University of Southern California Dissertations and Theses

On regularity and stability in fluid dynamics

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On regularity and stability in fluid dynamics

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Content ON REGULARITY AND STABILITY IN FLUID DYNAMICS by Fei Wang A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulllment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (MATHEMATICS) August 2017 Copyright 2017 Fei Wang Acknowledgments I am extremely grateful to my thesis advisor, Prof. Igor Kukavica, for his dedicated teaching, patient guidance, endless support, and valuable suggestions. His kind manner as a friend and great enthusiasm about research are crucial for me to keep going towards my academic achievement. Without him, this thesis would have been impossible. My sincere appreciation also extends to Prof. Amjad Tuaha, Prof. Vlad Vicol, and Prof. Mohammed Ziane for the fruitful discussion, their important suggestions, and the help in expanding my mathematical horizons. Working with them has been a great pleasure. I am very much thankful to Prof. Juhi Jang, Prof. Sergey Lototsky, and Prof. Paul Newton for their inspiration and useful comments and suggestions. ii Table of Contents Acknowledgments ii Abstract v Chapter 1: Introduction 1 1.1 Weighted decay of the SQG equations . . . . . . . . . . . . . . . . . . . . . 1 1.2 The active scalar equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Persistence of the Boussinesq equations . . . . . . . . . . . . . . . . . . . . 6 1.4 Blow-up of the Prandtl equations under general boundary value . . . . . . . 8 1.5 Free interface 2D Euler equation . . . . . . . . . . . . . . . . . . . . . . . . 10 Chapter 2: Weighted Decay for the Surface Quasi-Geostrophic Equation 13 2.1 Main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2 Proof of the main theorem for the case b2 (0; 1) . . . . . . . . . . . . . . . 14 2.3 Proof of the main theorem for the case b2 (1; 1 +) . . . . . . . . . . . . . 25 Chapter 3: On the ill-posedness of active scalar equations with odd singular kernels 30 3.1 The linearized problem and main results . . . . . . . . . . . . . . . . . . . . 30 3.2 Proof of linear ill-posedness . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.3 Proof of nonlinear Lipschitz ill-posedness . . . . . . . . . . . . . . . . . . . . 41 Chapter 4: Persistence of regularity for solutions of the Boussinesq equations in Sobolev spaces 44 4.1 On almost persistence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 4.2 A commutator lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 4.3 An L 1 bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.4 The proof of local persistence . . . . . . . . . . . . . . . . . . . . . . . . . . 54 4.5 Persistence for qs> 2 and for the intersection spaces . . . . . . . . . . . . . 59 4.6 Persistence with periodic boundary conditions . . . . . . . . . . . . . . . . . 64 Chapter 5: The van Dommelen and Shen singularity in the Prandtl equations 65 5.1 Finite time blowup for Prandtl . . . . . . . . . . . . . . . . . . . . . . . . . 65 5.2 Proof of Theorem 5.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 5.2.1 Local existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 5.2.2 Restriction of Prandtl dynamics on the y-axis . . . . . . . . . . . . . 69 iii 5.2.3 A shift of the boundary conditions . . . . . . . . . . . . . . . . . . . 70 5.2.4 Minimum principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 5.2.5 Blowup of a Lyapunov functional . . . . . . . . . . . . . . . . . . . . 74 5.2.5.1 Properties of the weight function w . . . . . . . . . . . . . 74 5.2.5.2 Evolution of the Lyapunov functionalG . . . . . . . . . . . 77 5.2.5.3 Bound for I 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 77 5.2.5.4 Bound for I 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 77 5.2.5.5 Bound for I 3 . . . . . . . . . . . . . . . . . . . . . . . . . . 78 5.2.5.6 Bound for I 4 . . . . . . . . . . . . . . . . . . . . . . . . . . 79 5.2.5.7 The lower bound for the growth of the Lyapunov functional 80 5.2.6 Conclusion of the proof of Theorem 5.1.1 . . . . . . . . . . . . . . . 80 5.3 Properties of the boundary condition lift . . . . . . . . . . . . . . . . . . 81 5.3.1 Proof of Lemma 5.2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . 81 5.4 Construction of a weight function w for the Lyapunov functional . . . . . . 83 5.4.1 Condition (5.40) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 Chapter 6: On the existence for the free interface 2D Euler equation with a localized vorticity condition 88 6.1 The main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 6.2 Preliminary lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 6.3 Proof of the main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 6.3.1 Div-curl estimates for , v, and a . . . . . . . . . . . . . . . . . . . . 95 6.3.2 Pressure estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 6.3.3 Tangential estimates and the conclusion . . . . . . . . . . . . . . . . 104 Bibliography 107 iv Abstract We address the regularity and stability problems of the following partial dierential equa- tions related to uid dynamics: the surface quasi-geostrophic (SQG) equation, the Boussi- nesq equations, the active scalar equations, the Euler equations, and the Prandtl boundary layer equations. We consider the SQG equation, whose rst moment decaykjxjk L 2 was obtained by M. and T. Schonbek. We obtain the decay rates ofkjxj b k L 2 for any b2 (0; 1) and the rate of increase of this quantity for b2 [1; 1 +) using dierent method under natural assumptions on the initial data. We investigate the global regularity of solutions to the Boussinesq equations with zero diusivity in two spatial dimensions. Previously, the persistence in the space H 1+s (R 2 ) H s (R 2 ) for all s 0 has been obtained. We address the persistence in general Sobolev spaces in which the behaviors of solutions turn out to be very dierent from that in Hilbert space, establishing it on a time interval which is almost independent of the size of the initial data. We also address the active scalar equations with constitutive laws that are odd and very singular, in the sense that the velocity eld loses more than one derivative with respect to the active scalar. We provide an example of such a constitutive law for which the equation is ill-posed: Either Sobolev solutions do not exist, from the Gevrey-class datum, or the solution map fails to be Lipschitz continuous in the topology of a Sobolev space, with respect to Gevrey class perturbations in the initial datum. Another model is the two dimensional incompressible Euler equations on a moving boundary, with no surface tension, under the Rayleigh-Taylor stability condition. The main feature of the result is the local regularity assumption on the initial vorticity, namely H 1:5+ Sobolev regularity in the vicinity of the moving interface in addition to the global regularity assumption on the initial uid velocity in the H 2+ space. We use a special change of variables and derive a priori estimates, establishing the local-in-time existence in v H 2+ . The assumptions on the initial data constitute the minimal set of assumptions for the existence of solutions to the rotational ow problem to be established in 2D. In 1980, van Dommelen and Shen provided a numerical simulation that predicts the spontaneous generation of a singularity in the Prandtl boundary layer equations from a smooth initial datum, for a nontrivial Euler background. We provide a proof of this numer- ical conjecture by rigorously establishing the nite time blowup of the boundary layer thickness. vi Chapter 1 Introduction This thesis focuses on partial dierential equations arising from uid dynamics, including the Navier-Stokes, Euler, Prandtl, and the active scalar equations, all of which have received considerable attention recently. More specically, our results address asymptotic properties of solutions, ill-posedness, blow-up, free surface uid evolution, boundary layers, Boussinesq system, Surface Quasi-Geostrophic (SQG) equations. 1.1 Weighted decay of the SQG equations The 2D SQG equation is given by t +ur + 2 = 0; (1.1) u =R ? = (R 2 ;R 1 ); (1.2) with the initial condition u(0) =u 0 ; (1.3) where R i is the i-th Riesz transform, 2 (0; 1], and = p 4 is the square root of the negative Laplacian. Here the nonlocal operator is dened by ( f) ^ () =jj ^ f(); (1.4) where ^ f() = (2) 1 R f(x)e ix dx is the Fourier-transform of f. The scalar function in the above equation represents the potential temperature, and u stands for the velocity. In particular, when = 1=2, the SQG equation describes the temperature distribution on the 2D boundary of a rapidly rotating uid with small Rossby and Ekman numbers [CW99]. Besides the physical interpretation of the SQG equation, it also serves as a simplied model for the 3D Navier-Stokes equations. The equation (2.1) is referred to as subcritical when 1=2<< 1, critical when = 1=2, and supercritical when 0 < < 1=2. Recently, this equation has received a considerable interest from many mathematicians. The global weak solutions and classical solutions 1 locally in time to the SQG equation are known to exist for 2 (0; 1] [Res95]. Furthermore the maximum principle is provided for this equation in the same reference as well. In [CW99], Constantin and Wu proved that the solutions are smooth on the whole space under the subcriticality assumption and obtained the decay result kk L 2C(1 +t) 1=2 ; (1.5) for 2 (0; 1). Later, M. and T. Schonbek obtained in [SS03] the decay of higher order derivatives k k L 2C(1 +t) (1+)=2 ; (1.6) for in a certain range. Additionally, they established a weighted decay estimate on kjxjk L 2. For other results on the well-posedness of the SQG equation, see [Wu01, Wu05, CV10, CTV15, KNV07, KN09, CCW01, CL03, CC04, CW08, CW09, Kis09, Kis11, Sil10, DKSV14, DKV12] and reference therein. Motivated by [SS05], we address here the weighted decay for solutions of subcritical quasi-geostrophic equations for more general weightskjxj b k L 2 for b2 [0; 1 +). (Note that the technique used in [SS05] depends essentially on the fact that the power of x is an integer.) For technical reasons we need to divide the proof into two cases: 0 < b < 1 and 1 < b < 1 +. Combined with the case b = 1 covered in [SS05] and b = 0 in [CW99], this gives a complete range of weights 0b 1 +. We obtain decay with the exponent (1b)=2 whenb2 (0; 1) and an increase with the exponent (b1)=2 whenb2 (1; 1+). For 0<b< 1, the main diculty we are faced with is thatx appears in the denominator when taking the derivative ofjxj b . In order to overcome this problem, we introduce the weight (x;t) = (jxj 2 + (1 +t) 1= ) 1=2 , for which the bound jr b (x;t)jC(1 +t) (1b)=2 (1.7) is available. Another diculty is that we face the commutatork b ( b )k L p, which we estimate here using the representation formula for . The third diculty is the decay property ofk k L q for q2 [1; 2); the only available estimate is for for q 2 in [SS03] (cf. Lemma 2.2.4 below). We obtain this in Lemma 2.2.5 below using ideas from [Kuk01]. At last, applying Gronwall's lemma directly does not give the desired result. Instead, we conclude using the integral representation of the solution. For the range 1 < b < 1 +, it seems dicult to nd a commutator estimate fork 2b ( 2b )k L q directly. We 2 instead consider R 2b ( 2b ) dx for (x) = (jxj 2 + 1) 1=2 (cf. Lemma 2.3.1 below). 1.2 The active scalar equations In Chapter 3, we address the well-posedness of the Cauchy problem for active scalar equa- tions @ t +ur = 0 (1.8) ru = 0 (1.9) (x; 0) = 0 (x) (1.10) posed onT 2 [0;1) = [;] 2 [0;1) with a certain constitutive law for the incompressible drift u =T; to be specied precisely below (cf. (1.16) below). The datum 0 and the solution are taken to have zero mean onT 2 . The study of active scalar equations of type (3.1){(3.2) is motivated by several important uid models: When T =r ? () 1 , the system becomes the vorticity formulation of the 2D Euler equations; the caseT =r ? () 1=2 corresponds to the surface quasi-geostrophic equation [CMT94]; the constitutive lawT =@ 1 r ? () 1 models ow in an incompressible porous medium with Darcy's law [CGO07]; while T =r ? B, where B is a scalar bounded Fourier multiplier, appears in magneto-geostrophic dynamics [FV11a]. The well-posedness theory of active scalar equations (either inviscid or with fractional dissipation) has attracted considerable attention in the last two decades. We refer the readers to [CV10, CCW01, CIW08, CMT94, CGO07, CTV15, CW99, CC04, CGO07, DKSV14, Don10, FV11a, HKZ15, Ju07, Kis11, KN09, KNV07, KW15, Miu06, Res95, SS03, Sil10, Wu01, Wu07] and the references therein. Recently, these equations have been considered with velocity elds determined by a singular constitutive law (i.e., the map T : 7! u is unbounded) which is also odd (by 3 which we mean the corresponding Fourier multiplier is an odd function of frequency). In particular, in [CCC + 12] the authors considered the velocity eld given by u =R ? ; (1.11) where 2 (0; 1], = p ,R ? =r ? 1 , and showed the local existence and uniqueness of solutions for (3.1){(3.3) with (1.11), in the Sobolev space H 4 . The two key ingredients in their proof were an estimate for the commutatork[@ i s ;g]fk L 2 and the identity Z fAfg = 1 2 Z f[A;g]f (1.12) which holds for smooth f;g, where A = @ i is an odd operator. This result was later sharpened in [HKZ15] where the local existence was established in H with > 2 +, using an improved commutator estimate. In contrast when the constitutive law is even and singular there are some ill-posedness results available for these equations. In [FGSV12, FV11b], the authors established ill- posedness for even singular constitutive laws, i.e., for operators T that have an even and unbounded Fourier multiplier symbol. The main ingredients in these works were a linear ill-posedness result (severe linear instability) in the spirit of [FSV97, MS61], and a classical linear implies nonlinear ill-posedness argument [Ren09, Tao06]. On the other hand, we are not aware of any ill-posedness result for (3.1){(3.3) with an odd constitutive law. This is due to the special cancellation property (1.12), which technically reduces the order of the constitutive law by one. The main purpose of Chapter 3 is to establish an ill-posedness result for (3.1){(3.3) with an odd constitutive law u =T =R ? M (1.13) where 1< 2 andM is a zero-order, scalar, even Fourier multiplier operator with symbol m. That is, d M(k) =m(k) ^ (k), where the specic Fourier multiplier symbolm we consider is given by m(k) =k 2 1 jkj 2 (k) (1.14) 4 for all k2Z 2 , where we dene (k) = 8 < : 1; k 2 =1; 1; otherwise: (1.15) For simplicity of notation, we denote by (ir) the Fourier multiplier operator with symbol (k). We may then rewrite (1.13){(1.15) concisely as u =R ? R 2 1 (ir) (1.16) where we recall that 2 (1; 2] throughout this paper. Our main result (Theorem 3.1.2 below) is to prove that the active scalar equation (3.1){ (3.3) with constitutive law (1.16) is ill-posed in any Gevrey spaceG s withs> (4)=(( 1)(3)). Here, by ill-posedness we mean that: either, given an initial datum 0 in G s , there is no local in time Sobolev solution 2L 1 t H =2+1+ x , where > 0 is arbitrary; or, the solution map 0 ! is not Lipschitz continuous in H =2+1+ , with respect to G s perturbations in the initial datum (see Denitionx 3.1 below). We note that the condition > 1 is strictly necessary for our result to hold. Indeed, for 2 (0; 1], by the results in [CCC + 12, HKZ15], we know that for the system (3.1){(3.3) and (1.16) we have local existence and uniqueness in the Sobolev space H with> 2 +. Additionally, using the identity (1.12) and the aforementioned commutator estimates, one may show that for2 (0; 1] the equations are locally Lipschitz (H =2+1 ;H +3 ) well-posed, in the sense of Denitionx 3.1 below. Thus, the requirement 2 (1; 2] in this paper is necessary and sucient. The proof of our main result is in the spirit of the earlier Lipschitz ill-posedness works [FGSV12, FV11b] with the main dierence being that as opposed to these works the eigenfunctions are constructed as sums of sines and cosines; this interplay leads to an eigenvalue problem involving a continued fraction with all positive signs (cf. (3.23) below). Such continued fractions require a dierent treatment than [FGSV12, FV11b] due to non- monotonicity of the corresponding approximation sequence. Finally, we need a way to bound the Sobolev and Gevrey norms ensuring that the L 2 norm of the solutions grow arbitrarily fast in any short period of time. 5 Chapter 3 is organized as follows. In Sectionx 3.1, we linearize the active scalar equation around a steady solution and state our main linear and nonlinear results. In Sectionx 3.2, we give the proof of the ill-posedness in Sobolev spaces and Gevrey classes for the linear equation. Finally, in Sectionx 3.3, we conclude the ill-posedness for the non-linear problem in the corresponding Sobolev and Gevrey spaces using a perturbation argument. 1.3 Persistence of the Boussinesq equations Consider the 2D Boussinesq equations with zero diusivity @u @t +uru u +rp =e 2 (1.17) divu = 0 (1.18) @ @t +ur = 0 (1.19) in general Sobolev spaces. Here, u is the velocity solving the 2D Navier-Stokes equation ([CF88, DG95, FMT88, Rob01, Tem97, Tem95]) driven by , which represents the density or temperature of the uid, depending on the physical context, and e 2 = (0; 1) T . This system arises in several dierent physical scenarios. One situation is the limiting case of small diusion Rayleigh-B enard problem, where represents the temperature. Also, the Boussinesq system serves as a simplied model for the 3D Navier-Stokes equations since it incorporates a vortex stretching eect. Due to the smoothing eect, the Boussinesq system with viscosity and non-zero diusion @ @t +ur = (1.20) replacing (4.3) is easier to treat than the same system without the diusion. The global well-posedness of the system (4.1){(4.3) with (4.3) replaced by (1.20), where > 0, was obtained in [CD80]. In 2006, Chae [Cha06] proved the global existence and uniqueness of solutions to the system (4.1){(4.3), proving that if (u 0 ; 0 )2 H m (R 2 )H m (R 2 ), then the solution (u;) stays in H m (R 2 )H m (R 2 ) for all t> 0, where m is an integer greater than or equal to 3. The main idea in the proof was to show that R T 0 krk L 1dt <1. At the same time, Hou and Li [HL05] proved the persistence of regularity for the solutions in H m (R 2 )H m1 (R 2 ) also for integer m greater than or equal to 3, with the key ingredient in their proof being an upper bound forkruk L 1. The persistence results in the class H s (R 2 )H s1 (R 2 ) for a larger range ofs were obtained more recently. Fors = 1, one may 6 refer to [DP08, LLT10], in which the global existence and uniqueness results were proven for the two-dimensional non-diusive Boussinesq system with viscosity only in the horizontal direction. Recently, the case s2 (1; 3) was resolved in [HKZ13, HKZ15]. The key step in the rst paper was an estimate forrB(u;v) while a borderline commutator estimate fork[ s @ j ;v]k L 2 (R 2 ) was obtained in the second paper. Note that the global existence for another extreme case when (4.1) is replaced by @u @t +uru +rp =e 2 (1.21) is still an open problem. In Chapter 4, we consider the persistence of the system (4.1){(4.3) in the Sobolev space W 1+s;q (R 2 )W s;q (R 2 ) where s2 (0; 1) and q2 [2;1). In comparison with the L 2 based result, we are faced with the following diculties. First, the velocity does not need to belong to L 2 (R 2 ) and thus we can not use the energy inequality. In order to avoid this problem, we couple the L p estimate for the velocity and that of the vorticity. Second, the available commutator estimates and the Kato-Ponce inequality are inadequate for our purpose. What we need here is the estimate fork[ s @ j ;g]fk L q (R 2 ) . As in [KP88], we write this commutator as a Coifman-Meyer operator in the frequency regions G and and use the complex interpolation in the remaining region. Also, the Brezis-Gallouet and related Beale-Kato-Majda inequalities ([BKM84]) do not apply in our situation. Instead we prove kuk L 1C 1 + log(1 +k s !k L r) 1+1=q (1 +k!k L q +kr(j!j q=2 )k 2=q L 2 ) (1.22) (cf. Lemma 4.3.1 below). Since the power of the logarithm is too high, we may only obtain a local persistence result, with the existence time depending on the initial data logarithmically. As we may see, a huge diculty to get a global result is the estimate of k!k L 1. If we furthermore assume that sq> 2, then we may prove k!