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Analyticity and Gevrey-class regularity for the Euler equations
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Analyticity and Gevrey-class regularity for the Euler equations
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ANALYTICITY AND GEVREY-CLASS REGULARITY FOR THE EULER EQUATIONS by Vlad Cristian Vicol A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (MATHEMATICS) August 2010 Copyright 2010 Vlad Cristian Vicol Acknowledgments I am infinitely grateful to my thesis advisor and mentor, Prof. Igor Kukavica, for his inspiration, patience, dedicated teaching, and contagious enthusiasm for research. Without his guidance this thesis would have been next to impossible to complete. I am also deeply indebted to Prof. Susan Friedlander and Prof. Mohammed Ziane for their endless support, stimulating discussions, and for their help in expanding my mathematical horizons. In addition, I would like to thank Prof. Sergey Lototsky and Prof. Paul Newton for their encouragement and insightful comments. ii Table of Contents Acknowledgments ii List of Figures v Abstract vi Chapter 1: Introduction 1 1.1 The Euler equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Analytic and Gevrey-class functions . . . . . . . . . . . . . . . . . . 3 1.3 Persistence of smooth regularity for the Euler equations . . . . . . . . 4 Chapter 2: The periodic domain 9 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 V orticity formulation and functional setting . . . . . . . . . . . . . . 10 2.3 The analyticity theorem . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.4 Proof of the main lemma . . . . . . . . . . . . . . . . . . . . . . . . 15 Chapter 3: The half-space 22 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.2 Main theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.3 Proofs of the main theorems . . . . . . . . . . . . . . . . . . . . . . 26 3.4 The commutator estimate . . . . . . . . . . . . . . . . . . . . . . . . 32 3.5 The pressure estimate . . . . . . . . . . . . . . . . . . . . . . . . . . 37 Chapter 4: The bounded domain 52 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.2 Notation and preliminary remarks . . . . . . . . . . . . . . . . . . . 54 4.2.1 Local change of coordinates . . . . . . . . . . . . . . . . . . 54 4.2.2 Gevrey-class norms . . . . . . . . . . . . . . . . . . . . . . . 57 4.3 Short time local Gevrey-class a priori estimates . . . . . . . . . . . . 58 4.4 The velocity commutator estimate . . . . . . . . . . . . . . . . . . . 65 4.5 The pressure estimate . . . . . . . . . . . . . . . . . . . . . . . . . . 71 iii 4.5.1 Semi-norms and a decomposition of the pressure term . . . . 72 4.5.2 The elliptic Neumann problem for the pressure . . . . . . . . 74 4.5.3 The interiorH 2 -regularity estimate . . . . . . . . . . . . . . 75 4.5.4 The estimation of tangential derivatives . . . . . . . . . . . . 76 4.5.5 The transfer of normal to tangential derivatives . . . . . . . . 79 4.5.6 Bounds forP 0 ;P 1 , andP 2 . . . . . . . . . . . . . . . . . . . 81 4.5.7 Gevrey-class estimates for the pressure . . . . . . . . . . . . 86 4.5.8 Proof of Lemma 4.3.3 . . . . . . . . . . . . . . . . . . . . . 90 4.6 Global Gevrey-class persistence . . . . . . . . . . . . . . . . . . . . 92 4.7 Appendix I: Combinatorial identities . . . . . . . . . . . . . . . . . . 99 4.8 Appendix II: Bounding the derivative of a supremum . . . . . . . . . 101 References 106 iv List of Figures 4.1 The local coordinate charts and the flow induced by t on the straight- ened domain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 4.2 The lower bound on theG s -radius obtained by patching local in time bounds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 v Abstract The Euler equations are the classical model for the motion of an incompressible invis- cid homogenous fluid. This thesis addresses geometric qualitative properties of smooth solutions to the Euler equations, namely the persistence of analyticity and Gevrey-class regularity on domains with smooth boundary. The structure of the spectrum of solutions to the three-dimensional Euler equations is a problem of fundamental interest in turbulence theory. The size of the uniform real- analyticity radius of the solution provides an estimate for the scale below which the Fourier coefficients decay exponentially; it moreover gives the rate of this exponential decay. This thesis also addresses the problem of finding sharp lower bounds for the uniform real-analyticity and Gevrey-class regularity radius of the solutions. We prove that the rate of decay of the radius depends at most exponentially on the supremum of the velocity gradient, and algebraically on the higher Sobolev norms of the solution. vi Chapter 1: Introduction 1.1 The Euler equations LetD be an open connected Gevrey-class s domain in R d , where d 2 f2; 3g and s 1. The motion of an incompressible inviscid homogenous fluid inD is classically modeled by the Euler equations for the unknown velocity vector field u(x;t) :D [0;1)7!R d and the scalar pressure fieldp(x;t) :D[0;1)7!R. The Euler system arises in continuum mechanics as an expression of the conservation of momentum and of the incompressibility condition, in the infinite Reynolds number limit: @ t u + (ur)u +rp = 0; (1.1) ru = 0; (1.2) inD [0;1). In the presence of physical boundaries, the equations are supplemented with the classical no-normal flow (free slip) boundary condition un = 0; (1.3) 1 on@D [0;1) , wheren is the outward unit normal to@D. IfD = T d = [0; 2] d we impose periodic boundary conditions instead of (1.3). We consider the initial value problem associated to (1.1)–(1.3) with a Gevrey-classs initial datum u(; 0) =u 0 ; (1.4) inD, satisfying the compatibility conditions arising from (1.2) and (1.3). We refer the interested reader for instance to Chorin and Marsden [17, Chapter 1] for more details on the physical derivation of the Euler equations (see also [16, 48, 50]). The local existence and uniqueness of SobolevH r (D) solutions, withr>d=2+1, on a maximal time interval [0;T ) is classical (cf. [15, 24, 34, 55, 56]). While in the two-dimensional case a smooth initial data yields global solutions (cf. [34, 56]), in the three-dimensional case ifu 0 2 H r (D), withr > 5=2, the maximal time of existence of the Sobolev solution, T , might be a priori finite. If T <1, the vorticity must accumulate in the sense that lim T%T R T 0 k curlu(;t)k L 1 (D) =1 (cf. [10, 25]). In the two-dimensional case the global existence of solutions comes with the fol- lowing a priori bounds on the solution kru(;t)k L 1 (D) C exp(Ct) and respectively ku(;t)k H r (D) C exp(C exp(Ct)) for t 0, where C is a sufficiently large constant. The sharpness of these bounds is an open question. In three dimensions, while no global in time upper bounds on 2 the solution are available, the bounds onku(;t)k H r (D) are exponentially worse than those onku(;t)k W 1;1 (D) due to the log-Sobolev inequality. This motivates one of the main aspects of the present work: we obtain lower bounds on the Gevrey-class radius which depend algebraically (instead of exponentially) on the high Sobolev norm of the solution. We refer the reader to the surveys [8, 16, 19, 48, 50] for the precise formulation of the above statements and for further results cornering the Euler equations. 1.2 Analytic and Gevrey-class functions We recall (cf. [29, 35, 47]) the definition of the Gevrey-classs, denoted in the following byG s . 1.2.1 Definition. A functionv2 C 1 (D) is said to be uniformly of Gevrey-classs on D, wheres 1, writtenv2G s , if there exist positive constantsM; > 0 such that k@ vk L 1 (D) M jj! s jj (1.5) for all multi-indices2N d 0 , wherejj = 1 +::: + d . We refer to the constant in (1.5) as the radius of Gevrey-class regularity ofv, or simply as theG s -radius ofv. Fors 1 the classG s is closed under multiplication, differentiation, and composi- tion. As opposed to the class of real analytic functionsG 1 , functions inG s withs> 1 may have compact support, they may vanish of infinite order at a point, and there exist G s partitions of unity (cf. [35]). WhenD = T d (orD = R d , with suitable modifications) we have the following classical elegant characterization of functions inG s in terms of their Fourier series (cf. [28, 41]). 3 1.2.1 Lemma. Fors 1 andr 0,v2G s (T d ) if and only if there exists > 0 such that v2D () r=2 e () 1=2s = ( v2H r (T d ) :k() r=2 e () 1=2s vk 2 L 2 = (2) d X k22Z d jkj 2r e 2jkj 1=s j^ v k j 2 <1 ) ; where H r (T d ) = v(x) = X k22Z d ^ v k e ikx : ^ v 0 = 0; ^ v k = ^ v k ; kvk 2 H r = (2) d X k22Z d (1 +jkj 2 ) r j^ v k j 2 <1 is the periodic Sobolev space. The exponential decay of the Fourier coefficients of functions inG s , and the explicit rate of this decay, follows immediately from Lemma 1.2.1. The above characterization of functions inG s has been used by Foias and Temam to address the Gevrey-class regularity of the Navier-Stokes equations, and by Levermore and Oliver to estimate the analyticity radius of solutions to the Euler equations. 