical variablez. Then: Z L 0 Z 0 −h ∂ x pvdzdx = Z L 0 ∂ x p Z 0 −h vdz dx = 0 (1.25) 14 for everyv∈V. • For the probabilistically ’strong’ solutions we consider, the stochastic basis is giveninadvance. Suchsolutionscanbeunderstoodpathwise. Thisisincontrast to the theory of Martingale solutions considered for many non linear systems where the underlying probability space is constructed as part of the solution. See [15] chapter 8 or [35]. • One has to check that each of the terms given in (2.44) are well defined. In par- ticular, the stochastic terms deserve special attention. Recall that the collection M 2 (0,T)ofcontinuoussquareintegrablemartingalesisaBanachspaceunder the norm: kXk M 2 = E sup 0≤t≤T |X(t)| 2 1/2 . Applying standard Martingale inequalities and making use of the uniform Lips- chitz assumption one establishes that: X k hg k (u,t),vidβ k for allv∈V converges in this space. See [27] or [15] for further details on the general construction of stochastic integrals. 1.4 TheGalerkinSystemsandAPrioriEstimates We now introduce the Galerkin systems associated to the original equation and estab- lish some uniform a priori estimates. 15 1.4.1Definition (The Galerkin System). An adapted processu (n) inC(0,T;H n ) is a solution to the Galerkin System of Ordern if for anyv∈H n : dhu (n) ,vi+hνAu (n) +B(u (n) ),vidt =hf,vidt+ ∞ X k=1 hg k (u (n) ,t),vidβ k hu (n) (0),vi =hu 0 ,vi. (1.26) These systems also be written as equations inH n : du (n) +(νAu (n) +P n B(u (n) ))dt =P n fdt+ ∞ X k=1 P n g k (u (n) ,t)dβ k u (n) (0) =P n u 0 . (1.27) We note that the second formulation (2.54) allows one to treatu (n) as a process in R n . As such one can apply the finite dimensional Itˆ o calculus to the Galerkin systems above. We next establish some uniform estimates onu (n) (independent ofn). To simplify notation we drop the(n) superscript for the remainder of the section. 1.4.1Lemma (A Priori Estimates). (i) Assume thatu is the solution of the Galerkin System of Ordern. Suppose that p≥ 2 and: g∈ Lip u (H,` 2 (H)) f ∈L p (Ω;L 2 (0,T;V 0 )) u 0 ∈L p (Ω,H) (1.28) then: E sup t∈[0,T] |u| p + Z T 0 kuk 2 |u| p−2 dt ! ≤C W (1.29) 16 and: E Z T 0 kuk 2 dt p/2 ≤C W (1.30) foranappropriateconstantC W =C W (p,ν,λ 1 ,E|u 0 | p ,T,|f| L p (Ω;L 2 (0,T;V 0 )) ,K H ), that does not depend onn. (ii) Given the additional assumptions on the data: g∈ Lip u (X,` 2 (X)) ∂ z f ∈L p (Ω;L 2 (0,T;V 0 )) ∂ z u 0 ∈L p (Ω;H) (1.31) then: E sup t∈[0,T] |∂ z u| p + Z T 0 k∂ z uk 2 |∂ z u| p−2 dt ! ≤C I (1.32) where C I = C I (p,ν,λ 1 ,E|∂ z u 0 | p ,T,|∂ z f| L p (Ω;L 2 (0,T;V 0 )) ,K X ), independent of n. (iii) Finally assume that in addition to (1.28): g∈ Lip u (V,` 2 (V)) f ∈L p (Ω;L 2 (0,T;H)) u 0 ∈L p (Ω,V). (1.33) Ifτ isastoppingtimetakingvaluesin[0,T],andM apositiveconstantsothat: Z τ 0 (kuk 2 +|∂ z u| 2 k∂ z uk 2 )dt≤M (1.34) then: E sup t∈[0,τ] kuk p + Z τ 0 |Au| 2 kuk p−2 dt ! ≤C S e C S M (1.35) whereC S =C S (p,ν,Eku 0 k p ,T,|f| L p (Ω;L 2 (0,T;H)) ,K V ). 17 Proof. By applying Itˆ o’s formula one finds a differential for|u| 2 and then fore φ |u| p : de φ |u| p +pνe φ kuk 2 |u| p−2 dt =pe φ hf,ui|u| p−2 dt+ p 2 e φ ∞ X k=1 |P n g k (u,t)| 2 |u| p−2 dt + p(p−2) 2 e φ ∞ X k=1 hg k (u,t),ui 2 |u| p−4 dt +pe φ ∞ X k=1 hg k (u,t),ui|u| p−2 dβ k +φ 0 |u| p e φ dt. (1.36) Here,φ is a non-positive element in C 1 (0,T) to be determined below. This function will be used to cancel off terms involving R T 0 e φ |u| p dt. For the deterministic external forcing term we estimate: Z T 0 pe φ hf,ui|u| p−2 dt ≤C(ν,p,λ 1 ) Z T 0 e φ |f| 2 V 0|u| p−2 dt+ νp 2 Z T 0 e φ kuk 2 |u| p−2 dt ≤C(ν,p,λ 1 ) sup t∈[0,T] (e (p−2)φ/p |u| p−2 ) ! Z T 0 |f| 2 V 0dt + νp 2 Z T 0 e φ kuk 2 |u| p−2 dt ≤ 1 4 sup t∈[0,T] (e φ |u| p )+ pν 2 Z T 0 e φ kuk 2 |u| p−2 dt +C(ν,p,λ 1 )|f| p L 2 (0,T;V 0 ) . (1.37) 18 Taking advantage of the Lipschitz condition assumed forg: ∞ X k=1 Z T 0 e φ |g k (u,t)| 2 |u| p−2 dt ≤K H Z T 0 e φ (1+|u| 2 )|u| p−2 dt ≤ 1 4 sup t∈[0,T] (e φ |u| p )+C(p,K H ,T) 1+ Z T 0 e φ |u| p dt . (1.38) Applying the above estimates, absorbing terms and rearranging one deduces: sup t∈[0,T] (e φ |u| p )+pν Z T 0 e φ kuk 2 |u| p−2 dt ≤C(p,ν,λ 1 ) |u 0 | p +|f| p L 2 (0,T;V 0 ) +1 +C 1 (p,K H ,T) Z T 0 e φ |u| p dt+2 Z T 0 φ 0 e φ |u| p dt +2p sup t∈[0,T] Z t 0 e φ ∞ X k=1 hg k (u,t),ui|u| p−2 dβ k . (1.39) For the final term involving the Itˆ o integral we apply the Burkholder-Davis-Gundy (BDG) inequality (see [27]). This yields the following: 2pE sup t∈[0,T] ∞ X k=1 Z t 0 e φ hg k (u,s),ui|u| p−2 dβ k ≤C(p)E Z T 0 e 2φ ∞ X k=1 hg k (u,t),ui 2 |u| 2(p−2) dt ! 1/2 ≤C(K H ,p)E Z T 0 e 2φ (1+|u| 2 )|u| 2(p−1) dt 1/2 ≤ 1 4 E sup t∈[0,T] (e φ |u| p ) ! +C 2 (p,T,K H )E 1+ Z T 0 e φ |u| p dt (1.40) 19 Note that the constants in estimates above are independent ofφ. Setφ(t) = −(C 1 + C 2 )t, whereC 1 ,C 2 are the constants arising in (1.39) and (2.124) respectively. This choice, used in conjunction with the preceding estimates implies (1.29). From (2.51) withp = 2 andφ = 0, one deduces that for anyr> 2: E Z T 0 kuk 2 dt r/2 ≤CE 1+|u 0 | r +|f| r L 2 (0,T;V 0 ) + sup t∈[0,T] |u| r ! +CE sup t∈[0,T] Z t 0 ∞ X k=1 hg k (u,t),uidβ k r/2 . (1.41) For the second term we again employ BDG and infer: CE sup t∈[0,T] ∞ X k=1 Z t 0 hg k (u,s),uidβ k r/2 ≤CE Z T 0 ∞ X k=1 hg k (u,t),ui 2 dt ! r/4 ≤CE sup t∈[0,T] |u| r +1 ! + 1 2 E Z T 0 kuk 2 r/2 . (1.42) Absorbing terms and applying estimates already obtained in (1.29) yields (1.30). Toobtainthedesiredestimatesin(ii)wemakeuseofthecommutativityofP n and ∂ zz onD(A) (see Remark 1.3.1). In particular, note that vertical cancellation property established in Lemma 1.3.1 along with this commutativity means that: hP n B(u),∂ zz ui =hB(u),∂ zz ui = 0. 20 Thus, when we apply Itˆ o’s formula for|∂ z v| 2 , the nonlinear term disappears as above. Bootstrapping top≥ 2 with a second application of Itˆ o we arrive at the differential: d|∂ z u| p +pνk∂ z uk 2 |∂ z u| p−2 dt =phf,∂ zz ui|∂ z u| p−2 dt + p 2 ∞ X k=1 |∂ z P n g k (u,t)| 2 |∂ z u| p−2 dt + p(p−2) 2 ∞ X k=1 hg k (u,t),−∂ zz ui 2 |∂ z u| p−4 dt +p ∞ X k=1 hg k (u,t),−∂ zz ui|∂ z u| p−2 dβ k . (1.43) We bound the first term on the right hand side of (1.43) as in (2.122). For the second termweutilizetheLipschitzconditionimposedin(1.31)andtheuniformbound(1.30) established in the previous case: p 2 E Z T 0 ∞ X k=1 |∂ z P n g k (u,t)| 2 |∂ z u| p−2 dt ≤C(p)E Z T 0 ∞ X k=1 |∂ z g k (u,t)| 2 |∂ z u| p−2 dt ≤C(K X ,p)E Z T 0 (kuk 2 +1)|∂ z u| p−2 dt ≤C(K X ,p)E Z T 0 (kuk 2 +1)dt p/2 + 1 8 E sup t∈[0,T] |∂ z u| p ! . (1.44) 21 The third term in (1.43) is estimated in the same manner. The final term is handled using BDG: pE sup t∈[0,T] ∞ X k=1 Z t 0 hg k (u,s),−∂ zz ui|∂ z u| p−2 dβ k ≤C(p)E Z T 0 ∞ X k=1 h∂ z g k (u,t),∂ z ui 2 |∂ z u| 2(p−2) dt ! 1/2 ≤C(p,K X )E Z T 0 (1+|u| 2 X )|∂ z u| 2(p−1) dt 1/2 ≤ 1 8 E sup t∈[0,T] |∂ z u| p ! +C(p,K X )E Z T 0 (1+kuk 2 )dt p/2 . (1.45) Forthe(iii),wedonothavecancellationinthenonlinearterm. Herethedifferential reads: de φ kuk p +pνe φ |Au| 2 kuk p−2 dt =pe φ hf−B(u),Auikuk p−2 dt + p 2 e φ ∞ X k=1 |∇P n g k (u,t)| 2 kuk p−2 dt + p(p−2) 2 e φ ∞ X k=1 (∇g k (u,t),∇u) 2 kuk p−4 dt +pe φ ∞ X k=1 (∇g k (u,t),∇u)kuk p−2 dβ k +φ 0 kuk p e φ dt. (1.46) 22 Once again φ will be a non-positive function chosen further on to cancel off terms. Integratinguptot∧τ,thentakingasupremumintandestimatingtermsaspreviously reveals: E sup t∈[0,τ] (kuk p e φ )+pν Z τ 0 e φ |Au| 2 kuk p−2 dt ! ≤CE(ku 0 k p +|f| p L 2 (0,T;H) ) +2E Z τ 0 e φ |(B(u),∂ xx u)|kuk p−2 dt +C 3 (p,K V )E Z τ 0 e φ kuk p dt+2E Z τ 0 φ 0 e φ kuk p dt. (1.47) We apply Lemma 1.3.1, (v) with =pν/4 inferring: Z τ 0 e φ |(B(u),∂ xx u)|kuk p−2 dt ≤ pν 4 Z τ 0 e φ |Au| 2 kuk p−2 dt +C 4 (ν,p) Z τ 0 e φ kuk p (kuk 2 +|∂ z u| 2 k∂ z uk 2 )dt. (1.48) Taking the previous estimates into account we set: φ(t) =−C 4 Z t 0 (kuk 2 +|∂ z u| 2 k∂ z uk 2 )−tC 3 whereC 3 andC 4 are the constants appearing in (1.47) and (1.48) respectively. Given the assumption (1.34), e φ(τ) ≥ C(ν,p,T)e −CM . With this we can apply (1.48) to (1.47) and conclude the final bound (1.35). 23 1.4.1Remark. Ifonecouldfindasubsequencen k ,astoppingtimeτ withP(τ > 0) = 1 and a positive constantM such that: sup n k Z τ 0 (ku (n k ) k 2 +|∂ z u (n k ) | 2 k∂ z u (n k ) k 2 )ds≤M a.s (1.49) then the existence of solutions taking values in L p (Ω;L 2 ([0,T];D(A)) ∩ L ∞ ([0,T];V)) would follow. We conclude this section with some comments concerning the existence and uniqueness of solutions to the Galerkin systems. Regarding existence one uses thatB is locally Lipschitz in conjunction with the a priori bounds established above. These propertiestakentogetherallowsonetoestablishglobalexistenceonanycompacttime interval via Picard iteration methods. See [20] for detailed proofs. Uniqueness is established as below for the full infinite dimensional system. 1.5 ExistenceandUniquenessofSolutions With the uniform estimates on the solutions of the Galerkin systems in hand, we pro- ceed to identify a (weak) limit u. This element is shown to satisfy a stochastic dif- ferential (1.54) with unknown terms corresponding to the nonlinear portions of the equation. Next we prove a comparison lemma that establishes a sufficient condition (1.67) for the identification of the unknown portions of the differential. This lemma, inconjunctionwithsomefurtherestimates,providesthefinalstepinthemaintheorem concerning existence below. 24 We will assume the following conditions on the data throughout this section: f,∂ z f ∈L p (Ω;L 2 (0,T;V 0 )) g∈ Lip u (H,` 2 (H))∩Lip u (X,` 2 (X)) u 0 ,∂ z u 0 ∈L p (Ω;H). (1.50) Here p ≥ 4 so that the sequence P n B(u (n) ) will have a weakly convergent subse- quence. These assumptions may be weakened slightly for several of the lemmas lead- ing up to the main result. In particular the limit system (1.54) can be obtained by merely assuming: u 0 ∈L p (Ω;H) f ∈L p (Ω;L 2 (0,T;V 0 )) g∈ Lip u (H,` 2 (H)). (1.51) 1.5.1 Lemma (Limit System). There exists adapted processesu,B ∗ andg ∗ with the regularity: u∈L p (Ω,L 2 (0,T;V)∩L ∞ (0,T;H)) ∂ z u∈L p (Ω,L 2 (0,T;V)∩L ∞ (0,T;H)) u∈C(0,T;H) a.s. (1.52) and: B ∗ ∈L 2 (Ω;L 2 (0,T;V 0 )) g ∗ ∈L 2 (Ω;L 2 (0,T;` 2 (H))) (1.53) such thatu,B ∗ andg ∗ satisfy: dhu,vi+hνAu+B ∗ ,vidt =hf,vidt+ ∞ X k=1 hg ∗ k (t),vidβ k hu(0),vi =hu 0 ,vi (1.54) for any test functionv∈V. 25 1.5.1 Remark. We use the following elementary facts regarding weakly convergent sequences in the proof below. (i) SupposeB 1 ,B 2 are Banach spaces and thatL : B 1 → B 2 is a bounded linear mapping. Ifx n *x inB 1 thenLx n *Lx inB 2 . (ii) Forp∈ [1,∞) take: L 1 (w)(t) = Z t 0 wds w∈L p (Ω×[0,T]). Ifx n *x inL p (Ω×(0,T)) and thenL 1 (x n )*L 1 (x) in the same space. (iii) Take: L 2 (v)(t) = X k Z t 0 v k dβ k for v = {v k } ∈ L 2 (Ω,L 2 (0,T;` 2 )). Given that v n * v in this space then L 2 (v n )*L 2 (v) inL 2 (Ω;L 2 (0,T)) (Proof- Lemma 1.5.1). Applying the estimate (1.29) with Alaoglu’s theorem we deduce the existence of a subsequence of Galerkin elements u (n) and an element u∈L p (Ω,L 2 (0,T;V)∩L ∞ (0,T;H)) so that: u (n) *u inL p (Ω;L 2 (0,T;V)) (1.55) and: u (n) * ∗ u inL p (Ω;L ∞ (0,T;H)). (1.56) 26 Todeducethedesiredregularityfor∂ z uweapplytheuniformestimatesgivenin(1.32) andthinoursubsequenceagainsothat∂ z u∈L p (Ω,L 2 (0,T;V)∩L ∞ (0,T;H))with: ∂ z u (n) *∂ z u inL p (Ω,L 2 (0,T;V)) (1.57) as well as: ∂ z u (n) * ∗ ∂ z u inL p (Ω,L ∞ (0,T;H)). (1.58) By an application of (1.13): E Z T 0 |P n B(u (n) )| 2 V 0dt ≤CE " sup t∈[0,T] (|u (n) | 4 +|∂ z u (n) | 4 )+ Z T 0 ku (n) k 2 dt 2 # . (1.59) The later quantity is uniformly bounded as a consequence of (1.30) and (1.32). Thin- ningu (n) as necessary we find an elementB ∗ ∈L 2 (Ω;L 2 (0,T;V 0 )) so that: 2 P n B(u (n) )*B ∗ inL 2 (Ω;L 2 (0,T;V 0 )). (1.60) Finally the Lipschitz assumption along with the Poincar´ e inequality imply: E Z T 0 ∞ X k=1 |P n g k (t,u (n) )| 2 dt≤CE Z T 0 (ku (n) k 2 +1)dt . (1.61) 2 Given only the weaker assumptions on the initial data (1.51) one can still show that B ∗ ∈ L 4/3 (Ω;L 2 (0,T;V 0 )) with the estimate: E Z T 0 |P n B(u (n) ))| 4/3 V 0 dt≤CE   sup t∈[0,T] |u (n) | 2 + Z T 0 ku (n) k 2 dt ! 3/2   . 27 This gives the uniform bounds needed to infer g ∗ ∈ L 2 (Ω;L 2 (0,T;` 2 (H))) and implies that: P n g(u (n) )*g ∗ inL 2 (Ω;L 2 (0,T;` 2 (H))). (1.62) We next establish thatu satisfies (1.54). To this end, fix any measurable setE ⊂ Ω×[0,T] andv ∈ V. By applying (1.55) and using that eachu (n) is a solution to an associated Galerkin system we infer: E Z T 0 χ E (u,v)dt = lim n→∞ E Z T 0 χ E (u (n) ,v)dt = lim n→∞ E Z T 0 χ E (P n u 0 ,v)dt −E Z T 0 χ E Z t 0 hνAu (n) +P n B(u (n) )−P n f,vids dt +E Z T 0 χ E " ∞ X k=1 Z t 0 hP n g k (u (n) ,s),vidβ k (s) # dt =E Z T 0 χ E (u 0 ,v)− Z t 0 hνAu+B ∗ −f,vids dt +E Z T 0 χ E " ∞ X k=1 Z t 0 hg ∗ k (s),vidβ k (s) # dt (1.63) For the final equality above we use Remark 1.5.1. Observe that: hP n B(u (n) ),vi*hB ∗ ,vi inL 2 (Ω×[0,T]) (1.64) hAu (u) ,vi*hAu,vi inL 2 (Ω×[0,T]) (1.65) along with: [(P n g k (u (n) ),v)] k≥1 * [(g ∗ k ,v)] k≥1 inL 2 (Ω;L 2 ([0,T],` 2 )). (1.66) 28 SinceE is arbitrary in (1.63), the equality (1.54) follows up to a set of measure zero. Referring then to results in [46], chapter 2 we find that u has modification so that u∈C([0,T];H) a.s. With a candidate solution in hand, it remains to show thatB(u) = B ∗ andg ∗ = g(u). A sufficient condition for these equalities, at least up to a stopping time, is captured in the following: 1.5.2Lemma. If0≤τ ≤T is any stopping time such that: E Z τ 0 ku−u (n) k 2 dt→ 0 (1.67) then,λ×P-a.e. 3 : B(u(t))1 1 t≤τ =B ∗ (t)1 1 t≤τ (1.68) and for everyk: g k (u,t)1 1 t≤τ =g ∗ k (t)1 1 t≤τ . (1.69) Proof. LetE,ameasurablesubsetofΩ×[0,T]andv∈V begiven. Itissufficientto show that: E Z T 0 χ E h(B(u(t))−B ∗ (t))1 1 t≤τ ,vidt = 0 (1.70) and that for anyk: E Z T 0 χ E h(g k (u(t),t)−g ∗ k (t))1 1 t≤τ ,vidt = 0 (1.71) 3 On [0,T] we use the Lebesgue measureλ. 29 where χ E is the characteristic function of E. Due to the weak convergence (1.60) established above: E Z T 0 χ E h(P n B(u (n) (t))−B ∗ (t))1 1 t≤τ ,vidt→ 0 asn→∞. (1.72) Using that: |hB(u)−P n B(u (n) ),vi| ≤Ckuk 2 kQ n vk+C(kukku−u (n) k+ku (n) kku−u (n) k)kvk (1.73) we estimate: E Z T 0 χ E h(P n B(u (n) ,t)−B(u,t))1 1 t≤τ ,vids ≤Ck(I−P n )vkE Z T 0 kuk 2 ds +Ckvk E Z T 0 (kuk+ku (n) k) 2 ds 1/2 E Z τ 0 ku−u (n) k 2 ds 1/2 (1.74) which, also vanishes asn→∞. These estimates imply the first item. Fortheseconditem(1.71)weagainexploittheweakconvergencein(1.62)toinfer that for everyk: E Z T 0 χ E hP n (g k (u (n) ,t)−g ∗ k (t))1 1 t≤τ ,vidt→ 0 asn→∞. (1.75) Also: E Z T 0 χ E hQ n g(u,t)1 1 t≤τ ,vidt ≤K H |Q n v|E Z T 0 (1+|u|)dt→ 0 asn→∞. (1.76) 30 Finally by the assumption (1.67), the Lipschitz continuity of g and the Poincar´ e Inequality: E Z T 0 χ E hP n (g k (u (n) ,t)−g k (u,t))1 1 t≤τ ,vidt ≤C(K H ,T)|v| E Z τ 0 |u (n) −u| 2 dt 1/2 → 0 (1.77) asn→∞. Combining (1.75), (1.76) and (1.77) provides the second equality (1.69). Withthislemmainmindwecompareutothesequenceu (n) ofGalerkinestimates to show that (1.67) is satisfied for a sequence of stopping timesτ n . Since we are able to chooseτ n so thatτ n ↑ T a.s., this is sufficient to deduce the existence result. Here we are adapting techniques used in [5]. 1.5.1 Theorem (Existence of Weak-Strong Solutions) . Suppose that p ≥ 4 and f,∂ z f ∈ L p (Ω;L 2 (0,T;V 0 )),g ∈ Lip u (H,` 2 (H))∩Lip u (X,` 2 (X)) andu 0 ,∂ z u 0 ∈ L p (Ω;H). Then there exists anH continuous weak-strong solutionu with the addi- tional regularity: ∂ z u∈L p (Ω,L 2 (0,T;V)∩L ∞ (0,T;H)). (1.78) Proof. The needed regularity conditions for u follow from Lemma 1.5.1. Moreover we found thatu satisfies the differential (1.54). Thus it remains only to show that for almost everyt,ω: B(u) =B ∗ g k (u) =g ∗ k . (1.79) 31 To this end, forR> 0, define: τ R = inf r∈[0,T] ( sup s∈[0,r] |u| 2 + sup s∈[0,r] |∂ z u| 2 + Z r 0 kuk 2 +k∂ z uk 2 ds>R ) ∧T. (1.80) Notice thatτ R is increasing as a function ofR> 0 and that moreover: P(τ R ≤T)≤P sup s∈[0,T] |u| 2 + sup s∈[0,T] |∂ z u| 2 + Z T 0 kuk 2 +k∂ z uk 2 ds≥R ! . (1.81) Thus, as a consequence of (1.52)τ R →T, almost surely. We now fixR and show thatτ R satisfies (1.67). Since by the dominated conver- gence theorem: lim n→∞ Z τ R 0 ku−P n uk 2 dt = 0 (1.82) it is sufficient to compare P n u and u (n) . The difference of these terms satisfies the differential: d(P n u−u (n) )+[νA(P n u−u (n) )+P n B ∗ −P n B(u (n) )]dt = ∞ X k=1 P n (g ∗ k −g k (u (n) ))dβ k . By applying Itˆ o’s lemma we find that: d(|P n u−u (n) | 2 e ψ )+2νkP n u−u (n) k 2 e ψ dt =2hB(u (n) )−B ∗ ,P n u−u (n) ie ψ dt +2 ∞ X k=1 hg ∗ k (t)−g k (u n ,t),P n u−u (n) ie ψ dβ k + ∞ X k=1 |P n (g ∗ k (t)−g k (u n ,t))| 2 e ψ dt +ψ 0 |P n u−u (n) | 2 e ψ dt. (1.83) 32 As in the a priori estimates for Lemma 1.4.1,ψ is aC 1 function chosen further on to cancel off terms. The first term on the right hand side of (1.83) is estimated using the cancellation property, (1.13) and (1.15): hB(u (n) )−B ∗ ,P n u−u (n) i =hB(u (n) −P n u,P n u)+(B(P n u)−B(u))+(B(u)−B ∗ ),P n u−u (n) i ≤C(|u (n) −P n u|ku (n) −P n ukkP n uk +|u (n) −P n u| 1/2 ku (n) −P n uk 3/2 |∂ z P n u|) +(|B(P n u−u,P n u)| V 0 +|B(u,P n u−u)| V 0)kP n u−u (n) k +hB(u)−B ∗ ,P n u−u (n) i ≤νkP n u−u (n) k 2 +C 1 (ν)|u n −P n u| 2 (kuk 2 +|∂ z u| 4 ) +C 2 (ν)(|B(P n u−u,u)| 2 V 0 +|B(u,P n u−u)| 2 V 0) +hB(u)−B ∗ ,P n u−u (n) i. (1.84) For the last term involvingg andg ∗ , using the basic properties of the` 2 inner product and the Lipschitz conditions forg we infer: |P n (g ∗ −g(u (n) ))| 2 ` 2 (H) =2(g ∗ −g(u),P n (g ∗ −g(u (n) ))) ` 2 (H) +|P n (g(u)−g(u (n) ))| 2 ` 2 (H) −|P n (g ∗ −g(u))| 2 ` 2 (H) ≤2(g ∗ −g(u),P n (g ∗ −g(u (n) ))) ` 2 (H) +4K 2 H (|P n u−u (n) | 2 +|P n u−u| 2 ). (1.85) 33 With estimates (1.84) and (1.85) in mind we now take: ψ(t) =−C 1 Z t 0 (kuk 2 +|∂ z u| 4 )ds−4K 2 H t (1.86) where C 1 is the constant from the last inequality in (1.84) and K H is the Lipschitz constant associated withg. By examining (1.80) we notice: e ψ(τ R ) ≥e −C 1 (R+R 2 )−4K 2 H T a.s. (1.87) The estimates given in (1.84) and (1.85) can now be applied to (1.83). After inte- grating up to the stopping timeτ R , taking expectations and rearranging one finds: E Z τ R 0 kP n u−u (n) k 2 