(t)k L 1C p 1 +t (1.23) (cf. Lemma 4.5.2), allowing us to prove the global existence in W 1+s;q (R 2 )W s;q (R 2 ). In Theorem 4.5.3 below we also obtain the global persistence in the intersection space (W 1+s;q (R 2 )W s;q (R 2 ))\ (H 1+s (R 2 )H s (R 2 )) where s2 (0; 1) and q2 [2;1). As a corollary of this result, we obtain the global persistence in the spaceW 1+s;q (R 2 )W s;q (R 2 ) 7 for the initial data with compact support. The global existence in W 1+s;q (T 2 )W s;q (T 2 ) is also established in Theorem 4.6.1 below. 1.4 Blow-up of the Prandtl equations under general bound- ary value We consider the 2D Prandtl boundary layer equations for the unknown velocity eld (u;v) = (u(t;x;y);v(t;x;y)): @ t u@ yy u +u@ x u +v@ y u =@ x P E (1.24) @ x u +@ y v = 0 (1.25) uj y=0 =vj y=0 = 0 (1.26) uj y!1 =U E : (1.27) The domain we consider is TR + = f(x;y) 2 TR: y 0g, with corresponding periodic boundary conditions in x for all functions. The function U E = U E (t;x) is the trace at y = 0 of the tangential component of the underlying Euler velocity eld (u E ;v E ) = (u E (t;x;y);v E (t;x;y)), and P E = P E (t;x) is the trace at y = 0 of the Euler pressure p E =p E (t;x;y). They obey the Bernoulli equation @ t U E +U E @ x U E =@ x P E (1.28) for x2T and t 0, with periodic boundary conditions. In Chapter 5 we prove the formation of nite time singularities in the Prandtl boundary layer equations when the underlying Euler ow is not trivial, i.e., when U E 6= 0. For this purpose, we consider the Euler trace U E = sinx (1.29) @ x P E = 2 2 sin(2x) (1.30) 8 proposed by van Dommelen and Shen in [vDS80], where 6= 0 is a xed parameter. These are stationary solutions of the Bernoulli equation (5.5). Moreover, the functions U E and P E above arise as traces at y = 0 of the stationary 2D Euler solution u E (x;y) = sinx cosy v E (x;y) = cosx siny p E (x;y) = 2 4 (cos 2x + cos 2y): It is clear that the function (u E ;v E ) described above is divergence free, obeys the boundary condition v(x; 0) = 0, and yields a stationary solution of the Euler equations in TR + . Remark 1.4.1 (Numerical blowup is observed). The Lagrangian computation of van Dommelen and Shen [vDS80] was revisited and improved by many groups in the past decades [Cow83, CSW96, HH03, GSS09, GSSC14]. The consensus is that all the numerical SEPARATING BOUNDARY LAYERS 127 t-o t-4 FIG. I. The distortion of a typical Lagrangian grid with time do blow up, but in the Lagrangian description those balancing large terms are replaced by a single time derivative. Therefore, as will be substantiated by the numerical results to be presented, the solution is better behaved in Lagrangian coor- dinates than in Eulerian ones. The present work extends the earlier work by van Dommelen and Shen ] 131, in which the same case as presented here was calculated, but with different initial data, u(x, y, 0) =f(y) sin x, (5) where f(y) is the Hiemenz velocity profile [ 11, instead of the step function initial data of Eq. (3). From [ 131 we borrow Fig. 1, as it gives a beautiful picture of how the Lagrangian grid distorts with time and consequently, like a geometrical mapping, Figure 1.1: The distortion of a typical Lagrangian grid with time. The gure is from [vDS80, p. 127]. experiments indicate a singularity formation in nite time from a smooth initial datum. We 9 refer to the recent paper [CGSS15, Section 4.2] for a detailed discussion of the numerical singularity formation in the Prandtl system. 1.5 Free interface 2D Euler equation In Chapter 6, we establish the local-in-time solutions to the incompressible Euler equations for a free moving interface, with no surface tension, for rotational ows under minimal regularity assumptions on the initial data and the Rayleigh-Taylor stability condition. The Euler equations u t +uru +rp = 0 in (t) (0;T ) (1.31) divu = 0 in (t) (0;T ) (1.32) which describe the ow of an ideal inviscid incompressible uid with a velocity eld u(x;t) and a uid pressure p(x;t) on a moving domain have attracted considerable attention in the mathematical literature. They model propagation of shallow water waves under the in uence of gravity. The boundary of the domain (t) consists of a exible part 1 (t), which moves with the uid velocity, and a stationary part 0 . The boundary conditions imposed are un = 0 in 0 (0;T ) (1.33) p = in 1 (t) (0;T ) (1.34) where (x) is the mean curvature of the boundary and 0 1 is the surface tension. In this paper we are interested in the no surface tension case, = 0. In this context, the boundary condition p = 0 is also an idealization of the two phase problem involving the interaction of the water waves with the atmosphere. In fact, a relevant problem in this regard has been the study of the two phase problem and showing that the solutions to the system converge to the solutions of the idealized system as the density of the second phase goes to zero [Pus11]. Initial results on well-posedness of the system were obtained for small irrotational initial data by Nalimov in 2D [Nal74], Yoshihara [Yos82], and Craig [Cra85]. Later, the authors in [BHL93], established the local existence of solutions to the linearized system under the Taylor condition. The general well-posedness of the system was an open standing problem 10 until Ebin [Ebi87] showed the ill-posedness of the system for general initial data without the Rayleigh-Taylor sign condition on the initial data. The major development came in 1996 with the seminal work of Wu, who showed local- in-time existence of solutions for irrotational ows in 2D [Wu97] and 3D [Wu99]. She relied on Cliord analysis to study the evolution of the interface and obtained well-posedness for general smooth initial data under the Rayleigh-Taylor sign condition. The Rayleigh-Taylor sign condition on the initial pressure is given by @q @N (x; 0) 1 C 0 < 0; x2 1 (1.35) and always holds for irrotational ows for domains with innite depth or at bottom (rigid part of the boundary) as was shown by [Wu97], or in case of rigid bottom with small curvature [Lan05]. The Taylor-Rayleigh condition is a stability criterion used for the study of two phase ideal uids and is physically interpreted to mean that the lighter uid is situated above the heavier one. This condition is connected to a well-known physical phenomenon known as the Rayleigh-Taylor instability, which involves turbulent mixing of two uids when the heavier uid happens to be on top of the lighter uid. A special attention in the literature has been given to the emergence of singularities on the interface, cf. [CCF + 12] for instance. In this context, there has been some recent work on the irrotational case which highlights the dispersive nature of the interface and which relies on singular integrals and Strichartz estimates of the potential ow and stream functions [ABZ11, ABZ13]. A particular emphasis is given to the initial regularity of the interface, and the minimal regularity of the initial interface and initial velocity. The authors were able to establish local-in-time existence of solutions but not uniqueness under an initial interface regularity ofH d=2+3=21=12+ and initial interface velocity inH d=2+11=12+ , whered> 0 is the interface dimension and > 0. Unlike earlier regularity results in the literature, these results suggest that the interface need not have bounded curvature and that the initial velocity need only be Lipschitz continuous. These results, are as far as we know, pose the lowest regularity requirement on the initial data for irrotational ows. However, the tools of analysis in this approach do not apply to the more general case of a rotational ow. In our previous results, we have imposed the condition that the vorticity ! is smoother by 1=2 space derivative than the gradient of the velocity. In this paper, for the purpose of local existence, we prove that in fact it is sucient for the vorticity to be smoother only in a neighborhood of the top boundary. In particular, in the 2D result in [KT14], the smoother vorticity was used to deduce the regularity of the Lagrangian ow map which in 11 turn was connected to the evolution of the interface. However, it was observed that the use of the Lagrangian ow map on the whole domain can be dispensed with, since it is only the evolution of the interface which drives the tangential regularity of the velocity eld. In fact, the choice of the coordinate map away from the boundary is inessential, and it is sucient to impose the smoother condition on the vorticity only in a neighborhood of the evolving interface. Our result is then the local existence of solutions for the 2D free-surface Euler equations only assuming that ! 0 lies in H 1:5+ (U 0 ) and v 0 is in H 2+ ( ) with the Taylor condition @q @N (x; 0) 1 C 0 < 0; x2 1 (1.36) where C 0 > 0 is a constant, and where U 0 =fx2 : dist(x; 1 ) 0 g for some xed 0 > 0. Since we only work under a higher regularity of the vorticity around the top boundary, the usual global Lagrangian coordinate does not work in our scenario. What we need here is a new coordinate transform introduced in Sectionx 6.1 that captures the evolution of the moving interface and the adjacent ow, but coincides with the Eulerian coordinate system away from the interface. We also use the property that the vorticity ! remains constant along the Lagrangian trajectories in 2D ow. In order to carry out the div-curl estimate, we need a good control of the vorticity near the moving interface represented by the term!(), which requires a Sobolev type estimate for a composite of two functions. Another feature of the analysis is that the tangential estimates on the velocity are not global and do not hold for the whole domain due to the nonlocal nature of the pressure terms. In particular, the localized condition on the initial data, leads to a loss of regularity in the time derivative of the pressure q t by one space derivative below what is required for the global tangential estimates to hold as in [KT14]. We note that our regularity result implies that the initial interface velocity is only Lipschitz continuous by the Sobolev embedding theorem. While we consider a at initial surface to simplify the analysis, the same result can be obtained by a change of variable. In particular, the initial interfaceh(x;t) can be assumed to be a graph of a function of H 3+ (R) which also implies an initial interface of class C 5=2 . Throughout this paper, we only provide a priori estimates. The rigorous proof is obtained by smoothing the initial data, estimating under the mollied setting, and passing to the limit. 12 Chapter 2 Weighted Decay for the Surface Quasi-Geostrophic Equation In this chapter, we consider the 2D SQG equation t +ur + 2 = 0; (2.1) u =R ? = (R 2 ;R 1 ); (2.2) with the initial condition u(0) =u 0 ; (2.3) where R i is the i-th Riesz transform, 2 (0; 1], and = p 4 is the square root of the negative Laplacian. Here the nonlocal operator is dened by ( f) ^ () =jj ^ f(); (2.4) where ^ f() = (2) 1 R f(x)e ix dx is the Fourier-transform of f. 2.1 Main result First, we state the main result on the weighted decay for solutions of the SQG equations. Theorem 2.1.1. Assume that 0 2L 1 \H 1+ for some2 (0; 1). Furthermore, we suppose thatjxj b 0 2 L 1 \L 2 ; where b2 (0; 1 +). Let be a weak solution of (2.1){(2.3) with 2 (1=2; 1). Then kjxj b k L 2C(1 +t) (1b)=2 (2.5) for all b2 [0;b]. In the caseb = 1, the theorem was proved by M. and T. Schonbek in [SS05]. The proof is divided into two parts corresponding to the cases b2 (0; 1) and b2 (1; 1 +). 13 2.2 Proof of the main theorem for the case b2 (0;1) Lemma 2.2.1. Let (x;t) = (jxj 2 + (1 +t) 1= ) 1=2 ; for x2 R 2 , where t 0 is a xed parameter. Suppose 2S(R 2 ) and assume that 2 [0; 2) and b2 [0; 1) are constants. Then k b ( b )k L pC(1 +t) (1b)=2 krk L q +C(1 +t) (2b)=2 kk L q (2.6) where p;q2 (1;1) satisfy 1 + 1=p ==2 + 1=q. Proof of Lemma 2.2.1. Rewriting b ( b ) in the integral form, we have b (x) ( b )(x) =c 0 b (x) Z 4 y (y) jxyj dyc 0 Z 4 y ( b )(y) jxyj dy =c 0 b (x) Z 4 y (y) jxyj dy +c 0 Z 4 y ( b (y))(y) + 2r y b (y)r y (y) + b (y)4 y (y) jxyj dy; (2.7) where c 0 is the normalizing constant. After a rearrangement, we obtain b (x) ( b )(x) =c 0 Z ( b (y) b (x))4 y (y) jxyj dy + 2c 0 Z r y b (y)r y (y) jxyj dy +c 0 Z 4 y ( b (y))(y) jxyj dy =I 1 +I 2 +I 3 : (2.8) A direct computation shows that jr b (x;t)jC(1 +t) (1b)=2 (2.9) and j4 b (x;t)jC(1 +t) (2b)=2 (2.10) where, recall, 0 b < 1 and where C is a constant depending on b. Postponing the treatment of the term I 1 , note that jI 2 jC(1 +t) (1b)=2 Z jr y (y)j jxyj dy: (2.11) 14 By the Hardy-Littlewood inequality, we get kI 2 k L pC(1 +t) (1b)=2 krk L q; (2.12) where we choose p;q2 (1;1) so that 1 + 1=p ==2 + 1=q. Using the bound for4 b above, we similarly estimate the I 3 term as kI 3 k L pC(1 +t) (2b)=2 kk L q; (2.13) where p and q are same as above. We treat the term I 1 =c 0 Z ( b (y) b (x))4 y (y) jxyj dy (2.14) next. Let I 1 =c 0 Z jxyj ( b (y) b (x))4 y (y) jxyj dy: (2.15) Integrating by parts, we obtain I 1 =c 0 Z jxyj r y b (y) b (x) jxyj r y (y)dy c 0 Z jxyj= b (y) b (x) jxyj @ @n (y)d(y); (2.16) where n is the outer normal vector to the ball B(0;). By the mean value theorem, we obtain j b (y) b (x)j =j(jyj 2 + (1 +t) 1= ) b=2 (jxj 2 + (1 +t) 1= ) b=2 j =jb(jzj 2 + (1 +t) 1= ) (1b=2) z (yx)j b(1 +t) (1b)=2 jyxj; (2.17) where z = x +(yx) for a suitable 2 [0; 1]. Therefore, for t 0, the second term on the right hand side of (2.16) is bounded by C(1 +t) (1b)=2 2 ; (2.18) which converges to 0 as ! 0: Thus we obtain I 1 = lim !0 I 1 =c 0 P.V. Z r y b (y) b (x) jxyj r y (y)dy: (2.19) 15 Note that r y b (y) b (x) jxyj C(1 +t) (1b)=2 =jxyj (2.20) and thus jI 1 jC(1 +t) (1b)=2 Z jr y (y)j jxyj dy: (2.21) Using the Hardy{Littlewood inequality, we obtain kI 1 k L pC(1 +t) (1b)=2 krk L q: (2.22) Then the lemma follows by combining (2.12), (2.13), and (2.22). We state a Gronwall type lemma next. Lemma 2.2.2. Let 0 j ; j < 1 and j 0 for j = 1; 2;:::;m, where m2N. Suppose that a continuously dierentiable function satises F 0 (t)C m X j=1 F (t) j (1 +t) j t j (2.23) for all t where it is bounded, where C > 0 is a constant and F (0)<1. Then for = max 0; 1 1 1 1 1 ;:::; 1 m m 1 m ; (2.24) there exists K(C;m)> 0 such that F (t)K(1 +t) : Proof of Lemma 2.2.2. First, (2.23) implies F 0 (t)C m X j=1 F (t) j t j : (2.25) Sincet j are integrable around 0, we obtain using the Gronwall lemma a uniform bound for F (t) on [0; 1]. Shifting the initial time to 1, we may thus assume without loss of generality that F 0 (t)C m X j=1 F (t) j (1 +t) j j : (2.26) The rest then follows by a standard application of the Gronwall lemma. Next, we recall the following result due to Constantin and Wu on the decay ofL 2 norms of solutions of (2.1){(2.3). 16 Lemma 2.2.3 ([CW99]). Let 2 (0; 1] and 0 2 L 1 (R 2 )\L 2 (R 2 ). Then there exists a weak solution of the SQG equation (2.1){(2.3) such that k(;t)k L 2 (R 2 ) C(1 +t) 1=2 ; (2.27) where C is a constant depending on the L 1 and L 2 norms of 0 . We also need a result [SS03, Theorem 3.2]. Lemma 2.2.4 ([SS03]). Let 2 (1=2; 1] and m. Assume that is a solution of (2.1) with the initial datum 0 2L 1 \H m . Then k (t)k L 2C(1 +t) (+1)=2 ; 0m; t 0 (2.28) where C is a constant which depends only on the norms of the initial datum. In order to obtain an upper bound fork(;t)k L q, we need the following interpolation type lemma. Lemma 2.2.5. For q2 [1;1] and v2C 1 0 (R 2 [0;1)) for t> 0, we have krv(t)k L qC( sup s2[t=2;t] kv(s)k L q) 11=2 ( sup s2[t=2;t] kv t (s)+ 2 v(s)k L q) 1=2 +Ct 1=2 sup s2[t=2;t] kv(s)k L q (2.29) where 2 (1=2; 1]. The proof uses ideas from [Kuk01]. Proof of Lemma 2.2.5. Denotingf =v t + 2 v, we may writev in the integral representa- tion form v(x;t) = Z K (xy;tt 0 )v(y;t 0 )dy + Z t t 0 Z K (xy;ts)f(y;s)dyds; (2.30) where K is the kernel for the operator v t + 2 v. Here t 0 2 [t=2;t] will be determined below. Taking the derivative of v with respect to x, we get rv(x;t) = Z rK (xy;tt 0 )v(y;t 0 )dy + Z t t 0 Z rK (xy;ts)f(y;s)dyds: (2.31) 17 By Minkowski's and Young's inequalities, we obtain krv(t)k L qkrK (tt 0 )k L 1kv(t 0 )k L q + Z t t 0 krK (ts)k L 1kf(s)k L qds C(tt 0 ) 1=2 kv(t 0 )k L q +C Z t t 0 (ts) 1=2 kf(s)k L qds (2.32) where we also used the inequality kx @ j t @ x K (t)k L qCt (j jjj)=2j(q1)=q ; j j<jj + 2 maxfj; 1g; j = 0; 1; 2; 3::: (2.33) for 1q1, from [SS05, p. 1301]. Since t 0 2 [t=2;t] and 2 (1=2; 1], we get from (2.32) krv(t)k L qC(tt 0 ) 1=2 sup s2[t=2;t] kv(s)k L q +C(tt 0 ) 11=2 sup s2[t=2;t] kf(s)k L q: (2.34) Next we consider two cases. If t 2 sup s2[t=2;t] kv(s)k L q= sup s2[t=2;t] kf(s)k L q, then we let tt 0 = sup s2[t=2;t] kv(s)k L q sup s2[t=2;t] kf(s)k L q (2.35) and arrive at krvk L qC( sup s2[t=2;t] kv(s)k L q) 11=2 ( sup s2[t=2;t] kf(s)k L q) 1=2 : (2.36) On the other hand, ift< 2 sup s2[t=2;t] kv(s)k L q= sup s2[t=2;t] kf(s)k L q, we chooset 0 =t=2 and obtain krvk L qCt 1=2 sup s2[t=2;t] kv(s)k L q +Ct 11=2 sup s2[t=2;t] kf(s)k L q Ct 1=2 sup s2[t=2;t] kv(s)k L q: (2.37) The proof is completed by combining (2.36) and (2.37). By [SS05, p. 1287], we have k(;t)k L 1C (2.38) where C depends on the initial datum. The next lemma provides an upper bound for the derivative. 