1.3 Persistence of smooth regularity for the Euler equations In the present manuscript we address the persistence of Gevrey-class regularity of the solution of the Euler equations, i.e., we prove that ifu 0 2G s , then the unique Sobolev 4 solutionu(;t)2 C([0;T );H r (D)) is of Gevrey-classs for allt < T . Therefore, if the solution fails to be inG s at a timeT , then it must blow up inH r (D) at timeT . Moreover, we are interested in sharp lower bounds on the rate of decay of the radius of analyticity and Gevrey-class regularity of the solution. We emphasize that the size of the uniform Gevrey-class radius of the solution provides an estimate for the minimal scale in dissipative flows, that is, the scale below which the Fourier coefficients decay exponentially [33, 36]; it moreover gives the rate of this exponential decay [28, 33, 41] (feature which is numerically computable). We note that the shear flow example of Bardos and Titi [9] (cf. [23, 57]) may be used to construct explicit solutions to the three-dimensional Euler equations whose radius of analyticity, or even more generally the Gevrey-class radius, decays for all time (cf. Remark 1.3.1 below). We summarize the rich history of this problem. (i) The persistence ofC 1 regularity (cf. Foias, Frisch, and Temam [27]) and of real- analyticity (cf. Bardos and Benachour [6]) holds in both two and three dimen- sions. (ii) In the two-dimensional analytic case, Bardos, Benachour, and Zerner [7] show that the radius of analyticity(t) of the solutionu(;t) is bounded from below as (t) exp(C exp(Ct))=C, for some sufficiently large constantC depending on the initial data. Their elegant proof is based on analyzing the complexified equations in vorticity form. (iii) In the three-dimensional analytic case, the persistence of analyticity is proven by Bardos and Benachour [6] using an implicit argument. In [5, 11, 3, 4, 22] using a nonlinear variant of the Cauchy-Kowalevski theorem, the authors prove the local 5 in time existence of globally (in space) analytic solutions, with an explicit lower bound on the radius of analyticity which vanishes in finite time (independent of T ). See also [49, 54] for the dissipative Prandtl boundary layer equations. The proof of [6] may be modified to yield an explicit rate of decay of the radius of analyticity(t) which depends exponentially onku(;t)k H r. Using different methods, Alinhac and Metivier [2, 1] for the interior, and Le Bail [42] for the boundary value problem, obtain the short time propagation of local analyticity, with lower bounds for (t) that also decay exponentially inku(;t)k H r. Note that the lower bounds for(t) obtained in [2, 1, 6, 42] do not recover the lower bounds of [7] in the two-dimensional case, since the presently known upper bounds on high Sobolev norms of the solution increase as C exp(C exp(Ct)), for someC > 0. Moreover, the methods used in [2, 1, 6, 3, 4, 22, 42] explicitly use the special properties of complex holomorphic functions, and hence may not be applied to the non-analytic Gevrey-class case. (iv) For the non-analytic Gevrey-class case, on a periodic domain, in both two and three dimensions, the persistence of Gevrey-class regularity follows from the elegant proof of Levermore and Oliver [46]. Their proof builds on the Fourier- based method introduced by Foias and Temam [28] for the Navier-Stokes equa- tions. The lower bound for the radius of Gevrey-class regularity obtained in [46] also decays exponentially inku(;t)k H r. In [46, Remark 4] the authors pose the question of whether the Fourier-based method can be employed to recover the two-dimensional rate obtained by Bardos, Benachour, and Zerner. In Chap- ter 2 (see also [38]) we answer this question in the positive , by proving that the radius of Gevrey-class regularity decays algebraically in a high Sobolev norm 6 of the solution, and exponentially in R t 0 kru(;s)k L 1 ds. For further results on analyticity in fluid dynamics cf. [12, 13, 14, 20, 26, 31, 32, 36, 37, 52]. (v) In Chapter 3 (see also [39]) we obtain the first persistency result in the non- analytic Gevrey-class case on domains with boundary, forD being a half-space. As opposed to the periodic case, here the main difficulty arises from the equation for the pressure. The classical methods of [47, 51] are not sufficient to prove that the pressure has the same radius of Gevrey-class regularity as the velocity. We overcome this by defining suitable norms that combinatorially encode the transfer of normal to tangential derivative in the elliptic estimate for the pressure. (vi) Our half-space proof does not apply directly to the case whenD is a general boundedG s -domain. The main obstruction is that if s > 1, under compo- sition with a Gevrey-class (or even analytic) boundary straightening map, the Gevrey-class regularity radius of the velocity may deteriorate (cf. [21, 35] and Remark 4.2.1 below). As a consequence, we need to localize the equation using particle trajectories and define suitable Lagrangian Gevrey-class norms. This gives rise to additional difficulties because the pressure is the solution of an ellip- tic Neumann problem (cf. [55]), and hence is non-local. The resolution of this problem for the generic bounded domainD is given in Chapter 4 (see also [40]). 1.3.1 Remark. Note that there exist explicit examples of solutions to (1.1)–(1.4) whose radius of Gevrey-classs regularity, wheres 1, decays for all time, and vanishes as t!1. Namely, consider the three-dimensional shear flow example (cf. Bardos and Titi [9], DiPerna and Majda [23], Yudovich [57]) given by u(x;t) = (f(x 2 ); 0;g(x 1 tf(x 2 ))); (1.6) 7 which is divergence free and satisfies (1.1) for smooth functionsf andg. In the analytic category s = 1 we may let f(x) = sin(x) and g(x) = 1=( 2 0 + cos 2 (x)), where for simplicityD is the periodic box [0; 2] 3 . Substituting these par- ticular functionsf andg into (1.6) we obtain thatu(; 0) has radius of analyticity 0 , whileu(x;t) = (sin(x 2 ); 0; 1=( 2 0 + cos 2 (x 1 t sin(x 2 )) has uniform radius of ana- lyticity that decreases with the rate 1=t. For a similar example in the non-analytic Gevrey-classes, s > 1, letD = R 3 and define g(x) = exp jxj 1=(s1) cf. [45]. Note thatg(x) is of Gevrey-classs, but not analytic. 8 Chapter 2: The periodic domain 2.1 Introduction We briefly recall the main previously known results, and give the motivation for the below analysis (see Section 1.3 for more details). In three dimensions, Bardos and Benachour [6] proved the persistence of real-analyticity, while in the two-dimensional case, using the absence of the vorticity stretching term, Bardos, Benachour, and Zerner [7] established an explicit bound for the rate of decay of the analyticity radius, which is exp(C exp(Ct))=C, for a suitable positive constantC. Using a Fourier space method, Levermore and Oliver [46] proved that the uniform analyticity radius of the solution decays exponentially ink!(;t)k H r , wherer is large enough. In two dimensions, this radius decays exponentially faster than the radius obtained by Bardos and Benachour. In [46, Remark 4] the authors posed the question of whether the Fourier-based method can be employed to recover the 2D-rate obtained by Bardos, Benachour, and Zerner. In this chapter (see also [38]) we answer this question in the positive. This is achieved by obtaining lower bounds on the rate of decay of the uniform space analyticity radius that depend only algebraically on kcurlu(;t)k H r and exp( R t 0 kru(;s)k L 1 ds). Our results hold also for the non-analytic Gevrey classes, and in all space dimensionsd 2 (cf. Remark 2.3.3 below). 9 This chapter is organized as follows. In Section 2.2 we give the functional setting. Section 2.3 contains the statement and the proof of our main result, Theorem 2.3.1. The core of the proof of the main theorem is Lemma 2.3.1, whose proof is given in Section 2.4. 2.2 Vorticity formulation and functional setting On the periodic domain, or on the whole-space, it is common to write the initial value problem associated to the Euler equations in terms of the vorticity! = curlu as @ t ! +ur! =!ru; (2.1) u =K!; (2.2) !(0) =! 0 = curlu 0 ; (2.3) whereK is the periodic Biot-Savart kernel (cf. [8, 16, 48, 50] for details). Hereu and! areT 3 -periodic functions with R T 3 u = 0, whereT 3 = [0; 2] 3 . The case of the whole space can be treated with minor modifications. While all results below are stated for three space dimensions, they clearly also hold by restriction in the two-dimensional case. The functional setting for the present paper is as follows. For fixedr; 0 and m = 1; 2; 3, we define D( r m e m ) = !2H r (T 3 ) : div! = 0; r m e m ! 2 L 2 = (2) 3 X k2Z 3 jk m j 2r e 2jkmj j^ ! k j 2 <1 ; 10 where H r (T 3 ) = !(x) = X k2Z 3 ^ ! k e ikx : ^ ! 0 = 0; ^ ! k = ^ ! k ; k!k 2 H r = (2) 3 X k2Z 3 (1 +jkj 2 ) r j^ ! k j 2 <1 is the periodic Sobolev space. Forr; 0 define the normed spacesY r; X r; by X r; = 3 \ m=1 D( r m e m ); k!k 2 Xr; = 3 X m=1 r m e m ! 2 L 2 ; andY r; = X r+1=2; . In the following lemma we prove that the above defined spaces consist of real-analytic functions. 2.2.1 Lemma. If !2 X r; for r 0 and > 0, then ! is of Gevrey-class 1 (i.e., analytic), with uniform space analyticity radius at least=3. Proof. It is sufficient to show that P k2Z 3 e 2jkj=3 j^ ! k j 2 <1 (cf. [36, 46]). This follows from P k2Z 3 e 2jkj=3 j^ ! k j 2 k!k 2 Xr; , a direct consequence of the triangle inequality and the mean-zero condition. Similarly, one can show thatX r; is equivalent to the subspaceD(( p ) r e p ) of Gevrey-class 1 which was used in [46]. 2.3 The analyticity theorem The following is our main theorem. 