dt ≤CE Z τ R 0 hB(u)−B ∗ ,P n u−u (n) i+(g ∗ −g(u),P n [g ∗ −g(u n )]) ` 2 e ψ dt +CE Z τ R 0 (|Q n u| 2 +|B(P n u−u,u)| 2 V 0 +|B(u,P n u−u)| 2 V 0)dt (1.88) where the numerical constantC depends onR andT. Note that since: E e ψ(τ R ) Z τ R 0 kP n u−u (n) k 2 dt ≤E Z τ R 0 e ψ(t) kP n u−u (n) k 2 dt. (1.89) and taking into account (1.87) we see that the terme φ can be absorbed into the con- stants on the right hand side of (1.88). Due to (1.55) one infers that P n u−u (n) * ∗ 0 in L p (Ω;L 2 (0,T;V)). Similarly (1.62) implies that P n g ∗ −P n g(u (n) ) * 0 in the space L 2 (Ω;L 2 (0,T;` 2 (H))). As such,thefirsttermsontherighthandsideof(1.88)vanishinthelimitasn→∞. The 34 term involving|Q n u| approaches zero in this limit as a consequence of the dominated convergence theorem. For the final terms we apply the estimate onB inV 0 given in (1.15) and make further use ofτ R to conclude: E Z τ R 0 |B(P n u−u,u)| 2 V 0dt ≤CE Z τ R 0 (|∂ z (P n u−u)| 2 +|P n u−u| 2 )kuk 2 dt +CE Z τ R 0 |(P n u−u)| 2 k∂ z uk 2 dt. (1.90) Moreover: E Z τ R 0 |B(u,P n u−u)| 2 V 0dt ≤CE Z τ R 0 (|∂ z u| 2 +|u| 2 )|∂ x (P n u−u)| 2 +|u| 2 k∂ z (P n u−u)k 2 dt ≤CE ( sup t∈[0,τ R ] |∂ z u| 2 + sup t∈[0,τ R ] |u| 2 ) Z τ R 0 k(P n u−u)k 2 dt ! +CE sup t∈[0,τ R ] |u| 2 Z τ R 0 k∂ z (P n u−u)k 2 dt ! ≤C(R)E Z τ R 0 (kP n u−uk 2 +k∂ z (P n u−u)k 2 )dt . (1.91) Thus, again by the dominated convergence theorem, the final two terms also converge to zero asn→∞. We can now conclude thatτ R satisfies Lemma 1.5.2. Fixing arbitrary (t,ω) off of an appropriately chosen set of measure zero, there is an R so that t ≤ τ R (ω). As such (1.79) for holds the given pair, completing the proof. Having established existence we next turn to the question of uniqueness: 35 1.5.2Theorem(Uniqueness). Supposethatu 1 andu 2 areweak-strong solutionsinthe sense of Definition 2.2.3 and thatu 1 (0) =u 2 (0)a.s. inH. Assume moreover that: ∂ z u 2 ∈L 4 ([0,T];H) a.s. (1.92) Then: P(u 1 (t) =u 2 (t)∀t∈ [0,T]) = 1. (1.93) In particular, the solutions constructed in Theorem 1.5.1 are unique since they belong (almost surely) toL ∞ ([0,T];H) which is included inL 4 ([0,T];H) Proof. Subtracting we find theu 1 −u 2 satisfies the differential: dhu 1 −u 2 ,vi+hνA(u 1 −u 2 ),vidt =hB(u 1 ,u 2 −u 1 )+B(u 2 −u 1 ,u 2 ),vidt + X k hg k (u 1 ,t)−g k (u 2 ,t),vidβ k hu 1 (0)−u 2 (0),vi =hu 0,1 −u 0,2 ,vi. (1.94) We apply Itˆ o’s lemma and deduce: d(|u 1 −u 2 | 2 e φ )+2νku 1 −u 2 k 2 e φ dt =2hB(u 1 −u 2 ,u 2 ),u 2 −u 1 ie φ dt + X k hg k (u 1 ,t)−g k (u 2 ,t),u 1 −u 2 ie φ dβ k + X k |g k (u 1 ,t)−g k (u 2 ,t)| 2 e φ dt +φ 0 |u 1 −u 2 | 2 e φ dt. (1.95) 36 Once againφ will be chosen judiciously to cancel off terms below. Applying (1.13) with Young’s inequality one finds: |hB(u 1 −u 2 ,u 2 ),u 2 −u 1 i| ≤νku 1 −u 2 k 2 +C(ν)|u 1 −u 2 | 2 (ku 2 k 2 +|∂ z u 2 | 4 ). (1.96) TakingK H tobetheLipschitzconstantassociatedwithg andC(ν)fromthepreceding inequality, set: φ(t) =−C(ν) Z t 0 (ku 2 k 2 +|∂ z u 2 | 4 )dt−K 2 H t. (1.97) Given the regularity conditions assumed for u 2 we infer, as in Theorem 1.5.1 that e φ(t) > 0 almost surely. Integrating (1.95) up tot taking expected values and making use of the estimates onB andg one concludes that for anyt∈ [0,T]: E(|u 1 (t)−u 2 (t)| 2 e φ(t) )≤E(|u 1 (0)−u 2 (0)| 2 ) = 0. (1.98) This implies: P(u 1 (t) =u 2 (t),∀t∈ [0,T]∩Q) = 1. However since bothu 1 andu 2 take values inC([0,T];H) (1.93) follow immediately. 1.6 Appendix: PhysicalBackground In this final section we introduce some physical background for the Primitive Equa- tions of the ocean, sketching the physical derivation of this system. Accounting for 37 both temperature and salinity and working in the beta-plane approximation for non- equatorial regions this system takes the form: ∂ t u+u·∇ 2 u+u 3 ∂ z u+ 1 ρ 0 ∇ 2 p s =−2ωsinφe 3 ×u+μ u Δ 2 u+ν u ∂ zz u+F(T,S) ∂ z p =−ρg ∇ 2 ·u+∂ z u = 0 ∂ t T +(u·∇ 2 )T +u 3 ∂ z T =μ T Δ 2 T +ν T ∂ zz T ∂ t S +(u·∇ 2 )S +u 3 ∂ z S =μ S Δ 2 S +ν S ∂ zz S F(T,S) =g Z 0 z ∇ 2 ρd¯ z, ρ =ρ 0 (1−β T (T −T r )−β S (S−S r )). (1.99) The variablesu = (u 1 ,u 2 ),u 3 ,p,p s ,ρ,T andS represent the horizontal and vertical componentsoftheflow,thepressure,thesurfacepressure,thedensity,thetemperature and the salinity respectively. The earth’s rotation and the latitude of the domain are denoted byω andφ. The constantsμ u ,ν u ,μ u ,ν u ,μ u andν u are various diffusion or viscosity parameters which take eddy as well as molecular dissipation into considera- tion. For the final equation,β T andβ S are the coefficients of thermal expansion and saline contraction, while T r and S r are reference temperatures and saline concentra- tions. 38 1.6.1 FundamentalEquations The basic equations governing fluid flow, are the fully compressible Navier-Stokes equations: ρ(∂ t v +(v·∇)v +2ω×v) =D(v)−∇p−ρge 3 (1.100a) ∂ t ρ+∇·(ρv) = 0 (1.100b) ρ =f(T,S,p). (1.100c) This system expresses the conservation of momentum and of mass in the convenient rotating reference frame of the earth’s surface. Here v = (u 1 ,u 2 ,u 3 ), p, ω and g describe the velocity, the pressure, the earth’s angular velocity and the gravitational constant respectively. The termD(u) is the molecular dissipation. The density ρ is a functionf of the temperatureT, salinityS and the pressure. Equations governing temperature and salinity are described below (c.f. (1.109) and (1.110)). While(1.100)istheusualstartingpointforthedescriptionoffluidflow,forpracti- cal purposes, these fundamental equations are intractable, both theoretically and com- putationally. Beginning with the pioneering work of Richardson in the 1920s [45] and later in the 1940s with the work of Von Neumann and his collaborators simplified models were derived on a combined basis of scale analysis and meteorological data. Thesenewmodelswereusedtoperformthefirstnumericalsimulationsofatmospheric processes. Todaythesystem(1.99)formsthedynamicalcoreoflargescaleglobalcir- culation models under development at the National Center for Atmospheric Research and other centers. 39 1.6.2 TheBoussinesqApproximation In oceanic systems the fluctuations ρ 0 in the density of the fluid usually are usually small in comparison to the mean densityρ 0 . We have: ρ =ρ 0 +ρ 0 , ρ 0 ρ 0 . (1.101) Working from this observation one finds that the continuity equation (conservation of mass): ∂ t ρ+∇·(ρv) = 0, can be replaced by incompressibility condition: ∇·v = 0. (1.102) Dividingthemomentumequationsbyρ 0 wefindthatwecanneglectmostoftheterms involving fluctuations in the density since they are dominated by larger neighboring terms. AssumingthatthefluidisNewtoniananddenotingthedynamicviscositybyν, the momentum equations (1.100a) simplify to: ∂ t u 1 +(v·∇)u 1 +2ωcosφu 3 −2ωsinφu 2 =νΔu 1 − 1 ρ 0 ∂ x p (1.103a) ∂ t u 2 +(v·∇)u 2 +2ωsinφu 1 =νΔu 3 − 1 ρ 0 ∂ y p (1.103b) ∂ t u 3 +(v·∇)u 3 +2ωsinφu 1 =νΔu 3 − 1 ρ 0 ∂ z p− ρ 0 ρ 0 g. (1.103c) Notecarefullythatthedensityfluctuationsinthebuoyancytermin(1.103c)cannotbe neglected. Duetocancellationswiththehydrostaticpressurethereisnocorresponding dominating term for the the remaining portion of the this term(ρ 0 /ρ 0 )g. 40 1.6.3 ScaleAnalysisandTheHydrostaticApproximation Further significant simplifications of the governing equations can be justified by ana- lyzing the physical scales typical for oceanic processes. Most importantly we empha- size that the vertical scales of motion are much smaller than the horizontal: H L. Quoting Roisin and Beckers recent text [14]: The atmospheric layer that determines our weather is only about 10 km thick, yet cyclones and anticyclones spread over thousands of kilome- ters. Similarly,oceancurrentsaregenerallyconfinedtotheupperhundred metersofthewatercolumnbutextendovertensofkilometersormore,up to the width of the ocean basin. Also significant in this analysis is the Earth’s rotationΩ. For example we find that: T . 1 Ω U L . Ω. where T, U and H are a typical values for time, horizontal velocity and horizontal length. Applying these assumptions to the third momentum equation, (1.103c), we arrive at the hydrostatic approximation: ∂ z p =ρg. (1.104) Similar scale considerations justify the removal of the reciprocal Coriolis term: 2ωcosφu 3 . 41 from the first momentum equation,(1.103a), at non-equatorial latitudes. 1.6.4 EddyViscosity Even with the dramatic advances in modern computing power, numerical models for the oceanic systems are able to resolve only large scale processes. Complex, sub- grid scale phenomena are often accounted for by placing additional viscous terms in the momentum equations. Since, as emphasized previously, the scales of motion are vastlydifferentinthehorizontalandverticaldirectionsthemeshsizeinthehorizontal and vertical for numerical models will contrast as well. Thus, practical models will consider different viscosity parameters for the vertical directions of motion. 1.6.5 TemperatureandSalinity The equation for the temperature is derived from the conservation of energy: ∂ t E +(v·∇)E =Q−W. (1.105) ThatistherateofchangeoftheenergyE inaparceloffluidisequaltotherateofheat gain from the surroundings Q minus the mechanical work W done by the pressure. Since energy is really an expression of the agitation of the particles that make up the fluid we write: E =ρ 0 C v T, (1.106) whereC v is the heat capacity. Since heat flows through the medium via a process of diffusion: Q =k T ΔT, (1.107) 42 where k T gives the thermal conductivity of the fluid. Work is defined to be the dis- placement times the amount of force in the direction of this displacement. Noting that pressure is force per unit area, we infer W =p(∂ t V +(v·∇)V). (1.108) where V is the volume. However, since the density ρ 0 is mass per unit volume, the Boussinesq approximation implies that the internal work done by the fluid is negligi- ble. We conclude that: ∂ t T +(v·∇)T = k T C v ρ 0 ΔT. (1.109) For the salinity we assume that distribution of salt is governed by a process of diffusion: ∂ t S +(v·∇)S =k S ΔS, (1.110) wherek S is the coefficient of salt diffusivity. 1.6.6 TheEquationofState To complete the description of the fundamental equations for geophysical flows given by (1.100), (1.109) and (1.110) we need to establish the relationship between the den- sity of the fluid and its temperature, salinity and pressure. This relation is referred to as the Equation of State. Since seawater is relatively incompressible the dependence onthepressureisusuallyassumedtobenegligible. Onanempiricalbasis,therelation betweenρ and the remaining variables is often assumed to be linear: ρ =ρ 0 (1−β T (T −T r )−β S (S−S r )). (1.111) 43 Above, β T and β S are appropriate linearization constants and T r , S r are convenient reference values forS andT. 1.6.7 OtherConsiderations Wehaveimplicitlyassumedthebeta-planeapproximation,thatlocallytheearthisflat. This is a reasonable assumption for local (regional) atmospheric studies. For larger scale models one needs to account for the curvature of the earth and work in spherical coordinates. See [32]. The boundary conditions are another important consideration omitted in this review. See [50] or [42] for details. 44 Chapter2 StrongPathwiseSolutionsofthe StochasticNavier-StokesSystem 2.1 Introduction In this article we study the Navier-Stokes equations in space dimension d = 2,3, on a bounded domainM forced by a multiplicative white noise: ∂ t u+(u·∇)u−νΔu+∇p =f +g(u) ˙ W (2.1a) ∇·u = 0 (2.1b) u(0) =u 0 (2.1c) u |M = 0. (2.1d) The system (2.1) describes the flow of a viscous incompressible fluid. Here u = (u 1 ,...,u d ), p and ν represent the velocity field, the pressure and the coefficient of kinematic viscosity respectively. The addition of the white noise driven terms to the basic governing equations is natural for both practical and theoretical applications. Such stochastically forced terms are used to account for numerical and empirical uncertainties and have been proposed as a model for turbulence. ThemathematicalliteratureforthestochasticNavier-Stokesequationsisextensive and dates back to the early 1970’s with the work of Bensoussan and Temam [2]. For 45 the study of well-posedness new difficulties related to compactness often arise due to the addition of the probabilistic parameter. For situations where continuous depen- dence on initial data remains open (for example ind = 3 when the initial data merely takes values inL 2 ) it has proven fruitful to consider Martingale Solutions. Here one constructs a probabilistic basis as part of the solution. For this context we refer the reader to the works of Cruzeiro [13], Capinski and Gatarek [9], Flandoli and Gatarek [21] and of Mikulevicius and Rozovskii [38]. On the other hand, when working in spaces where continuous dependence on the initial data can be expected, existence of solutions can sometimes be established on a preordained probability space. Such solutions are often referred to as “strong” or “pathwise” solutions. In the two dimensional setting, Da Prato and Zabczyk [16] and later Breckner [5] as well as Menaldi and Sritharan [34] established the exis- tence of pathwise solutions where u takes values in L ∞ ([0,T],L 2 ). On the other hand, Bensoussan and Frehse [1] have established local solutions in 3-d for the class C β ([0,T];H 2s ) where 3/4 < s < 1 and β < 1−s. The existence of pathwise, global solutions for the two dimensional Primitive equations with multiplicative noise was recently established by Glatt-Holtz and Ziane in [23], for the case when u and its vertical gradient are initially in L 2 . In the works of Mikulevicius and Rozovsky [37] and of Brzezniak and Peszat [7] the case of arbitrary space dimensions for local solutions evolving in Sobolev spaces of type W 1,p for p > d is addressed. Despite these extensive investigations, to the best of our knowledge, no one has addressed the case of local, probabilistically strong H 1 (W 2,1 ) solutions for the 3-d Navier-Stokes equations with multiplicative noise. 46 The exposition is organized as follows. In the first section we review the basic setting, defining the relevant function spaces and introducing various notions of path- wise solutions. We then turn to the Galerkin scheme which we analyze by modifying a pairwise comparison technique first developed in [37]. Key estimates are achieved using decompositions into high and low modes. In this way we are able to extract a locally strongly convergent subsequence and surmount the difficult issue of compact- ness. In the third section we establish the existence and uniqueness of a local solution uevolvingcontinuouslyinH 1 uptoamaximalexistencetimeξ. Forsampleswhereξ is finite we show that, on the one hand, theL 2 norm remains bounded and that on the other hand theH 1 norm of the solution blows up. By showing that certain quantities areundercontrolinthetwodimensionalcaseweareabletousethislaterblow-upcri- teriatogiveanewprooffortheglobalexistenceofstrongsolutionsin2-d. Inthefinal section we formulate and prove some abstract convergence results used in the proof of the main theorem. We believe these results to be more widely applicable for the studyofwellposednessofothernonlinearstochasticpartialdifferentialequationsand therefore hold independent interest. 2.2 TheAbstractFunctionalAnalyticSetting We begin by reviewing some basic function spaces associated with (2.1). In what followsd is the spatial dimension, the physical casesd = 2,3, being the focus of our attention below. For simplicity, we assume that the boundary∂M is smooth. Let: V :={φ∈ (C ∞ 0 (M)) d :∇·φ = 0}. (2.2) 47 Define: H :=cl L 2 (M) V ={u∈L 2 (M) d :∇·u = 0,u·n = 0}. (2.3) Heren is the outer pointing normal to∂M. OnH we take theL 2 inner product and norm: (u,v) := Z M u·vdM, |u| := p (u,u). (2.4) The Leray-Hopf projector, P H , is defined as the orthogonal projection of H into L 2 (M) d . At the next order let: V :=cl H 1 (M) V ={u∈H 1 0 (M) d :∇·u = 0}. (2.5) On this set we use theH 1 norm and inner products: ((u,v)) := Z M ∇u·∇vdM, kuk := p ((u,u)). (2.6) NotethatduetotheDirichletboundarycondition(c.f. (2.1d))thePoincar´ einequality: |u|≤Ckuk ∀u∈V (2.7) holds, justifying (2.6) as a norm. TakeV 0 to be the dual ofV, relative toH with the pairing notated byh·,·i. We next define the Stokes operatorA. A is understood as a bounded linear map fromV toV 0 via: hAu,vi = ((u,v)) u,v∈V. (2.8) AcanbeextendedtoanunboundedoperatorfromH toH accordingtoAu =−P H Δu withthedomainD(A) =H 2 (M)∩V. Byapplyingthetheoryofsymmetric,compact, 48 operators for A −1 , one can prove the existence of an orthonormal basis{e k } for H of eigenfunctions of A. Here the associated eigenvalues {λ k } form an unbounded, increasing sequence: 0<λ 1 <λ 2 ≤...≤λ n ≤λ n+1 ≤.... (2.9) We shall also make use of the fractional powers of A. For u ∈ H we denote u k = (u,e k ). Givenα> 0 take: D(A α ) = ( u∈H : X k λ 2α k |u k | 2 <∞ ) (2.10) and define: A α u = X k λ α k u k e k , u∈D(A α ). (2.11) We equipD(A α ) with the norm: |u| α :=|A α u| = X k λ 2α k |u k | 2 ! 1/2 . (2.12) Define: H n =span{e 1 ,...,e n } (2.13) and take P n to be the projection from H onto this space. Let Q n = I −P n . The following extension of the Poincar´ e inequality will be used for the estimates below: 2.2.1Lemma. Suppose thatα 1 <α 2 . For anyu∈D(A α 2 ): |Q n u| α 1 ≤ 1 λ α 2 −α 1 n |Q n u| α 2 (2.14) 49 and: |P n u| α 2 ≤λ α 2 −α 1 n |P n u| α 1 . (2.15) Proof. Working from the definitions: |Q n u| 2 α 1 = ∞ X k=n+1 λ 2α 1 k |u k | 2 ≤ ∞ X k=n+1 λ 2(α 2 −α 1 ) k λ 2(α 2 −α 1 ) n λ 2α 1 k |u k | 2 = 1 λ 2(α 2 −α 1 ) n |Q n u| 2 α 2 . (2.16) Similarly: |P n u| 2 α 2 = n X k=1 λ 2α 2 k |u k | 2 ≤λ 2(α 2 −α 1 ) n n X k=1 λ 2α 1 k |u k | 2 =λ 2(α 2 −α 1 ) n |P n u| 2 α 1 . (2.17) The nonlinear portion of (2.1) is given by: B(u,v) :=P H (u·∇)v =P H (u j ∂ j v) u,v∈V. (2.18) Here and below we occasionally make use of the Einstein convention of summing repeated indices from 1 to d. For notational convenience we will sometimes write B(u) := B(u,u). B can be shown to be well defined as a map from V ×V to V 0 according to: hB(u,v),wi := Z M (u·∇)v·wdM = Z M u j ∂ j v k w k dM. (2.19) We shall need the following classical facts concerningB: 2.2.2Lemma. Supposed = 2 or3. 