18 Lemma 2.2.6. Assume that 0 2L 1 \H 1+ for some 2 (0; 1). Then we have k(;t)k L 1 C t 1=2 (2.39) for t> 0. Proof of Lemma 2.2.6. For t> 0, we have by (2.38) and Lemma 2.2.5 t 1=2 kr(;t)k L 1C( sup s2[t=2;t] k(s)k L 1) 11=2 ( sup s2[t=2;t] sk t (s) + 2 (s)k L 1) 1=2 +C sup s2[t=2;t] k(s)k L 1 C( sup s2[t=2;t] sk t (s) + 2 (s)k L 1) 1=2 +C: (2.40) Now, tk t + 2 k L 1tkuk L 1krk L 1Ctk 1 uk 1=2 L 2 k 1+ uk 1=2 L 2 krk L 1 Ctk 1 k 1=2 L 2 k 1+ k 1=2 L 2 krk L 1 Ct (1 +t) 1= krk L 1Ct 1=2 krk L 1 (2.41) where we used < 1 in the last step. Above we also employed the inequality kuk L 1Ck 1 uk 1=2 L 2 k 1+ uk 1=2 L 2 (2.42) which can be proven by rst establishing kuk L 1Ckuk L 2 +Ck 1 uk 1=2 L 2 k 1+ uk 1=2 L 2 (2.43) fromkuk L 1 Ckuk H 1+ by rescaling, and then removing the rst term on the right side of (2.43) again by rescaling. Therefore, for every T 0 we have sup s2[0;T ] s 1=2 kr(;s)k L 1C( sup s2[0;T ] sk t (s) + 2 (s)k L 1) 1=2 +C C( sup s2[0;T ] s 1=2 kr(;s)k L 1) 1=2 +C: (2.44) If the expressionF (T ) = sup s2[0;T ] s 1=2 kr(;s)k L 1 is nite, we may use 1=2< 1 in order to obtain F (T )C uniformly in T . For the general case, we approximate the initial data 19 0 with 0 for whichk 0 k L 1 <1 and apply the above argument on the approximating sequence. Under the conditions of Lemma 2.2.6, we now claim k(;t)k L q C (1 +t) (11=q)= ; 1q1 (2.45) for all t> 0. It is sucient to check (2.45) for q = 1 and q =1. For q = 1, this is simply (2.38), while for q =1, we write k(;t)k L 1k 1 k 1=2 L 2 k 1+ k 1=2 L 2 C (1 +t) 1= (2.46) for all t 0, where we used Lemma 2.2.4. Also, interpolating between (2.39) andkk L 2C=t 1= , which holds by Lemma 2.2.4, we obtain under the conditions of Lemma 2.2.6 k(;t)k L q C t (1=q1=2)= (t + 1) (22=q)= ; q2 [1; 2] (2.47) for all t> 0. Proof of main theorem for b2 (0; 1). Let (x;t) = (jxj 2 + (1 +t) 1= ) 1=2 (2.48) and F (t) = Z 2b (x;t) 2 (x;t)dx: (2.49) Taking the derivative, we obtain F 0 (t) = Z @ t ( 2b 2 )dx = Z ( 2b ) t 2 dx + Z 2b ( 2 ) t dx = Z ( 2b ) t 2 dx + 2 Z 2b (ur 2 )dx = Z ( 2b ) t 2 dx 2 Z 2b urdx 2 Z 2b 2 dx (2.50) 20 where we used the SQG equation in the third equality. Thus we obtain F 0 (t) + 2 Z j ( b )j 2 dx = Z ( 2b ) t 2 dx 2 Z 2b urdx + 2 Z b 2 ( b ) b 2 dx =I 1 +I 2 +I 3 : (2.51) A direct computation shows that @ @t 2b = 2 b @ @t b C b (1 +t) (1b=2) (2.52) where C is a constant depending on b and . Therefore, jI 1 jC(1 +t) (1b=2) Z b 2 dxC(1 +t) (1b=2) k b k L 2kk L 2 C(1 +t) (1b)=21 F 1=2 (t); (2.53) where we used Lemma 2.2.3 in the last inequality, and where C is a constant depending on k 0 k L 1 andk 0 k L 2. Integrating by parts, we get Z 2b urdx = 1 2 Z 2 ur( 2b )dx (2.54) where we used the divergence free condition. This implies I 2 = Z 2 ur( 2b )dxC(1 +t) (12b)=2 kk L 1kuk L 2kk L 2 C(1 +t) (12b)=21=21=21= =C(1 +t) (52b)=2 : (2.55) In the last inequality, we used H older's inequality, Lemma 2.2.4, and the bound jr( 2b )j C (jxj 2 + (1 +t) 1= ) 1=2b C (t + 1) (12b)=2 : (2.56) By H older's inequality and Lemma 2.2.1 used with p = 2 and q = 2 3 2 2 (1; 2); (2.57) 21 we have I 3 k b k L 2k 2 ( b ) b 2 k L 2 Ck b k L 2 (1 +t) (1b)=2 krk L q + (1 +t) (1b=2)= kk L q : (2.58) By (2.45) and (2.47), we get I 3 CF 1=2 (t) t (1=q1=2)= (1 +t) (21=2b=2) + (1 +t) (1=2b=2+1) : (2.59) Summarizing, we obtain F 0 (t)CF (t) 1=2 (1 +t) (1b)=21 +C(1 +t) (52b)=2 +CF 1=2 (t) t (1=q1=2)= (1 +t) (1+b)=2 + (1 +t) (1b=2)=(11=q)= : (2.60) Using Lemma 2.2.2, we arrive at F (t)C; t 0: (2.61) Next we improve this bound by writing an equation for b . We rst have ( b ) t + 2 ( b ) =( b ) t + b t + 2 ( b ) =( b ) t + b (ur 2 ) + 2 ( b ) (2.62) and then write the solution in a integral form as b (t)(t) =K ( b )j t=0 + Z t 0 K (ts) ( b ) t (s) + b (ur)(s) ds + Z t 0 K (ts) ( 2 ( b ) b 2 )(s)ds; (2.63) 22 where K is the kernel for the operator t + 2 . Taking the L 2 norm of both sides, we get k b (t)(t)k L 2kK (jxj b 0 )k L 2 + Z t 0 kK (ts) ( b ) t (s) k L 2ds + Z t 0 kK (ts) b (ur)(s) k L 2ds + Z t 0 kK (ts) 2 ( b ) b 2 (s)k L 2ds =J 1 +J 2 +J 3 +J 4 : (2.64) Using the assumption b j t=0 2L 1 and Young's inequality, we obtain the bound J 1 kK k L 2k b j t=0 k L 1Ct 1=2 ; (2.65) where we also used the inequality (2.33). By @ @t ( b ) C(jxj 2 + (1 +t) 1= ) b=2 C (1 +t) 1b=2 (2.66) and Young's inequality, we get J 2 C Z t 0 kK (ts)k L 2(1 +s) (1b=2) kk L 1ds C Z t 0 (ts) 1=2 (1 +s) (1b=2) dsCt (1b)=2 ; (2.67) where we also used (see [SS05, p. 1288]) Z t 0 ds (ts) a (1 +s) b 8 > > > < > > > : Ct a if b> 1 Ct a 1 + log(1 +t) if b = 1 Ct a (1 +t) 1b if 0<b< 1 (2.68) 23 provided 0 < a < 1. Rather than to deal with the term J 3 directly, we instead use the product rule to write J 3 = Z t 0 kK (ts) b (ur)(s) k L 2ds = Z t 0 k(rK )(ts) ( b u)(s)K (ts) ur b (s)k L 2ds Z t 0 krK (ts)k L 1k b k L 2kuk L 1ds +C Z t 0 kK (ts)k L 2kk 2 L 2 (1 +s) (1b)=2 ds; (2.69) from where we get, using (2.61) and (2.33), J 3 C Z t 0 (ts) 1=2 (1 +s) 1= ds +C Z t 0 (ts) 1=2 (1 +s) 1=(1b)=2 ds Ct 1=2 : (2.70) For the last term J 4 , we denote A = 2 ( b ) b 2 : (2.71) By Lemmas 2.2.1 and 2.2.5, we have kAk L p 0C(1+t) (1b)=2 krk L q +C(1+t) (1b=2)= kk L qCt 1=2 (1+t) (3b)=2+1=q ; (2.72) where 0< 1=q = 2 1=p< 1 and 1=p 0 + 1 = + 1=q: By Young inequality, Z t 0 kK (ts)Ak L 2ds Z t 0 kK k L rkAk L p 0ds Z t 0 C(ts) (2=r2)=2 s 1=2 (1 +s) (b3)=2+1=q ds C(1 +t) (1b)=2 : (2.73) Note this is possible since we choose r = 1 + 1 and q = 1 + 2 such that 1 + 1=2 = 1=r + 1=p 0 = 1=r + + 1=q 1 (2.74) holds, where 1 ; 2 > 0 are suciently small. 24 Summarizing, we obtain k b k L 2C(1 +t) (1b)=2 : (2.75) This proves the theorem for the case b = b. When 0 b < b, we proceed by using the H older inequality kjxj b k L 2kjxj b k b=b L 2 kk 1b=b L 2 C(1 +t) (1b)=2 : (2.76) The proof for the case b2 (0; 1) is thus complete. 2.3 Proof of the main theorem for the case b2 (1;1+) Next we turn to the proof of the main result for b2 (1; 1 +). In the proof we need a dierent commutator estimate which is stated next. Lemma 2.3.1. Let2 [1=2; 1) andb2 (1; 1 +). Denote (x) = (jxj 2 + 1) 1=2 and assume 2S(R 2 ). Let p and q be given by p = 2 (1)b + 1 (2.77) and q = 2 1 +b ; (2.78) then Z 2b ( 2b ) dx Ck b k L 2k k L q +Ck b k L 2k b k 11=b L 2 k k 1=b L p +Ck b k L 2kk L q +Ck b k L 2k b k 11=b L 2 kk 1=b L p (2.79) holds. 25 Proof. Rewriting the commutator in integral form, we get 2b ( 2b ) = 2b (x)c 0 P.V. Z (x)(y) jxyj 2+ dyc 0 P.V. Z 2b (x)(x) 2b (y)(y) jxyj 2+ dy =c 0 P.V. Z 2b (y) 2b (x) jxyj 2+ (y)dy; (2.80) where c 0 is the normalizing constant. Using 2b (y) 2b (x) Cjyxj( 2b1 (y) + 2b1 (x)); (2.81) we get j 2b (x) ( 2b )(x)jCP.V. Z 2b1 (y) + 2b1 (x) jxyj 1+ j(y)jdy =CP.V. Z 2b1 (y) jxyj 1+ j(y)jdy +CP.V. Z 2b1 (x) jxyj 1+ j(y)jdy =A 1 +A 2 : (2.82) Next we estimate A 1 and A 2 . Using b1 (y)C(j (y) (x)j b1 + b1 (x)), we obtain A 1 CP.V. Z j (y) (x)j b1 jxyj 1+ b (y)j(y)jdy +CP.V. Z b1 (x) jxyj 1+ b (y)j(y)jdy CP.V. Z b (y)j(y)j jxyj 2+b dy +CP.V. Z b (y)j(y)j jxyj 1+ b1 (x)dy =A 11 +A 12 (2.83) where we also used j (x) (y)jjxyj: (2.84) For the term A 11 , we have Z A 11 j jdxkA 11 k L q 0k k L qCk b k L 2k k L q (2.85) 26 where we used the Hardy{Littlewood inequality, and where q;q 0 2 (1;1) satisfy 1 = 1=q + 1=q 0 and 1 + 1=q 0 = (2 +b)=2 + 1=2, which giveq = 2=(1 +b): Now we estimate the term involving A 12 Z A 12 j jdx =C ZZ b (y)j(y)j jxyj 1+ dy b1 (x)j (x)jdx C Z b (y)j(y)j jxyj 1+ dy L 0 k b1 (x) (x)k L Ck b k L 2k b1 k L Ck b k L 2k b k 11=b L 2 k k 1=b L p ; (2.86) where we used the Hardy{Littlewood inequality with 1 + 1= 0 = 1=2 + (1 +)=2 with 0 denoting the conjugate exponent of , and H older's inequality with 1= = (b1)=2b+1=bp. A simple calculation shows that p = 2= (1)b + 1 : Summarizing, we get Z A 1 j jdxCk b k L 2k k L q +Ck b k L 2k b k 11=b L 2 k k 1=b L p : (2.87) In order to bound the term involving A 2 , we write the corresponding term in double integral form and use Fubini theorem Z A 2 j jdx =C ZZ 2b1 (x) jxyj 1+ j(y)jdyj (x)jdx =C ZZ 2b1 (x)j (x)j jxyj 1+ dxj(y)jdy: (2.88) Note that R A 2 j jdx has the same structure with R A 1 j jdx, and therefore, we get the estimate Z A 2 j jdxCk b k L 2kk L q +Ck b k L 2k b k 11=b L 2 kk 1=b L p : (2.89) We conclude the proof by combining (2.87) and (2.89). Since the case b = 1 has already been addressed in [SS03], we only need to consider b2 (1; 1 +). 27 Proof of main theorem for b2 (1; 1 +). Let F (t) =k b k 2 L 2 = R 2b 2 dx. Taking the derivative with respect to t, we get 1 2 dF dt = Z 2b t dx = Z 2b (ur 2 )dx = Z 2b (ur)dx Z 2b 2 dx: (2.90) Adding R 2b (x)j j 2 dx to both sides of the above equation, we obtain 1 2 dF dt + Z 2b j j 2 dx = Z 2b (ur)dx + Z 2b ( 2b ) dx =I 1 +I 2 : (2.91) Integrating by parts, we get I 1 = Z 2b (ur)dx = Z div( 2b u)dx = Z 2 u j @ j ( 2b )dx + Z 2b (ur)dx; (2.92) which implies I 1 = 1 2 Z 2 u j @ j ( 2b )dx =b Z 2b1 2 u j @ j dx: (2.93) Therefore, using H older's inequality, we obtain I 1 Ck( b ) (2b1)=b k L 2b=(2b1)kuk L 4bkjj 1=b k L 4b Ck b k (2b1)=b L 2 kuk L 4bkk 1=b L 4 =CF 11=2b kuk L 4bkk 1=b L 4 : (2.94) For I 2 , we use Lemma 2.3.1 in order to get I 2 Ck b k L 2k k L q +Ck b k L 2k b k 11=b L 2 k k 1=b L p +Ck b k L 2kk L q +Ck b k L 2k b k 11=b L 2 kk 1=b L p : (2.95) Using Young's inequality leads to Ck b k L 2k b k 11=b L 2 k k 1=b L p 1 4 k b k 2 L 2 +Ck b k 2b=(b+1) L 2 k k 2=(b+1) L p : (2.96) 28 Similarly, Ck b k L 2kk L q 1 4 k b k 2 L 2 +Ckk 2 L q (2.97) and Ck b k L 2k b k 11=b L 2 kk 1=b L p 1 4 k b k 2 L 2 +Ck b k 22=b L 2 kk 2=b L p : (2.98) Using (2.91) with (2.94){(2.98) and absorbingk b k 2 L 2 terms in (2.96){(2.98) by the left side of (2.91), we arrive at dF dt + 1 4 k b k 2 L 2 CF (2b1)=2b kuk L 4bkk 1=b L 4 +CF 1=2 k k L q +CF b=(b+1) k k 2=b+1 L p +Ckk 2 L q +CF 11=b kk 2=b L p (2.99) from where dF dt + 1 4 k b k 2 L 2 C(1 +t) 1=1=2b F (2b1)=2b (2.100) +Ct (b)=2 (1 +t) (1+b)= F 1=2 +Ct (1)(b+2)=2 (1 +t) (31)(1b+b)(2+b)=2b F b=(b+1) +C(1 +t) (1+b)= +C(1 +t) (1(1)b)=b F 11=b : (2.101) The conclusion then follows by using Lemma 2.2.2. 29 Chapter 3 On the ill-posedness of active scalar equations with odd singular kernels In this chapter, we address the well-posedness of the Cauchy problem for active scalar equations @ t +ur = 0 (3.1) ru = 0 (3.2) (x; 0) = 0 (x) (3.3) posed onT 2 [0;1) = [;] 2 [0;1) with a certain constitutive law for the incompressible drift u =T; specied in (1.16). The datum 0 and the solution are taken to have zero mean onT 2 . 3.1 The linearized problem and main results Consider a scalar function of the form (x;t) = (x 2 ), where (x 2 ) is a real function in T with zero mean. According to (1.16) we have that u =T = 0, in view of the dierentiation with respect to x 1 inherent in T . Thus, any such is a steady state of (3.1). We choose a particular function (x 2 ) = cos(x 2 ) (3.4) 30 and consider the linearization of the nonlinear term in (3.1) around it. Denote the corre- sponding linear operator by L = urur =R 3 1 (ir) @ 2 = sin(x 2 )R 3 1 (ir) : (3.5) We shall use the method of continued fractions (see e.g. [FGSV12, FSV97, FV11b, MS61]) to prove that the operatorL has a sequence of eigenvalues whose positive real parts diverge to innity (cf. Theorem 3.1.3 below). In order to state our main results, we recall here the denition of the Gevrey-space G s (cf. [LO97]). Denition (Gevrey-space). A function 2 C 1 (R 2 ) belongs to the Gevrey class G s , where s 1, if there exists a positive constant > 0, called the Gevrey-class radius, such that the G s -norm is nite, i.e., kk 2 G s = X k2Z 2 j ^ (k)j 2 jkj 4 e 2jkj 1=s <1: (3.6) With this denition, G s =f2 C 1 (T 2 ):kk G s <1g is an algebra for s 1; > 0, and we have G s = S >0 G s . Next we recall the denition of the Lipschitz well-posedness (cf. [FV11b, GN11]). Denition (Locally Lipschitz (X;Y ) well-posedness). Let Y X H =2+1+ be Sobolev spaces, where > 0 is arbitrary. We say that the Cauchy problem for the active scalar equation (3.1){(3.3) with (1.13) is locally Lipschitz (X;Y ) well-posed, if there exist continuous functions T;K : [0;1) 2 ! (0;1), so that for every pair of initial data (1) (0;); (2) (0;)2 Y there exist unique solutions (1) ; (2) 2 L 1 (0;T ;X) of the initial value problem associated to (3.1){(3.3) with (1.13), such that k (1) (t;) (2) (t;)k X Kk (1) (0;) (2) (0;)k Y (3.7) for every t 2 [0;T ], where T = T (k (1) (0;)k Y ;k (2) (0;)k Y ) and K = K(k (1) (0;)k Y ;k (2) (0;)k Y ). 31 The role of the space H =2+1+ in the above denition is to ensure that the ranges of the linear operator L dened in (5.25), and that of the nonlinear operator in (3.1), dened as N[] =Tr; lie in L 2 . Here, recall that u = T is given by (1.16). More precisely, by the Sobolev embedding we have kLk L 2Ckk H Ckk H =2+1+ (3.8) and kN[]k L 2 =kTrk L 2Ckk 2 H =2+1+ (3.9) for a suciently large constant C > 0. Note that we can set = 0 if < 2. The role of the two-spaces X and Y in Denitionx 3.1 is to allow the solution to lose regularity, as it is expected due to the derivative losses of order (from T ) and 1 (fromr) present in the nonlinearity. The rst main result in this paper asserts that singular odd active scalar equations are locally Lipschitz ill-posed in Sobolev spaces. Theorem 3.1.1. Assume2 (1; 2], let> 0, and dener ==2+1+. Then the system (3.1){(3.3) with (1.13) is locally Lipschitz (H r ;H s ) ill-posed for any s>r. In fact, in view of the bound (3.12), one may obtain a stronger ill-posedness result: The solution is not well-posed in a class of Gevrey spaces. Theorem 3.1.2. Let 2 (1; 2], r ==2 + 1 + for some > 0, and let s> (4)=( 1)(3). Then the system (3.1){(3.3) with (1.13) is locally Lipschitz (H r ;G s ) ill-posed for > 0. Both theorems above hold with = 0 in the case < 2. The main ingredient in the proofs of Theorems 3.1.1 and 3.1.2 is the following bound on the eigenvalues of the linearized operator L dened in (5.25). Theorem 3.1.3. The linearized operator L dened in (5.25) has a sequence of entire real- analytic eigenfunctionsf k g k1 with corresponding eigenvaluesf k g k1 , such that k C 1 0 k 1 (3.10) 32 for all k 1, where C 0 10 is a universal constant. Moreover, we may normalize k such that either one of the following statements holds: (a) Given s 0 k k k H s = 1 and k k k L 2C 1 s k s(4)=(3) (3.11) for all k 1, where C s 1 is a constant that depends only on s; (b) given s 1 and > 0 k k k G s = 1 and k k k L 2C 1 s; exp C s; k (4)=s(3) (3.12) for all k 1, where C s; 1 is a constant that depends only on s;. A standard perturbation argument (see e.g. [Ren09, Tao06]) then implies the ill- posedness (3.1){(3.3) with the constitutive law (1.16). 3.2 Proof of linear ill-posedness Proof of Theorem 3.1.3. Fix an integer k 1. We seek an eigenvalue-eigenfunction pair (;) = ( k ; k ) for L, with oscillating at frequency k with respect to x 1 , i.e., we are looking for of the form (x 1 ;x 2 ) =c 1 sin(kx 1 ) cos(x 2 ) + sin(kx 1 ) X odd n3 c n cos(nx 2 ) + cos(kx 1 ) X even n2 c n sin(nx 2 ) (3.13) where the real coecientsfc n g n1 are to be determined. Using (ir)(x 1 ;x 2 ) = sin(kx 1 ) X odd n1 c n cos(nx 2 ) + cos(kx 1 ) X even n2 c n sin(nx 2 ) 33 we get L =k 3 cos(kx 1 ) X odd n1 (n 2 +k 2 ) (3)=2 c n cos(nx 2 ) sin(x 2 ) k 3 sin(kx 1 ) X even n2 (n 2 +k 2 ) (3)=2 c n sin(nx 2 ) sin(x 2 ) = cos(kx 1 ) X odd n1 k 3 (n 2 +k 2 ) (3)=2 c n sin((n + 1)x 2 ) sin((n 1)x 2 ) 2 + sin(kx 1 ) X even n2 k 3 (n 2 +k 2 ) (3)=2 c n cos((n + 1)x 2 ) cos((n 1)x 2 ) 2 : (3.14) We introduce the positive coecients p n =p n;k = 2(n 2 +k 2 ) (3)=2 k 3 (3.15) omitting the k dependence for ease of notation. Then for all n 1 we have p n 2n 3 k 3 ; (3.16) where by assumption 3 1. With (3.15) we simplify (3.14) as L = cos(kx 1 ) X even n2 c n1 p n1 c n+1 p n+1 sin(nx 2 ) c 2 p 2 sin(kx 1 ) cos(x 2 ) + sin(kx 1 ) X even n3 c n1 p n1 c n+1 p n+1 cos(nx 2 ): (3.17) 34 The equation =L thus becomes c 1 sin(kx 1 ) cos(x 2 ) + sin(kx 1 ) X odd n3 c n cos(nx 2 ) + cos(kx 1 ) X even n2 c n sin(nx 2 ) = cos(kx 1 ) X even n2 c n1 p n1 c n+1 p n+1 sin(nx 2 ) c 2 p 2 sin(kx 1 ) cos(x 2 ) + sin(kx 1 ) X even n3 c n1 p n1 c n+1 p n+1 cos(nx 2 ): (3.18) From (3.18) we obtain the recurrence relationship c n1 p n1 c n+1 p n+1 =c n ; n 2; (3.19) c 2 p 2 =c 1 ; n = 1: (3.20) Denoting n = c n p n c n1 p n1 1 ; n 2 the recurrence relation (3.19){(3.20) becomes 1 n n+1 =p n ; n 2 (3.21) 2 =p 1 : (3.22) A real number yields a solution of (3.21){(3.22) if and only if it solves the continued fraction equation p 1 = 1 p 2 + 1 p 3 + : (3.23) Note that here the coecientsfp n g n1 are given by (3.15), and that is the only unknown. Next we show the existence of a solution > 0 of (3.23). For this purpose, we need to prove rst that the continued fraction on the right side of (3.23) converges (note that since 35 the associated sequence is not monotone, this is not immediately clear), and then establish that it denes a continuous function of on (0;1). For n2f2; 3;g and m2f0; 1; 2; 3;g, dene F n;m () = 1 p n + 1 + 1 p n+m : It is clear that F n;m ()> 0 for 2 (0;1). From the identity F n;m () = 1 p n +F n+1;m1 () valid for n2f2; 3; 4;g and m2f1; 2; 3;g, we deduce F n;m+1 ()F n;m () = 1 p n +F n+1;m () 1 p n +F n+1;m1 () = F n+1;m1 ()F n+1;m () (p n +F n+1;m ())(p n +F n+1;m1 ()) from where we obtain jF n;m+1 ()F n;m ()j 1 (p n ) 2 jF n+1;m ()F n+1;m1 ()j: Using induction on m and the bound jF l;1 ()F l;0 ()j = 1 p l + 1 p l+1 1 p l 1 3 p 2 l p l+1 which is valid for all l 2, we obtain jF n;m+1 ()F n;m ()j 1 (p n ) 2 1 (p n+1 ) 2 1 (p n+m1 ) 2 1 3 p 2 n+m p n+m+1 = 1 2m+3 p n+m+1 (p n p n+1 :::p n+m ) 2 k 6m+9 2 2m+3 2m+3 (n 1)! 2 (n +m)!(n +m + 1)! 3 36 where used (3.16) in the last inequality. Hence, jF 2;m+l ()F 2;m ()j l1 X j=0 k 6m+6j+9 2 2m+2j+3 2m+2j+3 1 (2 +m +j)!(3 +m +j)! 3 : Since 3 1, the sequenceF 2;m () is uniformly Cauchy on every compact subset of (0;1). Therefore, as m!1 the sequence F 2;m () converges uniformly on compact subsets to a continuous function F 2 () = 1 p 2 + 1 p 3 + : Since F 2 () 1 p 2 (3.24) we moreover have lim !1 F 2 () = 0. The function G() =F 2 ()p 1 (3.25) is continuous on (0;1) and satises lim !1 G() =1: (3.26) On the other hand, we have the lower bound G() =F 2 ()p 1 1 p 2 + 1 p 3 p 1 = 1 2 p 1 p 2 p 1 p 3 p 2 + 1 p 3 : (3.27) Since p 3 >p 1 , it follows that there exists 0 > 0, such that G( 0 )> 0: (3.28) 37 From (3.26), (3.28), and the intermediate value theorem, we conclude that there exists = k 2 ( 0 ;1) such that G( k ) = 0 (3.29) providing a solution of (3.23). Note that (3.27) and (3.29) imply 0 1 2 k p 1 p 2 p 1 p 3 (3.30) providing a lower bound k p 1p 1 =p 3 p p 1 p 2 : Recalling (3.15), we obtain k k 3 q 1 (1 8=(9 +k 2 )) (3)=2 2(4 +k 2 ) (3)=2 for all k 1. An explicit computation shows that lim k!1 1 k 1 0 @ k 3 q 1 (1 8=(9 +k 2 )) (3)=2 2(4 +k 2 ) (3)=2 1 A = p 3 1 for any 2 (1; 2]. Using (1x) a 1x for 0a;x 1, we further obtain k C 1 k 1 (3.31) for any k 1 and any 2 (1; 2], where C 1 is a universal constant. In particular, k !1 as k!1 at least as fast as the power law k 1 . Having found a root = k of (3.23), we next need to estimate the size of the coecients c n dening in (3.13), and verify that they decay suciently fast so that is real-analytic. The function F n () = 1 p n + 1 p n+1 + ; 38 obeys the recurrence relationship F n () = 1 p n +F n+1 () ; or equivalently. F n+1 () =p n + 1 F n () : (3.32) Since also F 2 ( k ) = k p 1 ; (3.33) by the denition of k , we get, comparing the recurrence relations (3.21){(3.22) for n and (3.32){(3.33) for F n ( k ), that F n ( k ) = n : Using c n = (p n =p n1 )c n1 n for n 2 and F n ( k ) 1 k p n ; n 2 we get c n = n 2 p n c 1 p 1 =F n ( k )F 2 ( k ) p n c 1 p 1 1 k p n 1 k p n1 1 k p 2 p n c 1 p 1 = c 1 n1 k p n1 :::p 1 showing rapid convergence of c n to 0. Without loss of generality we choose c 1 = 1: Then, by (3.16) and (3.31) we have c n C n 1 k 1 n1 (k 3 ) n1 1 ((n 1)!) 3 C n k 4 n1 ((n 1)!) 