2.3.1 Theorem. Ifu 0 is divergence-free, and! 0 = curlu 0 is real-analytic onT 3 , then the unique solution !2 C(0;T ;H r (T 3 )), with r > 7=2, to the vorticity equations 11 (2.1)–(2.3) is real-analytic for all t < T , where T 2 (0;1] is the maximal time of existence. Furthermore, the uniform space analyticity radius (t) of the solution !(;t) satisfies: (t)C 1 exp C 2 Z t 0 kru(;s)k L 1 ds 1 +t 2 1 ; (2.4) whereC 2 > 0 is a constant depending only onr, andC 1 > 0 has additional depen- dence on! 0 (cf. (2.9) below). 2.3.1 Remark. The theorem remains valid in any dimensiond 2, with the modifica- tionr> (d + 4)=2. This is due to the fact that ford = 2 the term!ru vanishes, and that ford 4 the vorticity formulation of the Euler equations is similar to (2.1)–(2.3). 2.3.2 Remark. In dimension 2, we can takeT arbitrarily large and therefore the solu- tions remain analytic for all time. In this casekru(;t)k L 1 increases with a rate at mostC exp(Ct) for some positive constantC, whilek!(;t)k H r increases with a rate at most C exp(C exp(Ct). This allows us to recover the 2D-rate of decay given by Bardos, Benachour, and Zerner [6, 7]. 2.3.3 Remark. By working in X r;;s = 3 T m=1 D( r m e 1=s m ), for s > 1, one can show that the radius of Gevrey-classs regularity (cf. [28, 46] for a definition of the Gevrey classes) of the smooth solution to (2.1)-(2.3) satisfies the same lower bound (2.4) as in Theorem 2.1, given that the initial datum is of Gevrey-classs. As in Foias and Temam [28] the proof carries over directly from the analytic cases = 1 and relies on the fact that fors we havejk +jj 1=s jkj 1=s +jjj 1=s , and fors> 1 we additionally use jl m j 1=s jk m j 1=s Cjjl m jjk m jj 1 jl m j 1(1=s) +jk m j 1(1=s) ; 12 where C is a positive constant depending only on s, and m2f1; 2; 3g. The latter inequality is for instance used to estimate the termT 2 defined in (2.15) below. Identity (2.13) below still needs to be used in the Gevrey case. The following lemma is needed to prove Theorem 2.3.1. 2.3.1 Lemma. Letm = 1; 2; 3 and!2Y r; , wherer > 7=2. Ifu =K!, whereK is the periodic Biot-Savart kernel, then (ur!; 2r m e 2m !) + (!ru; 2r m e 2m !) C kruk L 1 + 2 k!k H r + 2 k!k Xr; k!k Yr; r+1=2 m e m ! L 2 +C kruk L 1 k!k Xr; + (1 +)k!k 2 H r r m e m ! L 2 ; (2.5) where the positive constantC depends only onr. We note that Lemma 2.3.1 is an improvement of Lemma 8 in [46]. In the first term on the right of (2.5), the lowest power of is paired with the better behaved quantitykruk L 1 , whilek!k H r is paired with 2 . This implies algebraic rather than exponential dependence of(t) on theH r -norm of!. We prove Theorem 2.3.1 by showing that if the initial datum is of Gevrey-class 1, the solution remains in this class as long as it exists. In the followingC denotes a generic positive constant depending onr. Proof of Theorem 2.3.1. We note that if the initial datum! 0 is real-analytic with radius of analyticity at least (0), with > 1, then ! 0 2 H r and e (0) p ! 0 H r <1 (cf. [36, 46]). Therefore ! 0 2 X r;(0) . We now prove that for all 0 t < T the H r -solution of (2.1)–(2.3) satisfies!(;t)2 X r;(t) , for an appropriate function(t). When no ambiguity arises, we suppress the time dependence of and! ont. 13 By taking the L 2 -inner product of (2.1) with 2r m e 2m !, where m = 1; 2; 3, we obtain 1 2 d dt r m e m ! 2 L 2 = _ r+1=2 m e m ! 2 L 2 (ur!; 2r m e 2m !) + (!ru; 2r m e 2m !): (2.6) The constant C in Lemma 2.3.1 can be taken large enough so thatk!(;t)k 2 H r k! 0 k 2 H r g(t) for all 0t<T , whereg(t) = exp C R t 0 kru(;s)k L 1 ds : In order to conclude the proof, we sum overm = 1; 2; 3 in (2.6) and use the estimate (2.5). We obtain 1 2 d dt k!k 2 Xr; C kruk L 1 k!k Xr; + (1 +)k!k 2 H r k!k Xr; + _ +Ckruk L 1 +C 2 k!k H r +C 2 k!k Xr; k!k 2 Yr; : If is such that the second term on the right of the above is negative, then is decreas- ing and d dt k!k Xr; Ckruk L 1 k!k Xr; +C(1 +(0))k!k 2 H r : By Gronwall’s inequality this implies k!(;t)k X r;(t) g(t) k! 0 k X r;(0) +C(1 +(0)) Z t 0 k!(;s)k 2 H r g(s) 1 ds =A(t): (2.7) 14 A sufficient condition for the above to hold is that _ +Ckruk L 1 +C 2 k!k H r +C 2 A(t) 0; for allt 0. It suffices to set (t) =g(t) 1 (0) 1 +C Z t 0 (k!(;s)k H r +A(s))g(s) 1 ds 1 : (2.8) In particular, sincek!(;t)k 2 H r k! 0 k 2 H r g(t), we obtain (t)g(t) 1 C 0 +C 00 t 2 1 ; (2.9) whereC 0 = 2=(0) and the constantC 00 depends onr;(0);k! 0 k H r , andk! 0 k X r;(0) . 2.4 Proof of the main lemma Before we start the proof of Lemma 2.3.1, we introduce the operators f(x) = X k2Z 3 jkj 1 ^ f k e ixk and H m f(x) = X k2Z 3 sgn(k m ) ^ f k e ixk ; m = 1; 2; 3; for allf2 H 1 (T 3 ). Herejkj 1 denotesjk 1 j +jk 2 j +jk 3 j. The followingL 2 -estimates follow directly from Plancherel’s theorem and the proofs are thus omitted. 15 2.4.1 Lemma. Let!2X r; , for 0 andr 1. Then form = 1; 2; 3 we have k r m !k L 2 r1 m ! L 2 Ck!k H r and rH m r1 m e m ! L 2 r1 m e m ! L 2 Ck!k Xr; : Since u = K!, an immediate consequence of the above is thatk r+1 m uk L 2 k r m uk L 2 Ck!k H r , for a positive constantC. Proof of Lemma 2.3.1. Letm2f1; 2; 3g. In order to estimatej(ur!; 2r m e 2m !)j, we appeal to the cancellation property (ur r m e m !; r m e m !) = 0. Using Plancherel’s theorem we obtain (ur!; 2r m e 2m !) = (ur!; 2r m e 2m !) (ur r m e m !; r m e m !) =i(2) 3 X j+k+l=0 (jl m j r e jlmj jk m j r e jkmj )^ u j k ^ ! k jl m j r e jlmj ^ ! l =i(2) 3 X j+k+l=0 (jl m j r jk m j r )e jkmj ^ u j k ^ ! k jl m j r e jlmj ^ ! l +i(2) 3 X j+k+l=0 (e jlmj e jkmj )jl m j r ^ u j k ^ ! k jl m j r e jlmj ^ ! l =T 1 +T 2 ; (2.10) 16 withj;k;l2Z 3 . Recall that ^ ! 0 = ^ u 0 = 0. The first term on the far right side of the above is rewritten using the mean value theorem as T 1 =ir(2) 3 X j+k+l=0 (jl m jjk m j) ( m;k;l jl m j + (1 m;k;l )jk m j) r1 jk m j r1 e jkmj ^ u j k ^ ! k jl m j r e jlmj ^ ! l +ir(2) 3 X j+k+l=0 (jl m jjk m j)jk m j r1 e jkmj ^ u j k ^ ! k jl m j r e jlmj ^ ! l ; (2.11) for some m;k;l 2 (0; 1). Sincej +k +l = 0, we have (jl m jjk m j) ( m;k;l jl m j + (1 m;k;l )jk m j) r1 jk m j r1 Cjj m j 2 (jj m j r2 +jk m j r2 ): (2.12) The exponential factor is bounded as e jkmj e + 2 jk m j 2 e jkmj , andj^ u j kj Cj^ u j jjkj 1 , for a positive constantC. To estimate the second term on the right of (2.11) we use the decomposition jj m +k m jjk m j =j m sgn(k m ) + 2(j m +k m ) sgn(j m ) fsgn(km+jm) sgn(km)=1g : (2.13) (A version of the latter identity was also used by Lemari´ e-Rieusset in [43, 44] for provingL p -analyticity of solutions to the Navier-Stokes equations.) The first term in the decomposition (2.13) is treated using the Fourier inversion theorem. On the region 17 fsgn(k m +j m ) sgn(k m ) =1g we have 0jk m jjj m j and thus in this region jl m jjk m j jk m j r1 Cjj m j r . We have thus proven jT 1 jC X j+k+l=0 (jj m j r +jj m j 2 jk m j r2 )(e + 2 jk m j 2 e jkmj )j^ u j jjkj 1 j^ ! k jjl m j r e jlmj j^ ! l j +Cj(@ m urH m r1 m e m !; r m e m !)j C k!k 2 H r +kruk L 1 k!k Xr; r m e m ! L 2 +C 2 k!k H r k!k Yr; r+1=2 m e m ! L 2 : (2.14) In the second inequality we have appealed to the estimates in Lemma 2.4.1,r > 7=2, andjk m j 1=2 jj m j 1=2 +jl m j 1=2 . Returning to (2.10) we writeT 2 as T 2 =i(2) 3 X j+k+l=0 e (jlmjjkmj) 1(jl m jjk m j) jl m j r1=2 e jkmj ^ u j k ^ ! k jl m j r+1=2 e jlmj ^ ! l +i(2) 3 X j+k+l=0 (jl m jjk m j)jk m j r1=2 e jkmj ^ u j k ^ ! k jl m j r+1=2 e jlmj ^ ! l +i(2) 3 X j+k+l=0 (jl m jjk m j)(jl m j r1=2 jk m j r1=2 )e jkmj ^ u j k ^ ! k jl m j r+1=2 e jlmj ^ ! l : (2.15) The three terms on the right are treated as follows. Sinceje x 1xj x 2 e jxj , for allx2R, andjl m j r1=2 C jj m j r1=2 +jk m j r1=2 , we obtain that the first term is bounded by C 2 k!k Xr; k!k Yr; r+1=2 m e m ! L 2 : (2.16) 18 The second term in the definition ofT 2 above is treated using the decomposition (2.13). Note that in the regionfsgn(k m +j m ) sgn(k m ) =1g we have 0jk m jjj m j and 0jl m j 2jj m j, and hencee jkmj 1 +jj m je jjmj . Therefore, the second term in (2.15) is bounded by C X j+k+l=0 jj m j r+1=2 (1 +jj m je jjmj )j^ u j jjkj 1 j^ ! k jjl m j r+1=2 e jlmj j^ ! l j +Cj(@ m urH m r1=2 m e m !; r+1=2 m e m !)j Ck!k 2 H r r m e m ! L 2 +C(kruk L 1 + 2 k!k H r )k!k Yr; r+1=2 m e m ! L 2 : (2.17) Usinge jkmj 1 +jk m je jkmj , the mean value theorem, and the triangle inequality, we obtain that the third term on the right side of (2.15) is bounded by Ck!k 2 H r r m e m ! L 2 +C 2 k!k H r k!k Yr; r+1=2 m e m ! L 2 : (2.18) Collecting (2.16)–(2.18) and the estimate onT 1 obtained earlier, we have proven that the termj(ur!; 2r m e 2m !)j is bounded by the right of (2.5). The vorticity stretching term (!ru; 2r m e 2m !) is treated similarly. We do not use the cancelation property, but instead subtract (!r r m e m u; r m e m !)