50 (i) B is continuous fromV ×V toV 0 with: hB(u,v),vi = 0 (2.20) and: hB(u,v),wi≤C                |u| 1/2 kuk 1/2 kvk|w| 1/2 kwk 1/2 ind = 2 |u| 1/2 kuk 1/2 kvkkwk ind = 3 kukkvk|w| 1/2 kwk 1/2 ind = 3 (2.21) for allu,v,w∈V. (ii) B is also continuous fromV ×D(A) toH. Ifu ∈ V,v ∈ D(A) andw ∈ H then: (B(u,v),w)≤C        |u| 1/2 kuk 1/2 kvk 1/2 |Av| 1/2 |w| ind = 2 kukkvk 1/2 |Av| 1/2 |w| ind = 3. (2.22) (iii) Ifu∈D(A) thenB(u)∈V and: kB(u)k 2 ≤C        kuk 2 |Au| 2 +|u| 1/2 kuk|Au| 5/2 +|u||Au| 3 ind = 2 kuk|Au| 3 +|u| 1/2 |Au| 7/2 ind = 3. (2.23) Proof. The items (i) and (ii) are classical and are easily established using H¨ older’s inequality and the Sobolev embedding theorem (See [49] or [12]). 51 For item (iii) fixu∈V. We have: kB(u)k 2 ≤ Z M |∂ m (u j ∂ j u k )∂ m (u l ∂ l u k )|dM. (2.24) Ford = 2 we recall, in particular the Sobolev embedding: |φ| L ∞ ≤C|Aφ| 1/2 |φ| 1/2 , φ∈D(A). (2.25) Using this and the embedding ofH 1/2 intoL 4 we estimate: kB(u)k 2 ≤C(|∇u| 4 L 4 +|u| L ∞|Au||∇u| 2 L 4 +|u| 2 L ∞|Au| 2 ) ≤C(kuk 2 |Au| 2 +|u| 1/2 kuk|Au| 5/2 +|u||Au| 3 ). (2.26) Ford = 3 we have: |φ| L ∞ ≤C|Aφ| 3/4 |φ| 1/4 , φ∈D(A). (2.27) This estimate and the embedding ofH 1 inL 6 implies: kB(u)k 2 ≤C(|∇u| 3 L 6kuk+|Au||∇u| 2 L 6|u| L 6 +|u| 2 L ∞|Au| 2 ) ≤C(|Au| 3 kuk+|Au| 7/2 |u| 1/2 ). (2.28) The stochastically driven term in (2.1) can be written formally in the expansion: g(u) ˙ W = X k g k (u) ˙ β k (2.29) whereβ k areindependentstandardBrownianmotions. Tomakethisrigorouswerecall some definitions: 52 2.2.1Definition. A stochastic basis: S := (Ω,F,{F t } t≥0 ,P,{β k } k≥1 ) (2.30) consists of a probability space (Ω,F,P) equipped with a complete, right-continuous filtration, namely: P(A) = 0⇒A∈F 0 , F t =∩ s>t F s (2.31) and a sequence of mutually independent, standard, Brownian motions β k relative to this filtration. We also need to define a class of spaces forg ={g k } k≥1 : 2.2.2 Definition. SupposeU is any (separable) Hilbert space. We define` 2 (U) to be the set of all sequenceh ={h k } k≥1 of elements inU so that: |h| 2 ` 2 (U) := X k |h k | 2 U <∞. ForanynormedspaceY,wesaythath :Y×[0,T]×Ω→` 2 (U)isuniformlyLipschitz with constantK Y if for allx,y∈Y: |h(x,t,ω)−h(y,t,ω)| ` 2 (U) ≤K Y |x−y| Y , (2.32) and |h(x,t,ω)| ` 2 (U) ≤K Y (1+|x| Y ). (2.33) We denote the collection of all such mappingsLip u (Y,` 2 (U)). 53 For the analysis below we shall assume that: g ={g k } : Ω×[0,∞)×H →` 2 (H) (2.34) and that: g∈Lip u (H,` 2 (H))∩Lip u (V,` 2 (V))∩Lip u (D(A),` 2 (D(A))) (2.35) We shall assume moreover that ifu : [0,T]×Ω→ H is predictable 1 then so isg(u). Given anH-valued, predictable process u∈L 2 (Ω;L 2 (0,T;H)) the expansion (2.29) can be shown to be well defined as a stochastic integral and: Z τ 0 g(u)·dW,v = X k Z τ 0 g k (u)dβ k ,v ! = X k Z τ 0 (g k (u),v)dβ k , (2.36) for allv∈H and stopping timesτ. See [15] or [44] for detailed constructions. In order to show that the conditions imposed above forg are not overly restrictive we now consider some examples of stochastic forcing regimes satisfying (2.35). 2.2.1Example. • (Independently Forced Modes) Suppose (κ k (t,ω)) is any sequence uniformly bounded inL ∞ ([0,T]×Ω). We force the modes independently defining: g k (v,t,ω) =κ k (t,ω)(v,e k )e k . 1 For a given a stochastic basis,S let Φ = Ω× [0,∞) and takeG to be theσ-algebra generated by sets of the form: (s,t]×F, 0≤s<t<∞,F ∈F s ; {0}×F, F ∈F 0 Recall that a U valued process u is called predictable (with respect to the stochastic basisS) if it is (Φ,G)−(U,B(U)) measurable. 54 In this case the Lipschitz constants can be taken to be: K H =K V =K D(A) = sup ω,k,t |κ k (t,ω)| (2.37) • (Uniform Forcing) Given a uniformly square summable sequence a k (t,ω) we can take: g k (v,t,ω) =a k (t,ω)v withK H =K V =K D(A) = (sup t,ω P k a k (t,ω) 2 ) 1/2 as the Lipschitz constants. • (Additive Noise) We can also include the case when the noise term does not depend on the solution: g k (v,t,ω) =g k (t,ω). HeretheuniformconstantscanbetakentobeK U := sup t,ω ( P k |g k (t,ω)| 2 U ) 1/2 forU =H,V,D(A) as desired. With the above framework in place, we next give a variational definition for local pathwise solutions of the stochastic Navier-Stokes equations. Given a Hilbert space X, forp∈ [1,∞], we denote: L p loc ([0,∞);X) = \ T>0 L p ([0,T];X). (2.38) Similarly: C loc ([0,∞);X) = \ T>0 C([0,T];X). (2.39) 55 Finally: C w,loc ([0,∞);X) ={v∈L ∞ loc ([0,∞);X) : (v,x)∈C loc ([0,∞);R),∀x∈X} (2.40) 2.2.3 Definition (Weak and Strong Pathwise Solutions). Let S = (Ω,P,F,(F t ) t≥0 ,(β k )) be a fixed stochastic basis. Assume that u 0 is F 0 mea- surable with u 0 ∈ L 2 (Ω,V). Suppose that f and g are V 0 and ` 2 (H) valued, predictable processes respectively with: f ∈L 2 (Ω;L 2 ([0,∞);H)), g∈Lip u (H,` 2 (H))∩Lip u (V,` 2 (V))∩Lip u (D(A),` 2 (D(A))). (2.41) (i) We say that the pair (u,τ) is a local weak (pathwise) solution ifτ is a strictly positive stopping time andu(·∧τ) is a predictable process inV 0 so that: u(·∧τ)∈L 2 (Ω;C w,loc ([0,∞);H)), u1 1 t≤τ ∈L 2 (Ω;L 2 loc ([0,∞);V)), (2.42) and so that for anyt> 0: u(t∧τ)+ Z t∧τ 0 (νAu+B(u))dt =u(0)+ Z t∧τ 0 fdt+ Z t∧τ 0 g(u)·dW (2.43) inV 0 . This equality is equivalent to requiring that: hu(t∧τ),vi+ Z t∧τ 0 hνAu+B(u),vidt =hu(0),vi+ Z t∧τ 0 hf,vidt+ ∞ X k=1 Z t∧τ 0 hg k (u),vidβ k , (2.44) 56 for allv∈V. (ii) The pair (u,τ) is a local strong (pathwise) solution ifτ is strictly positive and u(·∧τ) is a predictable process inH with u(·∧τ)∈L 2 (Ω;C loc ([0,∞);V)), u1 1 t≤τ ∈L 2 (Ω;L 2 loc ([0,∞);D(A))), (2.45) and such thatu satisfies (2.43) as an equation inH. (iii) Suppose thatu is a predictable process inV 0 and thatξ is strictly positive stop- ping time. The pair (u,ξ) is said to be a maximal (pathwise) strong solution if there exists an increasing sequenceτ n with: τ n ↑ξ a.s. (2.46) such that each pair(u,τ n ) is a local strong solutions and so that: sup t≤ξ kuk 2 + Z ξ 0 |Au| 2 dt =∞ (2.47) on the set{ξ <∞}. If, in addition: sup t∈[0,τn] kuk 2 + Z τn 0 |Au| 2 ds =n (2.48) on the set{ξ <∞}, then we say that{τ n } announcesξ. 2.2.1Remark. • Forthe’pathwise’solutionsweconsider,thestochasticbasisisgiveninadvance. In particular solutions corresponding to different initial laws are shown to be 57 driven by the same underlying Wiener process. This is in contrast to the theory of Martingalesolutionsconsideredformanynonlinearsystems. Heretheunder- lying probability space is constructed as part of the solution. See [15], chapter 8 or [35] and the references in the introduction. Since the context is clear we will drop the ’pathwise’ designation for the remainder of the exposition. • If(u,τ) is a local strong solution then (2.45) implies that: E sup t∈[0,τ] kuk 2 + Z τ 0 |Au| 2 ds ! <∞. (2.49) Sofar,wearenotabletoshowthatEku(t)k 2 isfiniteforanyfixed(deterministic) t> 0. Thisisthecaseevenin2-dcasewhereweprovetheexistenceofaglobal strong solution (c.f. Proposition 2.4.5). • Note that the pressure term in (2.1) disappears in the variational formulation (2.44). Formally, ifwemultiplythistermanywithv∈V andintegrateoverM, then an integration by parts reveals: Z M ∇p·vdM =− Z M p∇·vdM = 0. (2.50) See [12] for details. 58 • Suppose that (u,τ) is a local strong solution. By applying an infinite dimen- sional version of the Itˆ o lemma (See [46] or [44]) one can show that on the interval[0,τ], for anyp≥ 2,|u| p satisfies: d|u| p +pνkuk 2 |u| p−2 dt =phf,ui|u| p−2 dt+ p 2 ∞ X k=1 |g k (u)| 2 |u| p−2 dt + p(p−2) 2 ∞ X k=1 hg k (u),ui 2 |u| p−4 dt +p ∞ X k=1 hg k (u),ui|u| p−2 dβ k . (2.51) NotethatthenonlineartermB dropsoutduetothecancellationproperty(2.20). Similarly forkuk p we have: dkuk p +pν|Au| 2 kuk p−2 dt =phf−B(u),Auikuk p−2 dt + p 2 ∞ X k=1 kg k (u)k 2 kuk p−2 dt + p(p−2) 2 ∞ X k=1 hg k (u),Aui 2 kuk p−4 dt +p ∞ X k=1 hg k (u),Auikuk p−2 dβ k . (2.52) 2.3 TheGalerkinSchemeandComparisonEstimates The first step to prove the existence of a solution is to approximate the full equations with a sequence of finite dimensional stochastic differential equations, the Galerkin systems: 59 2.3.1 Definition (The Galerkin System). An adapted process u n in C(0,T;H n ) is a solution to the Galerkin System of Ordern if for anyv∈H n : dhu n ,vi+hνAu n +B(u n ),vidt =hf,vidt+ ∞ X k=1 hg k (u n ),vidβ k , hu n (0),vi =hu 0 ,vi. (2.53) We can also write (2.53) as equations inH n ( ∼ =R n ): du n +(νAu n +P n B(u n ))dt =P n fdt+ ∞ X k=1 P n g k (u n )dβ k , u n (0) =P n u 0 :=u n 0 . (2.54) The existence of solutions to (2.53) is classical and relies on a priori bounds that are established using the cancellation property (2.20). See [20] for detailed proofs. Uniqueness is established as below for the full infinite dimensional system. We now proceed to main result of the section. Note that conditions established below are precisely those needed to apply Lemma 2.5.1 in Proposition 2.4.2 below. 2.3.1Proposition. Suppose thatd = 2,3 and let{u n } be the sequence of solutions of (2.53). We assume that for some0< ˜ M <∞: ku 0 k 2 ≤ ˜ M a.s. (2.55) and that: f ∈L 2 (Ω;L 2 ([0,T];H) g∈Lip u (H,` 2 (H))∩Lip u (V,` 2 (V))∩Lip u (D(A),` 2 (D(A))). (2.56) 60 Consider the collection of stopping times: T M,T n = ( τ ≤T : sup s≤τ ku n k 2 +ν Z τ 0 |Au n (s)| 2 ds 1/2 ≤M +ku n 0 k ) (2.57) and takeT M,T m,n :=T M,T m ∩T M,T n . (i) For anyT > 0 andM > 1: lim n→∞ sup m>n sup τ∈T M,T m,n E sup s≤τ ku m −u n k 2 +ν Z τ 0 |A(u m −u n )| 2 ds = 0. (2.58) (ii) Moreover: lim S→0 sup n sup τ∈T M,T n P sup s≤τ∧S ku n k 2 +ν Z τ∧S 0 |Au n | 2 ds>ku n 0 k 2 +(M−1) 2 = 0. (2.59) Proof. Givenm>n we subtractu n fromu m and observe that: d(u m −u n )+νA(u m −u n )dt =[P n B(u n )−P m B(u m )+(P m −P n )f]dt + ∞ X k=1 [P m g k (u m )−P n g k (u n )]dβ k (u m −u n )(0) = (P m −P n )u 0 . (2.60) 61 By applying the Itˆ o lemma to (2.60) we derive an evolution system for theV norm of this difference: dku m −u n k 2 +2ν|A(u m −u n )| 2 dt =2hP n B(u n )−P m B(u m ),A(u m −u n )idt +2h(P m −P n )f,A(u m −u n )idt + ∞ X k=1 kP m g k (u m )−P n g k (u n )k 2 dt +2 ∞ X k=1 hP m g k (u m )−P n g k (u n ),A(u m −u n )idβ k . (2.61) Fix arbitraryτ ∈ T M,T m,n . Given stopping timesτ a andτ b with 0 ≤ τ a ≤ τ b ≤ τ, we integrate (2.61) fromτ a tor and take a supremum over [τ a ,τ b ]. After taking expected values we obtain: E sup t∈[τa,τ b ] ku m −u n k 2 +2ν Z τ b τa |A(u m −u n )| 2 dt ≤Eku m (τ a )−u n (τ a )k 2 +2E Z τ b τa |h(P m −P n )f,A(u m −u n )i|dt +2E Z τ b τa |hP m B(u m )−P n B(u n ),A(u m −u n )i|dt +E Z τ b τa ∞ X k=1 kP m g k (u m )−P n g k (u n )k 2 dt +E sup r∈[τa,τ b ] 2 ∞ X k=1 Z r τa hP m g k (u m )−P n g k (u n ),A(u m −u n )idβ k ! . (2.62) With the aim of employing Lemma 2.5.4 we estimate each of terms on the right side of (2.62). For the first term we merely split: |2h(P m −P n )f,A(u m −u n )i|≤ ν 2 |A(u m −u n )| 2 +C ν |Q n f| 2 . (2.63) 62 Using the Lipschitz conditions in (2.56) and Lemma 2.2.1 we next estimate: ∞ X k=1 kP m g k (u m )−P n g k (u n )k 2 ≤C ∞ X k=1 kg k (u m )−g k (u n )k 2 + ∞ X k=1 kQ n g k (u n )k 2 ! ≤C K V ku m −u n k 2 + 1 λ n ∞ X k=1 |Ag k (u n )| 2 ! ≤C K V ,K D(A) ku m −u n k 2 + 1 λ n (1+|Au n | 2 ) . (2.64) For the stochastically forced term we apply the Burkholder-Davis-Gundy inequality (c.f. [27]): E sup r∈[τa,τ b ] 2 ∞ X k=1 Z r τa hP m g k (u m )−P n g k (u n ),A(u m −u n )idβ k ! ≤CE Z τ b τa ∞ X k=1 hP m g k (u m )−P n g k (u n ),A(u m −u n )i 2 ds ! 1/2 ≤CE Z τ b τa ku m −u n k 2 ∞ X k=1 kP m g k (u m )−P n g k (u n )k 2 ds ! 1/2 ≤C K V ,K D(A) E Z τ b τa ku m −u n k 2 ku m −u n k 2 + 1 λn (1+|Au n | 2 ) ds 1/2 ≤ 1 2 E sup t∈[τa,τ b ] ku m −u n k 2 ! +C K V ,K D(A) E Z τ b τa ku m −u n k 2 + 1 λn (1+|Au n | 2 ) ds. (2.65) 63 It remains to study the nonlinear term which we split as follows: hP m B(u m )−P n B(u n ),(u m −u n )i =hB(u m −u n ,u m )+B(u n ,u m −u n )+(P m −P n )B(u n ),A(u m −u n )i :=T 1 +T 2 +T 3 . (2.66) ForT 1 in (2.66) we apply (2.22). In eitherd = 2,3 we estimate: T 1 ≤Cku m −u n k|Au m ||A(u m −u n )| ≤ ν 6 |A(u m −u n )| 2 +C ν ku m −u n k 2 |Au m | 2 . (2.67) Regarding theT 2 , (2.22) yields: T 2 ≤ku n kku m −u n k 1/2 |A(u m −u n )| 3/2 ≤ ν 6 |A(u m −u n )| 2 +C ν ku m −u n k 2 ku n k 4 (2.68) forbothd = 2,3. ForthefinaltermT 3 ,weapply(2.23)conjunctionwithLemma2.2.1 and infer: T 3 ≤ ν 6 |A(u m −u n )| 2 +C ν |(P m −P n )B(u)| 2 ≤ ν 6 |A(u m −u n )| 2 + C ν λ n kQ n B(u n )k 2 ≤ ν 6 |A(u m −u n )| 2 + C ν λ n (ku n k|Au n | 3 +|u n | 1/2 |Au n | 7/2 ) ≤ ν 6 |A(u m −u n )| 2 + C ν λ 1/4 n ku n k 2 |Au n | 2 (2.69) 64 which is once again valid for eitherd = 2,3. Combining the estimates (2.67),(2.68) and (2.69) and using thatτ a ,τ b ∈T M,T m ∩T M,T n (c.f. (2.57)) we estimate: 2E Z τ b τa |hP m B(u m )−P n B(u n ),A(u m −u n )i|dt ≤νE Z τ b τa |A(u m −u n )| 2 ds +C ν E Z τ b τa ku m −u n k 2 (|Au m | 2 +ku n k 4 )+ 1 λ 1/4 n ku n k 2 |Au n | 2 ds ≤νE Z τ b τa |A(u m −u n )| 2 ds +C ν,M, ˜ M E Z τ b τa ku m −u n k 2 (1+|Au m | 2 )+ 1 λ 1/4 n (1+|Au n | 2 ) ds. (2.70) Applyingtheestimates(2.63),(2.64),(2.65)and(2.70)to(2.62)andrearranginggives: E sup t∈[τa,τ b ] ku m −u n k 2 +ν Z τ b τa |A(u m −u n )| 2 dt ! ≤CE ku m (τ a )−u n (τ a )k 2 + Z τ b τa 1 λ 1/4 n (1+|Au n | 2 )+|Q n f| 2 dt +CE Z τ b τa (1+|Au m | 2 )ku m −u n k 2 dt . (2.71) Note thatC =C ν,M, ˜ M,K V ,K D(A) does not depend on the particular choice ofτ a andτ b . Also: Z τ 0 1+|Au m | 2 dt≤ (M + ˜ M) 2 +T a.s. (2.72) Applying Lemma 2.5.4 we finally obtain: E sup t∈[0,τ] ku m −u n k 2 +ν Z τ 0 |A(u m −u n )| 2 dt ! ≤CE kQ n u 0 k 2 + Z T 0 |Q n f| 2 dt+ 1 λ 1/4 n . (2.73) 65 Since the constantC = C ν,M, ˜ M,K V ,K D(A) ,T does not depends onn,m or the choice of τ ∈T M,T m ∩T M,T n , (2.73) implies (2.58). For item (ii) we apply the Itˆ o calculus to find an evolution equation forku m k 2 : dku m k 2 +2ν|Au m | 2 dt = 2hf−B(u m ),Au m i+ ∞ X k=1 kP m g k (u m )k 2 ! dt +2 ∞ X k=1 hg k (u m ),Au m idβ k . (2.74) Fixτ ∈T M,T n andS > 0. Integrating (2.74) up toτ∧S yields: sup r≤S∧τ ku m k 2 + Z S∧τ 0 2ν|Au m | 2 dr ≤ku m 0 k 2 + Z S∧τ 0 2|hf−B(u m ),Au m i|dr + Z S∧τ 0 ∞ X k=1 kP m g k (u m )k 2 dr + sup r≤S∧τ ∞ X k=1 Z r 0 2((g k (u m ),u m ))dβ k . (2.75) Applying (2.22) we see that in bothd = 2,3: hB(u m ),Au m i≤ku m k 3/2 |Au m | 3/2 ≤C ν kuk 6 + ν 4 |Au| 2 . (2.76) Using this observation and the Lipschitz conditions imposed forg one find that: sup r≤S∧τ ku m k 2 + Z S∧τ 0 ν|Au m | 2 dr ≤ku m 0 k 2 +C ν,Kv Z S∧τ 0 (|f| 2 +ku m k 6 +ku m k 2 +1)dr + sup r≤S∧τ Z r 0 2 ∞ X k=1 ((g k (u m ),u m ))dβ k . (2.77) 66 This implies: P sup s≤τ∧S ku n (s)k 2 +ν Z τ∧S 0 |Au n | 2 ds>ku n 0 k 2 +(M−1) 2 ! ≤P C ν,K V Z S∧τ 0 (|f| 2 +ku m k 6 +ku m k 2 +1)dr> (M−1) 2 2 +P sup r≤S∧τ Z r 0 ∞ X k=1 ((g k (u m ),u m ))dβ k > (M−1) 2 2 ! . (2.78) For the first term on right hand side of (2.78) Chebyshev’s Inequality and the fact that τ ∈T M,T n implies: P C ν,K V Z S∧τ 0 (|f| 2 +ku m k 6 +ku m k 2 +1)dr> (M−1) 2 2 ! ≤ 2C ν,K V (M−1) 2 E Z S∧τ 0 (|f| 2 +ku m k 6 +ku m k 2 +1)dr ≤C ν,K V ,M, ˜ M E Z S 0 (|f| 2 +1)dr . (2.79) By applying Doob’s Inequality for the second term we estimate: P sup r≤S∧τ ∞ X k=1 Z r 0 ((g k (u m ),u m ))dβ k > (M−1) 2 2 ! ≤ 4 (M−1) 2 E Z S∧τ 0 ku m k 2 ∞ X k=1 kg k (u m )k 2 dr ! ≤C M, ˜ M,K V S. (2.80) Given the integrability assumed for f and noting that right hand sides of (2.79) and (2.80) are independent ofτ we have now established the second item (3.3). 67 2.4 ExistenceandUniqueness With the comparison estimates for the Galerkin systems in hand we next turn to the questionsofexistenceanduniqueness. Sincewewillneedtosplittheprobabilityspace into pieces (see below) it is convenient to first address the question of uniqueness. 2.4.1Proposition (Uniqueness). Letτ > 0 be a stopping time. Suppose that (u (1) ,τ) and(u (2) ,τ)arerespectivelylocalstrongandweaksolutionstothestochasticNavier- Stokes equations in d = 2,3. Let u (1) 0 , u (2) 0 be the associated initial conditions and assume that: P(1 1 Ω 0 u (1) 0 = 1 1 Ω 0 u (2) 0 ) = 0 (2.81) for someΩ 0 ∈F 0 . Then P(1 1 Ω 0 u (1) (t∧τ) = 1 1 Ω 0 u (2) (t∧τ);t∈ [0,∞)) = 0. (2.82) Proof. Subtractingu (2) fromu (1) we find that forv∈V: dh(u (1) −u (2) ),vi =−[νhA(u (1) −u (2) ),vi+hB(u (1) )−B(u (2) ),vidt + ∞ X k=1 hg k (u (1) )−g k (u (2) ),vidβ k . (2.83) The Itˆ o lemma yields: d|u (1) −u (2) | 2 +2νk(u (1) −u (2) )k 2 dt =−2hB(u (1) )−B(u (2) ),u (1) −u (2) idt + ∞ X k=1 |g k (u (1) )−g k (u (2) )| 2 dt +2 ∞ X k=1 hg k (u (1) )−g k (u (2) ),u (1) −u (2) idβ k . (2.84) 68 Using the cancellation property (2.20) and (2.21): |hB(u (1) )−B(u (2) ),u (1) −u (2) i| =|hB(u (1) −u (2) ,u (1) ),u (1) −u (2) i| ≤|u (1) −u (2) | 1/2 ku (1) −u (2) k 3/2 ku (1) k ≤νku (1) −u (2) k 2 +C ν ku (1) k 4 |u (1) −u (2) | 2 . (2.85) ForR> 0, define the stopping times: σ R = inf t>0 {ku (1) (t)k 2 >R}∧τ. (2.86) FixR andintegrate(2.84)up toσ R ∧t. Afterapplying (2.85),(2.35), rearranging and taking expectation we have: E1 1 Ω 0 |u (1) (σ R ∧t)−u (2) (σ R ∧t)| 2 +ν Z σ R ∧t 0 k(u (1) −u (2) )k 2 ds ! ≤C ν,K V E1 1 Ω 0 Z σ R ∧t 0 (ku (1) k 4 +1)|u (1) −u (2) | 2 ds ≤C ν,K V ,R E1 1 Ω 0 Z σ R ∧t 0 |u (1) −u (2) | 2 ds. (2.87) By the Gr¨ onwall inequality: E1 1 Ω 0 |u (1) (σ R ∧t)−u (2) (σ R ∧t)| 2 ! = 0, (2.88) which implies: 1 1 Ω 0 |u (1) (σ R ∧t)−u (2) (σ R ∧t)| 2 = 0 a.s. (2.89) 69 Note that: P(σ R <τ)≤P sup s∈[0,τ] ku (1) k 2 ≥R ! ≤ 1 R E sup s∈[0,τ] ku (1) k 2 ! → 0. (2.90) Thus: P(1 1 Ω 0 |u (1) (t∧τ)−u (2) (t∧τ)| 2 6= 0) =P({σ R <τ}∩{1 1 Ω 0 |u (1) (t∧τ)−u (2) (t∧τ)| 2 6= 0}) +P({σ R =τ}∩{1 1 Ω 0 |u (1) (t∧τ)−u (2) (t∧τ)| 2 6= 0}) ≤P(σ R <τ)+P(1 1 Ω 0 |u (1) (σ R ∧t)−u (2) (σ R ∧t)| 2 6= 0) ≤P(σ R <τ). (2.91) So for anyt: 1 1 Ω 0 |u (1) (t∧τ)−u (2) (t∧τ)| 2 = 0 a.s. (2.92) which implies: P(1 1 Ω 0 u (1) (t∧τ) = 1 1 Ω 0 u (2) (t∧τ);t∈ [0,∞)∩Q) = 1. (2.93) Given the continuity assumption in (2.42) we finally conclude (2.82) from (2.93). 