3 (3.34) from where c n C n k 4 n (n!) 3 ; n 1: (3.35) 39 Furthermore, using n! (n=C) n for a suciently large C, we obtain c n exp n log Ck 4 n 3 (3.36) for all 2 (1; 2], n 1, and k 1. In particular, for any > 0, we see that for nn(;k;) :=Ce 4 k (4)=(3) ; (3.37) where C is suciently large, it holds that c n exp 2(n 2 +k 2 ) 1=2 : Using (3.13), we see that for ` = (` 1 ;` 2 )2Z 2 , we have j ^ k (`)j 2 = 8 < : c 2 n ; ` 1 =k;` 2 =n; 0; j` 1 j6=k: Therefore, k k k 2 G 1 = X n1 c 2 n (k 2 +n 2 ) 2 exp 2(k 2 +n 2 ) 1=2 n(;k;) X n=1 (k 2 +n 2 ) 2 exp 2(k 2 +n 2 ) 1=2 + 2n log 50k 4 n 3 + X n>n(;k;) (n 2 +k 2 ) 2 exp 2(n 2 +k 2 ) 1=2 <1 which shows that the function k whose Fourier seriesn-th coecients isc n , is in fact entire real-analytic. Next, we provide a proof of (3.11). For s 0, recalling (3.37) we estimate k k k 2 H s = X n1 (n 2 +k 2 ) s c 2 n 2 s n(1;k;) 2s n(1;k;) X n=1 c 2 n + X n>n(1;k;) (n 2 +k 2 ) s exp 2(n 2 +k 2 ) 1=2 C s k 2s(4)=(3) k k k 2 L 2 +C s exp k (4)=(3) 2C s k 2s(4)=(3) k k k 2 L 2 (3.38) 40 where C s > 0 is a suciently large constant that depends only on s, and is independent of 2 (1; 2]; we have also usedk k k 2 L 2 1, which follows from c 1 = 1. Since the equation L k = k k is linear, we may renormalize k to have the unit H s norm. Then, upon multiplying both sides of (3.38) with this constant, we obtain (3.11). The proof of (3.12) is similar. Given s 1 and > 0 we have k k k 2 G s = X n1 c 2 n (k 2 +n 2 ) 2 exp 2(k 2 +n 2 ) 1=2s n(;k;) 4 exp 4(n(;k;)) 1=s n(;k;) X n=1 c 2 n + X n>n(;k;) (k 2 +n 2 ) 2 exp 2(k 2 +n 2 ) 1=(2s) 2(n 2 +k 2 ) 1=2 2C s; exp C s; k (4)=s(3) k k k 2 L 2 (3.39) where we have also used that c 1 = 1. In order to conclude the proof we renormalize k to have unit G s norm, and then multiply both sides of (3.39) with this constant we obtain (3.12). 3.3 Proof of nonlinear Lipschitz ill-posedness Proof for Theorem 3.1.1. The proof of the above theorem follows directly from Theo- rem 3.1.3 and a classical perturbative argument (see, e.g. [FGSV12, FV11b, GN11, Ren09, Tao06]), so we only give here a sketch of these details. Fix (1) 0 (x 2 ) = (1) (x 2 ;t) = cos(x 2 ) a steady state of the system (3.1){(3.3) with (1.13). Clearlyk (1) 0 k H s = 1. For2 (0; 1], let (2;) 0 (x) = (1) 0 (x 2 )+ 0 (x) where 0 (x) is a smooth function such thatk 0 (x)k H s = 1, to be determined below. We havek (2;) 0 k H s 2 for all 2 (0; 1]. If the system (3.1){(3.3) with (1.13) would be locally Lipschitz well posed, then there would exist T 0 ;K 0 > 0 (uniform in ), such that the family of solutions (2;) (x;t) 2 L 1 (0;T ;H r ) of (3.1){(3.3) with (1.13) with the initial datum (2;) 0 (x) obey sup t2[0;T 0 ] k (2;) (t) (1) 0 k H rK 0 k (2;) 0 (1) 0 k H s =K 0 (3.40) 41 for all 2 (0; 1]. We dene (x;t) = 1 (2;) (x;t) (1) 0 ; to be the -perturbation of (2;) about (1) 0 . In view of (3.40) we have that is uniformly bounded in L 1 (0;T ;H r ), with the bound sup t2[0;T 0 ] k (t)k H rK 0 : (3.41) The equation obeyed by is @ t =L N[ ] so that by (3.8){(3.9) we have that@ t is uniformly bounded inL 1 (0;T ;L 2 ). Thus, Aubin{ Lions lemma ! in L 2 (0;T ;L 2 ), where 2L 1 (0;T ;H r ) with a bound inherited from (3.41), solves the equation @ t =L ; (0) = 0 : In order to conclude the proof, let 0 (x) = k (x), where k is an eigenfunction of L as constructed in Theorem 3.1.3, with eigenvalue k , and we choose k large enough such that exp T 0 k 1 =(2C 0 ) C s k s(4)=(3) 2K 0 whereC 0 is the constant from (3.10), andC s is the constant from (3.11). Then using (3.11) we obtain that k (T 0 =2)k L 2 = exp( k T 0 =2)k 0 k L 2 2K 0 which contradicts (3.41) since r> 0, thereby concluding the proof. Proof of Theorem 3.1.2. The proof is the same as the proof of Theorem 3.1.1, except that the eigenfunction 0 = k is normalized to have a unitG s norm, and we pickk large enough so that exp(T 0 k 1 =(2C 0 )) C s; exp(C s; k (4)=(s(3)) ) 2K 0 ; 42 where C 0 is the constant from (3.10), and C s; is the constant from (3.12). Finding such a values of k uses the restriction on s. Using (3.12) we then obtain thatk (T 0 =2)k L 2 2K 0 which yields a contradiction. 43 Chapter 4 Persistence of regularity for solutions of the Boussinesq equations in Sobolev spaces This chapter is devoted to the 2D Boussinesq equations with zero diusivity @u @t +uru u +rp =e 2 (4.1) divu = 0 (4.2) @ @t +ur = 0 (4.3) in general Sobolev spaces. Here, u is the velocity solving the 2D Navier-Stokes equation driven by , which represents the density or temperature of the uid, depending on the physical context, and e 2 = (0; 1) T . 4.1 On almost persistence In this section we prove that if the initial data (u 0 ; 0 ) belongs to W 1+s;q (R 2 )W s;q (R 2 ) where s2 (0; 1) and q2 (2;1), then (u(t);(t))2 W 1+s;q (R 2 )W s;q (R 2 ) for t2 [0;T ), whereT depends logarithmically on the size of the initial data. Further below we prove the global persistence in the intersection space (W 1+s;q (R 2 )W s;q (R 2 ))\(H 1+s (R 2 )H s (R 2 )). Before stating the rst main theorem, we assert the global persistence in W 1;q (R 2 ) L q (R 2 ). Proposition 4.1.1. Assume that, for some q 2, we have (u 0 ; 0 )2 W 1;q (R 2 )L q (R 2 ) with divu 0 = 0. Then there exists a unique solution (u;) to the equations (4.1){(4.3) such that u2C [0;1);W 1;q (R 2 ) and 2C [0;1);L q (R 2 ) . Moreover, with ! =ru, there exists C depending onku 0 k W 1;q andk 0 k L q such that kru(t)k L qC p t + 1; t2 [0;1) (4.4) 44 and Z t 0 kr(j!j q=2 )k 2 L 2 dsC(1 +t) q=2 (4.5) with k(t)k L q =k 0 k L q; t2 [0;1) (4.6) hold. The following is our rst main statement. Theorem 4.1.2. Let s 2 (0; 1) and q 2 (2;1). Assume that ku 0 k W 1+s;q M 0 with divu 0 = 0 andk 0 k W s;q M 1 , where M 0 ;M 1 1 are constants. Then there exists a constant C depending onku 0 k W 1;q andk 0 k L q such that for T = min 1 + 1 C(1 + log 1=q (1 +M 0 +M 1 )) 2=3 1; 1 C(1 + log 1=q (1 +M 0 +M 1 ))) 2q=(q2) ; 1 C(1 + log 1=q (1 +M 0 +M 1 ))) q=(q1) ; (4.7) there exists a unique solution (u;) to the equations (4.1){(4.3) such that u 2 C [0;T );W 1+s;q (R 2 ) and 2C [0;T );W s;q (R 2 ) . It is clear from the proof below that we can take C to be proportional toku 0 k W 1;q + k 0 k L q. Proof of Proposition 4.1.1. Since the case q = 2 is covered in [LLT10], we assume q > 2. We start by multiplying the equation (4.1) withjuj q2 u and integrating it with respect to x obtaining 1 q d dt kuk q L q + Z (uru)ujuj q2 dx Z uujuj q2 dx + Z rpujuj q2 dx = Z e 2 ujuj q2 dx: (4.8) Due to the divergence-free condition foru, we have R (uru)ujuj q2 dx = 0 while integrating by parts, we obtain Z uujuj q2 dx = Z @ j u k @ j u k juj q2 dx + (q 2) Z (u k @ j u k )(u l @ j u l )juj q4 dx; (4.9) 45 where the summation convention on repeated indices is used throughout. Denoting the right side of the above equality by D 0 , we get from (4.8) 1 q d dt kuk q L q +D 0 = Z rpujuj q2 dx + Z e 2 ujuj q2 dx krpk L qkuk q1 L q +kk L qkuk q1 L q : (4.10) Noting that D 0 0, we only need to estimate the pressure term. We take the divergence of the equation (4.1) and obtain p = div(uru)@ 2 : (4.11) Writing rp =r() 1 div(uru)@ 2 ; (4.12) we may apply the Calder on-Zygmund theorem in order to get krpk L qC(kuruk L q +kk L q)C(kuk L 1kruk L q +kk L q) C(kuk W 1;qkruk L q +kk L q)C(kuk L q +kruk L q)kruk L q +Ckk L q C(kuk L q +k!k L q)k!k L q +Ckk L q; (4.13) where ! = curlu is the vorticity and where the constant depends on q (considered xed). In order to get the bound forkk L q, we multiply the equation (4.3) byjj q2 and integrate with respect to x to get 1 q d dt kk q L q = 0; (4.14) where we used Z urjj q2 dx = 0: (4.15) Therefore, the L q norm of is preserved by this system, i.e., k(t)k L q =k 0 k L q; t> 0: (4.16) Combining (4.10) with (4.13) and (4.16) leads to 1 q d dt kuk q L q +D 0 C(kuk L qk!k L q +k!k 2 L q + 1)kuk q1 L q : (4.17) 46 Taking the curl of the equation (4.1), we get ! t +ur! ! = curl(e 2 ) =@ 1 : (4.18) We multiply both sides of the equation (4.18) byj!j q2 ! and integrate it to get an L q estimate of !, Z ! t j!j q2 !dx + Z ur!j!j q2 !dx Z !j!j q2 !dx = Z @ 1 j!j q2 !dx: (4.19) Due to the divergence-free condition, we have R ur!j!j q2 !dx = 0. Integrating by parts the third term on the left side of (4.19), we arrive at 1 q d dt k!k q L q + (q 1) 2 q 2 kr(j!j q=2 )k 2 L 2 =(q 1) Z j!j q2 @ 1 !dx: (4.20) By H older's inequality and using kj!j (q2)=2 @ 1 !k L 2 = 2 q k@ 1 (j!j q=2 )k L 2; (4.21) we have Z j!j q2 @ 1 !dx C q kk L qkr(j!j q=2 )k L 2k!k (q2)=2 L q : (4.22) We apply the inequality (4.22) to the equation (4.20) and use Young's inequality to get 1 q d dt k!k q L q + (q 1) 2 q 2 kr(j!j q=2 )k 2 L 2 1 2 (q 1) 2 q 2 kr(j!j q=2 )k 2 L 2 +Ckk 2 L qk!k q2 L q ; (4.23) where, recall, the constant C depends on q. Absorbing the rst term on the right side of the above inequality, we get d dt k!k q L q + 1 C kr(j!j q=2 )k 2 L 2 Ckk 2 L qk!k q2 L q : (4.24) The inequality (4.24), combined with (4.16), implies k!(t)k L qC p 1 +t; t2 [0;1); (4.25) 47 as well as Z t 0 kr(j!j q=2 )k 2 L 2 dsC(1 +t) q=2 ; (4.26) where C depends on the initial dataku 0 k W 1;q andk 0 k L q. Using (4.25) in (4.17) and applying the Gronwall's inequality then concludes the proof. 4.2 A commutator lemma The main step in the proof of Theorem 4.1.2 is a commutator estimate, which is stated next. Let = () 1=2 . Lemma 4.2.1. Let s2 (0; 1) and f;g2S(R 2 ). For 1<q<1 and j2f1; 2g, k[ s @ j ;g]fk L q (R 2 ) Ckfk L r 1k 1+s gk L e r 1 +Ck s fk L r 2kgk L e r 2 (4.27) holds, where r 1 ;e r 1 ;e r 2 2 [q;1] and r 2 2 [q;1) satisfy 1=q = 1=r 1 + 1=e r 1 = 1=r 2 + 1=e r 2 , and where C is a constant depending on r 1 ,e r 1 , r 2 ,e r 2 , s, and q. Recall that [ s @ j ;g]f = s @ j (gf)g s @ j f: (4.28) This lemma can be seen as an extension of the Kato-Ponce inequality since we allowe r 1 =1. The proof uses the ideas from [KP88]. Proof. Taking the Fourier transform of [ s @ j ;g]f, we get ([ s @ j ;g]f) ^ () =i Z R 2 (jj s j jj s () j ) ^ f()^ g()d: (4.29) Therefore, [ s @ j ;g]f =c 0 ZZ e ix (jj s j jj s () j ) ^ f()^ g()dd =c 0 ZZ e ix(+) (j +j s ( +) j jj s j ) ^ f()^ g()dd =c 0 3 X k=1 ZZ e ix(+) (j +j s ( +) j jj s j ) ^ f()^ g() k (jj=jj)dd (4.30) 48 where c 0 =i=4 2 , and where k :R! [0; 1] are C 1 cut-o functions such that 3 X k=1 k = 1 on [0;1) (4.31) with supp 1 [1=2; 1=2]; supp 2 [1=4; 3]; supp 3 [2;1): (4.32) Denote A k (;) = (j +j s ( +) j jj s j ) ^ f()^ g() k (jj=jj); k = 1; 2; 3: (4.33) The commutator (4.28) may be rewritten as [ s @ j ;g]f =c 0 3 X k=1 ZZ e ix(+) A k (;)dd: (4.34) First, we write A 1 as A 1 (;) = j +j s ( +) j jj s j jj 1+s 1 jj jj ^ f()( 1+s g)^() = 1 (;) ^ f()( 1+s g)^(): (4.35) It is easy to check that j 1 jC; (4.36) where C is a constant and where we used 2jjjj on supp 1 . Taking the rst derivative with respect to l , where l = 1; 2, we obtain @ 1 @ l C jj C jj +jj (4.37) while dierentiating with respect to l for l = 1; 2 leads to @ 1 @ l C jj C jj +jj : (4.38) Continuing by induction, we get j@ @ 1 jC(jj;jj)(jj +jj) (jj+jj) ; ;2N 2 0 : (4.39) 49 By the Coifman-Meyer theorem, we get ZZ e ix(+) A 1 (;)dd L q Ckfk L r 1k 1+s gk L e r 1 (4.40) where 1=q = 1=r 1 + 1=e r 1 . Postponing the treatment of A 2 , we deal with A 3 next. We rewrite this symbol as A 3 (;) =j +j s j 3 jj jj ^ f()^ g() + (j +j s jj s ) j 3 jj jj ^ f()^ g() =i j +j s jj s 3 jj jj ( s f)^()(@ j g)^() + (j +j s jj s ) j jj s 3 jj jj ( s f)^()^ g() = 3;1 (;)( s f)^()(@ j g)^() + (j +j s jj s ) j jj s 3 jj jj ( s f)^()^ g(): (4.41) Note that in the region 3 > 0, we havejj> 2jj. For 31 it is easy to check that j@ @ 31 jC(jj +jj) (jj+jj) ; ;2 (Z + ) 2 : (4.42) In order to estimate the second term on the far right side of (4.41), we write j (j +j s jj s ) = j Z 1 0 d dr (j +rj s )dr = j Z 1 0 sj +rj s2 ( +r)dr; (4.43) which implies that the second term may be rewritten as i 3 jj jj ( s f)^()(@ k g)^() Z 1 0 sj +rj s2 ( k +r k ) j jj s dr = 32 (;)( s f)^() (rg)^(): (4.44) A direct computation shows that j@ @ 32 (;)jC(jj +jj) (jj+jj) ; ;2N 2 0 : (4.45) 50 Therefore, we obtain that ZZ e ix(+) A 3 dd L q Ck s fk L r 2krgk L e r 2 ; (4.46) where 1=q = 1=r 2 + 1=e r 2 . Finally we treat the second term of the sum on the right side of (4.34), in which casejj andjj are comparable. Note that A 2 (;) =j +j s ( +) j 2 jj jj ^ f()^ g()jj s j 2 jj jj ^ f()^ g() =A 21 A 22 : (4.47) First we deal with the simpler term A 22 , which may be written as A 22 (;) = jj s j jj s jj 2 jj jj ( s f)^()(g)^(): (4.48) Applying the Coifman-Meyer theorem we get ZZ e ix(+) A 22 (;)dd L q Ck s fk L r 2kgk L e r 2 : (4.49) In order to conclude the lemma we only need to get a similar estimate for A 21 , to which the above method does not apply. The main reason is that when we take the derivative, the factor ofj +j appears in the denominator. Thus, as in [KP88], we use the complex interpolation to avoid this diculty. First, we have ZZ e ix(+) A 21 (;)dd L q C(b)k s fk L r 2kgk L e r 2 ; (4.50) where s =a +ib, with a a suciently large positive number and b2R, and where C(b) is a polynomial. Next we consider the case s =ib with b2R. In order to get an estimate for RR e ix(+) A 21 dd L q , we rewrite ZZ e ix(+) A 21 (;)dd = ZZ e ix(+) j +j s ( +) j ^ f()^ g() 2 dd = ZZ e ix(+) j +j s ( +) j jj ^ f()(g)^() 2 dd = Z e ix jj s Z j jj ^ f()(g)^() 2 dd: (4.51) 51 Using the H ormander-Mikhlin multiplier theorem for the symboljj s =jj ib , we have ZZ e ix(+) A 21 (;)dd L q C(b) Z e ix Z j jj ^ f()(g)^() 2 dd L q =C(b) ZZ e ix(+) ( +) j jj 2 ^ f()(g)^()dd L q : (4.52) Since @ @ ( +) j jj 2 jj jj C(jj +jj) (jj+jj) ; ;2N 2 0 ; (4.53) using the Coifman-Meyer theorem, we have ZZ e ix(+) A 21 (;)dd L q C(b)kfk L r 2kgk L e r 2 : (4.54) Thus, using the complex interpolation inequality, we get ZZ e ix(+) A 21 (;)dd L q C(b)k s fk L r 2kgk L e r 2 (4.55) for any s such thatRe(s)2 (0; 1). Taking s2 (0; 1), sinceIm(s) = 0, we have ZZ e ix(+) A 21 (;)dd L r 2 Ck s fk L qkgk L e r 2 (4.56) and the proof is completed. 4.3 An L 1 bound The following statement provides L 1 bounds on the velocity and the vorticity in terms of L q norms of the derivatives of the vorticity. Lemma 4.3.1. Assume thatu2S(R 2 ) 2 is a vector eld satisfying divu = 0. Let 0<s<1 and 1r1 be such that rs> 2. For 2q<1, with ! = curlu, the inequalities kuk L 1C 1 + log(1 +k s !k L r) 1+1=q (1 +k!k L q +kr(j!j q=2 )k 2=q L 2 ) (4.57) and k!k L 1C 1 + log(1 +k s !k L r) 1=q (1 +k!k L q +kr(j!j q=2 )k 2=q L 2 ) (4.58) 52 hold, where C =C(r;s;q). Proof. First we prove that kfk L 1C(kfk L b +k s fk L r); b 1; f2S(R 2 ) (4.59) where the constant C depends only on r and s, i.e., it is independent of b. Without loss of generality, we may assumekfk L b +k s fk L r = 1. Using the standard Littlewood-Paley notation, we have kfk L 1kS 0 fk L 1 + 1 X j=0 k j fk L 1: (4.60) By Bernstein's inequality, we bound the rst term on the right side as kS 0 fk L 1CkS 0 fk L bC: (4.61) In order to estimate the second term on the right side of (4.60), we write k j fk L 1C2 2j=r k j fk L rC2 2j=rjs k s j fk L rC2 j(2=rs) ; (4.62) where we used Bernstein's inequality in the rst step. Since rs > 2, we may sum up the terms and obtainkfk L 1 C, thus leading to (4.59). By rescaling (4.59), we obtain the Gagliardo-Nirenberg inequality kfk L 1Ckfk 1a L b k s fk a L r (4.63) where a = 2 2 +b(s 2=r) : (4.64) Applying the above inequality to u with bq to be determined, we get kuk L 1Ckuk 1a L b k 1+s uk a L rCbk!k 1a L b k s !k a L rCb(1 +k!k L b)k s !k a L r; (4.65) 53 where the parameters r, s, and q are xed, while we need to track the dependence of the constants on b. Also, we have k!k L b =kj!j q=2 k 2=q L 2b=q C s b q kj!j q=2 k q=b L 2 kr(j!j q=2 )k 1q=b L 2 ! 2=q Cb 1=q (kj!j q=2 k L 2 +kr(j!j q=2 )k L 2) 2=q Cb 1=q (k!k L q +kr(j!j q=2 )k 2=q L 2 ): (4.66) Using (4.66) in (4.65), we get kuk L 1Cb 1+1=q (1 +k!k L q +kr(j!j q=2 )k 2=q L 2 )k s !k a L r: (4.67) Therefore, by setting b =q + log(1 +k s !k L r), we obtain kuk L 1C 1 + log(1 +k s !k L r) 1+1=q (1 +k!k L q +kr(j!j q=2 )k 2=q L 2 ): (4.68) Indeed, note thatk s !k a L r = exp(a logk s !k L r), which is bounded by a constant by our choice of b and (4.64). Thus (4.57) is established. Similarly to (4.65), we get k!k L 1Ck!k 1a L b k s !k a L r: (4.69) Repeating the above procedure, we get k!k L 1C 1 + log(1 +k s !k L r) 1=q (1 +k!k L q +kr(j!j q=2 )k 2=q L 2 ); (4.70) which gives (4.58). 4.4 The proof of local persistence We are now ready to prove Theorem 4.1.2. Since the persistence in W 1;q (R 2 )L q (R 2 ) has been addressed in Proposition 4.1.1, we focus on the estimate of the highest derivatives of u , namely onk s !k L q andk s k L q. Since the cancellation property is not available, we obtain an extra term involvingkuk L 1. Using results in Sectionx 4.3, we may bound it by log 1+1=q (1 +k s !k L r)(1 +k!k L q + kr(j!j q=2 )k 2=q L 2 ), leading to logarithmic dependence on the norms. Note that since 1 + 1=q > 1, we do not obtain the global existence (note, however, the global results in Sectionsx 4.5 andx 4.6). 54 Proof of Theorem 4.1.2. Applying the operator s to the equation (4.18), multiplying it withj s !j q2 s !, and integrating in x, we get 1 q d dt k s !k q L q Z s !j s !j q2 s !dx + Z s (ur!)j s !j q2 s !dx = Z s @ 1 j s !j q2 s !dx: (4.71) Next, we integrate by parts in the second term on the left side of the above equation and move the third term to the right side in order to get 1 q d dt k s !k q L q + (q 1) 2 q 2 kr(j s !j q=2 )k 2 L 2 = Z s @ 1 j s !j q2 s !dx Z s (ur!)j s !j q2 s !dx =I 1 +I 2 ; (4.72) where we also used Z j s r!j 2 j s !j q2 dx = 2 q 2 kr(j s !j q=2 )k 2 L 2 : (4.73) For the term I 1 , we integrate by parts and use H older's inequality to obtain I 1 = (q 1) Z s (e 2 )j s !j q2 s @ 1 !dx Ck s k L qk s r!j s !j (q2)=2 k L 2kj s !j (q2)=2 k L 2q=(q2) =Ck s k L qkr(j s !j q=2 )k L 2k s !k (q2)=2 L q : (4.74) Since I 2 = Z s (ur!)u s r! j s !j q2 s !dx; (4.75) we have by H older's inequality and Lemma 4.2.1 I 2 k s (ur!)u s r!k L qk s !k q1 L q C(k s !k L qkuk L 1 +k!k L 1k 1+s uk L q)k s !k q1 L q Ck s !k q L q (kuk L 1 +k!k L 1): (4.76) 55 Using (4.74) and (4.76) in (4.72), we get 1 q d dt k s !k q L q + (q 1) 2 q 2 kr(j s !j q=2 )k 2 L 2 Ck s k L qkr(j s !j q=2 )k L 2k s !k (q2)=2 L q +Ck s !k q L q (kuk L 1 +k!k L 1) Ck s k q L q + q 1 q 2 kr(j s !j q=2 )k 2 L 2 +Ck s !k q L q +Ck s !k q L q (kuk L 1 +k!k L 1); (4.77) where we used Young's inequality in the last step. Absorbing the second term on the right side into the left side of the above inequality, we obtain d dt k s !k q L q +kr(j s !j q=2 )k 2 L 2 Ck s k q L q +Ck s !k q L q +Ck s !k q L q (kuk L 1 +k!k L 1): (4.78) Adding the inequalities (4.17) and (4.78) and omitting the D 0 term, we arrive at d dt (kuk q L q +k s !k q L q ) +kr(j s !j q=2 )k 2 L 2 C(kuk L qk!k L q +k!k 2 L q + 1)kuk q1 L q +Ck s k q L q +Ck s !k q L q +Ck s !k q L q (k!k L 1 +kuk L 1): (4.79) Next, we need an appropriate estimate fork s k L q andk!k L 1 +kuk L 1. We apply the operator s to the equation (4.3), multiply it withj s j q2 s , and integrate it with respect to x to get 1 q d dt k s k q L q = Z s (ur)j s j q2 s dx = Z s div(u)j s j q2 s dx; (4.80) where we used the divergence-free condition for u. Since Z s div(u)j s j q2 s dx = Z s div(u)ur s j s j q2 s dx (4.81) 56 we obtain by Lemma 4.2.1 Z s div(u)j s j q2 s dxk s div(u)ur s k L qk s k q1 L q Ck s k L qkuk L 1k s k q1 L q +Ckk L r 1k 1+s uk L s 1k s k q1 L q ; (4.82) where r 1 = 2q=(2s) and s 1 = 2q=s. Denoting I = 8 < : [q; 2q=(2qs)]; if qs< 2 [q;1); otherwise; (4.83) we observe that 0 2W s;q implies 0 2L r for r2I, and thus k(t)k L r 1 =C(M 1 ;r 1 ); t> 0: (4.84) Fork 1+s uk L s 1 , by the Sobolev embedding theorem and the Gagliardo-Nirenberg inequality, we have k 1+s uk L s 1Ck s !k L s 1Ckj s !j q=2 k 2=q L 2s 1 =q Ckj s !j q=2 k 2=q(1) L 2 krj s !j q=2 k 2=q L 2 =Ck s !k 1 L q krj s !j q=2 k 2=q L 2 ; (4.85) where = 1s=2. Therefore, combining (4.80) and (4.82) with (4.85) leads to 1 q d dt k s k q L q Ckuk L 1kk q W s;q +Ckk L r 1 (kuk W 1+s;q +k s !k 1 L q krj s !j q=2 k 2=q L 2 )k s k q1 L q Ckuk L 1kk q W s;q +Ckuk q W 1+s;q +Ck s !k q L q +Ck s k q L q + q 1 q 2 krj s !j q=2 k 2 L 2 ; (4.86) where we used Young's inequality and (4.84). Adding the above estimate to (4.79) and setting F (t) =kuk q L q +k s !k q L q +kk q L q +k s k q L q , we get 1 q F 0 (t) + (q 1) 2 q 2 kr(j s !j q=2 )k 2 L 2 C 1 +F (t) +CF (t)k!k L q +CF (t)(kuk L 1 +k!k L 1): (4.87) 57 Finally, we handle the termskuk L 1 andk!k L 1. By Lemma 4.3.1, we get from (4.87) 1 q F 0 (t) + (q 1) 2 q 2 kr(j s !j q=2 )k 2 L 2 C 1 +F (t) +CF (t)k!k L q +CF (t) 1 +kr(j!j q=2 )k 2=q L 2 1 + log 1+1=q (1 +k s !k L r) =C 1 +F (t) +CF (t) p 1 +t +CF (t)X(t) 1 + log 1+1=q (1 +k s !k L r) ; (4.88) where we chose r> max (2=s;q) and denoted X(t) = 1 +kr(j!j q=2 )k 2=q L 2 . By (4.26) Z t 0 X q (s)dsC(1 +t) q=2 (4.89) holds. As for the termk s !k L r, we have by the Gagliardo-Nirenberg and Young's inequal- ities, k s !k L r =kj s !j q=2 k 2=q L 2r=q (kj s !j q=2 k 1 L 2 kr(j s !j q=2 )k L 2 ) 2=q k s !k L q +kr(j s !j q=2 )k 2=q L 2 k s !k L q +C + q 1 q 2 kr(j s !j q=2 )k 2 L 2 : (4.90) Now we conclude the proof by applying the following Gronwall type lemma to (4.88). Lemma 4.4.1. Assume thatF (t) is a continuously dierentiable function in a neighborhood of 0 and F (0)M. Let X(t);Y (t) and D(t): [0;1)! [0;1) be such that Z t 0 X(s) q dsC(1 +t) q=2 (4.91) for someq 2 andY (t)D(t)=2+F (t)+M, whereM andM are constants. Furthermore, assume that F (t) satises F 0 (t) +D(t)C(1 +F (t)) +CF (t) p 1 +t +CF (t)X(t) 1 + log(1 +Y (t)) (4.92) for some > 0. With T =1 if 0< 1 and T = min 1 + 1 C(1 + log 1 (M + 1)) 2=3 1; 1 C(1 + log 1 (M + 1)) 2q=(q2) ; 1 C(1 + log 1 (M + 1)) q=(q1) (4.93) 58 for some constant C if > 1, we then have F (t)<1; t2 [0;T ) (4.94) where C is a constant depending only on M, M, and T . Proof. For the case 2 (0; 1], we have a basic inequality FX(1 + log(1 +Y )) FX(1 + log(1 +Y ))FX + Y C +FX log(1 +CFX): (4.95) Hence, we have F 0 (t) +D(t)C(1 +F (t)) +Y (t) +CF (t)X(t) +CF (t) p 1 +t +CF (t)X(t) log(1 +CF (t)X(t)) C +CF (t) + D(t) 2 +CF (t)X(t) +CF (t) p 1 +t +CF (t)X(t) log(1 +F (t)) +CF (t)X(t) log(1 +CX(t)): (4.96) The desired result is then obtained by the usual Gronwall lemma. If on the other hand > 1, we get FX(1 + log (1 +Y ))FX + Y C +CFX log (1 +CFX): (4.97) Similarly to (4.96), we arrive at F 0 (t) +D(t)C +CF (t) + D(t) 2 +CF (t)X(t) +CF (t) p 1 +t +CF (t)X(t) log (1 +F (t)) +CF (t)X(t) log (1 +CX(t)) (4.98) and the lemma is established by the classical Gronwall's inequality. 