+( r m e m ! ru; r m e m !). By H¨ older’s inequality and Lemma 2.4.1 we have (!r r m e m u; r m e m !) + ( r m e m !ru; r m e m !) Ckruk L 1 k!k Xr; r m e m ! L 2 ; 19 for a positive constant C depending only on r. Thus in order to estimate the term (!ru; 2r m e 2m !) it is sufficient to bound (!ru; 2r m e 2m !) (!r r m e m u; r m e m !) ( r m e m !ru; r m e m !) =i(2) 3 X j+k+l=0 (jl m j r e jlmj jk m j r e jkmj jj m j r e jjmj )^ ! j k ^ u k jl m j r e jlmj ^ ! l ; (2.19) wherej;k;l2Z 3 . We rewrite the left side of (2.19) as i(2) 3 X j+k+l=0 (jl m j r jj m j r )(e jlmj e jkmj )^ ! j k ^ u k jl m j r e jlmj ^ ! l +i(2) 3 X j+k+l=0 (jl m j r jk m j r jj m j r )e jkmj ^ ! j k ^ u k jl m j r e jlmj ^ ! l +i(2) 3 X j+k+l=0 jj m j r (e jlmj e jjmj )^ ! j k ^ u k jl m j r e jlmj ^ ! l = e T 1 + e T 2 + e T 3 : The above terms are estimated in absolute value as follows. The mean value theorem ande x 1 +xe x , forx 0, imply (jl m j r jj m j r )(e jlmj e jkmj ) C(jk m j r jj m j +jk m jjj m j r )e jjmj e jkmj : (2.20) Combined withe x 1 +xe x , for allx 0, and the triangle inequality, (2.20) gives j e T 1 jCk!k 2 H r r m e m ! L 2 +C 2 (k!k H r +k!k Xr; )k!k Yr; r+1=2 m e m ! L 2 : 20 Similarly, by the mean value theorem we have j(jl m j r jk m j r )jj m j r jCjj m j(jj m j r1 +jk m j r1 ) +jj m j r : Sincee x e +x 2 e x , for allx 0, the above implies j e T 2 jCk!k 2 H r r m e m ! L 2 +C 2 k!k H r k!k Yr; r+1=2 m e m ! L 2 : The third term e T 3 is bounded similarly to the first one, but instead of (2.20) we use the estimateje jlmj e jjmj jCjk m je jjmj e jkmj and obtain j e T 3 jCk!k 2 H r r m e m ! L 2 +C 2 (k!k H r +k!k Xr; )k!k Yr; r+1=2 m e m ! L 2 : This concludes the proof of the lemma. 21 Chapter 3: The half-space 3.1 Introduction The Euler equations on a half space for the velocity vector field u(x;t) = (u 1 (x;t);u 2 (x;t);u 3 (x;t)) and the scalar pressure fieldp(x;t) are given by @ t u + (ur)u +rp = 0; (3.1) ru = 0; (3.2) where (x;t)2H (0;1) =fx2R 3 : x 3 > 0g (0;1). Since@H =fx2R 3 : x 3 = 0g, we have thatn = (0; 0;1) is the outward unit normal to@H, and hence the boundary condition becomes u 3 (;t) = 0; (3.3) on@H (0;1). We consider the initial value problem associated to (3.1)–(3.3) with a divergence free, Gevrey-classs (wheres 1), initial datum u(; 0) =u 0 ; (3.4) inH, which satisfies the compatibility condition at the boundary, i.e.,u 03 = 0 on@H. 22 We recall that the persistence of analyticity was proven by Bardos and Benachour [6] (see Section 1.3 for more details). We note that the interior analyticity in the case of the half-space, for short time (independent ofT ), was treated in [54]. The persistence of Gevrey-class regularity on domains with boundary remained however open. In this chapter (see also [39]) we prove that the results obtained in the periodic setting (cf. Chapter 2, and [38]) also hold if the domain has a genuine boundary. In the analytic category we improve the previously known lower bounds on the radius of real-analyticity by proving that the radius decays algebraically in the Sobolev norm k curlu(;t)k H r, and exponentially in R t 0 kru(;s)k L 1 ds, for allt<T . Additionally we prove the persistence of sub-analytic Gevrey-class regularity up to the boundary for the Euler equations on the half space. To the best of our knowledge this was only known for the periodic domain (cf. [38, 46]), but not for a domain with boundary. The methods of [2, 3, 6, 42, 54] rely essentially on the special structure of the complex holomorphic functions, and do not apply to the non-analytic Gevrey-class setting. The presence of the boundary creates several difficulties that do not arise in the periodic setting. In particular we cannot use Fourier-based methods, nor can we use the vorticity formulation of the equations. Instead we need to estimate the pressure, which satisfies (cf. [55]) the elliptic Neumann problem p =@ j u i @ i u j ; inH (0;1); (3.5) @p @n = (ur)un = 0; on@H (0;1); (3.6) sincen = (0; 0;1), where the summation convention on repeated indices was used in (3.5). In order to close our argument we need to show that the pressure has the same analyticity radius as the velocity, and so we cannot appeal to the inductive argument 23 of Lions and Magenes [47] or Morrey and Nirenberg [51]. Moreover, the nature of the elliptic/hyperbolic boundary value problem imposes certain restrictions on the weights of the Sobolev norms that comprise the analytic norm. The analytic norm we define (cf. Section 3.3) respects the symmetries of the problem and is adequate to account for the transfer of derivatives arising in the higher regularity estimates for the pressure. The chapter is organized as follows. In Section 3.2 we state our main results, Theo- rems 3.2.1 and 3.2.2. In Section 3.3 we prove the main theorem assuming two key esti- mates on the convection term and the pressure term, Lemma 3.3.1 and Lemma 3.3.2. Section 3.4 contains the proof of the commutator estimate Lemma 3.3.1, and lastly, the higher regularity estimates for the pressure and the proof of Lemma 3.3.2 are given in Section 3.5. 3.2 Main theorems The following statement is our main theorem addressing the analyticity of the solution onH. Theorem 3.2.2 below concerns the Gevrey-class persistence. 3.2.1 Theorem. Fix r > 9=2. Let u 0 2 H r (H) be divergence-free and uniformly real-analytic inH. Then the unique solutionu(;t)2 C(0;T ;H r (H)) of the initial value problem associated to the Euler equations (3.1)–(3.4) is real-analytic for all time t < T , whereT 2 (0;1] denotes the maximal time of existence of theH r -solution. Moreover, the uniform radius of space analyticity(t) ofu(t) satisfies (t) 1 C 0 (1 +t) exp C Z t 0 kru(s)k L 1ds ; (3.7) whereC > 0 is a constant that depends only onr, whileC 0 has additional dependence onu 0 as described in (3.17) below. 24 3.2.1 Remark. The lower bound (3.7) improves the rate of decay from Bardos and Benachour [6] on a bounded domain (which can be inferred to be proportional to exp R t 0 ku(s)k H rds), and it matches the rate of decay we obtained in [38] on the peri- odic domain. 3.2.2 Remark. The proof of Theorem 3.2.1 also works in the case of the half-plane (recall that in two dimensions T may be taken arbitrarily large, cf. [50, 56]) with the same lower bound on the radius of analyticity of the solution. Since in two dimensionskru(t)k L 1 grows at a rate ofC exp(Ct), for some positive constantC, the estimate (3.7) shows that the rate of decay of the analyticity radius is at least exp(C exp(Ct))=C, for someC > 0 (see also [7]). 3.2.3 Remark. It would be interesting if one could prove a similar lower bound to (3.7) but where the quantity R t 0 kru(s)k L 1ds is replaced by R t 0 k curlu(s)k L 1ds. In particular, such an estimate would imply in two dimensions that the radius of analyt- icity decays as a single exponential in time. Recall (cf. Section 1.2) that a smooth function v is uniformly of Gevrey-class s, withs 1, if there existM; > 0 such that j@ v(x)jM jj! s jj ; (3.8) for allx2H and all multi-indices2N 3 0 . We call the constant in (3.8) the radius of Gevrey-class regularity. The following theorem shows the persistence of the Gevrey- class regularity for the Euler equations in a half-space. 3.2.2 Theorem. Fixr> 9=2. Letu 0 be uniformly of Gevrey-classs onH, withs> 1. Then the uniqueH r -solutionu(;t) of the initial value problem (3.1)–(3.4) on [0;T ) 25 is of Gevrey-classs, for allt < T , and the radius(t) of Gevrey-class regularity of the solution satisfies the lower bound (3.7). 3.3 Proofs of the main theorems For a multi-index = ( 1 ; 2 ; 3 ) inN 3 0 , we denote 0 = ( 1 ; 2 ). Define the Sobolev and Lipshitz semi-normsjj m andjj m;1 by jvj m = X jj=m M k@ vk L 2; (3.9) and jvj m;1 = X jj=m M k@ vk L 1; where M = j 0 j! 0 ! = 1 + 2 1 : (3.10) The need for the binomial weightsM in (3.9) shall be evident in Section 3.5 where we study the higher regularity estimates associated with the Neumann problem (3.5)– (3.6). Fors 1 and > 0, define the space X =fv2C 1 (H) :kvk X <1g; 26 where kvk X = 1 X m=3 jvj m m3 (m 3)! s : Similarly letY =fv2C 1 (H) :kvk Y <1g, where kvk Y = 1 X m=4 jvj m (m 3) m4 (m 3)! s : 3.3.1 Remark. The above defined spacesX andY can be identified with the classical Gevrey-s classes as defined in [47]. On the full space or on the torus, the Gevrey-s classes can also be identified withD(() r=2 exp (() 1=2s )) (cf. [28, 38, 46]). We shall prove Theorems 3.2.1 and 3.2.2 simultaneously by looking at the evolu- tion equation in Gevrey-s classes with s 1. If u 0 is of Gevrey-class s in , with s 1, then there exists (0) > 0 such that u 0 2 X (0) , and moreover (0) can be chosen arbitrarily close to the uniform real-analyticity radius ofu 0 , respectively to the radius of Gevrey-class regularity. Let u(t) be the classical H r -solution of the initial value problem (3.1)–(3.4). With the notations of Section 3.2 we have an a priori estimate d dt ku(t)k X (t) = _ (t)ku(t)k Y (t) + 1 X m=3 d dt ju(t)j m (t) m3 (m 3)! s : (3.11) Fixm 3. In order to estimate (d=dt)ju(t)j m , for eachjj =m we apply@ on (3.1) and take theL 2 -inner product with@ u. We obtain 1 2 d dt k@ uk 2 L 2+<@ (uru);@ u> +<r@ p;@ u>= 0: (3.12) 27 On the second term on the left, we apply the Leibniz rule and recall that < u r@ u;@ u >= 0. For the third term on the left of (3.12) we note that since n = (0; 0;1) and un = 0 on @H, we have that @ un = 0 for all such that 3 = 0. Together withru = 0 inH this implies that<r@ p;@ u>= 0 whenever 3 = 0. Using the Cauchy-Schwarz inequality and summing overjj = m we then obtain d dt juj m X jj=m X ;6=0 M k@ ur@ uk L 2 + X jj=m; 3 6=0 M kr@ pk L 2: Combined with (3.11), the above estimate shows that d dt ku(t)k X (t) _ (t)ku(t)k Y (t) +C +P; (3.13) where the upper bound on the commutator term is given by C = 1 X m=3 X jj=m X ;6=0 M k@ ur@ uk L 2 m3 (m 3)! s ; and the upper bound on the pressure term is P = 1 X m=3 X jj=m; 3 6=0 M kr@ pk L 2 m3 (m 3)! s : In order to estimateC we use the following lemma, the proof of which is given in Section 3.4 below. 3.3.1 Lemma. There exists a sufficiently large constantC > 0 such that CC (C 1 +C 2 kuk Y ); 28 where C 1 =juj 1;1 juj 3 +juj 2;1 juj 2 +juj 2;1 juj 3 ; and C 2 =juj 1;1 + 2 juj 2;1 + 3 juj 3;1 + 3=2 kuk X : The following lemma shall be used to estimateP. The proof is given in Section 3.5 below. 