2.4.1 ExistenceofLocalStrongSolutions 2.4.2Proposition (Local Existence). Suppose thatd = 2,3 and assume that: u 0 ∈L 2 (Ω;V), f ∈L 2 (Ω;L 2 ([0,∞);H)), g∈Lip u (H,` 2 (H))∩Lip u (V,` 2 (V))∩Lip u (D(A),` 2 (D(A))). (2.94) 70 Then there exists a local strong solution(u,τ). Proof. We proceed in two steps. First we assume thatku 0 k ≤ ˜ M, almost surely so that the estimates in Lemma 2.3.1 apply. Take{u n } to be the associated sequence of Galerkin solutions. Due to (2.58) and (3.3), the assumptions for Lemma 2.5.1, (i) are satisfied for the Banach spacesD(A) ⊂ V. We infer the existence of a subsequence {u n 0 },astrictlypositivestoppingtimeτ ≤T andaprocessu(·) =u(·∧τ),continuous inV, such that: sup t∈[0,τ] ku n 0 −uk 2 +ν Z τ 0 |A(u n 0 −u)| 2 ds→ 0 a.s. (2.95) Notice, moreover that u n 0 satisfies the conditions for Lemma 2.5.1, (ii) for any p ∈ (1,∞). Thus for any suchp: u(·∧τ)∈L p (Ω;C loc ([0,∞);V)) (2.96) and: u1 1 t≤τ ∈L p (Ω;L 2 loc ([0,∞);D(A))). (2.97) From Lemma 2.5.1, (ii) we also obtain a collection of measurable setsΩ n 0 ∈F with: Ω n 0 ↑ Ω (2.98) such that (c.f. (2.160)): sup n 0 E " sup t∈[0,T] ku n 0 (t∧τ)1 1 Ω n 0 k 2 +ν Z τ 0 |Au n 0 1 1 Ω n 0 | 2 ds # p/2 <∞. (2.99) 71 Given (2.95), (2.98) and (2.99) we next apply Lemma 2.5.2 and infer that: 1 1 Ω n 0,t≤τ u n 0 * 1 1 t≤τ u inL p (Ω;L 2 ([0,T];D(A))) (2.100) and that: 1 1 Ω n 0 u n 0 (·∧τ)* ∗ u inL p (Ω;L ∞ ([0,T];V)). (2.101) For the nonlinear term we apply (2.21) and estimate for bothd = 2,3: |hP n 0B(u n 0 )−B(u),vi| ≤|hB(u n 0 −u,u n 0 ),P n 0vi|+|hB(u,u n 0 −u),P n 0vi|+|hB(u),Q n vi| ≤C(ku n 0 −uk(ku n 0 k+kuk)kvk+kuk 2 |Q n v| 1/2 kvk 1/2 ) ≤C ku n 0 −uk(ku n 0 k+kuk)kvk+ 1 λ 1/4 n 0 kuk 2 kvk ! (2.102) for anyv∈V. By applying (2.95) with (2.102) we infer that: 1 1 t≤τ hP n B(u n ),vi→ 1 1 t≤τ hB(u),vi (2.103) 72 for almost every (ω,t)∈ Ω×[0,T]. Furthermore, making use of the uniform bound (2.99) withp = 4, one finds: sup n 0 E 1 1 Ω n 0 Z τ 0 |P n 0B(u n 0 )| 2 ds ≤Csup n 0 E 1 1 Ω n 0 Z τ 0 ku n 0 k 3 |Au n 0 |ds ≤Csup n 0 E 1 1 Ω n 0 sup t∈[0,τ] ku n 0 k 2 Z τ 0 |Au n 0 | 2 ds ! ≤Csup n 0 E1 1 Ω n 0 sup t∈[0,τ] ku n 0 k 4 + Z τ 0 |Au n 0 | 2 ds 2 ! <∞. (2.104) With(2.103)and(2.104)wemakeanotherapplicationofLemma2.5.2andgatherthat: 1 1 Ω n 0,t≤τ P n 0B(u n 0 )* 1 1 t≤τ B(u) inL 2 (Ω;L 2 ([0,T];H)). (2.105) For the stochastic terms the Lipschitz assumptions imply that: X k |P n 0g k (u n 0 )−g k (u)| 2 1/2 ≤ X k |P n 0g k (u n 0 )−P n 0g k (u)| 2 ! 1/2 + X k |Q n 0g k (u)| 2 ! 1/2 ≤C K V ku n 0 −uk+ 1 λ n 0 (1+kuk)ds . (2.106) With this estimate (2.95) implies: 1 1 t≤τ P n 0g(u n 0 )→ 1 1 t≤τ g(u) (2.107) 73 in` 2 (H) for almost every(ω,t)∈ Ω×[0,T]. On the other hand: sup n 0 E " 1 1 Ω n 0 Z τ 0 X k |P n 0g k (u n 0 )| 2 ds # ≤Csup n 0 E 1 1 Ω n 0 Z τ 0 (1+ku n 0 k 2 )ds <∞ (2.108) which means that: 1 1 Ω n 0,t≤τ P n 0g(u n 0 )* 1 1 t≤τ g(u) (2.109) inL 2 (Ω;L 2 ([0,T];` 2 (H))). We now apply Lemma 2.5.3 to the weakly convergent sequences (2.100), (2.105) and (2.109). For any fixedv∈H we deduce: 1 1 Ω n 0 Z t∧τ 0 (P n B(u n 0 ),v)ds* Z t∧τ 0 (B(u),v)ds 1 1 Ω n 0 Z t∧τ 0 (Au n 0 ,v)ds* Z t∧τ 0 (Au,v)ds 1 1 Ω n 0 X k Z t∧τ 0 (P n 0g k (u n 0 ),v)dβ k * X k Z t∧τ 0 (g k (u),v)dβ k (2.110) 74 allinthespaceL 2 (Ω×[0,T]). IfK ⊂ Ω×[0,T]isanymeasurablesetthenby(2.101) and (2.110): E Z T 0 χ K (u(t),v)dt = lim n→∞ E Z T 0 (1 1 Ω n 0 u n 0 (t∧τ),χ K v)dt = lim n→∞ E Z T 0 χ K 1 1 Ω n 0 (P n 0u 0 ,v)dt −E Z T 0 χ K 1 1 Ω n 0 Z t∧τ 0 (νAu n 0 +P n 0B(u n 0 )−P n 0f,v)ds dt +E Z T 0 χ K 1 1 Ω n 0 " X k Z t∧τ 0 (P n 0g k (u n 0 ),v)dβ k # dt ! =E Z T 0 χ K (u 0 ,v)− Z t∧τ 0 (νAu+B(u)−f,v)ds dt +E Z T 0 χ K " X k Z t∧τ 0 (g k (u),v)dβ k # dt. (2.111) Sincev,K above are arbitrary, (2.43) holds as an equation inH. The regularity con- ditions are given in (2.97), completing the proof in this case. NowsupposemerelythatEku 0 k 2 <∞. Fork≥ 0defineu k S ,τ k S to be the solution corresponding to the initial datau 0 1 1 k≤ku 0 k<k+1 . Let: u = ∞ X k=0 u k S 1 1 k≤ku 0 k<k+1 , τ = ∞ X k=0 τ k S 1 1 k≤ku 0 k<k+1 . (2.112) 75 We now show that (u,τ) is a local strong solution with initial data u 0 . Since u k S ∈ C([0,τ k S ],V)a.s.,weinferthatu∈C([0,τ];V)a.s. Moreoverbyapplying(2.158)we deduce: sup t∈[0,τ] kuk 2 +ν Z τ 0 |Au| 2 ds = ∞ X k=0 1 1 k≤ku 0 k<k+1 " sup t∈[0,τ] kuk 2 +ν Z τ 0 |Au| 2 ds # = N X k=0 1 1 k≤ku 0 k<k+1 " sup t∈[0,τ k S ] ku k S k 2 +ν Z τ k S 0 |Au k S | 2 ds # ≤C N X k=0 1 1 k≤ku 0 k<k+1 (M 2 +ku 0 k 2 ) ≤C(M 2 +ku 0 k 2 ). (2.113) Taking expectations one infers (2.45). Using that Eku 0 k 2 < ∞ one infers 1 = P ∞ k=0 1 1 k≤ku 0 k<k+1 a.s. Thus: u(t∧τ) = ∞ X k=0 1 1 k≤ku 0 k<k+1 u k S (t∧τ k S ) = ∞ X k=0 1 1 k≤ku 0 k<k+1 " u 0 − Z t∧τ k S 0 (νAu k S +B(u k S )−f)dt # + ∞ X k=0 1 1 k≤ku 0 k<k+1 " Z t∧τ k S 0 g(u k S )·dW # = ∞ X k=0 1 1 k≤ku 0 k<k+1 u 0 − Z t∧τ 0 (νAu+B(u)−f)dt+ Z t∧τ 0 g(u)·dW =u(0)− Z t∧τ 0 (νAu+B(u)−f)dt+ Z t∧τ 0 g(u)·dW, (2.114) where all equalities are inH. 76 2.4.2 MaximalExistenceTimeandBlow-Up In this section we establish the existence of a maximal strong solution (u,ξ). In addi- tionweshowthat(u,ξ)isaweaksolutionevenuptotheblowuptimeξ. Thefirststep is to show that if a strong solution exists up to timeτ one can (uniquely) extend this solution up to some stopping timeσ>τ. This is captured in the following Lemma: 2.4.3 Proposition. Assume that (u,τ) is a local strong solution of the stochastic Navier-Stokes equations and that τ is finite almost surely. Then there exists a local strong solution(u e ,σ) such thatσ>τ a.s. and such that: P(u e (t∧τ) =u(t∧τ);t∈ [0,∞)) = 1. (2.115) Proof. Define the stochastic basis: ˜ S := ( ˜ Ω, ˜ F,{ ˜ F t } t≥0 , ˜ P,{ ˜ β k (t)} k≥1 ) = (Ω,F,{F t+τ } t≥0 ,P,{β k (t+τ)} k≥1 ) (2.116) and let: ˜ u 0 =u(τ), ˜ f(t) =f(t+τ), ˜ g(·,t) =g(·,t+τ). (2.117) Observe that the data in (2.117) satisfies (2.94) for the basis ˜ S. As such, according to Proposition 2.4.2 there exists a local strong solution(˜ u,˜ τ) relative to ˜ S. Define: u e (t) =        u(t) fort≤τ ˜ u(t−τ) fort>τ. (2.118) Itisdirecttocheckthat(u e ,τ+˜ τ)isalocalstrongrelativetotheoriginalbasisS. 77 The next lemma establishes some further estimates on weak solutions (u,τ) in terms of the dataf,g andu 0 but that does not depend onτ. Note that this lemma will also be employed in the next section for the proof of global existence ind = 2. 2.4.1 Lemma. Suppose that in addition to the assumption imposed for Proposi- tion 2.4.2 that forp≥ 2: u 0 ∈L p (Ω;H) f ∈L p (Ω;L 2 loc ([0,∞);V 0 )). (2.119) If(u,τ) is local weak solution, then for anyT > 0: E " sup t∈[0,τ∧T] |u| p + Z τ∧T 0 kuk 2 |u| p−2 dt # ≤CE 1+|u 0 | p + Z T 0 |f| 2 V 0ds p/2 ! , (2.120) where the constantC :=C ν,K H ,p,T does not depend onτ. Proof. By (2.51), for any pair of stopping times0≤σ a ≤σ b ≤τ∧T: E sup s∈[σa,σ b ] |u| p +pν Z σ b σa kuk 2 |u| p−2 dt ≤C p E |u(σ a )| p + Z σ b σa hf,ui|u| p−2 dt+ Z σ b σa ∞ X k=1 |g k (u)| 2 |u| p−2 dt ! +C p E sup s∈[σa,σ b ] ∞ X k=1 Z s σa hg k (u),ui|u| p−2 dβ k ! . (2.121) 78 The first term on the right hand side of (2.121) is estimated by: C p Z σ b σa hf,ui|u| p−2 dt ≤C ν,p Z σ b σa |f| 2 V 0|u| p−2 dt+ νp 2 Z σ b σa kuk 2 |u| p−2 dt ≤C ν,p sup t∈[σa,σ b ] |u| p−2 ! Z σ b σa |f| 2 V 0dt+ νp 2 Z σ b σa kuk 2 |u| p−2 dt ≤ 1 6 sup t∈[σa,σ b ] |u| p + pν 2 Z σ b σa kuk 2 |u| p−2 dt+C ν,p Z σ b σa |f| 2 V 0ds p/2 . (2.122) Next, using the Lipschitz assumption forg we have: C p ∞ X k=1 Z σ b σa |g k (u)| 2 |u| p−2 dt≤C K H ,p Z σ b σa (1+|u| 2 )|u| p−2 dt ≤ 1 6 sup t∈[0,T] |u| p +C p,K H ,T 1+ Z σ b σa |u| p dt . (2.123) Finally: C p E sup s∈[σa,σ b ] ∞ X k=1 Z s 0 hg k (u),ui|u| p−2 dβ k ! ≤C p E Z σ b σa ∞ X k=1 hg k (u),ui 2 |u| 2(p−2) dt ! 1/2 ≤C K H ,p E Z σ b σa (1+|u| 2 )|u| 2(p−1) dt 1/2 ≤ 1 6 E sup t∈[σa,σ b ] |u| p ! +C p,T,K H E 1+ Z σ b σa |u| p dt . (2.124) 79 By applying the estimates (2.122), (2.123) and (2.124) to (2.121) we infer: E " sup t∈[σa,σ b ] |u| p + Z σ b σa kuk 2 |u| p−2 dt # ≤C ν,p,T,K H E 1+|u(σ a )| p + Z σ b σa |u| p dt+ Z σ b σa |f| 2 V 0ds p/2 ! . (2.125) We now employ Lemma to conclude (2.120). We now have everything we need to establish the existence of a maximal solution and a sequence of stopping times announcing any finite time blow-up. The argument is adapted from Jacod [26] and Mikulevicius and Rozovskii [37]. 2.4.4 Proposition. Given the assumptions in Proposition 2.4.2 there exists a unique maximal solution (u,ξ) and a sequenceρ n announcingξ. In addition, the pair (u,ξ) is a weak solution. Proof. Consider the setL of all stopping times such that τ ∈ L if and only if there exists a processu such that(u,τ) is a local strong solution. Notice that: σ 1 ,σ 2 ∈L⇒σ 1 ∨σ 2 ∈L (2.126) and that: σ∈L⇒ρ∧σ∈L, (2.127) for any stopping time ρ. Let ξ = supL (see [19], Chapter 5, Section 18). Using (2.126) we can choose an increasing sequenceσ k ∈L such that: σ k →ξ a.s. (2.128) 80 For eachσ k denote byu k the process such that (u k ,σ k ) is a local strong solution. Let: Ω k,k 0 ={u k (t∧σ k ∧σ k 0) =u k 0(t∧σ k ∧σ k 0);t∈ [0,∞)}. (2.129) By (2.127) and uniqueness (c.f. Proposition 2.4.1) we have that: ˜ Ω = \ k,k 0 Ω k,k 0 (2.130) is a set of full measure. For fixedω on this set, for everyt > 0 (simultaneously) the sequence{u k (t∧σ k )1 1 t<ξ } is Cauchy inV. Let: ˜ u(t) = lim k→∞ u k (t∧σ k )1 1 t<ξ a.s. (2.131) By Lemma 2.4.1 and the Monotone Convergence theorem, for anyT > 0: E " sup t∈[0,ξ∧T] |˜ u| 2 + Z ξ∧T 0 k˜ uk 2 dt # <∞. (2.132) We are therefore justified to define: hu(t),vi =hu(0),vi− Z t∧ξ 0 hνA˜ u+B(˜ u)−f,vidt+ ∞ X k=1 Z t∧ξ 0 hg k (˜ u),vidβ k , (2.133) for anyt > 0, v ∈ V. It is direct to check that fort < ξ(ω), u(t,ω) = ˜ u(t,ω) and that u is weakly continuous (almost surely) in H. These observations, with (2.132) and (2.133), implies that(u,ξ) is a local weak solution. ForR> 0 define the stopping time: ρ R = inf t≥0 ( sup s∈[0,t] kuk 2 + Z t 0 |Au| 2 ds>R ) ∧ξ. (2.134) 81 Clearly,(u,ρ R ) is a localstrong solution for anyR> 0. Suppose, toward a contradic- tion, that for someR,T sufficiently large,P(ρ R ∧T =ξ)> 0. By Lemma 2.4.3 this would imply the existence of an elementζ > ρ R ∧T a.s. withζ ∈L. But sinceξ is the supremal element ofL, we have our desired contradiction. We see moreover that {ρ R } R≥0 announcesξ. 2.4.3 GlobalExistenceforDimensionTwo In this section we prove that in d = 2 the maximal solution established in Proposi- tion 2.4.4 is global. 2.4.5 Proposition. Suppose that in addition to the assumptions imposed for Proposi- tion 2.4.2 that for somep≥ 4: u 0 ∈L p (Ω;H), f ∈L p (Ω;L 2 loc ([0,∞);V 0 )). (2.135) Ind = 2 the maximal solution(u,ξ) is global in the sense thatξ =∞ a.s. Proof. Let ρ n be an increasing sequence of stopping times announcing ξ. Observe that: {ξ <∞} = ∞ [ T=1 {ξ≤T} = ∞ [ T=1 ∞ \ n=1 {ρ n ≤T}. (2.136) Using thatρ n is increasing we have: P ∞ \ n=1 {ρ n ≤T} ! = lim N→∞ P N \ n=1 {ρ n ≤T} ! = lim N→∞ P(ρ N ≤T). (2.137) Thus to establish the desired result, it is sufficient to show that for any fixedT <∞: P(ρ N ≤T)→ 0 asN →∞. (2.138) 82 ForM > 0, define the stopping time: γ M := inf t≥0 Z t∧ξ 0 kuk 2 |u| 2 ds>M ∧T. (2.139) For any suchM we have: P(ρ N ≤T) ≤P sup t∈[0,ρ N ∧T] kuk 2 + Z ρ N ∧T 0 |Au| 2 ds≥N ! ≤P ( sup t∈[0,ρ N ∧T] kuk 2 + Z ρ N ∧T 0 |Au| 2 ds≥N ) ∩{γ M >T} ! +P(γ M ≤T) ≤P sup t∈[0,ρ N ∧γ M ] kuk 2 + Z ρ N ∧γ M 0 |Au| 2 ds≥N ! +P(γ M ≤T). (2.140) FixT,M,N and a pair of stopping timesτ a ≤τ b ≤ρ N ∧γ M . Integrating (2.52) for the casep = 2, and performing some standard manipulations using the weighted Young inequality, one finds: E sup t∈[τa,τ b ] kuk 2 +ν Z τ b τa |Au| 2 ds ≤C K V ,ν E ku(τ a )k 2 + Z τ b τa (|f| 2 +|hB(u),Aui|+kuk 2 +1)ds +E sup t∈[τa,τ b ] 2 ∞ X k=1 Z t τa ((g k (u),u))dβ k ! . (2.141) By making use of (2.22) ind = 2: hB(u),Aui≤|u| 1/2 kuk|Au| 3/2 ≤C ν |u| 2 kuk 4 + ν 2 |Au| 2 (2.142) 83 For last term in (2.141), the Burkholder-Davis-Gundy inequality implies: E sup t∈[τa,τ b ] 2 ∞ X k=1 Z t τa ((g k (u),u))dβ k ! ≤CE ∞ X k=1 Z τ b τa ((g k (u),u)) 2 dt ! 1/2 ≤C K V E Z τ b τa (1+kuk 2 )kuk 2 dt 1/2 ≤E 1 2 sup t∈[τa,τ b ] kuk 2 +C K V ,ν Z τ b τa (1+kuk 2 )dt ! . (2.143) Applying (2.142) and (2.143) to (2.141) and rearranging: E sup t∈[τa,τ b ] ku(t)k 2 +ν Z τ b τa |Au| 2 ds ! ≤C ν,K V E ku(τ a )k 2 + Z τ b τa (|u| 2 kuk 2 +1)kuk 2 +|f| 2 +1 ds . (2.144) The constant on the right hand side of (2.144) can be chosen independently ofτ a ,τ b . Also, by definition: Z γ M 0 |u| 2 kuk 2 ds≤M a.s. (2.145) Lemma 2.5.4 implies: E sup t∈[0,ρ N ∧γ M ] ku(t)k 2 +ν Z ρ N ∧γ M 0 |Au| 2 ds ! ≤C ν,Kv,M,T E ku 0 k 2 + Z T 0 (|f| 2 +1)ds . (2.146) Note thatC ν,Kv,M,T does not depend onN. Using (2.140) with (2.146) we infer: P(ρ N <T)≤ C N E ku 0 k 2 + Z T 0 (|f| 2 +1)ds +P(γ M ≤T). (2.147) 84 Thus, for any fixedM: lim N→∞ P(ρ N <T)≤P(γ M ≤T). (2.148) Finally, by applying Lemma 2.4.1 we find that asM →∞: P(γ M ≤T)≤P Z T∧ξ 0 kuk 2 |u| 2 dt≥M ≤ 1 M E Z T∧ξ 0 kuk 2 |u| 2 dt → 0. (2.149) 2.5 AbstractResults In this section we formulate and prove a collection of abstract lemmas which are employed above to circumvent the key difficulties related to compactness for the Galerkin scheme. As such we believe that these results could prove useful for the study of local well posedness for other nonlinear stochastic partial differential equa- tions. 2.5.1 APairwiseComparisonTheorem We shall make use of the following abstract comparison lemma. The formulation and proof extends Lemma 20 in [37]. Let(Ω,F,(F t ) t≥ ,P)beafixed,filteredprobabilityspace. SupposethatB 1 andB 2 are Banach Spaces withB 2 ⊂B 1 . We denote the associated norms by|·| i . Let: E(T) :=C([0,T];B 1 )∩L 2 ([0,T];B 2 ) (2.150) 85 with the norm: |Y| E(T) = sup t∈[0,T] |Y(t)| 2 1 + Z T 0 |Y(t)| 2 2 dt ! 1/2 . (2.151) LetX n be a sequenceB 2 valued random variables so that for everyT > 0: X n ∈E(T) a.s. (2.152) ForM > 1,T > 0 define the collection stopping times: T M,T n := τ ≤T :|X n | E(τ) ≤M +|X n (0)| 1 (2.153) and letT M,T n,n 0 :=T M,T n ∩T M,T n 0 . 2.5.1Lemma. (i) Suppose that forM > 1 andT > 0: lim n→∞ sup n 0 ≥n sup τ∈T M,T n 0 ,n E|X n −X n 0| E(τ) = 0 (2.154) and that: lim S→0 sup n sup τ∈T M,T n P |X n | E(τ∧S) >|X n (0)| 1 +M−1 = 0. (2.155) Then there is a stopping timeτ with: P(0<τ ≤T) = 1 (2.156) 86 and processX(·) =X(·∧τ)∈E(τ) with: |X n l −X| E(τ) → 0 a.s. (2.157) for some subsequencen l ↑∞. Moreover: |X| E(τ) ≤M +sup n |X n (0)| 1 . (2.158) (ii) If, in addition to the conditions imposed in (i): sup n E|X n (0)| q 1 <∞, (2.159) for some1≤q<∞. Then there exists of sequence of setΩ l ↑ Ω such that: sup l E1 1 Ω l |X n l | q E(τ) <∞ (2.160) and: E|X| q E(τ) ≤C q M q +sup n E|X n (0)| q 1 . (2.161) Proof. For (i), due to (2.154), a subsequencen l can be chosen in such a way that: sup τ∈T M,T n l+1 ,n l E|X n l −X n l+1 | E(τ) ≤ 2 −2l . (2.162) Define: τ l := inf t>0 |X n l | E(t) >|X n l (0)| 1 +(M−1+2 −l ) ∧T. (2.163) 87 Noting thatτ l ∈T M,T l we have: P |X n l −X n l+1 | E(τ l ∧τ l+1 ) ≥ 2 −(l+2) ≤ 2 l+2 E|X n l −X n l+1 | E(τ l ∧τ l+1 ) ≤ 2 −(l−2) . (2.164) Take: Ω N = ∞ \ l=N |X n l −X n l+1 | E(τ l ∧τ l+1 ) < 2 −(l+2) . (2.165) By the Borel-Cantelli lemma ˜ Ω := lim N Ω N is a set of full measure. Observe that forl ≥ N, τ l ≤ τ l+1 on the set Ω N . Indeed on{τ l+1 > τ l }∩ Ω N , τ l <T meaning that: |X n l+1 | E(τ l ∧τ l+1 ) >|X n l | E(τ l ∧τ l+1 ) −2 −(l+2) =|X n l | E(τ l ) −2 −(l+2) =|X n l (0)| 1 +(M−1+2 −l )−2 −(l+2) >|X n l+1 (0)| 1 +(M−1+2 −l )−2·2 −(l+2) =|X n l+1 (0)| 1 +(M−1+2 −(l+1) ). (2.166) On the other hand: |X n l+1 | E(τ l ∧τ l+1 ) ≤|X n l+1 | E(τ l+1 ) ≤|X n l+1 (0)|+(M−1+2 −(l+1) ), (2.167) which is a contradiction. Defineτ = lim l τ l . ForT >> 0 we have: {τ l <}⊂{|X n l | E(τ l ∧) >|X n l (0)| 1 +(M−1)} (2.168) 88 which implies that: P(τ <) =lim l P(τ l <) ≤limsup l P(τ l <) ≤limsup l P(|X n l | E(τ l ∧) >|X n l (0)| 1 +(M−1)). (2.169) Making use of the condition imposed in (2.155): P(τ = 0) =P(∩ >0 {τ <}) = lim ↓0 P(τ <) = 0 (2.170) which is equivalent to (2.156). Combining the above observations we see thatX n l is Cauchy inE(τ) a.s. As such there exists a processX(·) =X(·∧τ)∈E(τ) such that: lim l→∞ |X n l −X| E(τ) = 0 a.s. (2.171) Toestablish(2.158),(2.160)and(2.161)takeΩ l asin(2.165). Since,asestablished above,τ l ≥τ l+1 ≥τ onΩ l : 1 1 Ω l |X n l | E(τ) ≤ 2 −(l+2) +1 1 Ω l |X n l+1 | E(τ) ≤|X n l+1 (0)| 1 +M ≤ sup n |X n (0)| 1 +M (2.172) which yields the first item. The second inequality in (2.172) implies: E 1 1 Ω l |X n l | q E(τ) ≤C p (M q +E|X n l (0)| q 1 ). (2.173) 89 The bound (2.160) follows as a consequence of (2.159). By applying Fatou’s lemma we infer (2.161). 2.5.2 WeakConvergenceLemmas We next recall several elementary results concerning weak convergence that are used to uniquely identify certain limits that one infers from Lemma 2.5.1. The next result relies on Egoroff’s Theorem and is similar to [18], pg. 266. 2.5.2 Lemma. Suppose thatX is a separable Banach space and letX ∗ be the dual. Denote the dual pairing betweenX andX ∗ byh·,·i. Assume that (E,E,μ) is a finite measure space and that p ∈ (1,∞). Given elements u,u n ∈ L p (E,X ∗ ) with{u n } uniformly bounded inL p (E,X ∗ ) and: hu n ,yi→hu,yi μ−a.e. (2.174) for ally∈X. Then: u n * ∗ u (2.175) inL p (E,X ∗ ). Proof. Let {y k } be countable, dense subset of X. Consider any subsequence n 0 . Alaoglu’s theorem implies the existence of a further subsequencen 00 and an element u 00 ∈L p (E,X ∗ ) with: u n 00 * ∗ u 00 inL p (E,X ∗ ). (2.176) This implies: hu n 00 ,y k i*hu 00 ,y k i inL p (E). (2.177) 90 Fix> 0. By Egoroff’s theorem, for eachk we can choose a setE k, , so that: μ(E k, )≤ 2 k (2.178) with sup ω∈(E k, ) C |hu n (ω)−u(ω),y k i|→ 0, asn→∞. (2.179) Furthermore since bothu,u 00 ∈L p (E,X ∗ ) we adjustE k, so that: sup ω∈(E k, ) C |hu(ω)−u 00 (ω),y k i|<∞. (2.180) TakeE = ∪ k E k, . Due to (2.174) and (2.180) along with the integrability assumed foru−u 00 : lim n 00 →∞ Z (E) C hu n 00 ,y k ih(u−u 00 ),y k idμ = Z (E) C hu,y k ih(u−u 00 ),y k idμ. (2.181) On the other hand (2.180) implies that for any fixedk,h(u−u 00 ),y k iχ E C ∈ L p 0 (E) wherep 0 = p p−1 istheconjugateofp. Assuchtheweakconvergencein(2.177)assures that: lim n 00 →∞ Z (E) C hu n 00 ,y k ih(u−u 00 ),y k idμ = Z (E) C hu 00 ,y k ih(u−u 00 ),y k idμ. (2.182) Combining (2.181) and (2.182) and adjustingE on a set of measure zero: hu(ω)−u 00 (ω),y k i = 0, (2.183) 91 for everyω ∈ E C , k ≥ 0. TakeE = ∩ n E 1/n . By continuity, E is a set of measure zero. By(2.183)u =u 00 offofthisset. Sincetheoriginalsubsequencen 0 wasarbitrary, the proof is complete. The second lemma concerns the preservation of weak limits under bounded linear transformations: 2.5.3Lemma. (i) SupposeB 1 ,B 2 are Banach spaces and thatL : B 1 → B 2 is a bounded linear mapping. Ifx n *x inB 1 thenLx n *Lx inB 2 . (ii) Forp∈ (1,∞) take: L 1 (w)(t) = Z t 0 wds w∈L p (Ω×[0,T]) Ifx n *x inL p (Ω×(0,T)) and thenL 1 (x n )*L 1 (x) in the same space. (iii) Take: L 2 (v)(t) = X k Z t 0 v k dβ k for v = {v k } ∈ L 2 (Ω,L 2 (0,T;` 2 )). Given that v n * v in this space then L 2 (v n )*L 2 (v) inL 2 (Ω;L 2 (0,T)) 2.5.3 AGr¨ onwallLemmaForStochasticProcesses The following simple lemma is used several times above in an analogous manner to the classical Gr¨ onwall Lemma: 2.5.4Lemma. FixT > 0. Assume that: X,Y,Z,A : [0,T)×Ω→R (2.184) 92 are real valued, non-negative stochastic processes. Let τ < T be a stopping time so that: E Z τ 0 (AX +Z)ds<∞. (2.185) Assume, moreover that for some fixed constantM: Z τ 0 Ads<M a.s. (2.186) Suppose that for all stopping times0≤τ a <τ b ≤τ: E sup t∈[τa,τ b ] X + Z τ b τa Yds ! ≤C 0 E X(τ a )+ Z τ b τa (AX +Z)ds (2.187) whereC 0 is a constant independent of the choice ofτ a ,τ b . Then: E sup t∈[0,τ] X + Z τ 0 Yds ! ≤CE X(0)+ Z τ 0 Zds (2.188) WhereC =C(C 0 ,T,M). Proof. Choose a (finite) sequence of stopping times: 0 =τ 0 <τ 1 <...