4.5 Persistence for qs> 2 and for the intersection spaces Given initial datum (u 0 ; 0 ) 2 W 1+s;q W s;q we do not know in general whether the solution (u;) stays in the space W 1+s;q W s;q for all time. However, we can prove that this is so if sq > 2 or if the initial data (u 0 ; 0 ) belongs to the intersection space X = (H 1+s H s )\ (W 1+s;q W s;q ). We consider the case sq > 2 in the next theorem, while the intersection space is addressed in Theorem 4.5.3 below. 59 Theorem 4.5.1. Let s 2 (0; 1) and q 2 [2;1) be such that sq > 2. Assume that ku 0 k W 1+s;q M 0 with divu 0 = 0 and k 0 k W s;q M 1 , where M 0 ;M 1 1 are con- stants. Then there exists a unique solution (u;) of the equations (4.1){(4.3) such that u2C [0;1);W 1+s;q (R 2 ) and 2C [0;1);W s;q (R 2 ) . The key ingredient making the global persistence work is the following uniform bound on the vorticity. Lemma 4.5.2. Assume that q > 2 and ! 0 2 W 1;q (R 2 ), and let 2 L 1 (0;1;L q (R 2 )\ L 1 (R 2 )). Then the unique solution! of the equation (4.18) with the initial data! 0 satises !2C([0;1);L 1 (R 2 )). Moreover for any T > 0 the inequality k!(t)k L 1C(kk L 1 (0;1;L q \L 1 ) ;k! 0 k W 1;q) p 1 +T; t2 [0;T ] (4.99) holds. Proof. Fix T > 0. By (4.25) we have k!(t)k L qC(kk L 1 (0;1;L q ) ) p 1 +T; t2 [0;T ]: (4.100) For pq, denote p = R j!j p . Using the estimate Z j!j 2p2 @ 1 !dx C p kk L 1kr(j!j p )k L 2k!k p1 L 2p2 ; (4.101) we obtain from (4.20) that 1 2p 0 2p + 2p 1 2p 2 kr(j!j p )k 2 L 2 Cpkk 2 L 1 Z ! 2p2 Cpkk 2 L 1( p ) 2=p ( 2p ) (p2)=p : (4.102) Applying the Gagliardo-Nirenberg inequalitykfk L 2Ckfk 1=2 L 1 krfk 1=2 L 2 tof =j!j p leads to 2p C p kr(j!j p )k L 2; (4.103) which combined with (4.102) implies 1 2p 0 2p + 2p 1 Cp 2 2p p 2 Cpkk 2 L 1( p ) 2=p ( 2p ) (p2)=p : (4.104) 60 Note that if 2p C p=(p+2) p 2p=(p+2) kk 2p=(p+2) L 1 (2p+2)=(p+2) p ; (4.105) for a suciently large constant C, the second term on the left side dominates the term on the right and thus 0 2p 0: (4.106) Denoting K p =k!k L 1 (0;T ;L p (R) 2 ) ; (4.107) 2p may be estimated by 2p (t) max 2p (0);C 1=2p p 1=p kk 1=p L 1 K (p+1)=(p+2) p ; t2 [0;T ]: (4.108) Taking the supremum in t, we obtain K 2p max Ck! 0 k W s;q;C 1=2p p 1=p kk 1=p L 1 K (p+1)=(p+2) p : (4.109) By induction and (4.100), we arrive at K 2 n+1 q maxfCk! 0 k W s;q;K q gC P 1 i=1 1=2 i q 1 Y i=0 (2 i q) 1=2 i q (1 +kk L 1 (0;1;L 1 ) ) P 1 i=0 1=2 i q C(kk L 1 (0;1;L q \L 1 ) ;k! 0 k W s;q) p 1 +T; t2 [0;T ]: (4.110) Since the constant C does not depend on n, we may pass to the limit n!1 to conclude the proof. Next we prove Theorem 4.5.1. Proof of Theorem 4.5.1. Since the persistence in H 1+s H s was already addressed in [HKZ15], we only need to consider the case q > 2. The proof is similar to that of Theo- rem 4.1.2 except that we use a dierent estimate forkuk L 1 andk!k L 1. Recall that 1 q F 0 (t)+(q1) 2 q 2 kr(j s !j q=2 )k 2 L 2 C 1+F (t) p 1 +t+CF (t)(kuk L 1+k!k L 1); (4.111) where F (t) =kuk q L q +k s !k q L q +kk q L q +k s k q L q . By Lemma 4.5.2 , we have k!k L 1C p 1 +t; t 0: (4.112) 61 From (4.65), we deduce kuk L 1Cbk!k 1a L b k s !k a L rCbk!k q(1a)=b L q k!k (1q=b)(1a) L 1 k s !k a L r Cb(1 +k!k L q +k!k L 1)k s !k a L r; (4.113) where rs> 2, bq, and a = 2=(2 +b(s 2=r)). Similarly as in the proof of Lemma 4.3.1, we set b =q + log(1 +k s !k L r) in order to get kuk L 1C 1 + log(1 +k s !k L r) (1 +k!k L q +k!k L 1): (4.114) Therefore, noting thatk!(t)k L qC p 1 +t, we obtain 1 q F 0 (t) + (q 1) 2 q 2 kr(j s !j q=2 )k 2 L 2 C 1 +F (t) 1 + log(1 +k s !k L r) p 1 +t: (4.115) We conclude the proof by using Lemma 4.4.1. Denoting k(u;)k X = max kuk H 1+s;kuk W 1+s;q;kk H s;kk W s;q ; the global persistence result in the intersection space X reads as follows. Theorem 4.5.3. Let s2 (0; 1) and q 2 [2;1). Assume thatk(u 0 ; 0 )k X M where M is an arbitrary positive constant. There exists a unique solution (u;) to the equations (4.1){(4.3) such that (u;)2C [0;1);X). Moreover, k(u(t);(t))k X C(M;T ); t2 [0;T ] (4.116) for any xed T > 0. In the proof, we need the following modication of Lemma 4.3.1. Lemma 4.5.4. Assume that u2S(R 2 ) 2 . Also, let 0<s<1 and 1r1 be such that rs> 2. For 2q<1, with ! = curlu, we have the inequalities kuk L 1C 1 + log(1 +k 1+s uk L r) 1=2 (1 +kuk H 2) (4.117) and k!k L 1C 1 + log(1 +k s !k L r) 1=2 (1 +k!k H 1): (4.118) 62 where C =C(r;s;q). Proof of Lemma 4.5.4. By (4.65), we have kuk L 1Ckuk 1a L b k 1+s uk a L rC(1 +kuk L b)k 1+s uk a L r (4.119) where b 2 and a is given in (4.64). Also, since as in (4.66), kuk L bCb 1=2 (kuk L 2 +k 2 uk L 2)Cb 1=2 kuk H 2 (4.120) we get kuk L 1Cb 1=2 (1 +kuk H 2)k 1+s uk a L r (4.121) Then choose b = 2 + log(1 +k 1+s uk L r) and (4.117) follows. The inequality (4.118) is proven analogously. Proof of Theorem 4.5.3. The proof is similar to that of Theorem 4.1.2 except that we use a dierent estimate forkuk L 1 andk!k L 1. Recall that we have 1 q F 0 (t)+(q1) 2 q 2 kr(j s !j q=2 )k 2 L 2 C 1+F (t) p 1 +t+CF (t)(kuk L 1+k!k L 1); (4.122) where F (t) =kuk q L q +k s !k q L q +kk q L q +k s k q L q . By Lemma 4.5.4 and the Calder on- Zygmund inequality, we have kuk L 1C 1 + log(1 +k 1+s uk L r) 1=2 (1 +kuk H 2) C 1 + log(1 +k s !k L r) 1=2 (1 +kuk H 2) (4.123) and k!k L 1C 1 + log(1 +k s !k L r) 1=2 (1 +k!k H 1) C 1 + log(1 +k s !k L r) 1=2 (1 +kuk H 2): (4.124) Therefore, we obtain 1 q F 0 (t) + (q 1) 2 q 2 kr(j s !j q=2 )k 2 L 2 C 1 +F (t) p 1 +t +CF (t)(1 +kuk H 2) 1 + log(1 +k s !k L r) 1=2 : (4.125) 63 From [HKZ15] we recall that Z T 0 ku(t)k 2 H 2 dt<1: (4.126) Therefore, together with (4.90), we conclude the proof by using Lemma 4.4.1. Corollary 4.5.5. Let (u 0 ; 0 )2W 1+s;q (R 2 )W s;q (R 2 ) be compactly supported and assume that s and q are the same as in the above theorem. There exists a unique solution (u;) to the equations (4.1){(4.3) such that (u;)2C [0;1);W 1+s;q (R 2 )W s;q (R 2 ) . Moreover, ku(t)k W 1+s;q (R 2 ) ;k(t)k W s;q (R 2 ) C; t2 [0;T ] (4.127) for any xed T > 0, where C depends on the initial data and T . Proof. Since (u 0 ; 0 ) 2 X by the assumptions, the assertion follows by applying Theo- rem 4.5.3. 4.6 Persistence with periodic boundary conditions Using the above theorem, we may also get the global persistence in the periodic domain. Theorem 4.6.1. Letku 0 k W 1+s;q (T 2 ) ;k 0 k W s;q (T 2 ) M where M is an arbitrary positive constant and where 0<s< 1 and q2 [2;1). Then there exists a unique solution (u;) to the equations (4.1){(4.3) such that (u;)2C [0;1);W 1+s;q (T 2 )W s;q (T 2 ) . Moreover, ku(t)k W 1+s;q (T 2 ) ;k(t)k W s;q (T 2 ) C(M;T ); t2 [0;T ] (4.128) for any xed T > 0. Proof. Since H 1+s (T 2 ) ,! W 1+s;q (T 2 ) and H s (T 2 ) ,! W s;q (T 2 ), the theorem follows by applying Theorem 4.5.3 provided we can prove Lemma 4.2.1 for the periodic domainT 2 . In fact the Coifman-Mayer estimate still holds for the periodic domain as shown in [Wor10]. Then the same argument as in the proof of Lemma 4.2.1 shows that k[ s @ j ;g]fk L q (T 2 ) Ckfk L r 1 (T 2 ) k 1+s gk L e r 1 (T 2 ) +Ck s fk L r 2 (T 2 ) kgk L e r 2 (T 2 ) (4.129) holds, wheref,g,s,j,q,r 1 ,e r 1 ,r 2 , ande r 2 are as in the statement of Lemma 4.2.1, with the only dierences replacing 2R 2 and 2R 2 by discrete variables m2T 2 and n2T 2 . 64 Chapter 5 The van Dommelen and Shen singularity in the Prandtl equations Recall the 2D Prandtl boundary layer equations for the unknown velocity eld (u;v) = (u(t;x;y);v(t;x;y)): @ t u@ yy u +u@ x u +v@ y u =@ x P E (5.1) @ x u +@ y v = 0 (5.2) uj y=0 =vj y=0 = 0 (5.3) uj y!1 =U E : (5.4) The domain we consider isTR + =f(x;y)2TR: y 0g, with corresponding periodic boundary conditions in x for all functions. The function U E =U E (t;x) obey the Bernoulli equation @ t U E +U E @ x U E =@ x P E (5.5) for x2T and t 0, with periodic boundary conditions. In this chapter we consider U E = sinx (5.6) @ x P E = 2 2 sin(2x) (5.7) proposed by van Dommelen and Shen in [vDS80], where 6= 0 is a xed parameter. 5.1 Finite time blowup for Prandtl The following is the main result of this chapter. 65 Theorem 5.1.1 (Finite time blowup for Prandtl). Consider the Cauchy problem for the Prandtl equations (5.1){(5.4), with boundary conditions aty =1 matching (5.6){(5.7), with6= 0. There exists a large class of initial conditions (u 0 ;v 0 ) which are real-analytic in x and y, such that the unique real-analytic solution (u;v) to (5.1){(5.4) blows up in nite time. Remark 5.1.2 (Inviscid limit). For a real-analytic initial datum in the Navier-Stokes, Euler, and Prandtl equations, it was shown in [SC98b] that the inviscid limit of the Navier- Stokes equations is described to the leading order by the Euler solution outside of a bound- ary layer of thickness p t, and by the Prandtl solution inside the boundary layer (see also [Mae14] for initial vorticity supported away from the boundary). Indeed, if any series expansion (in ) of the Navier-Stokes solution holds, then the leading order term near the boundary must be given by the Prandtl solution. The result in [SC98b] states that the inviscid limit holds on a time interval on which the Prandtl solution does not lose real- analyticity. Our result in Theorem 5.1.1 shows that this time interval cannot be extended to be arbitrarily large, and thus the Prandtl expansion approach to the inviscid limit should only be expected to hold on nite time intervals. Remark 5.1.3 (The case = 0). In the case of a trivial Euler ow U E and a trivial Euler pressure P E , i.e., for = 0 in (5.6){(5.7), the emergence of a nite time singularity for the Prandtl equations was established in [EE97]. There, the initial datum is taken to have compact support in y and be large in a certain nonlinear sense. It is shown that either the solution ceases to be smooth, or that the solution along with its derivatives does not decay suciently fast as y!1. The proof given in [EE97] does not appear to handle the case 6= 0 treated in this paper. Indeed, here the initial datum does not need to have compact support in y and the pressure gradient is not trivial. Note that it is shown in [GSS09, Appendix] that from the numerical point of view the structure of the singularity for = 0 is dierent from the complex structure of the singularities in [vDS80]. Remark 5.1.4 (Boundary layer separation and the displacement thickness). The displacement thickness (cf. [Sch60, vDS80, CM07]) is dened as (t;x) = Z 1 0 1 u(t;x;y) U E (t;x) dy = Z 1 0 1 u(t;x;y) sin(x) dy: (5.8) Physically, it measures the eect of the boundary layer on the inviscid ow [CM07]. As long as (t;x) remains bounded, the Prandtl layer remains of thickness proportional to p , i.e., it remains a Prandtl layer. In turn, if the displacement thickness develops a singularity in 66 138 VAN DOMMELEN AND SHEN 1 -7-t x FIG. 11. The variation of the displacement thickness with x, for various instants, TABLE II Computed Values of Several Variables for Various Gridsizes Gridsize 19 x 9 37x 17 73 x 33 145 x 65 s at min(lgrad xi) for T= 1.5 2.002 1.954 1.943 1.939 u at min(lgrad.ui) for T= 1.5 -.3 17 -.298 -.276 p.274 T at separation 1.659 1.553 1.515 1.506 T for zero wall shear at .Y = R -1’ .3264 .3231 .3220 F” for x = E and T = 1.5 /I ii (-) indicates no value was determined. 1.130 1.1 125 1.1122 Indeed all calculations prior to the present one suffer from insufficient resolution in the x-direction to adequately resolve for the singularity. The present solution, however, has infinite resolution at the singularity because x is stationary.) One might be tempted to suppose that the differences between our and Cebeci’s solution, Figs. 8 and 9, are due to his insufficient resolution in x-direction. However, Figure 5.1: The variation of the displacement thickness (t;x) for various time instants. The gure is from [vDS80, p. 138]. nite time, this signals that a boundary layer separation has occurred, and after this point, the Prandtl expansion is not expected to hold anymore (see also[Gre00, GGN14c, GGN14b, GGN14a]). For a proof of a boundary layer separation for the stationary Prandtl equations, we refer to the recent paper [DM15]. The proof of Theorem 5.1.1 consists of showing that a Lyapunov functionalG(t) blows up in nite time. The functionalG is dened by G(t) = Z 1 0 ((t;y)@ x u(t; 0;y))w(y)dy where w(y) is an L 1 weight function and (t;y) is the solution of a nonhomogenous heat equation (see (5.19) and (5.35) below for details). We note that the Lyapunov functional was built to emulate a weighted iny version of (t;x)j x=0 . Indeed, as long asu is smooth near x = 0, we have that (t; 0) = lim x!0 (t;x) = Z 1 0 (@ x u(t; 0;y))dy: Remark 5.1.5 (More general Euler ows). The blowup result of Theorem 5.1.1 holds if the Euler ow dened (5.6){(5.7) is replaced by any smooth and odd function U E (x), upon dening P E to equal(U E (x)) 2 =2, which is in turn even in x. 67 Remark 5.1.6 (More general classes of initial conditions). We note that besides yield- ing the local existence of solutions (cf. [SC98a]), the analyticity of the initial datum is not required for proving Theorem 5.1.1. One may instead consider an initial datum that is merely Sobolev smooth with respect toy, analytic with respect tox, and decays suciently fast as y!1 (cf. [CLS01, KV13, IV15]). Alternatively Gevrey-class 7=4 regularity in x may be considered [GVM13], whose vorticity decays suciently fast with respect to y. On the other hand, in view of the oddness in x of the boundary condition (5.6), we cannot consider initial datum which is in the Oleinik class of monotone in y ows (uniformly with respect to x), although the local existence holds in this class [Ole66, MW14, AWXY15]. The datum in [XZ15] is also not allowed in view of the boundary conditions (5.6). The mixed analyticity near x = 0, and monotonicity away from the y-axis (of dierent signs) may however be treated, using the local existence result in [KMVW14]. Remark 5.1.7 (Ill-posedness for the Prandtl equations). The ill-posedness of the Prandtl equations was established at the linear level in [GVD10], and at the nonlinear level in [Gre00, GN11, GVN12]. We note that these results do not imply the nite time blowup from a given initial datum. Remark 5.1.8 (Finite time blowup for the hyrdostatic Euler equations). Here we note certain very interesting blowup results [CINT13, Won15] for the hydrostatic Euler equations (which has many analogies with the Prandtl equations). These equations are set in a nite strip = R [0;h], and have dierent boundary conditions (only v = 0 is imposed at the top and bottom boundaries). For these equations the local existence was established for convex [Bre99, MW12] or analytic [KTVZ11] data, as well as for the combination thereof [KMVW14]. These equations are at the same time severely unstable (i.e., ill-posed in Sobolev spaces) if convexity or analyticity is absent [Ren09]. In [CINT13] and [Won15] the nite blowup of odd solutions is established, by observing the behavior of u x =w z . The main dierence with [EE97] is the presence of the pressure, which is quadratic in u (cf. [CINT13] for further details). 5.2 Proof of Theorem 5.1.1 5.2.1 Local existence As an initial datum for the Prandtl equation we consider u 0 (x;y) =Erf y 2 sinx + u 0 (x;y) (5.9) 68 where u 0 (x;y) is a real-analytic function of x and y which is also odd with respect to x, and Erf(z) = 2 1=2 R z 0 exp( z 2 )d z is the Gauss error function. We assume that u 0 decays suciently fast (at least at an integrable rates) as y!1, and takes the value 0 at y = 0. Moreover, we assume that a 0 (y) = (@ x u 0 )(0;y) obeys a 0 (y)> 0 for all y> 0 and that G 0 = Z 1 0 a 0 (y)w(y)dy C ;w (5.10) where the weight w is as constructed in Sectionx 5.4, and C ;w 1 is a constant that depends on and on the L 1 (R + ) weight function w. For instance, we may consider u 0 (x;y) =Ay 2 exp(y 2 ) sinx (5.11) where A =A(;w)> 0 is a suciently large constant. This choice for u 0 (x;y) yields that a 0 (y) is a large constant multiple of '(y) = y 2 exp(y 2 ). The local in time existence of a unique real-analytic solution of the Prandtl system (5.1){(5.7) with initial datum given by (5.9){(5.11) follows from [SC98a]. We note though that the real-analyticity is not needed in the blowup proof. It is only used to ensure that we have the local in time existence and uniqueness of smooth solutions. Much more general classes of initial conditions u 0 may be considered as long as they are odd with respect to x, the Cauchy problem is locally well-posed (cf. Remark 5.1.6 above), and (5.10) holds. 5.2.2 Restriction of Prandtl dynamics on the y-axis Consider an initial datum u 0 (x;y) for the Prandtl equations that is odd in x (such as the one dened in (5.9)). Note that the boundary condition at y = 0 is homogenous and thus automatically odd in x, the boundary condition at y =1 given by the Euler trace in (5.6) is also odd in x, and the derivative of the Euler pressure trace (5.7) is odd in x as well. 69 Therefore, the unique classical solution u(t;x;y) of (5.1){(5.7) is also odd in x. Hence, as long as the solution remains smooth we have u(t; 0;y) = (@ y u)(t; 0;y) = (@ 2 x u)(t; 0;y) = 0: (5.12) Physically, this symmetry freezes the Lagrangian paths emanating from the y-axis, intro- ducing a stable stagnation point in the ow. As in [EE97] (see also [CINT13, Won15]) this allows one to consider the dynamics obeyed by the tangential derivative of u at x = 0, i.e., b(t;y) =(@ x u)(t;x;y)j x=0 : As long as the solution remains smooth (so that we may take traces at x = 0), using (5.12) one derives that the equation obeyed by b(t;y) is @ t b@ yy bb 2 +@ 1 y b@ y b = 2 (5.13) bj y=0 = 0 (5.14) bj y!1 =: (5.15) In (5.13) and throughout the paper, we denote the integration with respect to the vertical variable as @ 1 y '(t;y) = Z y 0 '(t;y 0 )dy 0 for any function '(t;y) which is integrable in y. In order to obtain (5.13), one applies@ x to (5.1) and then evaluates the resulting equation on the y-axis. Similarly, (5.15) follows upon taking a derivative with respect to x of (5.6) and setting x = 0. 