3.3.2 Lemma. There exists a sufficiently large constantC > 0 such that PC (P 1 +P 2 kuk Y ); where P 1 =juj 1;1 juj 3 +juj 2;1 juj 2 +juj 2;1 juj 3 + 2 juj 3;1 juj 3 ; and P 2 =juj 1;1 + 2 juj 2;1 + 3 juj 3;1 + 3=2 kuk X : 29 Letr > 9=2 be fixed. The Sobolev embedding theorem, the two lemmas above, and (3.13) imply d dt ku(t)k X (t) _ (t)ku(t)k Y (t) +Cku(t)k 2 H r(1 +(t) 2 ) +Cku(t)k Y (t) (t)kru(t)k L 1 + ((t) 2 +(t) 3 )ku(t)k H r +(t) 3=2 ku(t)k X (t) : (3.14) If(t) decreases fast enough so that for all 0t<T we have _ (t) +C(t)kru(t)k L 1 +C((t) 2 +(t) 3 )ku(t)k H r +C(t) 3=2 ku(t)k X (t) 0; (3.15) then (3.14) implies that d dt ku(t)k X (t) Cku(t)k 2 H r(1 +(0) 2 ); and therefore ku(t)k X (t) ku 0 k X (0) +C (0) Z t 0 ku(s)k 2 H rds =M(t); for all 0t<T , whereC (0) = 1+(0) 2 . Since must be chosen to be a decreasing function, a sufficient condition for (3.15) to hold is that _ (t) +C(t)kru(t)k L 1 +C(t) 3=2 C 0 (0) ku(t)k H r +M(t) 0; (3.16) 30 whereC 0 (0) =(0) 1=2 +(0) 3=2 . For simplicity of the exposition we denote G(t) = exp C Z t 0 kru(s)k L 1ds ; where the constantC > 0 is taken sufficiently large so thatku(t)k 2 H rku 0 k 2 H rG(t). It then follows that (3.16) is satisfied if we let (t) =G(t) 1=2 (0) 1=2 +C Z t 0 C 0 (0) ku(s)k H r +M(s) G(s) 1 ds 1=2 : The lower bound (3.7) on the radius of analyticity stated in Theorem 3.2.1 is then obtained by noting that (0) 1=2 +C Z t 0 C 0 (0) ku(s)k H r +M(s) G(s) 1 ds (0) 1=2 +C Z t 0 C 0 (0) ku 0 k H r +ku 0 k X (0) +sC (0) ku 0 k 2 H r ds C 0 (1 +t) 2 ; (3.17) and therefore (t)G(t) 1=2 C 0 1 +t : The last inequality in (3.17) above gives the explicit dependence of C 0 on u 0 . This concludes the a priori estimates that are used to prove Theorem 3.2.1. The proof can be made formal by considering an approximating solution u (n) , n2 N, proving the above estimates foru (n) , and then taking the limit asn!1. We omit these details. 31 3.4 The commutator estimate Before we prove Lemma 3.3.1 we state and prove two useful lemmas about multi- indexes, that will be used throughout in Sections 3.4 and 3.5 below. 3.4.1 Lemma. We have M M 1 M 1 jj jj (3.18) for all;2N 3 0 with. Proof. Using (3.10) we have that 0 0 M M 1 M 1 = j 0 j j 0 j ; and hence the left side of (3.18) is bounded by j 0 j j 0 j 3 3 : The lemma then follows from n i m j n +m i +j ; for anyn;m 0 such thatn i andm j, which in turn we obtain by computing the coefficient in front of x i+j in the binomial expansions of (1 +x) n (1 +x) m and (1 +x) m+n . The second lemma allows us to re-write certain double sums involving multi- indices. 32 3.4.2 Lemma. Letfx g 2N 3 0 andfy g 2N 3 0 be real numbers. Then we have X jj=m X jj=j; x y = 0 @ X jj=j x 1 A 0 @ X j j=mj y 1 A : (3.19) The proof of the above lemma is omitted: it consists of re-labeling of the terms on the left side of (3.19). Now we proceed by proving the commutator estimate. Proof of Lemma 3.3.1. We have C = 1 X m=3 m X j=1 C m;j ; where we denoted C m;j = m3 (m 3)! s X jj=m X jj=j; M k@ ur@ uk L 2: (3.20) We now split the right side of the above equality into seven terms according to the values ofm andj, and prove the following estimates. For lowj, we claim 1 X m=3 C m;1 Cjuj 1;1 juj 3 +Cjuj 1;1 kuk Y ; (3.21) 1 X m=3 C m;2 Cjuj 2;1 juj 2 +Cjuj 2;1 juj 3 +C 2 juj 2;1 kuk Y ; (3.22) for intermediatej, we have 1 X m=6 [m=2] X j=3 C m;j C 3=2 kuk X kuk Y ; (3.23) 1 X m=7 m3 X j=[m=2]+1 C m;j C 3=2 kuk X kuk Y ; (3.24) 33 and for highj, 1 X m=5 C m;m2 C 3 juj 3;1 kuk Y ; (3.25) 1 X m=4 C m;m1 Cjuj 2;1 juj 3 +C 2 juj 2;1 kuk Y ; (3.26) 1 X m=3 C m;m Cjuj 1;1 juj 3 +Cjuj 1;1 kuk Y : (3.27) Due to symmetry we shall only prove (3.21)–(3.23) and indicate the necessary modi- fications for (3.24)–(3.27). Proof of (3.21). The H¨ older inequality, (3.20), and Lemma 3.4.1 imply that 1 X m=3 C m;1 = X jj=3 X jj=1; M k@ uk L 1 M k@ ruk L 2 M M 1 M 1 + 1 X m=4 X jj=m X jj=1; M k@ uk L 1 M k@ ruk L 2 (m 3) m4 (m 3)! s M M 1 M 1 1 m 3 C X jj=3 X jj=1; M k@ uk L 1 M k@ ruk L 2 +C 1 X m=4 X jj=m X jj=1; M k@ uk L 1 M k@ ruk L 2 (m 3) m4 (m 3)! s m m 3 : (3.28) The first sum on the far right side of (3.28) can be estimated by Cjuj 1;1 jruj 2 Cjuj 1;1 juj 3 : 34 Sincem 4, Lemma 3.4.2 implies that the second term on the far right side of (3.28) is bounded by Cjuj 1;1 1 X m=4 jruj m1 (m 3) m4 (m 3)! s Cjuj 1;1 kuk Y ; concluding the proof of (3.21). Proof of (3.22). As in the proof of (3.21) above, we have 1 X m=3 C m;2 C X jj=3;4 X jj=2; m3 M k@ uk L 1 M k@ ruk L 2 M M 1 M 1 +C 1 X m=5 X jj=m X jj=2; M k@ uk L 1 M k@ ruk L 2 (m 4) m5 (m 4)! s M M 1 M 1 1 (m 4)(m 3) s 2 : (3.29) Using Lemma 3.4.2, the first sum on the right of (3.29) can be estimated from above by Cjuj 2;1 jruj 1 +Cjuj 2;1 jruj 2 Cjuj 2;1 juj 2 +Cjuj 2;1 juj 3 : On the other hand, sinces 1,jj = 2, andjj = m 5, we have by Lemma 3.4.1 that M M 1 M 1 1 (m 4)(m 3) s m 2 1 (m 4)(m 3) C: 35 By Lemma 3.4.2, the second sum on the right of (3.29) is thus bounded by C 2 1 X m=5 juj 2;1 jruj m2 (m 4) m5 (m 4)! s C 2 juj 2;1 kuk Y : This proves the desired estimate. Proof of (3.23). We first observe that the H¨ older inequality and the Sobolev inequality give k@ ur@ uk L 2Ck@ uk 1=4 L 2 k@ uk 3=4 L 2 kr@ uk L 2: Therefore we can bound the right hand side of (3.23) as follows 1 X m=6 [m=2] X j=3 C m;j 1 X m=6 [m=2] X j=3 X jj=m X jj=j; M k@ uk L 2 j3 (j 3)! s 1=4 3=2 A ;;s M k@ uk L 2 j1 (j 1)! s 3=4 M k@ ruk L 2 (mj 2) mj3 (mj 2)! s ; where A ;;s =M M 1 M 1 (j 3)! s=4 (j 1)! 3s=4 (mj 2)! s (m 3)! s (mj 2) : By Lemma 3.4.1, we have that form 6 and 3j [m=2] A ;;s C m j m 3 j 1 s 1 (mj 2)(j 1) s=4 (j 2) s=4 C m 3 j 1 s+1 1 j 1+s=2 : 36 Since s 1 the above chain of inequalities gives thatA ;;s C. Together with Lemma 3.4.2 and the discrete H¨ older inequality this shows that 1 X m=6 [m=2] X j=3 C m;j C 3=2 1 X m=6 [m=2] X j=3 juj j j3 (j 3)! s 1=4 juj j j1 (j 1)! s 3=4 jruj mj (mj 2) mj3 (mj 2)! s : The discrete Young and H¨ older inequalities then give 1 X m=6 [m=2] X j=3 C m;j C 3=2 kuk X kuk Y ; concluding the proof of (3.23). To prove (3.24)–(3.27) we proceed as in the proofs of (3.21)–(3.23) above, with the roles of j and m j reversed. Instead of estimatingk@ ur@ uk L 2 with k@ uk L 1k@ ruk L 2 we instead bound k@ ur@ uk L 2k@ uk L 2k@ ruk L 1: We omit further details. This concludes the proof of Lemma 3.3.1. 3.5 The pressure estimate In the proof of the Lemma 3.3.2 we need to use the following higher regularity estimate on the solution of the Neumann problem associated to the Poisson equation for the half-space. 37 3.5.1 Lemma. Assume thatp is a smooth solution of the Neumann problem p =v in ; @p @n = 0 on@ ; withv2C 1 ( ). Then there is a universal constantC > 0 such that k@ 3 @ pk L 2C X s;t2N 0 ;jj=m1 0 0 =(2s;2t) s +t s k@ vk L 2; (3.30) for anym 1 and any multiindex2N 3 0 withjj =m and 3 6= 0. Additionally, if 3 2 then k@ 1 @ pk L 2C X s;t2N 0 ;jj=m1 0 0 =(2s+1;2t) s +t s k@ vk L 2; (3.31) k@ 2 @ pk L 2C X s;t2N 0 ;jj=m1 0 0 =(2s;2t+1) s +t s k@ vk L 2; (3.32) whereC > 0 is a universal constant. We emphasize that the constantC in the above lemma is independent of andm. In (3.30) we have are summing over the set f2N 3 0 :jj =m 1;9s;t2N 0 such that 0 0 = (2s; 2t)g and similar conventions are used in (3.31), (3.32), and throughout this section. 38 Proof. In order to avoid repetition, we only prove (3.30) and indicate the necessary changes for (3.31) and (3.32). Let 0 =@ 11 +@ 22 be the tangential Laplacian. Using induction onk2N 0 we obtain the identity @ 2k+2 3 p = ( 0 ) k+1 p k X j=0 @ 2j 3 ( 0 ) kj v; and upon applying@ 3 to the above equation @ 2k+3 3 p =@ 3 ( 0 ) k+1 p k X j=0 @ 2j+1 3 ( 0 ) kj v: Therefore givenjj =m, with 3 = 2k + 1 1, we have @ 3 @ p =@ 2k+2 3 @ 0 p = ( 0 ) k+1 @ 0 p + k X j=0 (1) kj+1 @ 2j 3 (@ 11 +@ 22 ) kj @ 0 v; (3.33) and if 3 = 2k + 2 2, we have @ 3 @ p =@ 2k+3 3 @ 0 p =@ 3 ( 0 ) k+1 @ 0 p + k X j=0 (1) kj+1 @ 2j+1 3 (@ 11 +@ 22 ) kj @ 0 v: (3.34) Sincen = (0; 0;1), the functiong = ( 0 ) k @ 0 p satisfies the Neumann problem g = ( 0 ) k @ 0 v in ; @g @n = 0 on@ : 39 Using the classicalH 2 -regularity argument for the Neumann problem we then have k 0 gk L 2Ck( 0 ) k @ 0 vk L 2; and k@ 3 0 gk L 2Ck@ 3 ( 0 ) k @ 0 vk L 2; for a positive universal constantC. Combining the above estimates with (3.33), (3.34), and the identity (@ 11 +@ 22 ) m w = m X s=0 m s @ 2s 1 @ 2m2s 2 w; we obtain k@ 3 @ pk L 2C k X j=0 k@ 2j 3 (@ 11 +@ 22 ) kj @ 0 vk L 2 C k X j=0 kj X s=0 kj s k@ 2s+ 1 1 @ 2k2j2s+ 2 2 @ 2j 3 vk L 2 (3.35) if 3 = 2k + 1 1, and k@ 3 @ pk L 2C k X j=0 k@ 2j+1 3 (@ 11 +@ 22 ) kj @ 0 vk L 2 C k X j=0 kj X s=0 kj s k@ 2s+ 1 1 @ 2k2j2s+ 2 2 @ 2j+1 3 vk L 2 (3.36) if 3 = 2k + 2 2. To simplify (3.35) above, let t = kjs 0 and = (2s + 1 ; 2k 2j 2s + 2 ; 2j) = (2s + 1 ; 2t + 2 ; 3 1 2s 2t)2N 3 0 . Since 40 jj =m and 3 = 2k + 1, we havejj =m 1, and by re-indexing the sums, (3.35) can be re-written as k@ 3 @ pk L 2C X s;t2N 0 ;jj=m1 0 0 =(2s;2t) s +t s k@ vk L 2; The above estimate also holds for 3 = 2k+2 with the substitution = (2s+ 1 ; 2k 2j 2s + 2 ; 2j + 1), thereby simplifying the upper bound (3.36), and concluding the proof of (3.30). To prove (3.31) we proceed as above and obtain k@ 1 @ pk L 2 =k@ 1 +1 1 @ 2 2 @ 2k+2 3 pk L 2 C k X j=0 kj X s=0 kj s k@ 1 +2s+1 1 @ 2 +2k2j2s 2 @ 2j 3 vk L 2 (3.37) if 3 = 2k + 2 2, and k@ 1 @ pk L 2 =k@ 1 +1 1 @ 2 2 @ 2k+3 3 pk L 2 C k X j=0 kj X s=0 kj s k@ 1 +2s+1 1 @ 2 +2k2j2s 2 @ 2j+1 3 vk L 2 (3.38) if 3 = 2k + 3 3. In (3.37) we let t = kjs 0 and = ( 1 + 2s + 1; 2 + 2t; 2j) = ( 1 + 2s + 1; 2 + 2t; 3 2 2s 2t), since 3 = 2k + 2 andjj = m. Similarly in (3.38) we let = ( 1 + 2s + 1; 2 + 2t; 2j + 1) = ( 1 + 2s + 1; 2 + 2t; 3 2 2s 2t), since 3 = 2k + 3 andjj = m. The above substitutions and re-indexing prove (3.31). Upon permuting the first and second coordinates, this also proves (3.32). 