<τ N <τ N+1 =τ (2.189) so that: Z τ k τ k−1 Ads< 1 2C 0 a.s. (2.190) 93 Foreachpairτ k−1 ,τ k wetakeasupremumintimeforX onthelefthandsideof(2.187) and rearrange to deduce: E sup t∈[τ k−1 ,τ k ] X + Z τ k τ k−1 Yds ! ≤CEX(τ k−1 )+CE Z τ k τ k−1 Zds. (2.191) Assuming that: E sup t∈[0,τ j ] X + Z τ j 0 Yds ! ≤CEX(0)+CE Z τ j 0 Zds (2.192) then: E sup t∈[0,τ j+1 ] X + Z τ j+1 0 Yds ! ≤CEX(0)+CE Z τ j 0 Zds +CE sup t∈[τ j ,τ j+1 ] X + Z τ j+1 τ j Yds ! ≤CEX(0)+CE Z τ j+1 0 Zds+CEX(τ j ) ≤CEX(0)+CE Z τ j+1 0 Zds. (2.193) 94 Chapter3 SingularPerturbationSystemswith SmallStochasticForcingandthe RenormalizationGroupMethod 3.1 Introduction Perturbation theory has long played an important role in applied analysis. Perturbed systemscanprovidetherelevantsettingforthestudyofphysicalphenomenaexhibiting multiplespatialandtemporalscales. Inthefieldofclimatology,forexample,thebasic momentum equations naturally exhibit multiple orders of magnitude, with the largest order being driven by the Coriolis term arising from to the earth’s rotation. A basic challenge in the study of perturbed dynamical systems is that the unper- turbedproblemmayexhibitfundamentallydifferentquantitativeandqualitativebehav- iors,particularlyforlargetimescalesornearcertainboundaries. Suchdifficultieswere appreciated as early as the 18th century in the study of multi-body problems in celes- tial mechanics. Needless to say such “singular” systems are notoriously challenging to analyze. A variety of asymptotic methods have been developed, each germane to differ- ent types of singular perturbation problems. These methods seek to find approximate solutions to perturbed systems that separate scales and remain valid over long time 95 intervals. See the recent texts of Verhulst [51], [52] for a broad introductory survey for deterministic systems. For the stochastic setting see Freidlin and Wentzell [22] or Skorokhod, Hoppensteadt and Habib [47] for an overview. One approach, the renormalization group technique has enjoyed considerable suc- cess in recent investigations of singularly perturbed systems. The method was first developedbyChen,GoldenfeldandOonointhecontextofperturbativequantumfield theory. See [10] or [11] and references therein. Subsequently work of Ziane [53] and later joint work of DeVille, Harkin, Holzer, Josi´ c, and Kaper [17] put the subject on a firm mathematical foundation. Using this technique, Moise, Simonnet, Temam, and Ziane [39] conducted a series of Numerical simulations of ordinary differential equations arising in geophysical fluid dynamics. In the context of partial differential equations the method has been applied to Navier-Stokes type systems by Moise and Ziane[41]andMoiseandTemam[40]. The2-dNavier-Stokesequationsperturbedby a small additive white noise was addressed using this method by Bl¨ omker, Gugg, and Maier-Paape [3]. In any physical system uncertainties (measurement error, unresolved scales or interactions,numericalinaccuracies,etc.) arisethatarehardtoaccountforinthebasic model. One would like to be able to quantify the robustness of a model, particularly one involving singular perturbations, in the presence of these uncertainties. As such, itisnaturaltoincorporatesmallwhitenoisedrivenperturbationsintoexistingsingular models. As far as we are aware [3] is the only investigation to apply Renormalization group techniques to a stochastically driven system. In particular the crucial case of highly oscillatory systems remains unaddressed up to the present. 96 In this work we investigate a class of stochastic equations taking the form: dX + 1 AX dτ =F(X )dτ + α G(τ,τ/)dW. (3.1) The goal is to study the behavior of the system when becomes small. A is a linear operatorwhichisassumedtobeeithersymmetricpositivesemidefiniteorantisymmet- ric. F is a nonlinear operator, but may exhibit important cancellation properties that arise physically. dW is a white noise process in the appropriate sense. The parameter α determines the strength of the noise term. In most cases we restrictα > 0. In the case of strictly dissipative systems we are also able to address the case of a noise term of moderate strength,−1/2<α≤ 0. Following the classical techniques we attempt to find an approximate solution via a naive perturbation expansion of the solution, setting u ≈ u (0) +u (1) . This can (and usually does) break down due to resonances between the nonlinear termF and the semi-group generated by A. Additionally we face the new difficulty of accounting forintermediatescalediffusionintroducedbythesmallnoiseterm. Tocompensatefor this we derive the renormalized system: dV =R(V )dτ + α H(τ,τ/)dW. (3.2) This system greatly simplifies the original equations. The structure ofH depends on A and the desired rate of convergence for the approximate solution. In particular, we show that we can take H to be identically zero at the cost of a reduced rate of convergence. EvenforcaseswhereH isnon-zero,sending tozeroreducestoamore tractable small noise asymptotic problem. 97 The solution V of (3.2) defines an approximate solution ¯ X = e −τ/A V . Note thatinourformulationweshowthattheoscillatorycorrectiontermintroducedinother investigations ([53], [41], [39], [40] etc.) is unnecessary. We prove below that if the behavior ofV is reasonable, namely that for someK > 0 P sup τ∈[0,T] kV (τ)k>K ! →0 −−→ 0, (3.3) then ¯ X is a valid approximation in the sense that: P sup τ∈[0,T] k ¯ X (τ)−X (τ)k>C γ ! →0 −−→ 0. (3.4) Here, γ > 0, and dependsα as well as the structure ofA andH. Note that beyond establishing convergence in probability, typical for such results, we also have also managed to establish a rate of convergence. One interesting consequence of these results is that when A ≡ 0 in the original system (3.1), (3.4) can be interpreted as a small noise asymptotic result. While such results are classical for systems with Lipschitz nonlinear terms (see [22] or [15]) our result allows us to address the physically important case when one can only expect cancellations in the nonlinear portion of the equation. We next turn to some concrete examples. In particular we consider stochastic ver- sions of a meteorological model developed by Lorenz in [33] and of a simple model for turbulent flow proposed by Temam in [48]. In each case we exhibit the renormal- ization group which decouples the two scale inherent in the original system. Current workinpreparationbythefirstauthorseekstoapplytheresultsinthisworktoaseries of numerical studies of these and other systems. 98 The final section collects some general results concerning slowly varying stochas- tic processes. The estimates that we establish in this section form the analytical core of the main approximation results and may hold independent interest for other stud- iesofstochasticsingularperturbationsystems. Appendicescollectfurthersmallnoise asymptoticresultsandaswellassomemostlyclassicalestimatesonstochasticconvo- lutions terms arising in the proof of the main theorems. 3.2 ADerivationoftheRenormalizationGroup Forα∈Rconsiderthesingularperturbationsystemgivenbythestochasticdifferential equation: dY (τ)+ 1 ¯ AY (τ)dτ = ¯ F(Y (τ))dτ + α ¯ G(τ,τ/)dW Y (0) =Y 0 . (3.5) We assume that ¯ A is a diagonalizable matrix. Below, we will analyze the cases when ¯ A is antisymmetric and when ¯ A is positive semidefinite. The nonlinear term ¯ F, is a polynomial.W = (W 1 ,...,W m ) ⊥ isastandardmdimensionalBrownianmotionrel- ative to some underlying filtered probability space (Ω,F,(F τ ) τ≥0 ,P). ¯ G takes values inM n×m and is bounded in the Frobenius norm independently ofτ andτ/: sup τ≥0 k ¯ G(τ,τ/)k 2 = sup τ≥0 X j,k | ¯ G j,k (τ,τ/)| 2 <∞. (3.6) Further assumptions will be imposed on ¯ G when we consider the case when ¯ A is positive semidefinite. Throughout what followsT is a fixed large time. 1 The goal will 1 This time interval can be improved slightly with no additional mathematical complications. See Remark 3.3.1 99 be to study the behavior ofX on this time interval when is small and in the limit as → 0. For the analysis below we shall work in a different basis. LetQ be an orthogonal matrix diagonalizing ¯ A. Take: X (τ) :=QY (τ), F(x) :=Q ¯ F(Q ∗ x), G(τ,τ/) :=Q ¯ G(τ,τ/). (3.7) Multiplying (3.5) byQ gives the following evolution system forX : dX (τ)+ 1 AX (τ)dτ =F(X (τ))dτ + α G(τ,τ/)dW X (0) =X 0 . (3.8) Note that in this basisX may evolve inC n . The results below are easily translated back to the original coordinate frame for the system (3.5). See Remark 3.2.1. The next step is to writeX in a naive perturbation expansion in powers of. The goal is to derive a reduced system that approximates (3.8) on large time intervals and that separates scales. To this end, we consider (3.8) on a new time scalet =τ/: dX (t)+AX (t)dt =F(X (t))dt+ α+1/2 G(t,t)d ˜ W X (0) =X 0 . (3.9) Below we set β = α + 1/2. Note that ˜ W, also a standard Brownian motion, is a rescaling ofW given by: ˜ W(t) = 1 √ W(t) = 1 √ W(τ). (3.10) 100 Whenβ < 2, the perturbation expansion has the form 2 : X (t) =X (0) (t)+ β X (β) +X (1) (t)+O( 2 ) X 0 =X (0) (0). (3.11) We plug (3.11) into (3.9) and match power of to formally derive the coupled system of equations: dX (0) =−AX (0) dt; X (0) (0) =X 0 dX (β) =−AX (β) dt+Gd ˜ W; X (β) (0) = 0 dX (1) = −AX (1) +F(X (0) ) dt; X (1) (0) = 0 which admits the solution: X (0) (t) =e −At X 0 X (β) (t) = Z t 0 e −A(t−s) G(s,s)d ˜ W X (1) (t) = Z t 0 e −A(t−s) F(e −As X 0 )ds. (3.12) We have therefore derived the approximate solution: ¯ X (t) =e −At X 0 + Z t 0 e As F(e −As X 0 )ds+ β Z t 0 e As G(s,s)d ˜ W +O( 2 ) =e −At X 0 + Z t 0 e As F(e −As X 0 )ds + β Z t 0 e −A(t−s) G(s,s)d ˜ W +O( 2 ). (3.13) 2 Note however that no such restriction on β is needed for Proposition 3.3.1 or Proposition 3.3.2 below. 101 Theapproximation(3.13)assumesthatX (1) remainO(1)withhighprobabilityonthe timescale1/. 3 ThismaybreakdowninX (1) duetoresonanceswithF,asanalyzedin [53]. In contrast to previous investigations we also need to account for the role of the noisetermin(3.13). Asweshallseebelow,whenthenoiseis“small”(i.e. whenα> 0 intheoriginalsystem(3.5)or(3.8))anapproximatesolutionthatcompletelydropsthe stochastic terms can be shown to be valid on the time scale 1/. In the case of strictly dissipativesystemswhereAis(strictly)positivedefiniteweshallseethatthenoisecan be even taken to be of intermediate strength (for exampleα = 0). On the other hand wealsoshowthatincludingsomeorallofthenoisetermsintheapproximatesolution leads to a faster rate of convergence between the real and the approximate solutions. 3.2.1 ResonanceAnalysisfortheNonlinearTerm WenowsplitF intoitsresonantandnon-resonantpartswithrespecttothesemi-group generated byA. SinceF is a polynomial we can write: F(u) = n X j=1   X |α|≤m C j α u α   b j . 3 We formalize this statement with the following rigorous definition: Definition. Suppose that: δ 1 (·),δ 2 (·) : [0, 0 ]→ [0,∞) (3.14) are monotonically decreasing functions. We say that a collection of processes: X (·) : Ω×[0,∞)→C n (3.15) areO(δ 1 ()) with high probability on time scale 1/δ 2 () if for allT > 0 there exists aK > 0 so that: P sup t∈[0,T/δ2()] kX (t)k>Kδ 1 () ! ↓0 − − → 0 (3.16) 102 Hereb j thejthstandardbasiselementinC n . LetΛ ={λ 1 ,...,λ n }betheeigenvalues ofA. Given any multindexα we adopt the notation(Λ,α) := P n i=1 α i λ i . Let: N j r ={α∈N n :|α|≤m, (Λ,α) =λ j }. (3.17) The resonant terms inF are given by: R(u) = n X j=1   X α∈N j r C j α u α   b j . (3.18) Notice that for anyu∈C n e At R(e −At u) =R(u). (3.19) Define: F NR (u) =F(u)−R(u) = n X j=1   X α6∈N j r C j α u α   b j (3.20) and take S(t,u) =e tA F NR (e −tA u) = n X j=1   X α6∈N j r C j α e t(λ j −(Λ,α)) u α   b j . (3.21) We have derived the decompositions: e At F(e −At u) =R(u)+S(t,u), F(u) =e −At R(e At u)+F NR (u) =R(u)+F NR (u). (3.22) 103 Applying (3.22) to (3.13) we find: ¯ X =e −At X 0 +tR(X 0 )+ Z t 0 S(s,X 0 )ds+ β Z t 0 e As G(s,s)d ˜ W +O( 2 ) =e −At X 0 +tR(X 0 )+ Z t 0 S(s,X 0 )ds + β Z t 0 e −A(t−s) G(s,s)d ˜ W +O( 2 ). (3.23) Suppose that β > 1/2 (equivalently α > 0 in the original time scale). For γ < β−1/2, Lemma 3.6.1 implies that: P sup t∈[0,T/] β Z t 0 e −A(t−s) G(s,s)d ˜ W > γ ! ≤CT 2(β−γ)−1 ↓0 −→ 0. (3.24) As such β R t 0 e −A(t−s) G(s)d ˜ W isO( γ ) with high probability on the time scale 1/. Referring back to (3.23) we arrive at the renormalized system: dV (t) =R(V )dt V (0) =V 0 , (3.25) with the associated approximate solution taking the form: ¯ X (t) =e −At V (t)+ Z t 0 S(s,V (s))ds . (3.26) Thisisexactlytheformtheapproximatesolutiontakesinotherworks([53],[41],[39], [3] etc.) However by applying Proposition 3.5.1, one can show that: Z t 0 e −At S(s,V )ds = Z t 0 e −A(t−s) F NR (e −sA V )ds (3.27) 104 remains O(1) with high probability on time scale 1/ as long as V remains O(1) with high probability on this time scale. We are therefore justified in simplifying the approximate solution to: ¯ X (t) =e −At V (t) (3.28) This approximate solution satisfies: d ¯ X +A ¯ X dt =R( ¯ X )dt ¯ X (0) = ¯ X 0 =V 0 . (3.29) In the original time scale the renormalization group takes the form: dV (τ) =R(V (τ))dτ V (0) =V 0 , (3.30) and the approximate solution is given by: ¯ X (τ) =e −(τ/)A V (τ) (3.31) which solves: d ¯ X + 1 A ¯ X dτ =R( ¯ X )dτ ¯ X (0) = ¯ X 0 =V 0 . (3.32) Note that the approximate solution in (3.31) splits the dynamics into two scales with the (possibly) intermediate scale introduced by the diffusion term of no consequence totheasymptoticvalidityoftheapproximation. Employingtheestimate(3.24)weare 105 able to establish that ¯ X converges toX with high probability on time scale 1 with a convergence rate γ for anyγ <α. More precisely we show that: P sup τ∈[0,T] k ¯ X (τ)−X (τ)k>C γ ! →0 −−→ 0, (3.33) foranysuchγ <α. Inorderimprovetheseconvergencerateswemaywishtoinclude some stochastic terms in the renormalization group. The analysis must consider the antisymmetric and positive semidefinite cases separately. 3.2.2 TheAntisymmetricCase: ImprovedConvergenceRates SupposethatAisantisymmetricalongwiththestandingassumptionthatβ > 1/2(i.e. α> 0). If we include all of the noise in the renormalized system we obtain: dV (t) =R(V )dt+ β e tA G(t,t)d ˜ W, V (0) =V 0 , (3.34) which reads as: dV (τ) =R(V )dτ + α e (τ/)A G(τ,τ/)dW, V (0) =V 0 , (3.35) in the original time scale. Since A is antisymmetric (3.6) implies that sup τ>0 ke (τ/)A G(τ,τ/)k < ∞. As such the small noise asymptotic results estab- lished below (c.f. Remark 3.3.2) show that for a large class of physically motivated systems, if the deterministic renormalized system in (3.30) isO(1) with high prob- ability on time scale 1 then so is the solution of (3.35). We are justified, as in the previous case, in assuming thatV is a slowly varying process in the sense of (3.110), 106 (3.111) and (3.112). Once again Proposition 3.5.1 justifies simplifying the approxi- matesolution ¯ X accordingto(3.28)(or(3.31)intheoriginaltimescale). Inthiscase ¯ X satisfies: d ¯ X +A ¯ X dt =R( ¯ X )dt+ β G(t,t)d ˜ W ¯ X (0) = ¯ X 0 =V 0 (3.36) or in time scaleτ: d ¯ X + 1 A ¯ X dτ =R( ¯ X )dτ + α G(τ,τ/)dW ¯ X (0) = ¯ X 0 =V 0 (3.37) Whencomparingthissystemto(3.8),thestochastictermscancelexactly. Theapprox- imate solution thus enjoys a faster rate of convergence to the original system (3.8). In particular we prove below that: P sup τ∈[0,T] k ¯ X (τ)−X (τ)k>C ! →0 −−→ 0. (3.38) 3.2.1Remark. Insomeapplications,particularlyfortheantisymmetriccase,thegov- erning equations (3.5) are given in a non-diagonalized basis, relative to the dominant linear term. The renormalized systems defined by (3.30) or (3.35) can be translated back to this original basis. Let: U =Q ∗ V , ¯ R(·) =Q ∗ R(Q·), ¯ Y =Q ∗ ¯ X . (3.39) 107 If we takeV according to (3.30) then in the original basisU solves: dU (τ) = ¯ R(U (τ))dτ U (0) =U 0 . (3.40) On the other hand if we include stochastic forcing in the renormalized system as in (3.35) then: dU (τ) = ¯ R(U (τ))dτ + α e τ/ ¯ A ¯ G(τ,τ/)dW U (0) =U 0 . (3.41) In either case, the approximate solution is given by: ¯ Y (τ) =e −(τ/) ¯ A U (τ). (3.42) In the first case ¯ Y is the solution of: d ¯ Y + 1 ¯ A ¯ Y dτ = ¯ R( ¯ Y )dτ ¯ Y (0) = ¯ Y 0 =U 0 , (3.43) whereas: d ¯ Y + 1 ¯ A ¯ Y dτ = ¯ R( ¯ Y )dτ + α ˜ G(t,t)dW ¯ Y (0) = ¯ Y 0 =U 0 , (3.44) inthesecondcase. TheresultsinProposition3.3.1applywhenwereplaceV ,X and ¯ X withU ,Y and ¯ Y respectively. 108 3.2.3 The Positive Semidefinite Case: Improved Convergence RatesandtheCaseofLargeNoise We next consider the case whenA is positive semidefinite. Here, beyond the uniform time bounds in (3.6) we assume that G is inM n×n and diagonal. LetM,N be the projections onto ker(A) and ker(A) ⊥ respectively. This applying this decomposition to (3.23) gives: ¯ X =e −At X 0 +tR(X 0 )+ Z t 0 S(s,X 0 )ds+ β Z t 0 MG(s,s)d ˜ W + β Z t 0 e −A(t−s) NG(s,s)d ˜ W +O( 2 ) (3.45) Suppose that β > 0 (i.e. α > −1/2 in the original time scale). By applying Lemma 3.6.2, we have the estimate: P sup t∈[0,T/] β Z t 0 e −A(t−s) NG(s,s)dW > γ ! ≤C(q)T q(β−γ)−1 , (3.46) for any 0 < γ < β. By choosing q > max{(β − γ) −1 ,2} we see that β R t 0 e −A(t−s) G(s)dW remainsO( γ ) with high probability on time scale 1/. With this in mind we define the renormalization group: dV (t) =R(V (t))dt+ β MG(t,t)d ˜ W, V (0) =V 0 , (3.47) which gives: dV (τ) =R(V (τ))dτ + α MG(τ,τ/)dW, V (0) =V 0 , (3.48) 109 in the original time scale. Note that in order to apply either Proposition 3.5.1 or Remark 3.3.2 to this renormalized system we must require either thatM = 0 (equiv- alently thatA is strictly positive definite) or thatα > 0. The approximate solution is defined in either case by ¯ X (t) =e −tA V which is the solution of: d ¯ X +A ¯ X dt =R( ¯ X )dt+ β MG(t,t)d ˜ W ¯ X (0) = ¯ X 0 =V 0 (3.49) Returning to the original time scale we have ¯ X (τ) =e −(τ/)A V (τ) which solves: d ¯ X + 1 A ¯ X dτ =R( ¯ X )dτ + α MG(τ,τ/)dW ¯ X (0) = ¯ X 0 =V 0 (3.50) Relying on the stronger estimates available for the stochastic convolution due to the dissipation in the system (c.f. (3.46) above) we are able to show that: P sup τ∈[0,T] k ¯ X (τ)−X (τ)k>C γ ! →0 −−→ 0, (3.51) whereγ <α+1/2. 