5.2.3 A shift of the boundary conditions In order to homogenize the boundary condition at y =1 when t = 0, we add to b a lift (t;y) dened as the solution of the nonhomogenous heat equation @ t @ yy = 2 (5.16) j y=0 = 0 (5.17) j y!1 = + 2 t (5.18) 70 with an initial datum that we may choose, as long as it obeys compatible boundary condi- tions. We consider 0 (y) =Erf y 2 ; so that the solution of (5.16){(5.18) is explicit (t;y) =Erf y p 4(t + 1) ! + 2 t y 2 2t Erf y p 4t 1 + Erf y p 4t + y p t exp y 2 4t : (5.19) In Sectionx 5.3 we prove a number of properties (such as 0 and that @ y 0) of the function dened in (5.19). Letting a(t;y) =b(t;y) +(t;y) (5.20) the system (5.13){(5.15) becomes @ t a@ yy a = (a) 2 @ 1 y (a)@ y (a) (5.21) aj y=0 = 0 (5.22) aj y!1 = 2 t: (5.23) The equation (5.21) is similar to the one obtained in [EE97] for = 0, except for two additional terms on the right side: a forcing term F (t;y) =(t;y) 2 @ 1 y (t;y)@ y (t;y) (5.24) and a linear term L[a](t;y) =2a(t;y)(t;y) +@ 1 y (t;y)@ y a(t;y) +@ 1 y a(t;y)@ y (t;y): (5.25) 71 The forcing term is explicit in view of (5.19), while the linear operator L[a] has nice coef- cients given in terms of . With the notation (5.24){(5.25), the evolution equation for a becomes @ t a@ yy a =a 2 @ 1 y a@ y a +L[a] +F (5.26) aj y=0 = 0 (5.27) aj y!1 = 2 t: (5.28) In order to prove Theorem 5.1.1, we show that the solution of (5.26){(5.28) blows up in nite time from a very large class of smooth initial data a 0 . 5.2.4 Minimum principle The main purpose of shifting the functionb up by is so that the resulting functiona obeys a positivity principle. Lemma 5.2.1. Assume thata 0 =a(0;y) is such thata 0 (y)> 0 for ally2 (0;1). Consider a smooth solutiona(t;y) of the initial value problem associated with (5.26){(5.28) and initial condition a 0 , on a time interval [0;T ]. Then we have that a(t;y) 0 for all y 0 and t2 [0;T ]. Before proving Lemma 5.2.1, we need to establish certain positivity properties concern- ing the function . Lemma 5.2.2. Let (t;y) be as dened in (5.19) with > 0. Then we have (t;y) 0 (5.29) (t;y)C (1 +t) (5.30) @ y (t;y) 0 (5.31) @ yy (t;y) 0 (5.32) for all t 0 and all y 0, where C > 0 is a constant that depends only on . Moreover, the inequality in (5.29) is strict for y> 0. The proof of Lemma 5.2.2 is given in Sectionx 5.3 below. 72 Proof of Lemma 5.2.1. We argue by contradiction. Since the solution is classical on [0;T ] and decays suciently fast as y!1, in order to reach a strictly negative value in [0;T ] (0;1) there must exist a time t 0 and an interior point y 0 > 0, such that a(t 0 ;y 0 ) = 0 a(t 0 ;y) 0 for all y2R + (@ t a)(t 0 ;y 0 ) 0: As is classical for the heat equation the contradiction arises by computing the time derivative ofa at the point (t 0 ;y 0 ) and showing that it is strictly positive, contradicting the minimality of t 0 . In order to bound (@ t a)(t 0 ;y 0 ) from below we use (5.26). Since a(t 0 ;) has a global minimum at the interior point y 0 , we have (@ y a)(t 0 ;y 0 ) = 0 (@ yy a)(t 0 ;y 0 ) 0: Since by assumption a(t 0 ;y 0 ) = 0, it follows that (@ yy a +a 2 @ 1 y a@ y a)(t 0 ;y 0 ) 0. More- over, since is a non-negative non-decreasing function, we immediately obtain from (5.25) that L[a](t 0 ;y 0 ) 0. We conclude the proof by showing that F (t 0 ;y 0 ) > 0. Indeed, by (5.29) and (5.32) we have that @ y F =@ y @ 1 y @ yy @ y = 1 2 @ y ( 2 ) and thus F (t;y) = Z y 0 @ y F (t; y)d y 1 2 Z y 0 @ y ( 2 )(t; y)d y = 1 2 ((t;y)) 2 > 0 (5.33) whenever y> 0, in view of Lemma 5.2.2. In order to fully justify this argument, we apply the proof to e a(t;y) = a(t;y) +", for which the equation reads @ t e a@ yy e a =e a 2 @ 1 y e a@ y e a +L[e a] +F 2e a +y@ y e a + 2 L[] (5.34) 73 and show thate a(t;y) remains non-negative for every"> 0. The latter requires an additional observation L["](t;y) =2"(t;y) +"y@ y (t;y) =" Z y 0 ( y@ yy (t; y)@ y (t; y))d y 0; which holds in view of Lemma 5.2.2. This concludes the proof of the minimum principle. 5.2.5 Blowup of a Lyapunov functional Motivated by the displacement thickness (cf. (5.8)) we consider the evolution of the weighted average ofa(t;y) onR + . For a suitable weightw(y) to be dened below and a non-negative solution a of (5.26){(5.28), we dene the Lyapunov functional G(t) = Z 1 0 a(t;y)w(y)dy: (5.35) Note that since a 0 as long as a remains smooth (cf. Lemma 5.2.1), we have that G(t) 0 for all t 0. Our goal is to establish an inequality of the type dG dt 1 C G 2 C(1 +t)(1 +G) (5.36) for a constantC 1. Choosing a suitable initial datum, we then conclude thatG blows up in nite time. The rst step is to present the properties of the weight w in (5.35) which are needed in the proof of (5.36). 5.2.5.1 Properties of the weight function w We consider a weight function w(y) such that: w2W 2;1 (R + ) w 0 wj y=0 =wj y!1 = 0 w2L 1 (R + ): 74 The weight is given by glueing two functions f and g, i.e., w(y) = 8 < : f(y); 0yQ; g(y);yQ; where Q> 0; the function f is such that f(0) = 0 (5.37) f 0; on [0;Q] (5.38) f 00 0; on [0;Q] (5.39) f 00 (y)c f f(y); for all y2 [0;Q]; where c f > 0 (5.40) yf 0 (y) c f f(y); for all y2 [0;Q]; where c f > 0 (5.41) and the function g, which we extend to be dened on [M;1) for some M2 (0;Q) obeys lim y!1 g(y) = 0 = lim y!1 g 0 (y) (5.42) g(y)> 0; for all yM (5.43) g 0 (y)< 0; for all yM (5.44) g 00 (y)> 0; for all yM (5.45) g 0 (y) 2 g(y)g 00 (y) < 1; for all yQ: (5.46) Also let (z) be a smooth non-decreasing cuto function such that (z) = 0 for all z 0, (z) = 1 for all z 1, andj 0 (z)j 2 for all y2 (0; 1). Dene the cuto function (y) = yM QM : (5.47) Note that vanishes identically on [0;M] and equals 1 on [Q;1). Its derivative localizes to [M;Q] and obeys 0 0 (y) 2 QM 75 for all y2 (M;Q). Lastly, we require that the functions f and g obey the compatibility conditions (y) g 0 (y) 2 f(y)g 00 (y) < 1; for all y2 (M;Q) (5.48) 2 0 (y)jg 0 (y)j(y)g 00 (y)f 00 (y); for all y2 (M;Q): (5.49) The construction of two functions f and g that obey the properties (5.37){(5.49) listed above is provided in Sectionx 5.4. A sketch of the graph of the resulting weight w(y) is given in Figure 5.2 below. Throughout paper, we shall denote derivatives of the functions w;f;g with primes, as they are only functions of the variable y. 0.5 1.0 1.5 2.0 2.5 3.0 3.5 y 0.005 0.010 0.015 0.020 W(y) f M Q " g Figure 5.2: Graph of the weight function w(y). The weight is a linear function on [0;"], a quadratic function on [";Q], and an shifted negative power of y on [Q;1). 76 5.2.5.2 Evolution of the Lyapunov functionalG We use (5.26){(5.28) and the boundary values of w, given by (5.37) and (5.42), to deduce d dt G = Z 1 0 @ yy a +a 2 @ 1 y a@ y a +L[a] +F wdy = Z 1 0 @ y aw 0 dy + 2 Z 1 0 a 2 wdy + Z 1 0 a@ 1 y aw 0 dy + Z 1 0 L[a]wdy + Z 1 0 Fwdy Z 1 0 aw 00 dy + 2 Z 1 0 a 2 wdy 1 2 Z 1 0 (@ 1 y a) 2 w 00 dy + Z 1 0 L[a]wdy =I 1 + 2I 2 1 2 I 3 +I 4 : (5.50) The above integrations by parts are justied by the fact thata andw are suciently smooth, w obeys the Dirichlet boundary conditions,a and@ 1 y a vanish aty = 0, while@ y w vanishes as y!1. Here we have used that by (5.33) we have F (t;y) 0, which combined with w(y) 0 shows that the forcing is non-negative. We now bound each of the terms in (5.50) separately. 5.2.5.3 Bound for I 1 To boundI 1 we use the convexity ofw on [Q;1) (cf. (5.45)) and the fact that (5.40) holds. We deduce that I 1 Z Q 0 af 00 c f Z Q 0 afc f G: (5.51) 5.2.5.4 Bound for I 2 By the Cauchy-Schwartz inequality, and denoting c 1 =kwk L 1 (R + ) <1; (5.52) we obtain G = Z 1 0 awdy = Z 1 0 a p w p wdyka p wk L 2k p wk L 2 =c 1=2 1 I 1=2 2 and thus I 2 1 c 1 G 2 : (5.53) 77 5.2.5.5 Bound for I 3 We proceed to bound the dicult termI 3 . Let(y) be the cuto function dened in (5.47). First we have I 3 = Z 1 0 (@ 1 y a) 2 w 00 = Z 1 0 (@ 1 y a) 2 w 00 + Z 1 0 (1)(@ 1 y a) 2 f 00 Z 1 0 (@ 1 y a) 2 g 00 + Z Q M (@ 1 y a) 2 (f 00 g 00 ) + Z Q M (1)(@ 1 y a) 2 f 00 =J + Z Q M (@ 1 y a) 2 (f 00 g 00 ) (5.54) where we have used (5.39) in the second to last step to bound Z M 0 (1)(@ 1 y a) 2 f 00 0: The term J above is the leading one and it shall be bounded using integration by parts, which is justied since g 0 vanishes as y!1. Since @ 1 y aj y=0 , we obtain from (5.42) and (5.44) that J = Z 1 0 (@ 1 y a) 2 g 00 =2 Z 1 0 a(@ 1 y a)g 0 Z 1 0 @ y (@ 1 y a) 2 g 0 = 2 Z 1 0 a(@ 1 y a)jg 0 j + Z 1 0 @ y (@ 1 y a) 2 jg 0 j: Further, recalling that is supported on [M;1), by appealing to (5.46) and (5.48) we get J 2 Z 1 0 (@ 1 y a) 2 g 00 1=2 Z 1 0 a 2 (g 0 ) 2 g 00 1=2 + Z 1 0 0 (@ 1 y a) 2 jg 0 j 2 p Z 1 0 (@ 1 y a) 2 g 00 1=2 Z 1 0 a 2 w 1=2 + Z 1 0 0 (@ 1 y a) 2 jg 0 j = 2 p J 1=2 I 1=2 2 + Z 1 0 0 (@ 1 y a) 2 jg 0 j: 78 Using the inequality 2xyx 2 =2 + 2y 2 , we further estimate J J 2 + 2I 2 + Z 1 0 0 (@ 1 y a) 2 jg 0 j which in turn yields J 4I 2 + 2 Z 1 0 0 (@ 1 y a) 2 jg 0 j: (5.55) Combining (5.54) and (5.55), we obtain I 3 4I 2 + 2 Z Q M 0 (@ 1 y a) 2 jg 0 j + Z Q M (@ 1 y a) 2 (f 00 g 00 ): (5.56) By appealing to (5.49) we then arrive at the bound I 3 4I 2 (5.57) which is a convenient estimate when < 1. 5.2.5.6 Bound for I 4 Consider I 4 = Z 1 0 L[a]w =2 Z 1 0 aw + Z 1 0 @ y a@ 1 y w + Z 1 0 @ y @ 1 y aw: (5.58) Using that (5.29){(5.31) hold (cf. Lemma 5.2.2), upon integrating by parts in the second term on the far right side of (5.58), we have that I 4 3 Z 1 0 aw Z 1 0 a@ 1 y w 0 3C (1 +t)G Z Q 0 a@ 1 y (w 0 ) + where (w 0 ) + = maxfw 0 ; 0g. In the last step we used that w is decreasing on [Q;1). Since the bounds (5.38), (5.41), and (5.30) hold on [0;Q], we arrive at @ 1 y (w 0 ) + y(Q)(w 0 ) + C (1 +t) c f w 79 for y2 [0;Q]. We thus obtain I 4 (3 + c f )C (1 +t)G: (5.59) 5.2.5.7 The lower bound for the growth of the Lyapunov functional Combining (5.50), (5.51), (5.53), (5.57), and (5.59), with the assumption that < 1, we arrive at d dt Gc f G + 2I 2 (1) (3 + c f )C (1 +t)G (c f + (3 + c f )C )(1 +t)G + 2(1) c 1 G 2 C ;w (1 +t)G + 1 C ;w G 2 (5.60) for some suciently large positive constant C ;w 1, which only depends on the choice of and the weight w. Note that < 1 is essential here. 5.2.6 Conclusion of the proof of Theorem 5.1.1 Therefore, if we ensure thatG 0 =Gj t=0 is suciently large, the solutionG(t) of (5.60) blows up in nite time. Quantitatively, it is sucient to let G 0 4C 2 ;w : (5.61) The condition (5.61) may be achieved by a smooth initial datum. For instance we may let a 0 (y) = a(t;y)j y=0 be given by a large amplitude Gaussian bump, i.e., a 0 (y) = A'(y), where' is as in (5.11) above, andA> 0 is suciently large. In view of Remark 5.1.6, more general classes of functions '(y) may be considered, including those with compact support in y (cf. [CLS01, KV13]). 80 5.3 Properties of the boundary condition lift It is easy to verify that the function dened in (5.19), i.e., (t;y) =Erf y p 4(t + 1) ! + 2 t y 2 2t Erf y p 4t 1 + Erf y p 4t + y p t exp y 2 4t =Erf y p 4(t + 1) ! + 2 t 2z(t;y) 2 (Erf (z(t;y)) 1) + Erf (z(t;y)) + 2z(t;y) p exp z(t;y) 2 (5.62) obeys the non homogenous heat equation (5.16){(5.18), with initial value 0 (y) =Erf(y=2), where z(t;y) =y= p 4t is the heat self-similar variable. 5.3.1 Proof of Lemma 5.2.2 Proof of (5.31). First note that for t > 0 and y > 0, the function @ y obeys the heat equation, i.e., (@ t @ yy )(@ y ) = 0. Using the exact formula (5.19) we obtain the initial and boundary values for the quantity @ y . Taking the y derivative of gives @ y = p hti exp y 2 4hti + 2 y Erf y p 4t 1 + y 2 p 4t exp y 2 4t +t 2 p t exp y 2 4t y 2 2t p t exp y 2 4t wherehti =t + 1. Sending y! 0 and y!1 we obtain @ y j y=0 = p hti + 2 2 p t p > 0 @ y j y=1 = 0 for all t> 0. Taking the limit t! 0, we arrive at @ y j t=0 = p exp y 2 4 0: 81 The fact @ y (t;y) 0 for t;y 0 now follows from the parabolic maximum principle. For the sake of completeness we repeat this classical argument. We consider the nonnegative C 2 function (x) = 8 < : x 4 ; x 0 0; x 0: Taking the t derivative of the quantity R 1 0 (@ y (t;y))dy, upon integrating by parts in y and using the boundary conditions for @ y we arrive at @ t Z 1 0 (@ y )dy = Z 1 0 0 (@ y )@ t @ y dy = Z 1 0 0 (@ y )@ 3 y dy = Z 1 0 00 (@ y )(@ 2 y ) 2 dy 0 (@ y j y=0 )@ yy j y=0 = Z 1 0 00 (@ y )(@ 2 y ) 2 dy 0; from where we deduce that R 1 0 (@ y (t;y))dy = 0 for t 0 since R 1 0 (@ y 0 (y))dy = 0: Therefore, we obtain @ y (t;y) 0, concluding the proof. Proof of (5.29). Since (t; 0) = 0, for all t > 0, the non-negativity of follows from the fundamental theorem of calculus and the above established monotonicity property @ y 0. Proof of (5.30). The proof follows from (5.62) since 2z 2 (Erf (z) 1) + Erf(z) + 2 p z exp(z 2 )C 0 for all z 0, for some universal constant C 0 > 0. We may then take C = maxf;C 0 2 g. Proof of (5.32). Taking the second derivative of dened in formula (5.19), we arrive at @ yy = y p 4hti 3 exp y 2 4hti 2 t 2y p t 3 exp y 2 4t y 3 4 p t 5 exp y 2 4t + 2 Erf y p 4t 1 + 2y p t exp y 2 4t y 3 4 p t 3 exp y 2 4t : (5.63) 82 From this expression we get the initial and boundary values for @ yy as @ yy j y=0 = 2 0; for t> 0 @ yy j y!1 = 0; for t> 0; @ yy j t=0 = y 2 p exp y 2 4 0; for y> 0: An argument similar to the one above shows that by the parabolic maximum principle we have @ yy (t;y) 0 for all t;y 0. 5.4 Construction of a weight function w for the Lyapunov functional FixQ = 1, and letr> 1 be a free parameter, to be chosen below. In terms of thisr we shall pick 1=2 < M = M(r) < 1, 0 < = (r) < 1, and B = B(r) > 1 so that the conditions (5.37){(5.49) hold. Dene f(y) = 2B +r B r y B +r B r y 2 (5.64) Therefore, f(0) = 0 and f 0 (y) = 2B +r B r 2(B +r) B r y with f 00 (y) = 2(B +r) B r : It thus follows that f 2 W 2;1 ([0; 1]), f 0, and f 00 0 on (0; 1), so that (5.37){(5.39) hold. Moreover, (5.41) holds with c f = 1. Note however that (5.40) does not hold in a neighborhood of the origin, since f(0) = 0. Instead, the function f dened in (5.64) needs to be modied in a small neighborhood near the origin so that it is linear there (see Remarkx 5.4.1 below). Next, we dene g(y) = B (y +B 1) r 83 set initially for all y 1, but it is a well-dened function on yM as long as M +B > 1. Note that r> 1 implies that (5.52) holds. We have that g 0 (y) = rB (y +B 1) r+1 < 0 and g 00 (y) = r(r + 1)B (y +B 1) r+2 > 0: Thus, the properties (5.42){(5.45) hold for this function g. Moreover, g 0 (y) 2 g(y)g 00 (y) = r r + 1 < 1 for all y> 0. This veries that condition (5.46) holds for any 2 [r=(r + 1); 1). Note that at y =Q = 1 we have f(1) =B 1r =g(1) and f 0 (1) =rB r =g 0 (1) and thus f and g may be glued together at Q = 1 to yield a W 2;1 function onR. In order to assure that (5.48) holds, it is sucient to verify that r r + 1 yM 1M g(y)f(y) (5.65) holds for all y2 [M; 1], where r=(r + 1) < < 1 is arbitrary. The condition (5.65) holds automatically with = 2r + 1 2r + 2 if we ensure that sup y2[M;1] g(y) f(y) 2r + 1 2r : (5.66) 84 In view of the continuity of the above functions, (5.66) holds if we choose M 2 (0; 1) suciently close to 1. We need though to be more precise on this choice of M. Indeed, (5.66) holds for y2 [M; 1] if we impose that B r+1 (y +B 1) r 2r + 1 2r y ((2B +r) (B +r)y) which is a consequence of B r+1 (M +B 1) r 2r + 1 2r MB: Assuming that 1M 1=(4r + 2), the above follows from 1 (1 (1M)=B) r 4r + 1 4r which holds provided 4r 4r + 1 1=r 1 1M B : The last condition may be written as 1MB 1 4r 4r + 1 1=r ! : (5.67) Therefore, (5.67) holds if we choose 1M = min ( 1 4r + 2 ;B 1 4r 4r + 1 1=r !) = 1 4r + 2 (5.68) as long as B 1 (4r + 2) 1 (4r=(4r + 1)) 1=r : (5.69) This ensures the validity of the condition (5.65), and thus also (5.48) holds. We nally verify that (5.49) holds, or equivalently 2 1M 0 yM 1M rB (y +B 1) r+1 yM 1M r(r + 1)B (y +B 1) r+2 + 2(B +r) B r : (5.70) 85 Sincej 0 j 2 and 0, the above condition holds on [M; 1] once we ensure that 4 1M rB (y +B 1) r+1 2(B +r) B r which is implied by B r+1 (M +B 1) r+1 (1M)(B +r) 2r : The above condition holds if we take B suciently large, depending only on r. More precisely, since M obeys (5.68), letting B obey (5.69) and also B 2r(4r + 2) 4r + 1 4r (r+1)=r ; (5.71) we complete the proof of (5.70). In summary, the conditions (5.37){(5.49), except for (5.40), are obeyed once we set r = 2 = 5 6 < 1 M = 9 10 <Q = 1 B = 50: 5.4.1 Condition (5.40) In order to ensure that (5.40) holds, we need to tweak the functions f andg dened above. Let 0<"< 1=2 be a small parameter, to be determined. We then have f(") = 2B +r B r " B +r B r " 2 > 0 f 0 (") = 2B +r B r 2(B +r) B r "> 0: Therefore, we can extend f(y) by the linear function h " (y) =f 0 (")(y") +f(") = 2B +r B r 2(B +r) B r " (y") + 2B +r B r " B +r B r " 2 86 on the interval [y " ;"], where y " = f(") f 0 (") " 0: Note that h " (y " ) = 0, h " (") = f("), and h 0 " (") = f 0 ("). Therefore, glueing h " with f at y =", and then f with g at y =Q = 1, yields a W 2;1 function w " (y) = 8 > > > < > > > : h " (y); y2 [y " ;"] f(y); y2 ["; 1] g(y); y 1 which obeys all the properties (5.37){(5.49), but on the interval [y " ;1) instead of [0;1). Here we used that " is suciently small so that " M=4. Also, (5.40) holds trivially on [y " ;") since on this interval h 00 " (y) = 0h " (y). Then in view of the continuity of f 00 and f, and the fact that f only vanishes at y = 0, on the compact ["; 1] we have that jf 00 (y)j=f(y)c f for some constant c f 1. Therefore, to complete the construction of the weight function w, let w(y) =w " (y +y " ) for a xed 0<" 1, where the values ofQ andM are themselves shifted byy " . In view of the smoothness of all parameters on " this is possible without aecting (5.46){(5.48), upon slightly increasing the value of (so that it remains less than 1). 87 Chapter 6 On the existence for the free interface 2D Euler equation with a localized vorticity condition In this chapter, we establish the local-in-time and global solutions to the incompressible Euler equations for a free moving interface, with no surface tension, for rotational ows under minimal regularity assumptions on the initial data and the Rayleigh-Taylor stability condition. Recall the Euler equations u t +uru +rp = 0 in (t) (0;T ) (6.1) divu = 0 in (t) (0;T ) (6.2) which describe the ow of an ideal inviscid incompressible uid with a velocity eld u(x;t) and a uid pressure p(x;t) on a moving domain have attracted considerable attention in the mathematical literature. The boundary of the domain (t) consists of a exible part 1 (t), which moves with the uid velocity, and a stationary part 0 . 