41 3.5.1 Remark. We note that Lemma 3.5.1 does not give an estimate fork@ 1 @ pk L 2 andk@ 2 @ pk L 2 if 3 = 1. In this case we note that the functiong =@ 0 p satisfies the Neumann problem g =@ 0 v in ; @g @n = 0 on@ : The classicalH 2 -regularity argument then gives k@ 1 @ pk L 2 =k@ 1 @ 3 @ 0 pk L 2Ck@ 0 vk L 2; and k@ 2 @ pk L 2 =k@ 2 @ 3 @ 0 pk L 2Ck@ 0 vk L 2; for a positive universal constantC > 0. We note that Lemma 3.5.1 is different from the classical higher regularity estimates (cf. [30, 47, 55]) for the Neumann problem in the fact that the constantC in (3.30)– (3.32) does not increase withm. The dependence onm is encoded in the sums with binomial weights on the right side of (3.30)–(3.32). The following lemma shows that only a factor of m is lost in the above higher regularity estimates if eachk@ vk L 2 term is paired with a proper binomial weight. This explains the definition of the homogeneous Sobolev normsjj m in (3.9). 42 3.5.2 Lemma. There exists a positive universal constantC such that [ 1 =2] X s=0 [ 2 =2] X t=0 1 + 2 2s 2t 1 2s s +t s Cm 1 + 2 1 (3.39) for anym 3 and any multi-index = ( 1 ; 2 ;m 1 1 2 )2N 3 0 . Additionally, if 1 1 we have [( 1 1)=2] X s=0 [ 2 =2] X t=0 1 + 2 2s 1 2t 1 2s 1 s +t s Cm 1 + 2 1 ; (3.40) while if 2 1 we have [ 1 =2] X s=0 [( 2 1)=2] X t=0 1 + 2 2s 2t 1 1 2s s +t s Cm 1 + 2 1 ; (3.41) whereC is a universal constant. We note that in particular the constantC is independent ofm and. Proof. Due to symmetry we only give the proof of (3.39). Estimates (3.40) and (3.41) are proven mutatis-mutandi. First we recall that given; 2N 3 0 , with , we have jj j j : Using the above inequality we get [ 1 =2] X s=0 [ 2 =2] X t=0 1 + 2 2s 2t 1 2s s +t s 1 + 2 1 1 [ 1 =2] X s=0 [ 2 =2] X t=0 1 + 2 st 1 s 1 + 2 1 1 [ 1 =2] X s=0 [ 2 =2] X t=0 s +t s 1 : 43 The lemma is then proven if we find a constantC such that [ 1 =2] X s=0 [ 2 =2] X t=0 s +t s 1 C( 1 + 2 ): Without loss of generality we may assume that 1 ; 2 4. We split the above sum into [ 1 =2] X s=0 [ 2 =2] X t=0 s +t s 1 [ 2 =2] X t=0 t 0 1 + t + 1 1 1 + [ 1 =2] X s=0 s s 1 + s + 1 s 1 + [ 1 =2] X s=2 [ 2 =2] X t=2 s +t s 1 =T 1 +T 2 +T 3 : (3.42) It is clear that T 1 +T 2 C( 1 + 2 ): (3.43) We estimateT 3 by appealing to the Stirling estimate (cf. [53, p. 200]) e 7=8 p n n e n <n!<e p n n e n : This implies s!t! (s +t)! e 9=8 r st s +t 1 (1 +s=t) t 1 (1 +t=s) s : 44 Thus we obtain T 3 C [ 1 =2] X s=2 [ 2 =2] X t=2 p t 1 (1 +t=s) s : (3.44) Sinces 2, the Binomial Theorem implies 1 + t s s 1 + s 2 t s 2 ; and by (3.44) we have T 3 C [ 1 =2] X s=2 [ 2 =2] X t=2 p t 1 t 2 C 0 @ [ 1 =2] X s=2 1 1 A 1 X t=2 1 t 3=2 ! C 1 : Since 1 + 2 m 1, the above inequality, (3.42), and (3.43) complete the proof of the lemma. Proof of Lemma 3.3.2. First, note that sincep satisfies the elliptic Neumann problem (3.5)–(3.6) we may use Lemma 3.5.1 to estimate higher derivatives of@ 3 p as X jj=m; 3 6=0 M k@ 3 @ pk L 2C X jj=m; 3 6=0 X s;t2N 0 ;jj=m1 0 0 =(2s;2t) M s +t s k@ (@ i u k @ k u i )k L 2: By re-indexing the terms in the parenthesis, the right side of the above inequality may be re-written as X jj=m1 [ 1 =2] X s=0 [ 2 =2] X t=0 1 + 2 2s 2t 1 2s s +t s k@ (@ i u k @ k u i )k L 2: 45 Using the estimate (3.39) of Lemma 3.5.2 we bound the above expression by Cm X jj=m1 M k@ (@ i u k @ k u i )k L 2 and therefore 1 X m=3 0 @ X jj=m; 3 6=0 M k@ 3 @ pk L 2 1 A m3 (m 3)! s C 1 X m=3 X jj=m1 M k@ (@ i u k @ k u i )k L 2 m m3 (m 3)! s : (3.45) On the other hand, higher derivatives of@ 1 p are estimated using the decomposition X jj=m; 3 6=0 M k@ 1 @ pk L 2 = X jj=m; 3 =1 M k@ 1 @ pk L 2 + X jj=m; 3 2 M k@ 1 @ pk L 2: (3.46) By Remark 3.5.1, the first term on the right of (3.46) is bounded by C X jj=m; 3 =1 M k@ 0 (@ i u k @ k u i )k L 2 =C X jj=m1; 3 =0 M k@ (@ i u k @ k u i )k L 2: (3.47) Using estimate (3.31), the second term on the right side of (3.46) is estimated by C X jj=m; 3 2 0 B B @ X s;t2N 0 ;jj=m1 0 0 =(2s+1;2t) M s +t s k@ (@ i u k @ k u i )k L 2 1 C C A : 46 By re-indexing the above expression equals C X jj=m1; 1 1 [(1)=2] X s=0 [ 2 =2] X t=0 1 1 + 2 2s 2t 1 2s 1 s +t s k@ (@ i u k @ k u i )k L 2; and using (3.40) it is bounded from above by Cm X jj=m1; 1 1 M k@ (@ i u k @ k u i )k L 2: (3.48) Therefore, by (3.46), (3.47), and (3.48), we have 1 X m=3 0 @ X jj=m; 3 6=0 M k@ 1 @ pk L 2 1 A m3 (m 3)! s C 1 X m=3 X jj=m1 M k@ (@ i u k @ k u i )k L 2 m m3 (m 3)! s : (3.49) By symmetry, we also get 1 X m=3 0 @ X jj=m; 3 6=0 M k@ 2 @ pk L 2 1 A m3 (m 3)! s C 1 X m=3 X jj=m1 M k@ (@ i u k @ k u i )k L 2 m m3 (m 3)! s : (3.50) Combining (3.45), (3.49), (3.50), and the Leibniz rule we obtain PC 1 X m=3 X jj=m1 M k@ (@ i u k @ k u i )k L 2 m m3 (m 3)! s C 1 X m=3 m1 X j=0 P m;j ; (3.51) 47 where P m;j = m m3 (m 3)! s X jj=m1 X j j=j; M k@ @ i u k @ @ k u i k L 2: We split the right side of (3.51) into seven terms according to the values ofm andj. For lowj, we claim 1 X m=3 P m;0 Cjuj 1;1 juj 3 +Cjuj 1;1 kuk Y (3.52) 1 X m=3 P m;1 Cjuj 2;1 juj 2 +Cjuj 2;1 juj 3 +C 2 juj 2;1 kuk Y (3.53) 1 X m=5 P m;2 C 2 juj 3;1 juj 3 +C 3 juj 3;1 kuk Y (3.54) for intermediatej, we have 1 X m=8 [m=2]1 X j=3 P m;j C 3=2 kuk X kuk Y (3.55) 1 X m=6 m3 X j=[m=2] P m;j C 3=2 kuk X kuk Y (3.56) and for highj, we claim 1 X m=4 P m;m2 Cjuj 2;1 juj 3 +C 2 juj 2;1 kuk Y (3.57) 1 X m=3 P m;m1 Cjuj 1;1 juj 3 +Cjuj 1;1 kuk Y : (3.58) The above estimates are proven similarly to (3.21)–(3.27) in the proof of Lemma 3.3.1. Due to symmetry we have presented there the proofs of the estimates wherejmj. 48 For completeness of the exposition we provide the proofs of (3.56)–(3.58), where we havemj <j. Proof of (3.56). We proceed as in the proof of (3.23) in Section 3.4. First, the H¨ older and Sobolev inequalities imply that k@ @ i u k @ @ k u i k L 2Ck@ @ i u k k L 2k@ @ k u i k 1=4 L 2 k@ @ k u i k 3=4 L 2 : Therefore, 1 X m=6 m3 X j=[m=2] P m;j C 1 X m=6 m3 X j=[m=2] X jj=m1 X j j=j; M k@ @ i u k k L 2 (j 2) j3 (j 2)! s M k@ @ k u i k L 2 mj3 (mj 3)! s 1=4 M k@ @ k u i k L 2 mj1 (mj 1)! s 3=4 3=2 B ; ;s ; where B ; ;s =M M 1 M 1 m(j 2)! s (mj 3)! s=4 (mj 1)! 3s=4 (j 2)(m 3)! s : By Lemma 3.4.1 we have that form 6 and [m=2]jm 3 B ; ;s C m 1 j m 3 j 2 s m (j 2)(mj 1) s=4 (mj 2) s=4 C m 3 j 2 1s (mj) s=2 ; 49 since m1 j C m3 j2 , when j m=2. Therefore, B ; ;s C; hence, by Lemma 3.4.2 and the discrete H¨ older inequality, we have 1 X m=6 m3 X j=[m=2] P m;j C 3=2 1 X m=6 m3 X j=[m=2] j@ k u i j mj1 mj3 (mj 3)! s 1=4 j@ k u i j mj1 mj1 (mj 1)! s 3=4 j@ i u k j j (j 2) j3 (j 2)! s : The discrete Young and H¨ older inequalities then give 1 X m=6 m3 X j=[m=2] P m;j C 3=2 kuk X kuk Y ; concluding the proof of (3.56). Proof of (3.57). As above we use the H¨ older inequality and obtain 1 X m=4 P m;m2 1 X m=4 X jj=m1 X j j=m2; M k@ @ i u k k L 2k@ @ k u i k L 1 m m3 (m 3)! s C X jj=3 X j j=2; k@ @ i u k k L 2k@ @ k u i k L 1 +C 2 1 X m=5 X jj=m1 X j j=m2; M k@ @ i u k k L 2 m m5 (m 4)! s M k@ @ k u i k L 1 M M 1 M 1 1 (m 3) s : Using Lemma 3.4.1, Lemma 3.4.2, ands 1, this shows that the far right side of the above chain of inequalities is bounded by Cj@ i u k j 2 j@ k u i j 1;1 +C 2 j@ k u i j 1;1 1 X m=5 j@ i u k j m2 m m5 (m 4)! s Cjuj 2;1 juj 3 +C 2 juj 2;1 kuk Y ; 50 thereby proving (3.57). Proof of (3.58). By the H¨ older inequality we have 1 X m=3 P m;m1 1 X m=3 X jj=m1 M k@ @ i u k k L 2k@ k u i k L 1 m m3 (m 3)! s Cj@ i u k j 2 k@ k u i k L 1 +Ck@ k u i k L 1 1 X m=4 j@ i u k j m1 m m4 (m 3)! s Cjuj 1;1 juj 3 +Cjuj 1;1 kuk Y ; which gives the desired estimate. By symmetry, we may similarly prove (3.52)–(3.55), but in these cases we apply the H¨ older inequality as k@ @ i u k @ @ k u i k L 2k@ @ i u k k L 1k@ @ k u i k L 2; that is we reverse the roles ofj andmj. We omit further details. This concludes the proof of Lemma 3.3.2. 51 Chapter 4: The bounded domain 4.1 Introduction We recall the Euler equations @ t u + (ur)u +rp = 0; inD (0;1); (4.1) ru = 0; inD (0;1); (4.2) un = 0; on@D (0;1); (4.3) u(; 0) =u 0 ; inD; (E.4) whereD is an open bounded Gevrey-classs domain inR 3 , andn is the outward unit normal to@D. The following is our main theorem. 4.1.1 Theorem. Letu 0 be divergence-free and of Gevrey-classs onD, a Gevrey-class s, open bounded domain inR 3 , wheres 1, and letr 5. Then the unique solution u(;t)2C([0;T );H r (D)) to the initial value problem (4.1)–(E.4) is of Gevrey-classs 52 for allt<T , whereT 2 (0;1] is the maximal time of existence inH r (D). Moreover, the radius(t) of Gevrey-class regularity of the solutionu(;t) satisfies (t)C 0 exp C Z t 0 ku(s)k W 1;1 (D) ds 2 ! exp C 0 tCt 2 ku 0 k 2 H r (D) ; (4.4) for allt < T , whereC is a sufficiently large constant depending only on the domain D, 0 is the radius of Gevrey-class regularity ofu 0 , andC 0 has additional dependence on the Gevrey-class norm ofu 0 . 