3.3 RigorousApproximationResults Wenowrigorouslystateandprovethemainresultslegitimating ¯ X asanapproximate solution of (3.8). We start with the highly oscillatory case. 110 3.3.1 Proposition. Assume that X solves (3.8) with A antisymmetric and α > 0. Suppose thatV solves either (3.30) or (3.35) so that for anyT > 0 we can choose K > 0: P sup τ∈[0,T] kV (τ)k>K ! :=φ 1 () →0 −−→ 0. (3.52) Additionally, we posit that: P(kV 0 −X 0 k>K) :=φ 2 () →0 −−→ 0. (3.53) (i) If we takeV as in (3.30) and therefore ¯ X as in (3.31) then for anyγ <α: P sup τ∈[0,T] k ¯ X (τ)−X (τ)k>C γ ! →0 −−→ 0. (3.54) whereC =C(K,F). (ii) If, on the other hand,V solves (3.35) and ¯ X is given by (3.37) then: P sup τ∈[0,T] k ¯ X (τ)−X (τ)k>C ! →0 −−→ 0. (3.55) 3.3.1 Remark. As in [3], T can be replaced withT := T log(1/) in (3.52), (3.54) and (3.55) (or in (3.86), (3.88), (3.89) below) with no additional complications. Proof. Throughout the following we work on the time scalet. LetZ (t) =e At X (t). This process satisfies: dZ =e At F(e −At Z )+ β e At G(t,t)d ˜ W; Z (0) =X 0 . (3.56) 111 SinceA is antisymmetric: kV (t)−Z (t)k =k ¯ X (t)−X (t)k, t≥ 0. (3.57) Thus, we are justified to consider the error process: E (t) :=V (t)−Z (t). (3.58) For the first case, (i), using (3.22): E (t) =E 0 + Z t 0 R(V )−e As F(e −As Z ) ds− β Z t 0 e As G(s,s)d ˜ W =E 0 + Z t 0 e As F(e −As V )−e As F(e −As (V −E ) ds− Z t 0 S(s,V )ds − β Z t 0 e As G(s,s)d ˜ W. (3.59) In the case (ii) there is no stochastic integral term: E (t) =E 0 + Z t 0 R(V )−e As F(e −As Z ) ds =E 0 + Z t 0 e As F(e −As V )−e As F(e −As (V −E ) ds− Z t 0 S(s,V )ds. (3.60) Notice that in either caseV has the form (3.110) and satisfies the conditions (3.111) and (3.112). For α 6∈ N j R , let K α,j be the constant arising in Proposition 3.5.1 with f(u) =C j α u α . Define: C 1 = n X j=1 X α6∈N j r K α,j . (3.61) 112 Applying Proposition 3.5.1 we estimate: P sup t∈[0,T/] Z t 0 S(s,V )ds >C 1 ! ≤ X 1≤j≤n α6∈N j R P sup t∈[0,T/] Z t 0 e t(λ j −(Λ,α)) C j α (V ) α ds >K α,j ! ↓0 −→ 0. (3.62) For case (i) we need to estimate the stochastic integral term in (3.59). Lemma 3.6.1 implies: P sup t∈[0,T/] β Z t 0 e As G(s,s)d ˜ W > γ ! =P sup t∈[0,T/] Z t 0 e −A(t−s) G(s,s)d ˜ W > γ−β ! ≤CT 2(β−γ)−1 =CT 2(α−γ) ↓0 −→ 0. (3.63) By the mean value theorem: kF(e −At V )−F(e −At V (t)−e −At E (t))k ≤ sup σ∈[0,1] kD(F)(e −At V (t)−σe −At E (t))kkE (t)k. (3.64) Let: η (t) = sup σ∈[0,1] kD(F)(e −At V (t)−σe −At E (t))k. (3.65) Note that: Z t 0 e As F(e −As V )−F(e −As (V −E ) ds ≤ Ξ (t), (3.66) 113 where: Ξ (t) = Z t 0 η (s)kE (s)kds. (3.67) Define: G ={kE 0 k≤K}∩ ( sup t∈[0,T/] Z t 0 S(s,V )ds ≤C 1 ) ∩ ( sup t∈[0,T/] β Z t 0 e As G(s,s)d ˜ W ≤ γ ) (3.68) in case (i) and: G ={kE 0 k≤K}∩ ( sup t∈[0,T/] Z t 0 S(s,V )ds ≤C 1 ) (3.69) for (ii). Due to (3.53) and (3.62) and in the first case (3.63) we have: P(G C ) →0 −−→ 0. (3.70) Wecompletetheproofforeither(i)or(ii)usingamaximalargument. Wegivethe details for (i), the other case is nearly identical. LetC 2 = K +C 1 +1. We have for any sampleω coming fromG : kE (t,ω)k≤ γ C 2 +Ξ(t,ω). (3.71) Thus: dΞ (t,ω) dt =η (t,ω)kE (t,ω)k≤ γ (C 2 +Ξ (t,ω))η (t,ω). (3.72) 114 The Gr¨ onwall’s inequality implies: Ξ (t,ω)≤ γ C 2 exp Z t 0 γ η(s,ω)ds Z t 0 η(s,ω)ds. (3.73) Forω∈G , defineI (ω)⊂ [0,T/] to be the maximal interval so thatkE (s,ω)k≤ 1 foreverys∈I (ω). Notethatfor<K −1 thedefinitionofG insuresthatthisinterval is nontrivial for almost everyω∈G . Let: C 3 := sup t∈R + kxk≤K,kyk≤1 sup σ∈[0,1] kD(F)(e −At (x−σy))k ! . (3.74) By using the estimate (3.73) we obtain: kE (s,ω)k≤ γ (C 2 e γ TC 3 TC 3 )≤ γ C 4 , (3.75) for anys∈ I(ω). Note thatC 4 can be chosen independently ofω and. As such for < 1 C 4 1/γ it follows thatI = [0,T/]. The proof is complete. A special case of Proposition 3.3.1, (i) can be interpreted as a small noise asymp- totic result: 3.3.1Corollary. Suppose that for> 0,X solves: dX =F(X )dτ + α G(τ,τ/)dW; X (0) =X 0 . (3.76) and thatx is the solution of the associated deterministic system: dx =F(x )dτ; x (0) =x 0 . (3.77) 115 Assume thatG(·,·) is uniformly bounded as in (3.6) that: hF(u),ui≤ 0 for allu∈R n , (3.78) that: P(kX 0 −x 0 k>K α ) →0 −−→ 0. (3.79) and finally that: P(kx 0 k>K) →0 −−→ 0. (3.80) Then for anyγ <α: P sup τ∈[0,T] kX (τ)−x (τ)k>C γ ! →0 −−→ 0. (3.81) WhereC :=C(F,K) Proof. Notice that (3.77) is in the form of (3.5) 4 for A ≡ 0. Defining R according to (3.18) we find that F(u) = R(u) and that the renormalization group derived in Section 3.2.1 is given by (3.77). Due to (3.78) we have: kx (τ)k 2 =kx 0 k 2 +2 Z τ 0 hF(x ),x idτ ≤kx 0 k 2 . (3.82) By applying (3.80) we find that the condition (3.52) holds for x . The conclusion (3.81) therefore follows from Proposition 3.3.1, (i). 3.3.2Remark. 4 In this case, (3.5) and (3.8) are identical 116 (i) One important class of systems that satisfy (3.78) arise when: F(u) =Lu+B(u,u), (3.83) whereL is linear and either antisymmetric or positive semidefinite andB is a bilinear form satisfying the cancellation propertyhB(u,v),vi = 0. (ii) Small noise asymptotic results involving Lipschitz continuous nonlinear terms are classical and have been studied by many authors (see [22] and [15]). As noted, Corollary 3.3.1 covers the physically important case when we can only expect cancellations in the nonlinear portion of the equation. In Section 3.7 below we provide a different proof of Corollary 3.3.1 that covers the case of multiplicative noise and also establishes convergence to the deterministic limit system inL p (Ω) forp≥ 2. (iii) Corollary 3.3.1 can sometimes be used to verify (3.52) for V , the solution of (3.35). Suppose that according to (3.18) R(u) = Lu + B(u,u), so that hR(u),ui≤ 0. Take: dv =R(v )dτ; v (0) =v 0 . (3.84) IfV 0 −v 0 satisfies(3.79)andv 0 fulfills(3.80),eachwithconstantκ,thenCorol- lary 3.3.1 and the calculation in (3.82) imply: P sup τ∈[0,T] |V (τ)|>κ+1 ≤P sup τ∈[0,T] |V (τ)−v (τ)|> 1 ! +P sup τ∈[0,T] |v (τ)|>κ ! ↓0 −→ 0. (3.85) 117 In a similar manner one may verify (3.86) forV satisfying (3.48) in Proposi- tion 3.3.2 below. We next address the positive semidefinite case: 3.3.2Proposition. Assume thatX solves (3.8) withA positive semidefinite. Suppose that V is a solution of either (3.30) or (3.48) so that for T > 0 there is a constant K > 0 such that: P sup τ∈[0,T] kV (τ)k>K ! :=φ 1 () →0 −−→ 0. (3.86) Additionally, suppose that the initial dataV 0 approximatesX 0 with high probability for small: P(kV 0 −X 0 k>K) :=φ 2 () →0 −−→ 0. (3.87) (i) If α > 0 in (3.8) and V solves (3.30) then there is a positive constant C = C(K,F) so that wheneverγ≤α: P sup τ∈[0,T] k ¯ X (τ)−X (τ)k>C γ ! →0 −−→ 0, (3.88) where ¯ X (τ) :=e −( τ )A V (τ). (ii) Inadditiontotheuniformbound(3.6),supposethatG(·,·)isdiagonal. Ifα> 0 in (3.8) andV is the solution of (3.48) then: P sup τ∈[0,T] k ¯ X (τ)−X (τ)k>C γ ! →0 −−→ 0, (3.89) for anyγ <α+1/2. 118 (iii) If, moreover,A is (strictly) positive definite (i.e. σ(A) is strictly positive) then if α>−1/2, (3.89) holds forγ <α+1/2. Proof. As in the proof of Proposition 3.3.1 we work on time scalet. Here we define the error process byE (t) = ¯ X (t)−X (t). Subtracting (3.29) from (3.9) in case (i), or (3.49) from (3.9) in cases (ii) and (iii), we arrive at the system: dE +AE dt = R( ¯ X )−F(X ) dt− β PG(t,t)dW; E (0) =V 0 −X 0 . (3.90) Incase(i)P =I whileinthelatercasesP =N,theprojectiononker(A) ⊥ . Wehave: E (t) =e −At E 0 + Z t 0 e −A(t−s) R( ¯ X )−F(X ) ds− β Z t 0 e −A(t−s) PG(t,t)dW =e −At E 0 + Z t 0 e −A(t−s) e −As R(e As ¯ X )−F(X ) ds − β Z t 0 e −A(t−s) PG(t,t)dW =e −At E 0 + Z t 0 e −A(t−s) F( ¯ X )−F( ¯ X −E ) ds − Z t 0 e −A(t−s) F NR ( ¯ X )ds− β Z t 0 e −A(t−s) PG(t,t)dW. (3.91) First we estimate the term: Z t 0 e −A(t−s) F NR ( ¯ X )ds = n X k=1 X α6∈N j r Z t 0 e −λ j (t−s) C j α ( ¯ X ) α ds b j = n X k=1 X α6∈N j r Z t 0 e −λ j (t−s) e −s(Λ,α) C j α (V ) α ds b j . (3.92) 119 Here λ j = A jj ≥ 0 (since A is assumed diagonal, this is the jth eigenvalue of A) and (Λ,α) = P k α k λ k . Recalling (3.17) we have λ j 6= (Λ,α), for each α 6∈ N j r . Notice that for cases (i) and (iii)V takes the form (3.110) trivially since both (3.25) and (3.47) are deterministic systems (since σ(A) > 0 in the later case M = 0). On the other hand, for case (ii), β > 1/2 in (3.47). Combining these observations with the boundedness condition (3.86) we see that each of the terms in (3.92) satisfy Lemma 3.5.1 (ii). Similarly to (3.61) and (3.62) we infer a constantC 1 , depending on C j α u α andλ j in (3.92), so that: P sup t∈[0,T/] Z t 0 e −A(t−s) F NR ( ¯ X )ds >C 1 ! ↓0 −→ 0. (3.93) We next address the stochastic integral terms. For case (i) we estimate: P sup t∈[0,T/] β Z t 0 e −A(t−s) G(t,t)dW > γ ! ≤CT 2(β−γ)−1 ↓0 −→ 0, (3.94) using Lemma 3.6.1. On the other hand for cases (ii) and (iii) we apply Lemma 3.6.2, choosingq> max{(β−γ) −1 ,2} so that: P sup t∈[0,T/] β Z t 0 e −A(t−s) NG(t,t)dW > γ ! ≤C(q)T q(β−γ)−1 ↓0 −→ 0. (3.95) The final step is to estimate the term: Z t 0 e −A(t−s) F( ¯ X )−F( ¯ X −E ) ds. (3.96) 120 By the mean value theorem: kF( ¯ X )−F( ¯ X −E )k ≤ sup σ∈[0,1] kD(F)( ¯ X (s)−E (s))kkE (t)k. (3.97) letting: η (t) = sup σ∈[0,1] kD(F)(e −As V (s)−σE (s))k. (3.98) Similarly to the proof of Proposition 3.3.1 we define: Ξ (t) = Z t 0 η (s)kE (s)kds. (3.99) and: G = ( sup t∈[0,T/] ke −At E 0 k≤K ) ∩ ( sup t∈[0,T/] Z t 0 e −A(t−s) F NR ( ¯ X )ds ≤C 1 ) ∩ ( sup t∈[0,T/] β Z t 0 e −A(t−s) NG(s)d ˜ W ≤ γ ) , (3.100) or: G = ( sup t∈[0,T/] ke −At E 0 k≤K ) ∩ ( sup t∈[0,T/] Z t 0 e −A(t−s) F NR ( ¯ X )ds ≤C 1 ) ∩ ( sup t∈[0,T/] β Z t 0 e −A(t−s) G(s)d ˜ W ≤ γ ) (3.101) The maximal argument is then employed exactly as in Proposition 3.3.1 to complete the proof. 121 3.4 Applications: Some Examples from Fluid Dynam- ics Withtheresultsdevelopedaboveinhandwenowconsiderseveralexamplesofstochas- ticallyperturbedmulti-scalesystems. Thesesimplemodelswerefirstconsideredinthe deterministic setting in [39] and are motivated by the numerical study of geophysical fluid flow and turbulence. For each example below observe that the renormalized sys- temdecouplesthefastandslowcomponentsoftheoriginalsystem. Thestochastically forced renormalized systems are seen to satisfy (3.52) (or (3.86)) for a large class of initial conditions by Remark 3.3.2. We further remark that for the first two examples the perturbation systems are written in the non-diagonalized form (3.5). As such we exhibit the deterministic and stochastic renormalized system according to (3.40) and (3.41)respectively. SeeRemark3.2.1. Afuturearticlewhichdevelopsapplicationsof theresultsinthisworktothenumericalintegrationofsingularstochasticsystemswill consider the examples below in more detail. 3.4.1 Example (Linear Renormalization Group). For the first example we consider a nonlinear equation that exhibits a linear renormalization group: dY 1 +(λ 1 Y 1 − 1 Y 2 −Y 1 Y 3 )dτ = √ dW 1 dY 2 + λ 2 Y 2 + 1 Y 1 +Y 2 Y 3 dτ = √ dW 2 dY 3 + λ 3 Y 3 +Y 2 1 −Y 2 2 dτ = √ dW 3 (3.102) 122 where λ 1 , λ 2 , λ 3 are fixed constants. After some routine computations we conclude that renormalization group has the form: dU 1 + 1 2 (λ 1 +λ 2 )U 1 dτ = 0 dU 2 + 1 2 (λ 1 +λ 2 )U 2 dτ = 0 dU 3 +λ 3 U 3 dτ = 0 (3.103) The stochastic counterpart is given by: dU 1 + 1 2 (λ 1 +λ 2 )U 1 dτ = √ cos(τ/)dW 1 − √ sin(τ/)dW 2 dU 2 + 1 2 (λ 1 +λ 2 )U 2 dτ = √ sin(τ/)W 1 + √ cos(τ/)dW 2 dU 3 +λ 3 U 3 dτ = √ dW 3 . (3.104) 3.4.2 Example (The Lorenz System). We next consider a system advanced by E. Lorenz(see[33])withasmalladditivenoise. Letλ 1 ,λ 2 andbbefixedpositiveparam- eters. dY 1 = (−λ 1 Y 1 −Y 2 Y 3 +bY 2 Y 5 )dτ + √ dW 1 dY 2 = (−λ 1 Y 2 +2Y 1 Y 3 −2bY 1 Y 5 )dτ + √ dW 2 dY 3 = (−λ 1 Y 3 −Y 1 Y 2 )dτ + √ dW 3 dY 4 = −λ 2 Y 4 − 1 Y 5 dτ + √ dW 4 dY 5 = −λ 2 Y 4 + 1 Y 4 +bY 1 Y 2 dτ + √ dW 5 . (3.105) 123 In this case the renormalized systems are given by: dU 1 =(−λ 1 U 1 −U 2 U 3 )dτ dU 2 =(−λ 1 U 2 +2U 1 U 3 )dτ dU 3 =(−λ 1 U 3 −U 1 U 2 )dτ dU 4 =−λ 2 U 4 dτ dU 5 =−λ 2 U 5 dτ (3.106) and by: dU 1 =(−λ 1 U 1 −U 2 U 3 )dτ + √ dW 1 dU 2 =(−λ 1 U 2 +2U 1 U 3 )dτ + √ dW 2 dU 3 =(−λ 1 U 3 −U 1 U 2 )dτ + √ dW 3 dU 4 =−λ 2 U 4 dτ + √ cos(τ/)dW 4 − √ sin(τ/)dW 5 dU 5 =−λ 2 U 5 dτ + √ sin(τ/)dW 4 + √ cos(τ/)dW 5 . (3.107) 3.4.3Example(ASymmetricPositiveSemidefinite). Thefinalexampleaddressesthe dissipative case. dX 1 +(X 1 +X 1 X 2 )dτ = √ dW 1 dX 2 + 1 X 2 −X 2 1 = √ dW 2 . (3.108) 124 This system was studied in [48] as a simple model for the numerical simulation of turbulent fluid flows. Here we have replaced the external forcing terms with a small white noise. The renormalized system has the form: dV 1 +V 1 dτ = √ dW 1 dV 2 = 0. (3.109) 3.5 LargeTimeEstimatesforSlowlyVaryingProcesses Thefollowinggeneralresultestablishesthattimeintegralsofslowlyvaryingprocesses againstoscillatoryordissipativetermsremainO(1)withhighprobabilityonlargetime scales. In the other time scaleτ =t, result oscillation limit 3.5.1Proposition. Suppose that, for> 0,Z ∈C n solves: dZ =Φ(t,Z )dt+ β Ψ(t,Z )dW; Z (0) =Z 0 . (3.110) Here we suppose thatβ > 1/2, Φ and Ψ are uniformly bounded in time for bounded subsets ofC n in the sense that, for anyR> 0: sup t>0,kxk≤R kΦ(t,x)k<∞, sup t>0,kxk≤R kΨ(t,x)k<∞. (3.111) Assume that for some constantK > 0: P sup t∈[0,T/] kZ (t)k>K ! :=φ() ↓0 −→ 0. (3.112) Then: 125 (i) For anyσ∈R\{0} and any analytic functionf :C n →C: P sup t∈[0,T/] Z t 0 e iσs f(Z )ds >C ! ↓0 −→ 0, (3.113) whereC :=C(f,σ,T). (ii) Suppose thatZ take values inR n . Assume thatλ,σ ≥ 0 withσ 6=λ. Then for anyC 2 functionf :R n →R: P sup t∈[0,T/] Z t 0 e −λ(t−s) e −σs f(Z )ds >C ! ↓0 −→ 0, (3.114) whereC :=C(f,σ,T). Proof. In both cases the Itˆ o formula gives: df(Z ) = n X k=1 ∂f(Z ) ∂z k Φ k (t,Z )dt+ β n X k=1 ∂f(Z ) ∂z k (Ψ(t,Z )dW) k + 2β 2 n X k,l,j=1 ∂ 2 f(Z ) ∂z k ∂z l Ψ k,j (t,Z )Ψ l,j (t,Z ) dt :=H D (t,Z )dt+ β H S (t,Z )dW + 2β H C (t,Z )dt. (3.115) For (i), integrating by parts and using (3.115) we have: Z t 0 e iσs f(Z )ds = e iσt f(Z (t))−f(Z (0)) iσ + Z t 0 e iσs iσ (H D (s,Z )+ 2β−1 H C (s,Z ))ds+ β Z t 0 e iσs iσ H S (s,Z )dW. (3.116) 126 Define: C 1 := 2 |σ| sup kzk≤K |f(z)| C 2 := T |σ| sup t>0,kzk≤K (|H D (t,z)|+|H C (t,z)|) C 3 := 1 |σ| sup t>0,kzk≤K kH S (t,Z )k. (3.117) Note that (3.111) assures that these constants are finite. For the first term in (3.116): P sup t∈[0,T/] e iσt f(Z (t))−f(Z (0)) iσ >C 1 ! ≤φ(). (3.118) Next: P sup t∈[0,T/] Z t 0 e iσs iσ (H D (s,Z )+ 2β−1 H C (s,Z ))ds >C 2 ≤P sup t∈[0,T/] |σ| Z t 0 (|H D (s,Z )|+|H C (s,Z )|)ds>C 2 ! ≤P sup t∈[0,T/] T(|H D (t,Z (t))|+|H C (t,Z (t))|) |σ| >C 2 ! ≤φ(). (3.119) For the final term from (3.116), define the stopping time: ξ ,K := inf t≥0 {kZ (t)k>K}. (3.120) 127 Splitting the integral and making use of Doob’s inequality: P sup t∈[0,T/] β Z t 0 e iσs iσ H S (s,Z )dW(s) >C 3 ! ≤P sup t∈[0,T/] β Z t 0 e iσs iσ H S (s,Z )1 1 ξ ,K >s dW(s) >C 3 ! +P(ξ ,K ≤T/) ≤ 2β C 2 3 E Z T/ 0 1 1 ξ ,K >s e iσs iσ H S (s,Z ) 2 ds+ψ() ≤ 2β C 2 3 E Z T/ 0 1 1 ξ ,K >s C 2 3 ds+ψ 1 ()≤ 2β−1 T +ψ(). (3.121) Setting C = C 1 +C 2 +C 3 and applying (3.118), (3.119) and (3.121) with (3.116) implies the desired result. For item (ii), the integration by parts reveals: e −λt Z t 0 e (λ−σ)s f(Z )ds = e −σt f(Z (t))−e −λt f(Z (0)) λ−σ + Z t 0 e −λ(t−s) e −sσ λ−σ (H D (s,Z )+ 2β−1 H C (s,Z ))ds + β Z t 0 e −λ(t−s) e −sσ λ−σ H S (s,Z )dW. (3.122) We define constantsC 1 andC 2 similarly to the previous case: C 1 := 2 |λ−σ| sup kzk≤K |f(z)| C 2 := T |λ−σ| sup kzk≤K,t>0 (|H D (t,Z )|+|H S (t,Z )|). (3.123) 128 The estimates for the first two terms on the left hand side of (3.122) are carried out in the same manner as (3.118) and (3.119). For the stochastic integral terms we consider two cases. First whenλ> 0 we takeC 3 = 1. We have: P sup t∈[0,T/] β Z t 0 e −λ(t−s) λ−σ e −σs H S (s,Z )dW(s) > 1 ! ≤P sup t∈[0,T/] Z t 0 e −λ(t−s) λ−σ 1 1 ξ ,K >s e −σs H S (s,Z )dW(s) > −β ! +P(ξ ,K ≤T/) ≤CT 2β−1 +φ(), (3.124) where we apply Lemma 3.6.1 with G(t) = 1 1 ξ ,K >t e −σt H S (t,Z ) for the second inequality. On the other hand, if λ = 0 then σ > 0 and we take C 3 as in (3.117). The stochastic integral estimate can be made as in (3.121) 3.6 AppendixI:StochasticConvolutionEstimates The following Lemmas are use to estimate terms arising in the proof of Proposi- tion 3.3.1, (i) and Proposition 3.3.2 3.6.1 Lemma. Suppose thatH takes values inM n×m , thatW = (W 1 ,...,W m ) T is a standard m dimensional Brownian motion and that A inM n×n is symmetric non- negative definite or antisymmetric. Assume thatH is uniformly bounded in time: sup t>0 kH(t)k = sup t>0 X j,k |H k,j (t)| 2 ! 1/2 <∞ (3.125) 129 Given constantsK,T 0 > 0: P sup t∈[0,T 0 ] Z t 0 e −A(t−s) H(s)dW >K ! ≤ CT 0 K 2 (3.126) whereC :=C(kHk) Proof. Recall that: X(t) = Z t 0 e −A(t−s) H(s)dW (3.127) is the solution of: dX +AXdt =HdW; X(0) = 0. (3.128) A calculation employing Itˆ o’s lemma reveals: dkXk 2 +2hAX,Xi = 2 X j hH ·,j ,XidW j +kHk 2 dt. (3.129) Using the Burkholder-Davis-Gundy inequality (c.f. [27], Theorem 3.28) we estimate: E sup t∈[0,T 0 ] 2 X j Z t 0 hH ·,j ,XidW j ! ≤CE Z T 0 0 X j hH ·,j ,Xi 2 ds ! 1/2 ≤CE Z T 0 0 kHk 2 kXk 2 ds 1/2 ≤ 1 2 E sup t∈[0,T 0 ] kXk 2 +CE Z T 0 0 kHk 2 ds. (3.130) This estimate with (3.129) reveals that: E sup t∈[0,T 0 ] kXk 2 ! ≤C Z T 0 0 kHk 2 ds. (3.131) 130 The Chebyshev inequality therefore implies (3.126) completing the proof. Given additional assumptions on H one can improve on the estimates in (3.126) when A is non-negative-definite. The proof relies on the Factorization method. See Lemma 5.1 in [4] and also Theorem 5.9 in [15]. 