6.1 The main result Consider the Euler equation in the Lagrangian formulation, set in the domain =R 1 (0; 1)R 2 (6.3) with periodic boundary condition in x 1 with period 1. The top 1 =Rfx n = 1g (6.4) represents the free boundary, while the rigid bottom is given by 0 =Rfx n = 0g: (6.5) 88 We denote the -neighborhood of the top boundary by U =fx2 : dist(x; 1 )g; (6.6) where> 0. Fix a small constant 0 > 0 and introduce a smooth cut-o function =(x 2 ) such that (x 2 ) = 1; x 2 1 6 0 (6.7) (x 2 ) = 0; x 2 1 5 0 (6.8) with 0(x 2 ) 1 for 1 6 0 x 2 1 5 0 . We usee to denote the standard Lagrangian coordinate of the system, the equations for which read e t =u(e ) (6.9) e (0;x) =x; x2 ; (6.10) and to denote the coordinate system satisfying t = (1())u() (6.11) (0;x) =x; x2 : (6.12) Lete v(x;t) = (e v 1 ;e v 2 ) and v(x;t) = (v 1 ;v 2 ) represent the velocities in the Lagrangian and coordinates respectively, whilee q(x;t) andq(x;t) stand for the pressures respectively. Denote by a the inverse matrix ofr, a = (r) 1 (6.13) or in coordinates a k j @ k i = ij ; i;j = 1; 2: (6.14) Note that the summation convention on repeated indices is used throughout. The equations (6.1){(6.2) in the coordinates read v i t + a k j v j @ k v i +a k i @ k q = 0 in (0;T ); i = 1; 2 (6.15) a k i @ k v i = 0 in (0;T ) (6.16) 89 where (x;t) =((x;t)) (6.17) with the initial condition v(0) =v 0 : (6.18) The matrix a evolves according to a t =a :r t :a a(x; 0) =I; x2 (6.19) where the symbol : denotes the matrix multiplication. This follows froma :r =I by time dierentiation. On the top, which represents the free boundary, we impose q = 0 on 1 (0;T ) (6.20) while on the bottom boundary we assume v i N i = 0 on 0 (0;T ) (6.21) whereN = (N 1 ;N 2 ) stands for the outward unit normal. Since our domain (6.3) is assumed at, for simplicity, we have N = (0;1) on 0 and N = (0; 1) on 1 . The following is our main result. Theorem 6.1.1. Let > 0. Assume that v(; 0) = v 0 2 H 2+ ( ) is divergence-free with v 0 N = 0 on 0 and curlv 0 2H 1:5+ (U 10 0 ) (6.22) for some 0 > 0. Assume that the initial pressure q(; 0) satises the Rayleigh-Taylor con- dition @q @N (x; 0) 1 C 0 < 0; x2 1 (6.23) 90 where C 0 > 0 is a constant. Then there exists a solution (v;q;a;) to the free boundary Euler system with the initial condition v(0) =v 0 with v2L 1 ([0;T ];H 2+ ( ))\C([0;T ];H 1+ ( )) v t 2L 1 ([0;T ];H 1+ ( )) 2L 1 ([0;T ];H 2:5+ ( ))\C([0;T ];H 2+ ( )) a2L 1 ([0;T ];H 1:5+ ( ))\C([0;T ];H 1+ ( )) q2L 1 ([0;T ];H 2:5+ ( )) q t 2L 1 ([0;T ];H 1+ ( )) (6.24) on [0;T ], where T > 0 depends on the initial data. 6.2 Preliminary lemmas Before proving the main result, we state several lemmas needed in the proof. Denote a_b = maxfa;bg anddse = minfm2N :msg. Lemma 6.2.1. Let F :R n !R be a function inS(R n ) and let g : !R n be C 1 , where R n is a bounded open set. Assume that det(rg)6= 0. Then kFgk H s ( ) .k det(rg) 1 k 1=2 L 1 ( ) (1 +krgk H (s1)_(1+) ( ) ) dse kFk H s (R n ) (6.25) holds for any > 0. Proof. We show the proof for the ranges 0s 1 and 1s 2; the treatment for larger s is similar. First, assume 0s 1. Since kFgk 2 L 2 ( ) = Z jFgj 2 dx = Z g( ) jF (y)j 2 j det(rg) 1 jdy kFk 2 L 2 (R n ) k det(rg) 1 k L 1 ( ) (6.26) 91 and kFgk 2 H 1 ( ) .kFgk 2 L 2 ( ) + Z j(rF )gj 2 jrgj 2 dx .kFk 2 L 2 (R n ) k det(rg) 1 k L 1 ( ) +krgk 2 L 1 ( ) krFk 2 L 2 (R n ) k det(rg) 1 k L 1 ( ) . (1 +krgk 2 L 1 ( ) )kFk 2 H 1 (R n ) k det(rg) 1 k L 1 ( ) . (1 +krgk 2 H 1+ ( ) )kFk 2 H 1 (R n ) k det(rg) 1 k L 1 ( ) (6.27) hold, we get by complex interpolation for 0s 1 kFgk H s ( ) . (1 +krgk 2 H 1+ ( ) ) 1=2 k det(rg) 1 k 1=2 L 1 ( ) kFk H 1 (R n ) : (6.28) Next we consider the case 1s 2 (the case s> 2 is treated similarly). Using the above computation we have kFgk 2 H s.kFgk 2 H 1 +kr(Fg)k 2 H s1 .kFgk 2 H 1 +k(rF )gk 2 H s1 krgk 2 H (s1)_(1+) .k det(rg) 1 k L 1(1 +krgk 2 H 1+ )(1 +krgk 2 H (s1)_(1+) )kFk 2 H s; (6.29) where we used that H r is an algebra for r> 1 and kfgk H rkfk H rkgk H 1+: (6.30) for r2 [0; 1]. In the next lemma, we state a priori estimates for the coecient matrix a and for the particle map . Lemma 6.2.2. Assume thatkrvk L 1 ([0;T ];H 1+ ( )) M. For 0<T < 1=CM, where C is a suciently large constant, the following statements hold: (i)kr(;t)k H 1+ ( ) . 1 for t2 [0;T ], (ii)ka(;t)k H 1+ ( ) . 1 (and thus alsoka(;t)k L 1 ( ) . 1) for t2 [0;T ], (iii)ka t (;t)k L p ( ) .krv(;t)k L p ( ) for p2 [1;1] and t2 [0;T ], (iv)ka t (;t)k H r ( ) .krv(;t)k H r ( ) for r2 [0; 1 +) and t2 [0;T ], (v)ka tt (;t)k H ( ) .krv(;t)k H 1+ ( ) krv(;t)k H ( ) +krv t (;t)k H ( ) ; for all t2 [0;T ] 92 and all 2 [0;], (vi) For every 2 (0; 1] and all t2 [0;T 0 ], where T 0 T is suciently small, we have kRa j l jl k H 1+ ( ) (6.31) for j;l = 1; 2 and kRa j l a k l jk k H 1+ ( ) (6.32) for j;k = 1; 2, where R = 1 or R =J. (vii)kJk H 1+ ( ) . 1 for t2 [0;T ], (viii) 1=2J = det(r(x;t)) 2 for (x;t)2 [0;T ], (ix)k k H 2+ ( ) ;k t k H 2+ ( ) . 1 for t2 [0;T ], where =(), (x)kJ t k H 1+ ( ) .kvk H 1+ ( ) andkJ tt k H ( ) .kvk H 1+ ( ) (1 +kvk H 1+ ( ) ) +kv t k H ( ) for t2 [0;T ], (xi)ke k H 2+ ( ) . 1 for t2 [0;T ], (xii)k(;t)(; 0)k L 1 ( ) 0 for t2 [0;T ]. Proof. For the proof of (i){(vii), cf. [KT14] and [KT12], up to small adjustments. (viii) From [BG10] (cf. also [KT12]), we know that J t =Ja k i @ k t =Ja k i @ k ((1 )v i ) =Ja k i @ k v i ; (6.33) which implies 0J exp(Tkak L 1kr k L 1kvk L 1) exp(CTMkr k L 1) exp(CTM) (6.34) where we used (i). For T 1=CM with suciently large C, we obtain J 2. Integrating (6.33) directly gives jJ 1j 2Tkak L 1kr k L 1kvk L 1.TMkr k L 1.TM (6.35) providing the claimed lower bound for J. (ix) By Lemma 6.2.1, we have k k H 2+ .k det(r) 1 k 1=2 L 1 ( ) (1 +krk H 1+) 3 kk H 2+ (T(0;1)) . 1: (6.36) 93 Taking the time derivative of , we arrive at @ t = (r)() t : (6.37) Since 2C 1 and t 2H 2+ , we getk t k H 2+ ( ) . 1. (x) The rst inequality is a consequence of (6.33). Dierentiating (6.33), we get J tt =(J t a k i @ k v i +J@ t a k i @ k v i +Ja k i @ k t v i +Ja k i @ k v i t ): (6.38) From the above identity we obtain kJ tt k H .kvk 2 H 1+ ( ) +kvk 2 H 1+ ( ) +kvk H 1+ ( ) +kv t k H ( ) (6.39) providing the second inequality. (xi) From the equation e t =u(e ) (6.40) we obtain by Lemma 6.2.1 ke k H 2+ ( ) . 1 + Z t 0 ku(e )k H 2+ ( ) ds . 1 + Z t 0 kuk H 2+ ( (t)) ke k 3 H 2+ ( ) ds . 1 + Z t 0 kvk H 2+ ( ) k 1 k 3 H 2+ ( ) ke k 3 H 2+ ( ) ds: (6.41) Applying Gronwall inequality giveske k H 2+ ( ) . 1 provided T is small enough. (xii) Using the fundamental theorem of calculus, we obtain k(x;t)(x; 0)k L 1 Z t 0 k t (x;s)k L 1ds Z t 0 kvk L 1dsMt: (6.42) Choosing t 1=CM, where C is suciently large, we have proved (xi). We also need the following result from [BM01] addressing the Sobolev norm of the composite of two functions. Lemma 6.2.3. Given 1s<1 and 1<p<1. Let be a smooth bounded domain in R n and f2C m be such that f;f 0 ; ;f (m) 2L 1 where m =dse. Let M f (u) =fu: (6.43) 94 Then M f is a continuous map from W s;p ( )\W 1;sp ( ) to W s;p ( ). 6.3 Proof of the main result The proof of the main theorem is divided into three subsections: div-curl estimates, pressure estimate, and tangential estimates. 6.3.1 Div-curl estimates for , v, and a As in [KT14], we introduce the variable curl operator B a f =a k 1 @ k f 2 a k 2 @ k f 1 (6.44) and the variable divergence operator A a f =a k i @ k f i (6.45) where k;i2f1; 2g and f is a vector valued smooth function in R 2 . Observe that if a =I, then B I and A I agree with the usual curl and divergence operators. Dene the tangential derivative operator S = (I@ 2 1 ) (2+)=2 : (6.46) For technical reasons, we need another cut-o function : !R such that = 1; x 2 1 2 0 0 1; 1 3 0 x 2 1 2 0 = 0; x 2 1 3 0 : (6.47) Let = (). Denote by P a generic polynomials in the variableska t k H 1+,kk H 2:5+,kvk H 2+, and kak H 1:5+. Lemma 6.3.1. Assume that (v;q;a;) satises the Euler equation (6.11){(6.19) in [0;T ) and that we havekrvk L 1 ([0;T ];H 1+ ( )) M. Suppose that a satises the estimates in Lemma 6.2.2 for a suciently small constant > 0. Then we have kk H 2:5+ .kk L 2 + Z t 0 Pds +kS 2 k L 2 ( 1 ) +k! 0 k H 1:5+ (6.48) 95 with kak H 1:5+ .kk 4 H 2:5+ (6.49) and kvk H 2+ .kvk L 2 +kS(v)k L 2 +k! 0 k H 1+; (6.50) for t2 [0;T ]. Proof. We start with the estimate for curl. First, we have curl = 0; x2fx 2 1 6 0 g (6.51) since (6.11) implies that (x;t) =x; (x;t)2fx 2 1 6 0 g [0;T ]: (6.52) Therefore, we have k curlk H 1:5+ ( ) .k curlk H 1:5+ (U 7 0 ) .kB I rk H 0:5+ (U 7 0 ) +k curlk H 0:5+ (U 7 0 ) .kB I rB a rk H 0:5+ +kB a rk H 0:5+ +k curlk H 0:5+ .kIak H 1+krk H 1:5+ + Z t 0 kB at rk H 0:5+ds + Z t 0 kB a r t k H 0:5+ds +kk H 1:5+: (6.53) Since kB at rk H 0:5+ .ka t k H 1+krk H 1:5+; (6.54) we deduce Z t 0 kB at rk H 0:5+ds. Z t 0 ka t k H 1+krk H 1:5+ds: (6.55) Next we rewrite B a r t =B a r((1 )v) =B a ((1 )rv) +B a (r(1 ) v); (6.56) 96 from where we get Z t 0 kB a r t k H 0:5+ds . Z t 0 k1 k H 1+kB a rvk H 0:5+ +kak H 1+kr k H 1+krvk H 0:5+ +k1 k H 2+kB a vk H 0:5+ +kak H 1+kD 2 k H 0:5+kvk H 1:5+ds: (6.57) By the identity B a rv =r(B a v)B ra v; (6.58) we further obtain kB a rvk H 0:5+ .kr(B a v)k H 0:5+ +kB ra vk H 0:5+ .kB a vk H 1:5+ +krak H 0:5+krvk H 1+ .k!()k H 1:5+ +kak H 1:5+kvk H 2+; (6.59) where ! = curlu. By Lemma 6.2.2, we have k(x;t)xk L 1 0 (6.60) for small T . From Lemma 6.2.1 and 6.2.2, one gets k!()k H 1:5+ .k det(r) 1 k 1=2 L 1 (1 +krk H 1+) 2 k!k H 1:5+ ((U 7 0 )) .k!k H 1:5+ ((U 7 0 )) : (6.61) Noting(U 7 0 )fx :x 2 1 8 0 g withe 1 (U 7 0 )U 10 0 and using Lemma 6.2.1 again, we arrive at k!()k H 1:5+ .k det(re )k 1=2 L 1 (1 +kre 1 k H 1+) 2 k! 0 k H 1:5+ (e 1 (U 7 0 )) .k! 0 k H 1:5+ (U 10 0 ) ; (6.62) where we also used thatke k H 2+ . 1. In summary, we get k curlk H 1:5+ .kk H 2:5+ +kk H 1:5+ + Z t 0 Pds +k! 0 k H 1:5+ (U 10 0 ) : (6.63) 97 Next we will use similar method to get an estimate for div. k divk H 1:5+ .k(A I A a )rk H 0:5+ +kA a rk H 0:5+ +k divk H 0:5+ .kIak H 1+kk H 2:5+ +k divk H 0:5+ + Z t 0 kA at rk H 0:5+ +kA a r t k H 0:5+ds: (6.64) Note that A a r t =A a r((1 )v) =A a (r(1 ) v) +A a ((1 )rv): (6.65) We use product rule in order to get kA a r t k H 0:5+ .kak H 1+kr k H 1+krvk H 0:5+ +k1 k H 2+kA a vk H 0:5+ +kak H 1+kD 2 k H 0:5+kvk H 1:5+; (6.66) where we use that fact that A a rv = 0 because of the divergence free condition divu = 0. Substituting the above estimate into (6.64) gives k divk H 1:5+ .kk H 2:5+ +k divk H 0:5+ + Z t 0 Pds: (6.67) Resorting to the inequality kfk H s ( ) .kfk L 2 ( ) +k curlfk H s1 ( ) +k divfk H s1 ( ) +k@ 1 fNk H s1:5 (@ ) (6.68) where f is a vector valued function such that f2 H s ( ) for s > 1:5 and N is the outer unite normal vector (cf. [CS07, CS10]), we obtain kk H 2:5+ ( ) .kk L 2 ( ) +k curlk H 1:5+ ( ) +k divk H 1:5+ ( ) +kS 2 k L 2 ( 1 ) : (6.69) Now we replace curl and div by (6.63) and (6.67) in (6.69), absorbing the -term to the left side, to get kk H 2:5+ ( ) .kk L 2 + Z t 0 Pds +kk H 1:5+ +kS 2 k L 2 ( 1 ) : (6.70) 98 We note that by the Gagliardo-Nirenberg inequality and Young's inequality kk H 1:5+ .kk 1=(2:5+) L 2 kk (1:5+)=(2:5+) H 2:5+ .kk L 2 +kk H 2:5+; (6.71) which proves (6.48) by absorbing the term to the left side. Similarly, we estimatekvk H 2+ as kvk H 2+ ( ) .kvk L 2 ( ) +k curlvk H 1+ ( ) +k divvk H 1+ ( ) +k@ 1 vNk H 0:5+ ( 1 ) .kvk L 2 +k(B I B a )vk H 1+ +kB a vk H 1+ +k(A I A a )vk H 1+ +kA a vk H 1+ +k@ 1 v 2 k H 0:5+ ( 1 ) : (6.72) By the trace theorems, we are allowed to estimate k@ 1 v 2 k H 0:5+ ( 1 ) =k@ 1 (v 2 )k H 0:5+ ( 1 ) .k@ 1 (v 2 )k H 1+ .k@ 1 (v 2 )k L 2 +k@ 1 r(v 2 )k H .kS(v 2 )k L 2 +k@ 1 @ 2 (v 2 )k H : (6.73) A direct computation shows that @ 1 @ 2 (v 2 ) =@ 1 (@ 2 v 2 +@ 2 v 2 ) =@ 1 (@ 2 v 2 ) +@ 1 ((divv@ 1 v 1 )) =@ 1 (@ 2 v 2 ) +@ 1 ((A Ia v@ 1 v 1 )) (6.74) where we used that A a v = 0. Therefore, we obtain by Lemma 6.2.2 k@ 1 @ 2 (v 2 )k H .kvk H 1+ +kA Ia vk H 1+ +k@ 2 1 v 1 k H .kvk H 1+ +kvk H 2+ +k@ 2 1 v 1 k H : (6.75) By the identity @ 2 1 v 1 =@ 2 1 (v 1 ) (@ 2 1 v 1 + 2@ 1 @ 1 v 1 ) (6.76) it follows k@ 2 1 v 1 k H .kS(v)k L 2 +kvk H 1+: (6.77) 99 From (6.73) we further deduce that k@ 1 v 2 k H 0:5+ ( 1 ) .kS(v)k L 2 +kvk H 1+ +kvk H 2+: (6.78) On the other hand, by the Gagliardo-Nirenberg inequality and Young's inequality, we have kvk H 1+ .kvk 1=(2+) L 2 kvk (1+)=(2+) H 2+ .kvk L 2 +kvk H 2+: (6.79) Therefore, by absorbing the term to the left side, we obtain from (6.72) that kvk H 2+ .kvk L 2 +kS(v)k L 2 +k! 0 k H 1+; (6.80) which proves (6.50). To nish the proof, a is the only term left to treat. Noting that a = (r) 1 = 1 J Cof(r) (6.81) we get kak H 1:5+ . 1 J H 1:5+ kk 2 H 2:5+ .kJk H 1:5+kk 2 H 2:5+ .kk 4 H 2:5+ (6.82) where we used Lemma 6.2.3. In fact, since 1=2J 2, we have 1 J = (J) J (6.83) holds for a smooth cut-o function :R! [0; 1] such that () = 0; 1=4 or 4 () = 1; 2 [1=2; 2] 0() 1; otherwise: (6.84) Therefore, by Lemma 6.2.3,k1=Jk H 1:5+ =k(J)=Jk H 1:5+ .kJk H 1:5+ holds, which con- cludes the proof. 6.3.2 Pressure estimates In the following lemma, we obtain the elliptic estimates for the pressure q. 100 Lemma 6.3.2. For t2 [0;T ], the pressure q obeys kqk H 2:5+P +P Z t 0 kq t k L 2ds (6.85) and kq t k H 1+P +P Z t 0 kq t (s)k L 2ds (6.86) with kq t k H 2+ (U 4 0 ) P +P Z t 0 kq t (s)k L 2ds (6.87) where P is a polynomial inkvk H 2+,kv t k H 1+,kk H 2:5+, andkv 0 k H 2+. Proof. Applying a k i @ k operator to Equation (6.15) gives a k i @ k v i t +a k i @ k ( a k j v j @ k v i ) +a k i @ k (a k i @ k q) = 0; (6.88) from where we obtain @ kk q =@ t a k i @ k v i +a k i @ k ( a l j @ l v i v j ) + (a k i a l i kl )@ kl q +a k i @ k a l i @ l q: (6.89) By the boundary conditions (6.20){(6.21), we get that q satises q = 0 on 1 (0;T ) (6.90) and @ i qN i = 0 on 0 (0;T ); (6.91) where we also used that r =I on 0 (0;T ): (6.92) By the elliptic regularity estimate, we have kqk H 2:5+ .k@ t ak H 1+kvk H 2+ +ka k i @ k ( a l j @ l v i v j )k H 0:5+ +kIa :a T k H 1+kqk H 2:5+ +kak H 1+kak H 1:5+kqk H 2+: (6.93) 101 Noting the identity a k i @ k ( a l j @ l v i v j ) =a k i @ k ( a l j v j )@ l v i +a k i ( a l j v j )@ lk v i =a k i @ k ( a l j v j )@ l v i @ l a k i @ k v i ( a l j v j ) (6.94) where we used the divergence free condition (6.16), we arrive at ka k i @ k ( a l j @ l v i v j )k H 0:5+ .kak H 1+k a l j v j k H 1:5+kvk H 2+ +kak H 1:5+k k H 1+kak H 1+kvk 2 H 2+ .kak H 1:5+kak H 1+k k H 1+kvk 2 H 2+ : (6.95) Then from (6.93) and by Lemma 6.2.2, it follows kqk H 2:5+ .kvk H 2+ +kak H 1:5+k k H 1+kvk 2 H 2+ +kqk H 2:5+ +kak H 1:5+kqk H 2+; (6.96) which implies that kqk H 2:5+ .kvk H 2+ +kak H 1:5+kvk 2 H 2+ +kak H 1:5+kqk 0:5=(2:5+) L 2 kqk (2+)=(2:5+) H 2:5+ : (6.97) By Young's inequality, we further obtain kqk H 2:5+ .kvk H 2+ +kak H 1:5+kvk 2 H 2+ +kak 5+2 H 1:5+ kqk L 2 .P +P Z t 0 kq t k L 2ds (6.98) where P is a polynomial ofkk H 2:5+,kvk H 2+,kv 0 k H 2+ andkv t k H 1+. This accomplished the proof of (6.85). Next we achieve the global and interior estimate for q t . We rst get another equation for q as @ kk q =J@ t a k i @ k v i +Ja k i @ k ( a l j @ l v i v j ) +@ k (Ja k i a l i @ l q@ k q): (6.99) Taking the t derivative of the above equation we obtain @ kk q t =@ k @ t (@ t (Ja k i )v i ) +@ k @ t (Ja k i a l j @ l v i v j ) +@ k @ t ((Ja k i a l i @ l q kl )@ l q) (6.100) 102 where we used the divergence free condition (6.16) and the Piola identity @ k (Ja k i ) = 0; i = 1; 2: (6.101) Naturally, the boundary condition reads q t = 0 on 1 (0;T ) (6.102) and @ i q t N i = 0 on 0 (0;T ): (6.103) The elliptic estimate gives kq t k H 1+ .k@ t (@ t (Ja k i )v i )k H +k@ t (Ja k i a l j @ l v i v j )k H +k@ t ((Ja k i a l i @ l q kl )@ l q)k H .k@ tt (Ja k i )v i k H +k@ t (Ja k i )@ t v i k H +k@ t (Ja k i ) a l j @ l v i v j k H +kJa k i @ t ( a l j )@ l v i v j k H +kJa k i a l j @ t (@ l v i v j )k H +k@ t ((Ja k i a l i @ l q kl ))@ l qk H +k(Ja k i a l i @ l q kl )@ l q t k H : (6.104) Note by Lemma 6.2.2 k@ tt (Ja k i )k H .k@ tt Ja k i k H +k@ t J@ t a k i k H +kJ@ tt a k i k H .kvk H 1+(1 +kvk H 1+) +kv t k H +kvk 2 H 1+ +kvk H 2+kvk H 1+ +krv t k H (6.105) and k@ t (Ja k i )k H 1 .k@ t Jk H 1+kak H 1+ +kJk H 1+k@ t ak H 1+ .krvk H 1+ +kvk H 1+: (6.106) 103 We further get from (6.104) kq t k H 1+ . ((1 +kvk H 2+)kvk H 1+ +kv t k H 1+)kvk H 1+ +kvk H +kvk 2 H 2+ kvk H 1+ +kv t k H krvk H 1+ + (krvk H 1+ +kvk H 1+)kvk H 2+kvk H 1+ +k@ t k H 1+kk 2 H 2+ kvk 2 H 2+ +k k H 1+kk 2 H 2+ kvk 3 H 2+ +kk 2 H 2+ krv t k H +k@ t (Ja k i a l i kl )k H kqk H 1+ +kq t k H 1+: (6.107) From (6.15), it is easy to see by the algebra property of H 1+ that kv t k H 1+ .kvk H 2+ +k@ k qk H 1+: (6.108) Combining the estimate (6.104) and (6.108) and absorbing the term, we get kq t k H 1+ .P +P Z t 0 kq t k L 2ds (6.109) where P is a generic polynomial as above and which proves (6.86). Noting that = 0 on U 4 0 (0;T ), we apply an interior elliptic estimate to get kq t k H 2+ (U 4 0 ) .k@ t (J@ t a k i @ k v i )k H (U 4 0 ) +k@ t (Ja k i a l j @ l v i v j )k H 1+ (U 4 0 ) +k@ t ((Ja k i a l i @ l q kl )@ l q)k H 1+ (U 4 0 ) +kq t k H 1+ ( ) . krvk 2 H 1+ ( ) +krv t k H ( ) kvk H 2+ ( ) +k@ t (Ja k i a l i @ l q kl )k H 1+ (U 4 0 ) kqk H 2+ (U 2 0 ) +kq t k H 2+ (U 4 0 ) +kq t k H 1+: (6.110) The inequality (6.87) is obtained by absorbing the term in the above estimate. 6.3.3 Tangential estimates and the conclusion In this section, the tangential estimates are given. To be more specic, we obtain a bound forkS(v(t))k 2 L 2 , which already has appeared in the divergence-curl estimates and where S and are as in Subsectionx 6.3.1. The result is stated as follows. 104 Lemma 6.3.3. For t2 [0;T ], we have kS(v(t))k 2 L 2 +ka 2 l (t)S l (t)k 2 L 2 ( 1 ) Z t 0 P (kvk H 2+;kv t k H 1+;kqk H 2:5+;kq t k H 1+;kk H 2:5+)ds +Q(kv 0 k H 2+) (6.111) where P and Q are polynomials in indicated arguments. Proof. We begin with a simple observation that = 0 for t 2 [0;T ] due to the fact k(;t)(; 0)k L 1 ( ) 0 . In fact we have that = 0 onfx :x 2 1 4 0 g and = 0 on fx :x 2 1 4 0 g. We multiply (6.15) by in order to get (v i ) t +a k i @ k q = t v i : (6.112) Applying the operator S to the above equation, we get S(v i ) t +S(a k i @ k q) =S( t v i ): (6.113) We multiply this equation by S(v i ), integrating it and summing over i = 1; 2, to obtain 1 2 d dt kS(v)k 2 L 2 = Z S(a k i @ k q)S(v i )dx + Z S( t v i )S(v i )dx = Z S(a k i )@ k qS(v i )dx Z a k i S(@ k q)S(v i )dx Z (S(a k i @ k q)S(a k i )@ k qa k i S(@ k q))S(v i )dx + Z S( t v i )S(v i )dx: (6.114) For the last term on the right side, we have Z S( t v i )S(v i )dx.kS( t v i )k L 2kS(v i )k L 2.kvk 2 H 2+ ; (6.115) 105 where we used the algebra property of H 2+ and kk H 2+ . 1 by Lemma 6.2.1 and Lemma 6.2.2. The rest three terms are treated similarly as in [KT14] except one dier- ence. Namely, as in that paper, we get the boundary term Z t 0 Z 1 a 2 l S l a 2 i @ 2 qS(v i )dds. Z 1 a 2 l S l a 2 i S i d t Z 1 a 2 l S l a 2 i S i @ 2 qd 0 Z t 0 Z 1 @ t a 2 l S l a 2 i S i @ 2 qdds 1 2 Z t 0 Z 1 a 2 l S l a 2 i S i @ 2t qdds (6.116) where we used the fact = 1 on 1 and the Rayleigh-Taylor condition @q @N 1 C 0 < 0 on 1 : (6.117) Sinceq t is not inH 2+ ( ), we are not able to carry out the estimate in the same manner as in [KT14] for the last term. However, we have q t 2H 2+ (U 4 0 ), which allow us to estimate as Z t 0 Z 1 a 2 l S l a 2 i S i @ 2t qdds. Z t 0 kak 2 L 1 ( 1 ) kk 2 H 2+ ( 1 ) k@ 2t qk L 1 ( 1 ) ds . Z t 0 kk 2 H 2:5+ ( ) k@ t qk H 2+ (U 4 0 ) ds. Z t 0 Pds (6.118) whereP is a polynomial inkvk H 2+;kv t k H 1+;kqk H 2:5+;kq t k H 1+; andkk H 2:5+. The proof is completed. Now we conclude the proof of the main theorem by the same Gronwall type argument as in section 8 of [KT14]. 106 Bibliography [ABZ11] T. Alazard, N. Burq, and C. Zuily. On the water-wave equations with surface tension. Duke Math. J., 158(3):413{499, 2011. [ABZ13] T. Alazard, N. 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Abstract (if available)