4.1.1 Remark. In the proof of Theorem 4.1.1 we also address the local (in space) propagation of Gevrey-class regularity of the solution (cf. Theorem 4.3.1 below), in the interior of the smooth domain, or in the neighborhood of a point where@D is locally of Gevrey-classs. This extends the results of [2, 42] to the non-analytic Gevrey-classes. This chapter is organized as follows. In Section 4.2 we introduce the notation used to define the Lagrangian Gevrey-class norms. Section 4.3 consists of a priori estimates needed to prove the short time propagation of local analyticity (cf. Theorem 4.3.1). Lemmas 4.3.2 and 4.3.3 are proven in Sections 4.4 and 4.5 respectively. Lastly, in Section 4.6 we show how the local in space and time results may be patched together to obtain the global persistence of Gevrey-class regularity (cf. Theorem 4.6.1). 53 4.2 Notation and preliminary remarks The existence of a unique H r solution, where r > 5=2, on a maximal time interval [0;T ), whereT 2 (0;1], implies the existence and uniqueness of the particle trajec- tories (cf. [18, 50]), that is solutions to d dt X(t) =u(X(t);t) (4.5) X(0) =a; (4.6) where a2 D. For simplicity we denote by t (a) the solution of (4.5)–(4.6). It is well known that for all t < T the maps t :D 7!D, and t j @D : @D 7! @D are diffeomorphisms. 4.2.1 Local change of coordinates Fix x 0 2 @D. In a sufficiently small neighborhood of x 0 , the boundary @D is the graph of a Gevrey-class s function , i.e., for 0 < r 0 1 we haveD r 0 ;x 0 =D\ B r 0 (x 0 ) = fx 2 B r (x 0 ): x 3 > (x 1 ;x 2 )g. Moreover, since the Euler equations are invariant under rigid body rotations ofR 3 , modulo composition with a rigid body rotation aboutx 0 , we may assume thatk@ 1 k L 1 ( D 0 r 0 ;x 0 ) +k@ 2 k L 1 ( D 0 r 0 ;x 0 ) " 1, for r 0 sufficiently small, where " is a fixed, sufficiently small universal constant, to be chosen later. Here we have denotedD 0 r 0 ;x 0 =fx 0 : x2D r 0 ;x 0 g, where we write x 0 = (x 1 ;x 2 ) forx = (x 1 ;x 2 ;x 3 ). Define a boundary straightening map :R 3 !R 3 by (x 1 ;x 2 ;x 3 ) = (x 1 ;x 2 ;x 3 (x 1 ;x 2 )) = (y 1 ;y 2 ;y 3 ): (4.7) 54 Note that det(@=@x) = 1. By the construction of we have e D r 0 ;x 0 = (D r 0 ;x 0 ) = fy2(B r 0 (x 0 )): y 3 > 0g. Let =D\B r 0 =2 (x 0 )D r 0 ;x 0 be a neighborhood ofx 0 . Also let e =( ) and e t =( t ). There exists T 1 = T 1 (r 0 ;u) such that for all 0 = T 0 t T 1 we have t = t ( )D r 0 ;x 0 . The value of T 1 may be estimated from below by using the representation formula for solutions of (4.5)–(4.6). We have j t (a)aj Z t 0 ju( s (a);s)jds Z t 0 ku(;s)k L 1 (Dr 0 ;x 0 ) dsK(t); (4.8) where we set K(t) = Z t 0 ku(s)k W 1;1 (D) ds: (4.9) Therefore, it is sufficient to choseT 1 such thatK(T 1 ) dist( ;@B r 0 (x 0 )) =r 0 =2. On the closure of e D r 0 ;x 0 we let% be the Euclidean distance to the curved part of the boundary of e , that is,%(y) = 0 ify2 e c and%(y) = dist(y;@ e nfy 3 = 0g) ify2 e . As in [2, 1, 42], for all 0 < 0 , where 0 > 0 is sufficiently small, we define the set e =fy2 e : %(y)>g: (4.10) By the triangle inequality and the definition of% it follows thatjy (1) y (2) j r for ally (1) 2 e +r andy (2) 2 c . Also let = 1 ( e ), t; = t ( ) and e t; = ( t; ). Here 0 = 0 ( ) 1 is chosen small enough such that for all2 [0; 0 ), the set is a Gevrey-classs domain, i.e., it lies on one side of a Gevrey-class surface. 55 Ω Ω δ D D D r0,x0 D r0,x0 Ω t Ω δ,t ~ D r0,x0 ~ ~ Ω Ω δ ~ ~ ~ D r0,x0 Ωt Ω δ,t θ θ t θ t θ -1 Figure 4.1: The local coordinate charts and the flow induced by t on the straightened domain. Ify (1) 2 e +r;t andy (2) 2 e ;t , where;r +2 (0; 0 ), it follows by the mean value theorem that rj t 1 (y (1) ) t 1 (y (2) )jCjy (1) y (2) jkr t k L 1 (D) Cjy (1) y (2) j(1 +K 2 (t)); (4.11) whereC is a constant depending on. In (4.11) we have used thatr t is the inverse matrix ofr t (whose determinant is 1 since divu = 0), and the fact that the 2 2 minors of this matrix are bounded by 1 +K 2 . Therefore, by (4.11), we havejy (1) 56 y (2) j r=(C +CK 2 (t)). Hence there exists a smooth cut-off function such that 1 on e +r;t and 0 on e c ;t , with jrj C +CK 2 (t) r ; (4.12) for some positive constant C = C(D). We denotee u(y;t) = u(x;t) and similarly e p(y;t) =p(x;t). 4.2.2 Gevrey-class norms TheG s -norms used in this chapter are defined as follows. For a Gevrey-classs functione v(y;t) denote [e v(t)] m = X jj=m 3 sup 0< 0 m3 k@ e v(;t)k L 2 ( e t; ) ; (4.13) for allm 3. In this paper we work with the Lagrangian Gevrey-classs norm defined by ke v(t)k X (t) = 1 X m=3 [e v(t)] m (t) m3 (m 3)! s ; (4.14) wheres 1, and > 0. We also let ke v(t)k Y (t) = 1 X m=4 [e v(t)] m m(t) m4 (m 3)! s : (4.15) 4.2.1 Remark. Ifke u(y)k X <1, it follows from the Sobolev inequality thate u2G s and thate u(y) has Gevrey-class regularity radius at least. As opposed to the analytic case, ifs > 1, the map 1 : y7! x possibly shrinks the radius by a constant factor 57 0 < a 1, wherea = a ( ). This fact may be proven using the multi-dimensional generalization of the Fa´ a di Bruno formula (cf. [21, 35]). Thus, ife u(y) has Gevrey- class radius, thenu(x) hasG s -radius at leasta . Throughout this chapter we additionally use the following notation. When it is clear from the context that we are working with a function on the flattened domain, we simply write v instead ofe v. In the present paper we set n! = 1 whenever n 0. Also we use the notationkD k vk L p = P jj=k k@ vk L p and similarlykD 0k vk L p = P jj=k; 3 =0 k@ vk L p. Lastly, C denotes a sufficiently large positive constant which may depend on the domain. 4.3 Short time local Gevrey-class a priori estimates The proof of Theorem 4.1.1 consists of a priori estimates. These estimates can be made rigorous by noting thatu(;t)2 C 1 ( D) for allt < T (cf. [27]), and by performing all below estimates on truncated sums P q m=3 [e v] m m3 =(m 3)! s . For q 5 these estimates close, since the energy estimates for the Euler equations close in Sobolev spaces (cf. [55]), and are independent ofq, so we may letq!1. Letd + f(t)=dt = lim sup h!0+ (f(t +h)f(t))=h denote the right derivative of a functionf(t), which agrees with the usual derivative if the latter exists. Using the definitions (4.13)–(4.14) we obtain d + dt ke u(t)k X (t) _ (t)ke u(t)k Y (t) + 1 X m=3 0 @ X jj=m 3 d + dt sup 0< 0 m3 k@ e u(t)k L 2 ( e ;t ) 1 A (t) m3 (m 3)! s : (4.16) 58 In order to switch the d + =dt and the sup (cf. Lemma 4.8.3) we need upper bounds for (d=dt)k@ e u(t)k e ;t for alljj 3. The following lemma is a Lagrangian energy estimate in the straightened domain and provides the desired upper bound. 4.3.1 Lemma. For all2N 3 0 ;t> 0, and 0< 0 , we have d dt k@ e u(;t)k L 2 ( e ;t ) k[@ ;e u j @ j k @ k ]e u(;t)k L 2 ( e ;t ) +k@ (@ j k @ k e p(;t))k L 2 ( e ;t ) ; (4.17) where the bracket [;] represents a commutator, and we denote the right side of (4.17) byM (t). Proof. The standard Lagrangian energy estimate (cf. [2, Lemma 2.3] and [42, Section 2.b]) shows that a smooth solutionv to@ t v + (ur)v =g satisfies d dt kv(t;)k 2 L 2 ( ;t ) = d dt Z jv(t; t (x))j 2 dx = 2 Z ;t g(t;x)v(t;x)dx (4.18) Here we used the fact that divu = 0 implies det(@ t (x)=@x) = 1. Let y = (x) and denotee v(y) = v(x). Similarly definee u(y) = u(x) ande g(y) = g(x). Thene v solves the equation @ t e v +e u j @ j k @ k e v = e g, and since det(@=@x) = 1, we have ke v(t;)k L 2 ( e ;t ) =kv(t;)k L 2 ( ;t ) . The lemma follows from the above remarks with e v =@ e u ande g = [@ ;e u j @ j k @ k ]e u +@ (@ j k @ k e p), and the H¨ older inequality. Using the bound (4.17), from Lemma 4.8.3 we obtain d + dt sup 0< 0 m3 k@ e u(t)k e ;t sup 0< 0 m3 M (t): 59 Therefore, we may estimate the sum on the right of (4.16) as d + dt ke u(t)k X (t) _ (t)ke u(t)k Y (t) + 1 X m=3 0 @ X jj=m 3 sup 0< 0 m3 M (t) 1 A (t) m3 (m 3)! s _ (t)ke u(t)k Y (t) +C +P; (4.19) where C = 1 X m=3 0 @ X jj=m 3 sup 0< 0 m3 k[@ ;e u j @ j k @ k ]e u(;t)k L 2 ( e ;t ) 1 A (t) m3 (m 3)! s ; (4.20) and P = 1 X m=3 0 @ X jj=m 3 sup 0< 0 m3 k@ (@ j k @ k e p(;t))k L 2 ( e ;t ) 1 A (t) m3 (m 3)! s : (4.21) The estimates forC andP are given in the following two lemmas. 4.3.2 Lemma. If < , where is the Gevrey-class regularity radius of the bound- ary, then the following estimate holds CC(1 + 2 ) ke uk 2 W 2;1 ( e t) +ke uk 2 H 5 ( e t) +C kDe uk L 1 ( e t) + ( 2 + 3 ) ke uk W 2;1 ( e t) +ke uk H 5 ( e t) + ( 3=2 + (1 +K 3 ) 3 )ke uk X ! ke uk Y ; (4.22) whereC is a sufficiently large positive constant depending on , andK is as defined in (4.9). 60 The proof of Lemma 4.3.2 is given in Section 4.4, while the proof of Lemma 4.3.3 below is given in Section 4.5. 4.3.3 Lemma. For > 0 fixed, sufficiently small depending only on , if , where is the Gevrey-class regularity radius of the boundary, then we have PC(1 + 2 ) ke uk 2 W 2;1 ( e t) +ke uk 2 H 3 ( e t) + (1 +K 2 )ke pk H 4 ( e t) +ke pk W 3;1 ( e t) +C ke uk W 1;1 ( e t) + ( 2 + 3 )ke uk W 2;1 ( e t) + 2 ke uk H 5 ( e t) + ( 3=2 + (1 +K 3 ) 4 )ke uk X ! ke uk Y ; (4.23) for some sufficiently large constantC depending on , whereK is as defined in (4.9). By combining estimates (4.19), (4.22), and (4.23), with the Sobolev embedding, and the classical pressure estimate in Sobolev spaceskpk H m (D) Ckuk 2 H m1 (D) (cf. [55, Lemma 1.2]), we obtain forr 5 that d dt ke uk X C(1 + 2 )(1 +K 2 )kuk 2 H r (D) + _ +Ckuk W 1;1 (D) +C( 2 + 3 )kuk H r (D) +C( 3=2 + (1 +K 3 ) 4 )ke uk X ke uk Y : (4.24) for some fixed positive constantC depending on the domainD. Let M(t) =ku(;t)k H r (D) (4.25) 61 and N(t) =ku(;t)k W 1;1 (D) (4.26) for all 0t<T . Note thatK(t) = R t 0 N(s)ds. By possibly increasing the constant C =C(D), we have d dt ke uk X C(1 + 2 )(1 +K 2 )M 2 + _ +CN +C( 2 + 3 )M +C( 3=2 + (1 +K 3 ) 4 )ke uk X ke uk Y : (4.27) Let(t) be chosen such that(t) 0 , where is the radius of Gevrey-class regularity of the boundary, and for all 0 =T 0 tT 1 let(t) be the solution of _ +C 0 N +C 0 3=2 L = 0; (4.28) with the initial condition (0) = 0 , where C 0 is a sufficiently large fixed positive constant (for instanceC 0 =(2 + 2 )>C, the constant of (4.27)); we have denoted L(t) =C 0 M(t) + 1 +C 0 1 +K 3 (t) ke u 0 k X 0 +C 0 Z t 0 1 +K 2 (s) M 2 (s)ds : (4.29) Then is decreasing, and by (4.27) for short time we have ke u(t)k X (t) ke u 0 k X 0 +C 0 Z t 0 1 +K 2 (s) M 2 (s)ds: (4.30) 62 By (4.27), if (4.28) holds for all t2 [T 0 ;T 1 ], thene u(t)2 X (t) and (4.30) is also valid for allt2 [T 0 ;T 1 ]. The radius of Gevrey-class regularity(t) may be computed explicitly from (4.28) as (t) = exp C 0 K(t) 1=2 0 +C 0 Z t 0 L(s) exp C 0 K(s) ds ! 