3.6.2 Lemma. Suppose thatA andH(·) are diagonal with sup t>0 kH(t)k < ∞ and the spectrum ofA real and non-negative. For q≥ 2 we have: P sup t∈[0,T 0 ] Z t 0 e −A(t−s) NH(s)dW >K ! ≤ CT 0 K q (3.132) whereC :=C(q,H) andN is the projection ontoker(A) ⊥ Proof. Since the proof is very similar to Lemma 5.1 in [4] we shall be brief in details. The factorization method relies on the identity: Z t s (t−σ) α−1 (σ−s) −α dσ = π sin(πα) , (3.133) whichisvalidforα∈ (0,1)andforanys<t. From(3.133)andthestochasticFubini Theorem (see [15], Theorem 4.18) we infer: Z t 0 e −A(t−s) NH(s)dW s = sin(πα) π Z t 0 e −A(t−s) Z t s (t−σ) α−1 (σ−s) −α dσ NH(s)dW s = sin(πα) π Z t 0 (t−σ) α−1 e −A(t−σ) N Z σ 0 (σ−s) −α e −A(σ−s) NH(s)dW s dσ. (3.134) 131 Letλ mp = min{A jj > 0}. Byapplyingthedecomposition(3.134)withtheChebyshev and H¨ older inequalities we find that for anyq> 2: P sup t∈[0,T 0 ] Z t 0 e −A(t−s) NH(s)dW >K ! ≤C(α,q,λ mp ) T 0 K q sup σ>0 E Z σ 0 (σ−s) −α e −A(σ−s) NH(s)dW q , (3.135) where we can take: C(α,q,λ mp ) = sin(πα) π q Z ∞ 0 e −(q 0 λmp)σ σ q 0 (1−α) dσ q/q 0 . (3.136) Note that this constant is finite for any choice ofα> 1/q. To complete the proof we estimate the moments of: M(σ) := Z σ 0 (σ−s) −α e −A(σ−s) NH(s)dW. (3.137) This process is Gaussian with mean zero and covariance matrix: V α jk (σ) =δ jk Z σ 0 (σ−s) −2α e −2λ j (σ−s) N jj H jj (s) 2 ds. (3.138) As such we restrict α ∈ (1/q,1/2). Fixing some α in this range we take, V j (σ) = V α jj (σ). Notice, moreover: V ∗ j = sup σ>0 V j (σ)<∞. (3.139) 132 For appropriately small values ofs, the moment generating function ofM(σ) is given by: φ(s) =E(exp(skM(σ)k 2 ) = Y λ j 6=0 E(exp(s|M jj (σ)| 2 ) = Y λ j 6=0 (2πV j (σ)) −1/2 Z ∞ −∞ exp − 1−2V j (σ)s 2V j (σ) x 2 j dx j = exp − 1 2 n X j=1 ∞ X m=1 (2sV j (σ)) m m ! . (3.140) Differentiatingφ(s), we find: E(|M(σ)| 2k ) =φ k (0)≤C(k) n X j=1 V ∗ j ! k . (3.141) Applying (3.141) with2k>q to (3.135), one infers (3.132). 3.7 AppendixII:SmallNoiseAsymptoticResults In this section we present an alternative proof of the small noise asymptotic result in Corollary3.3.1. Notethatwhiletheapproachbelowiscloserinspirittoclassicalwork (see [22], for example) we are able to address the case of multiplicative noise. Consider stochastically perturbed system: dX +[LX +B(X ,X )]dt = α σ(t,,X )dW X (0) =X 0 (3.142) 133 whereα> 0. The related deterministic system is given by: dx dt +[Lx+B(x,x)] = 0 x(0) =x 0 . (3.143) We assume thatL is linear and either non-negative definite or anti-symmetric. B(.,.) is a bilinear form with the cancellation property: hB(y,x),xi = 0. (3.144) As in the previous sections, W = (W 1 ,...,W m ) is a standard Brownian motion relative to a filtered probability space (Ω,F,(F t ) t≥0 ,P). The diffusion termσ takes values inM n×m . OnM n×m we use the Frobenius norm |A| = P j,k A 2 j,k 1/2 and impose the uniform Lipschitz condition: |σ(t,,x)−σ(t,,y)|≤K|x−y| |σ(t,,x)|≤K(1+|x|). (3.145) Note thatK is assumed to be independent of both andt. 3.7.1Remark. UndertheconditionsgivenforL,B andσ andassumingthatX 0 isF 0 measurable (3.142) admits a unique continuous solution. If, in addition, we assume thatX 0 ∈L p (Ω) then: E Z T 0 |X (t)| p dt<∞ (3.146) See [20] for detailed proofs. We first establish sufficient conditions for convergence inL p (Ω): 134 3.7.1 Proposition (L p (Ω) Convergence). LetT > 0, p ≥ 2, X 0 ∈ L p (Ω) for > 0 andx 0 be a fixed element inR n . Assume that for some appropriate constantC 0 : E|X 0 −x 0 | p ≤ αp C 0 . (3.147) Then: E sup t∈[0,T] (1+|X (t)| 2 ) p/2 ≤C (3.148) and for> 0: E sup t∈[0,T] |X (t)−x(t)| p ≤ αp C (3.149) whereC =C(|x 0 |,T,L,B,K,p). For the proof of Proposition 3.7.1 we shall use the following Lemma that func- tions as a stochastic analogue of the Gr¨ onwall Lemma. See [23] for a more general formulation and the proof. 3.7.1Lemma. Suppose thatY,Z : [0,T]×Ω→R + are stochastic processes so that: E Z T 0 (Y +Z)dt<∞. (3.150) Assume that for any0≤S a ≤S b ≤T: E sup t∈[Sa,S b ] Y(t) ! ≤C 0 E Y(S a )+ Z S b Sa (Y +Z)dt (3.151) whereC 0 is independent of the choice ofS a ,S b . Then: E sup t∈[0,T] Y(t) ! ≤CE Y(0)+ Z T 0 Zdt (3.152) 135 whereC =C(C 0 ,T) We now turn to the proof of the Proposition: Proof - Proposition 3.7.1: We first establish (3.148). By applying Itˆ o’s lemma to 1+ |X (t)| 2 , using of the cancellation property for B and then applying Itˆ o’s lemma to (1+|X (t)| 2 ) p/2 we discover: d(1+|X | 2 ) p/2 =−p(1+|X | 2 ) p/2−1 hLX ,X idt + 2α p 2 (1+|X | 2 ) p/2−1 X j,k σ j,k (t,,X ) 2 dt + α p(1+|X | 2 ) p/2−1 X j,k X j σ j,k (t,,X )dW k + 2α p(p−2) 2 (1+|X | 2 ) p/2−2 X k X j X j σ j,k (t,,X ) ! 2 dt. (3.153) 136 Given the assumptions on L the first term on the right hand side of (3.153) is non- positive for allt > 0. Fix arbitrary 0≤ S a < S b ≤ T. Integrate (3.153) betweenS a andt and take a supremum over[S a ,S b ]. After taking an expected value we have: E sup t∈[Sa,S b ] (1+|X | 2 ) p/2 ≤E(1+|X (S a )| 2 ) p/2 +C(p)E Z S b Sa (1+|X | 2 ) p/2−1 X j,k σ j,k (s,,X ) 2 ds +C(p)E Z S b Sa (1+|X | 2 ) p/2−2 X k X j X j σ j,k (s,,X ) ! 2 ds +C(p)E sup t∈[Sa,S b ] X j,k Z t Sa (1+|X | 2 ) p/2−1 X j σ j,k (t,,X )dW k . (3.154) By applying the Cauchy-Schwartz inequality and using the Lipschitz assumption: (1+|X | 2 ) p/2−2 X k X j X j σ j,k (s,,X ) ! 2 ≤(1+|X | 2 ) p/2−2 |X | 2 X j,k σ j,k (s,,X ) 2 ≤C(1+|X | 2 ) p/2 . (3.155) 137 We estimate the stochastic integral terms using the Burkholder-Davis-Gundy inequal- ity (see [27]): E sup t∈[Sa,S b ] X j,k Z t Sa (1+|X | 2 ) p/2−1 X j σ j,k (t,,X )dW k ≤CE   Z S b Sa (1+|X | 2 ) p−2 X k X j X j σ j,k (t,,X ) ! 2 dt   1/2 ≤CE " sup t∈[Sa,S b ] (1+|X | 2 ) p/4 Z S b Sa (1+|X | 2 ) p/2 dt 1/2 # ≤ 1 2 E sup t∈[Sa,S b ] (1+|X | 2 ) p/2 ! +CE Z S b Sa (1+|X | 2 ) p/2 dt . (3.156) Using the observations in (3.155) and (3.156) with (3.154) and rearranging: E sup t∈[Sa,S b ] (1+|X | 2 ) p/2 ! ≤CE (1+|X (S a )| 2 ) p/2 + Z S b Sa (1+|X | 2 ) p/2 ds . (3.157) Note that the constant C = C(K,p) above is independent of S a ,S b . Applying Lemma 3.7.1 one infers: E sup t∈[0,T] (1+|X (t)| 2 ) p/2 ≤C(K,p,T)E(1+|X 0 | 2 ) p/2 . (3.158) Given the assumptions on the initial data (3.147) we have uniform bound: E(1+|X 0 | 2 ) p/2 ≤CE(1+|x 0 | p ) (3.159) which implies (3.148). 138 Toestablish(3.149)weagainapplyItˆ o’slemmatodetermineanevolutionequation for|X −x| p : d|X −x| p =−p|X −x| p−2 hL(X −x),X −xidt −p|X −x| p−2 hB(X ,X )−B(x,x),X −xidt + 2α p 2 |X −x| p−2 X j,k σ j,k (t,,X ) 2 dt + α p|X −x| p−2 X j,k (X j −x j )σ j,k (t,,X )dW k + 2α p(p−2) 2 |X −x| p−4 X k X j (X j −x j )σ j,k (t,,X ) ! 2 dt. (3.160) Using the cancellation assumption (3.144) one infers: hB(X ,X )−B(x,x),X −xi =hB(X −x,x),X −xi. (3.161) Assumption (3.144) also allows us to determine an a priori bound forx, the solution of (3.143). Taking inner products we have: d dt |x| 2 =−2hLx,xi≤ 0. (3.162) So, fort∈ [0,T]: |x(t)|≤|x 0 |. (3.163) 139 Now fix 0≤S a ≤S b ≤T. Integrate (3.160) fromS a tot, take a supremum over this time interval and then an expected value: E sup t∈[Sa,S b ] |X −x| p ≤E|X (S a )−x(S a )| p +C(p,B,|x 0 |)E Z S b Sa |X −x| p dt + 2α C(K,p)E Z S b Sa |X −x| p−2 (1+|X | 2 )dt + α C(p)E sup t∈[Sa,S b ] Z t Sa |X −x| p−2 X j,k (X j −x j )σ j,k (t,,X )dW k ! . (3.164) Once again we make use of the Burkholder-Davis-Gundy inequality: α CE sup t∈[Sa,S b ] X j,k Z t Sa |X −x| p−2 (X j −x j )σ j,k (t,,X )dW k ≤ α CE   Z S b Sa |X −x| 2(p−2) X k X j (X j −x j )σ j,k (t,,X ) ! 2 dt   1/2 ≤ α C(K)E Z S b Sa |X −x| 2(p−1) (1+|X | 2 )dt 1/2 ≤ 1 2 E sup t∈[Sa,S b ] |X −x| p ! + αp C(K,p) Z S b Sa (1+|X | 2 dt p/2 ≤ 1 2 E sup t∈[Sa,S b ] |X −x| p ! + αp C(K,p,T) Z S b Sa (1+|X | 2 ) p/2 dt. (3.165) 140 Using this estimate, (3.148) and rearranging we have: E sup t∈[Sa,S b ] |X −x| p ≤ 2E|X (S a )−x(S a )| p +C Z S b Sa |X −x| p dt+ αp C Z S b Sa (1+|X | 2 ) p/2 . (3.166) Noting,onceagainthattheconstantsC(K,p,T,|x 0 |)aboveareindependentofS a ,S a we apply Lemma 3.7.1. Making use of (3.147) and (3.148) we finally deduce: E sup t∈[0,T] |X −x| p ≤CE|X 0 −x 0 | p + αp C Z T 0 (1+|X | 2 ) p/2 ≤ αp C. (3.167) As a corollary of Proposition 3.7.1 we infer convergence ofX tox in probability along with a rate of convergence: 3.7.2Proposition (Convergence in Probability). LetT > 0,p≥ 2,X 0 ∈ L p (Ω) and x 0 be a fixed element inR n . Suppose that: E|X 0 −x 0 | p ≤ αp C 0 . (3.168) Then for anyγ <α: P sup t∈[0,T] |X −x|> γ ! ≤C (α−γ)p . (3.169) Proof. The desired result follows directly from (3.149) by applying Chebyshev’s Inequality. 141 References [1] A. Bensoussan and J. Frehse, Local solutions for stochastic Navier Stokes equa- tions, M2AN Math. Model. Numer. Anal. 34 (2000), no. 2, 241–273, Special issue for R. Temam’s 60th birthday. MR MR1765659 (2001g:35278) [2] A.BensoussanandR.Temam, ´ EquationsstochastiquesdutypeNavier-Stokes ,J. Functional Analysis13 (1973), 195–222. MR MR0348841 (50 #1336) [3] D.Bl¨ omker,C.Gugg,andS.Maier-Paape, Stochasticnavier-stokesequationand renormalization group theory, Physica D1 (2002), no. 173, 137–152. [4] D. Bl¨ omker, S. Maier-Paape, and G. Schneider, The stochastic Landau equation as an amplitude equation, Discrete Contin. Dyn. Syst. Ser. B 1 (2001), no. 4, 527–541. MR MR1877321 (2002j:60110) [5] H. Breckner, Galerkin approximation and the strong solution of the Navier- Stokes equation, J. Appl. Math. Stochastic Anal.13 (2000), no. 3, 239–259. MR MR1782683 (2001g:60154) [6] D. Bresch, A. Kazhikhov, and J. Lemoine, On the two-dimensional hydrostatic Navier-Stokesequations ,SIAMJ.Math.Anal.36(2004/05),no.3,796–814.MR MR2111916 (2006k:35210) [7] Z. Brze´ zniak and S. Peszat, Strong local and global solutions for stochastic Navier-Stokes equations , Infinite dimensional stochastic analysis (Amsterdam, 1999), Verh. Afd. Natuurkd. 1. Reeks. K. Ned. Akad. Wet., vol. 52, R. Neth. Acad. Arts Sci., Amsterdam, 2000, pp. 85–98. MR MR1831412 (2002c:76035) [8] C.CaoandE.Titi,Globalwell-posednessofthethree-dimensionalviscousprim- itiveequationsoflargescaleoceanandatmospheredynamics,Ann.ofMath.(2) 166 (2007), no. 1, 245–267. MR MR2342696 [9] M. Capi´ nski and D. G ‘ atarek, Stochastic equations in Hilbert space with appli- cation to Navier-Stokes equations in any dimension , J. Funct. Anal.126 (1994), no. 1, 26–35. MR MR1305062 (95j:60094) [10] L.-Y. Chen, N. Goldenfeld, and Y. Oono, Renormalization group theory for global asymptotic analysis, Phys. Rev. Lett. 73 (1994), no. 10, 1311–1315. MR MR1289625 (95d:81089) 142 [11] , Renormalization group and singular perturbations: Multiple scales, boundary layers, and reductive perturbation theory, Phys. Rev. E 54 (1996), no. 1, 376–394. [12] P. Constantin and C. Foias, Navier-Stokes equations , Chicago Lectures in Math- ematics, University of Chicago Press, Chicago, IL, 1988. MR MR972259 (90b:35190) [13] A. B. Cruzeiro, Solutions et mesures invariantes pour des ´ equations d’´ evolution stochastiques du type Navier-Stokes , Exposition. Math. 7 (1989), no. 1, 73–82. MR MR982157 (90c:35161) [14] B.Cushman-Roisin and J.-M. Beckers, Introduction to geophysical fluid dynam- ics: Physical and numerical aspects, To be published by Academic Press, 2007. [15] G.DaPratoandJ.Zabczyk,Stochasticequationsininfinitedimensions,Encyclo- pedia of Mathematics and its Applications, vol. 44, Cambridge University Press, Cambridge, 1992. MR MR1207136 (95g:60073) [16] ,Ergodicityforinfinite-dimensionalsystems ,LondonMathematicalSoci- etyLectureNoteSeries,vol.229,CambridgeUniversityPress,Cambridge,1996. MR MR1417491 (97k:60165) [17] R.E.L. DeVille, A. Harkin, M. K. Holzer, Josi´ c, and T. Kaper, Analysis of a renormalization group method for solving perturbed ordinary differential equa- tions, Physica D (2008), to appear. [18] E. DiBenedetto, Real analysis, Birkh¨ auser Advanced Texts: Basler Lehrb¨ ucher. [Birkh¨ auserAdvancedTexts: BaselTextbooks],Birkh¨ auserBostonInc.,Boston, MA, 2002. MR MR1897317 (2003d:00001) [19] J. L. Doob, Measure theory, Graduate Texts in Mathematics, vol. 143, Springer- Verlag, New York, 1994. MR MR1253752 (95c:28001) [20] F. Flandoli, An introduction to 3d stochastic fluid dynamics, SPDE in Hydrody- namic: RecentProgressandProspects,LectureNotesinMathematics,vol.1942, Springer Berlin / Heidelberg, 2008, pp. 51–150. [21] F. Flandoli and D. Gatarek, Martingale and stationary solutions for stochastic Navier-Stokes equations , Probab. Theory Related Fields102 (1995), no. 3, 367– 391. MR MR1339739 (96m:60137) [22] M. I. Freidlin and A. D. Wentzell, Random perturbations of dynamical sys- tems, second ed., Grundlehren der Mathematischen Wissenschaften [Funda- mental Principles of Mathematical Sciences], vol. 260, Springer-Verlag, New 143 York, 1998, Translated from the 1979 Russian original by Joseph Sz¨ ucs. MR MR1652127 (99h:60128) [23] N. Glatt-Holtz and M. Ziane, The stochastic primitive equations in two space dimensions with multiplicative noise, Discrete Contin. Dyn. Syst. Ser. B, (to appear). [24] F. Guill´ en-Gonz ´ alez, N. Masmoudi, and M. A. Rodr´ ıguez-Bellido, Anisotropic estimates and strong solutions of the primitive equations, Differential Integral Equations14 (2001), no. 11, 1381–1408. MR MR1859612 (2003b:76038) [25] C. Hu, R. Temam, and M. Ziane, Regularity results for linear elliptic prob- lems related to the primitive equations, Frontiers in mathematical analysis and numerical methods, World Sci. Publ., River Edge, NJ, 2004, pp. 149–170. MR MR2172554 [26] J. Jacod, Calcul stochastique et probl` emes de martingales, Lecture Notes in Mathematics, vol. 714, Springer, Berlin, 1979. MR MR542115 (81e:60053) [27] I. Karatzas and S. E. Shreve, Brownian motion and stochastic calculus, second ed., Graduate Texts in Mathematics, vol. 113, Springer-Verlag, New York, 1991. MR MR1121940 (92h:60127) [28] G. M. Kobelkov, Existence of a solution ‘in the large’ for the 3D large-scale ocean dynamics equations, C. R. Math. Acad. Sci. Paris343 (2006), no. 4, 283– 286. MR MR2245395 [29] I. Kukavica and M. Ziane, On the regularity of the primitive equations of the ocean, Nonlinearity20 (2007), no. 12, 2739–2753. MR MR2368323 [30] J.-L. Lions, R. Temam, and S. Wang, Models for the coupled atmosphere and ocean. (CAO I,II), Comput. Mech. Adv. 1 (1993), no. 1, 120. MR MR1252502 (94m:35247) [31] J.-L.Lions,R.Temam,andS.H.Wang, Newformulationsoftheprimitiveequa- tionsofatmosphereandapplications,Nonlinearity5(1992),no.2,237–288.MR MR1158375 (93e:35088) [32] ,Ontheequationsofthelarge-scaleocean ,Nonlinearity5(1992),no.5, 1007–1053. MR MR1187737 (93k:86004) [33] E.N.Lorenz,Ontheexistenceofaslowmanifold,J.AtmosphericSci.43(1986), no. 15, 1547–1557. MR MR855190 (87h:76037) [34] J.-L. Menaldi and S. S. Sritharan, Stochastic 2-D Navier-Stokes equation , Appl. Math. Optim.46 (2002), no. 1, 31–53. MR MR1922695 (2003k:35190) 144 [35] R. Mikulevicius and B. L. Rozovskii, Martingale problems for stochastic PDE’s, Stochastic partial differential equations: six perspectives, Math. Surveys Monogr., vol. 64, Amer. Math. Soc., Providence, RI, 1999, pp. 243–325. MR MR1661767 (2000c:60096) [36] ,Onequationsofstochasticfluidmechanics,Stochasticsinfiniteandinfi- nite dimensions, Trends Math., Birkh¨ auser Boston, Boston, MA, 2001, pp. 285– 302. MR MR1797092 (2002h:76043) [37] , Stochastic Navier-Stokes equations for turbulent flows , SIAM J. Math. Anal.35 (2004), no. 5, 1250–1310. MR MR2050201 (2005b:60156) [38] , Global L 2 -solutions of stochastic Navier-Stokes equations , Ann. Probab.33 (2005), no. 1, 137–176. MR MR2118862 (2005k:60198) [39] I.Moise,E.Simonnet,R.Temam,andM.Ziane, Numerical simulation of differ- entialsystemsdisplayingrapidlyoscillatingsolutions,J.Engrg.Math.34(1998), no. 1-2, 201–214, A celebration of the sixty-fifth anniversary of Pieter J. Zand- bergen: teacher and research leader in applied mathematics. MR MR1642864 (99d:65241) [40] I. Moise and R. Temam, Renormalization group method. Application to Navier- Stokesequation,DiscreteContin.Dynam.Systems6(2000),no.1,191–210.MR MR1739923 (2001k:76030) [41] I. Moise and M. Ziane, Renormalization group method. Applications to partial differential equations, J. Dynam. Differential Equations 13 (2001), no. 2, 275– 321. MR MR1829600 (2002h:35006) [42] J. Pedlosky, Geophysical fluid dynamics, Springer Verlag, 1982. [43] M. Petcu, R. Temam, and D. Wirosoetisno, Existence and regularity results for the primitive equations in two space dimensions, Commun. Pure Appl. Anal. 3 (2004), no. 1, 115–131. MR MR2033944 (2004m:76171) [44] C. Pr´ evˆ ot and M. R¨ ockner, A concise course on stochastic partial differential equations, LectureNotes in Mathematics, vol.1905, Springer, Berlin,2007. MR MR2329435 [45] L. F. Richardson, Weather prediction by numerical process, Cambridge Univer- sity Press; 2nd edition (August 13, 2007). [46] B. L. Rozovskii, Stochastic evolution systems, Mathematics and its Applications (Soviet Series), vol. 35, Kluwer Academic Publishers Group, Dordrecht, 1990, Lineartheoryandapplicationstononlinearfiltering,TranslatedfromtheRussian by A. Yarkho. MR MR1135324 (92k:60136) 145 [47] A. V. Skorokhod, F. C. Hoppensteadt, and H. Salehi, Random perturbation methods with applications in science and engineering, Applied Mathemati- cal Sciences, vol. 150, Springer-Verlag, New York, 2002. MR MR1912425 (2003m:34129) [48] R. Temam, Multilevel methods for the simulation of turbulence. A simple model, J. Comput. Phys.127 (1996), no. 2, 309–315. MR MR1412237 (97e:76068) [49] , Navier-Stokes equations , AMS Chelsea Publishing, Providence, RI, 2001, Theory and numerical analysis, Reprint of the 1984 edition. MR MR1846644 (2002j:76001) [50] R. Temam and M. Ziane, Some mathematical problems in geophysical fluid dynamics, Handbook of mathematical fluid dynamics. Vol. III, North-Holland, Amsterdam, 2004, pp. 535–657. MR MR2099038 (2005h:35275) [51] F. Verhulst, Nonlinear differential equations and dynamical systems, second ed., Universitext, Springer-Verlag, Berlin, 1996, Translated from the 1985 Dutch original. MR MR1422255 (97g:34003) [52] , Methods and applications of singular perturbations, Texts in Applied Mathematics, vol. 50, Springer, New York, 2005, Boundary layers and multiple timescale dynamics. MR MR2148856 (2006k:34001) [53] M.Ziane,Onacertainrenormalizationgroupmethod,J.Math.Phys.41(2000), no. 5, 3290–3299. MR MR1755506 (2001i:82033) 146