Abstract We address the regularity and stability problems of the following partial differential equations related to fluid dynamics: the surface quasi-geostrophic (SQG) equation, the Boussinesq equations, the active scalar equations, the Euler equations, and the Prandtl boundary layer equations. ❧ We consider the SQG equation, whose first moment decay $\Vert\, |x|\theta\Vert_{L^2}$ was obtained by M. and T. Schonbek. We obtain the decay rates of $\Vert |x|^{b}\theta\Vert_{L^2} $ for any $b\in(0, 1)$ and the rate of increase of this quantity for $b\in[1, 1 + α)$ using different method under natural assumptions on the initial data. ❧ We investigate the global regularity of solutions to the Boussinesq equations with zero diffusivity in two spatial dimensions. Previously, the persistence in the space $H^{1+s}(\mathbb{R}^2)\times H^{s}(\mathbb{R}^2)$ for all s ≥ 0 has been obtained. We address the persistence in general Sobolev spaces in which the behaviors of solutions turn out to be very different from that in Hilbert space, establishing it on a time interval which is almost independent of the size of the initial data. ❧ We also address the active scalar equations with constitutive laws that are odd and very singular, in the sense that the velocity field loses more than one derivative with respect to the active scalar. We provide an example of such a constitutive law for which the equation is ill-posed: Either Sobolev solutions do not exist, from the Gevrey-class datum, or the solution map fails to be Lipschitz continuous in the topology of a Sobolev space, with respect to Gevrey class perturbations in the initial datum. ❧ Another model is the two dimensional incompressible Euler equations on a moving boundary, with no surface tension, under the Rayleigh-Taylor stability condition. The main feature of the result is the local regularity assumption on the initial vorticity, namely $H^{1.5+\delta}$ Sobolev regularity in the vicinity of the moving interface in addition to the global regularity assumption on the initial fluid velocity in the H²⁺ᵟ space. We use a special change of variables and derive a priori estimates, establishing the local-in-time existence in H²⁺ᵟ. The assumptions on the initial data constitute the minimal set of assumptions for the existence of solutions to the rotational flow problem to be established in 2D. ❧ In 1980, van Dommelen and Shen provided a numerical simulation that predicts the spontaneous generation of a singularity in the Prandtl boundary layer equations from a smooth initial datum, for a nontrivial Euler background. We provide a proof of this numerical conjecture by rigorously establishing the finite time blowup of the boundary layer thickness.

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Creator Wang, Fei (author)

Core Title On regularity and stability in fluid dynamics

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Mathematics

Publication Date 07/19/2017

Defense Date 04/24/2017

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

Tag active scalar equations,Boussinesq equations,Euler equations,fluid dynamics,OAI-PMH Harvest,Prandtl boundary layer equations,regularity,stability,surface quasi-geostrophic (SQG) equation

Language English

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Advisor Kukavica, Igor (committee chair)

Creator Email feiw2662@gmail.com,wang828@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c40-404266

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