Abstract (if available)

Abstract This work collects three interrelated projects that develop rigorous mathematical tools for the study of the stochastically forced equations of geophysical fluid dynamics and turbulence. Since the presence of persistent random fluctuations are ubiquitous in fluid flow problems, the development of new analytical techniques in this setting are ultimately of paramount practical interest in the pursuit of more accurate models.

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Creator Glatt-Holtz, Nathan Edward (author)

Core Title Well posedness and asymptotic analysis for the stochastic equations of geophysical fluid dynamics

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Applied Mathematics

Publication Date 07/02/2008

Defense Date 05/07/2008

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag asymptotic analysis,geophysical fluid dynamics,Navier-Stokes equations,OAI-PMH Harvest,primitive equations,stochastic partial differential equations

Language English

Advisor Ziane, Mohammed (committee chair), Kukavica, Igor (committee member), Newton, Paul K. (committee member)

Creator Email glatthol@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-m1312

Unique identifier UC1200237

Identifier etd-GlattHoltz-20080702 (filename),usctheses-m40 (legacy collection record id),usctheses-c127-82250 (legacy record id),usctheses-m1312 (legacy record id)

Legacy Identifier etd-GlattHoltz-20080702.pdf

Dmrecord 82250

Document Type Dissertation

Rights Glatt-Holtz, Nathan Edward

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Repository Name Libraries, University of Southern California

Repository Location Los Angeles, California

Repository Email cisadmin@lib.usc.edu