2 ; (4.31) whereL is defined by (4.29). By further estimating the Sobolev norms in (4.31) using M 2 (t) =ku(t)k 2 H r (D) Cku 0 k 2 H r (D) exp C Z t 0 ku(s)k W 1;1 (D) ds =Cku 0 k 2 H r (D) exp CK(t) ; for some positive constantC = C(r;D), we obtain a more compact lower bound for (t) given by (t) 0 1 +Ctke u 0 k X 0 +Ct 2 ku 0 k 2 H r 1 +K 5 (t) 2 exp CK(t) 0 1 +Ctke u 0 k X 0 +Ct 2 ku 0 k 2 H r 2 exp CK(t) ; (4.32) we used (1 +x 5 ) 2 exp(2x) for allx 0, whereC = C(D;r) is a sufficiently large positive constant. Therefore we have proven the following theorem. 4.3.1 Theorem. Let u 0 be divergence-free and of Gevrey-class s, with s 1, on a Gevrey-class s, open, bounded domainD R 3 . Fix r 5, x 0 2 @D, and r 0 > 0 sufficiently small. Let be a neighborhood ofx 0 compactly embedded inB r 0 (x 0 )\D, and letT 1 be the maximal time such that t ( )B r 0 (x 0 )\D for all 0t<T 1 . Then the uniqueH r -solutionu( t ();t) to the initial value problem (4.1)–(E.4) is of Gevrey- classs for allt<T 1 . Moreover, there exist constants =(D), and = (D), such 63 that ife u(0)2 X 0 , and 0 , thene u(;t)2 X (t) for all t2 [0;T 1 ), where the Gevrey-class radius(t) of the solutionu( t ();t) satisfies (t) 0 1 +Ctke u 0 k X 0 +Ct 2 ku 0 k 2 H r 2 exp C Z t 0 ku(s)k W 1;1ds (4.33) for allt<T 1 , withC a sufficiently large constant depending only on the domainD. 4.3.1 Remark. Theorem 4.3.1 gives the local in time Gevrey-class persistence at the boundary ofD. The short-time Gevrey-class persistence in the interior ofD, with explicit bound on the radius of Gevrey-class regularity is obtained using similar argu- ments to the ones given in this section. Namely, givenx 0 2D andr 0 > 0 sufficiently small, we let be a Gevrey-class neighborhood ofx 0 , with B r 0 (x 0 )\D. The main step is to show that for all t > 0 such that t ( ) B r 0 (x 0 )\D, the analo- gous estimates to the ones given in Lemmas 4.3.2 and 4.3.3 hold. The bound on the velocity commutatorC is proven by repeating exactly the same estimates as in Sec- tion 4.4 below. Since we are away from the boundary, the bound for the pressure term P is obtained from classical interior elliptic estimates for the pressure (cf. (4.75)) and arguments parallel to the ones presented in Lemma 4.5.3. Since the interior pressure estimates are only simpler than the boundary case, we omit further details. It follows that the solutionu(;t) is of Gevrey-classs on t ( ), the radius of Gevrey-class regu- larity(t) satisfies the lower bound (4.33), and that the Gevrey-class norm is bounded from below by the right side of (4.30). 64 4.4 The velocity commutator estimate Since in this section we work only for a fixed timet and on the straightened domain, we suppress the time dependence and the tilde for all functions and domains. The goal of this section is to prove Lemma 4.3.2, that is to estimate C = 1 X m=3 X jj=m 3 sup 0< 0 m3 k[@ ;u j @ j k @ k ]uk L 2 ( ) m3 (m 3)! s : Proof of Lemma 4.3.2. The proof consists of two parts. First we estimate the @ j k coefficients from the definition ofC and exploit the commutator (cf. (4.39) below). Then we estimate the Gevrey-class norm ofu i @ j u k (cf. (4.48)–(4.51) below) for all 1i;j;k 3. The Leibniz rule and the fact sup P n x n; P n sup x n; for all sequencesx n; , imply C 1 X m=3 X jj=m X 0< X 0 3 k@ @ j k k L 1 ( ) sup 0< 0 m3 k@ u j @ @ k uk L 2 ( ) m3 (m 3)! s ; (4.34) where = S 0< 0 . Since the boundary is of Gevrey-classs, there exist constants C; > 0 such that X jj=n k@ D k k L 1 ( ) C (n 3)! s n ; (4.35) 65 for alln 0. Using = mk jk , we may rewrite the right side of (4.34) as CC 1 X m=3 m X j=1 j X k=0 m k (k 3)! s (mk 3)! s (m 3)! s k X jj=m X jj=j; X j j=k; m k 1 k@ Dk L 1 ( ) k (k 3)! s 3 mk3 (mk 3)! s mk jk sup 0< 0 m3 k@ uk L p ( ) k@ Duk L 2p=(p2) ( ) ; (4.36) wherep = 2 ifjk >mj, andp =1 ifjkmj. Observe that= < 1. Sinces 1, there exists a constantC such that m k (k 3)! s (mk 3)! s (m 3)! s C (mk + 1) s1 +C fk=0g (4.37) 66 for all 0 k m, where fk=0g = 1 ifk = 0, and fk=0g = 0 ifk 1. By (4.35), (4.37), Lemma 4.7.1, and using m k , we obtain CC 1 X m=3 m X j=1 j X k=0 k X jj=m X jj=j; X j j=k; k@ Dk L 1 ( ) k (k 3)! s mk3 (mk 3)! s mk jk 1 (mk + 1) s1 + fk=0g 3 sup 0< 0 m3 k@ uk L p ( ) k@ Duk L 2p=(p2) ( ) C 1 X m=3 m X j=1 j X k=0 k X jj=mk X jj=jk; mk3 (mk 3)! s mk jk 1 (mk + 1) s1 + fk=0g 3 sup 0< 0 m3 k@ uk L p ( ) k@ Duk L 2p=(p2) ( ) : (4.38) Due to the definition of the Gevrey-class norm, in (4.38) we need to consider the cases mk < 3 and mk 3 separately. We estimate the discrete convolution using Lemma 4.7.2 to obtain CCkuk W 1;1 ( ) kuk H 3 ( ) +Ckuk L 1 ( ) kuk Y +C 1 X m=3 m X j=1 m j m3 (m 3)! s X jj=m X jj=j; 3 sup 0< 0 m3 k@ uk L p ( ) k@ Duk L 2p=(p2) ( ) ; (4.39) 67 where C is a constant depending on the domain, and on = < 1. We rewrite the estimate (4.39) as CCkuk 2 H 3 ( ) +Ckuk 2 W 1;1 ( ) +Ckuk L 1 ( ) kuk Y +C (C 1 +C 2 +C low +C high +C 3 +C 4 +C 5 ); (4.40) where for 1j 2 we denoted C 1 = 1 X m=3 m X jj=m X jj=1; m3 (m 3)! s sup 0< 0 3 k@ uk L 1 ( ) 3 3 m3 k@ Duk L 2 ( ) ; (4.41) C 2 = 1 X m=4 m 2 X jj=m X jj=2; m3 (m 3)! s sup 0< 0 3 k@ uk L 1 ( ) 3 3 m4 k@ Duk L 2 ( ) ; (4.42) for 3jm 3 C low = 1 X m=6 [m=2] X j=3 m j X jj=m X jj=j; m3 (m 3)! s sup 0< 0 3 j1 k@ uk L 1 ( ) 3 3 mj2 k@ Duk L 2 ( ) ; (4.43) C high = 1 X m=7 m3 X j=[m=2]+1 m j X jj=m X jj=j; m3 (m 3)! s sup 0< 0 3 j3 k@ uk L 2 ( ) 3 3 mj k@ Duk L 1 ( ) ; (4.44) 68 and form 2jm C 3 = 1 X m=5 m m 2 X jj=m X jj=m2; m3 (m 3)! s sup 0< 0 3 m5 k@ uk L 2 ( ) 3 3 2 k@ Duk L 1 ( ) ; (4.45) C 4 = 1 X m=4 m m 1 X jj=m X jj=m1; m3 (m 3)! s sup 0< 0 3 m4 k@ uk L 2 ( ) 3 3 k@ Duk L 1 ( ) ; (4.46) C 5 = 1 X m=3 X jj=m m3 (m 3)! s sup 0< 0 3 m3 k@ uk L 2 ( ) kDuk L 1 ( ) : (4.47) These seven terms are bounded as in the proof of Lemma 3.3.1. Namely, letting = S 0< 0 , andj =jj, for the casesj = 1 andj =m we have C 1 +C 5 CkDuk L 1 ( ) kuk H 3 ( ) +CkDuk L 1 ( ) kuk Y ; (4.48) for the casesj = 2 andj =m 1 it holds that C 2 +C 4 CkD 2 uk L 1 ( ) kuk H 3 ( ) +C 2 kD 2 uk L 1 ( ) kuk Y ; (4.49) whenj =m 2 we have C 3 C 2 kuk 2 H 5 ( ) + 3 kuk H 5 ( ) kuk Y ; (4.50) and when 3jm 2 we have C low +C high C 3=2 + (1 +K 3 ) 3 kuk X kuk Y ; (4.51) 69 for some sufficiently large constantC, whereK is as defined in (4.9). We sketch the proof of theC low estimate and refer the reader to Chapter 3 for further details on the other five terms. Modulo multiplying by a smooth cut-off function supported on r and which is identically 1 on , (4.12) and the three-dimensional Agmon inequality give that kvk L 1 ( ) Ckvk 1=4 L 2 ( r ) kvk 3=4 L 2 ( r ) + C +CK 3 r 3=2 kvk L 2 ( r ) ; (4.52) whereC > 0 is a constant depending on the geometry of and on 0 , andK(t) = kuk L 1 t (0;t)W 1;1 x (D) is as in (4.9). Lettingr ==j, forj 3, we have sup 0< 0 3 j1 k@ uk L 1 ( ) C(1 +K 3 ) sup 0< 0 (j=) 3=2 3 (=j) j3 k@ uk L 2 ( =j ) 2 + sup 0< 0 3 (=j) j3 k@ uk L 2 ( =j ) 1=4 sup 0< 0 3 (=j) j1 k@ uk L 2 ( =j ) 3=4 1=2 (4.53) In the above inequality we used (1 + 1=(j 1)) j1 e for allj 1. By the H¨ older inequality, and Lemma 3.4.2, we obtain from the definition ofC low and the above inequality C low C 1 X m=6 [m=2] X j=3 m j [u] 1=4 j [u] 3=4 j+2 [u] mj+1 m3 (m 3)! s +C(1 +K 3 ) 1 X m=6 [m=2] X j=3 m j [u] j j 3=2 [u] mj+1 m3 (m 3)! s ; (4.54) 70 where we have denoted [v] m = X jj=m 3 sup 0< 0 m3 k@ vk L 2 ( ) (4.55) for all smoothv. In the above estimate (4.54) we used the fact that [Dv] m C[v] m+1 and [v] m C[v] m+2 , whereC > 0 may depend on, which is fixed. The right side of (4.54) is then bounded by C 3=2 + (1 +K 3 ) 3 kuk X kuk Y : Here we used the definitions (4.14)–(4.15), the discrete Young and H¨ older inequalities, and the combinatorial estimate m j (j 3)! s=4 (j 1)! 3s=4 (mj 2)! s (m 3)! s (mj + 1) + m j (j 3)! s j 3=2 (mj 2)! s (m 3)! s (mj + 1) C; (4.56) which holds for allm 6, 3j [m=2], ands 1, whereC > 0 is a dimensional constant. By reversing the roles ofj andmj, similar estimates give the bound on C high , thereby proving (4.51). This concludes the proof of the Lemma 4.3.2. 4.5 The pressure estimate The goal of this section is to prove Lemma 4.3.3. This is achieved in several steps: First we use an H 2 regularity estimate on the flattened domain to estimate all tangential derivatives of the pressure; next, we obtain a recursion formula to bootstrap to an estimate with higher number of normal derivatives, which leads to an estimate in terms 71 of the velocity; lastly, we prove a product-type estimate for the Lagrangian Gevrey- class norms defined in Section 4.2 which concludes the proof. For the rest of this section all functions depend ony = (x), hence we shall fur- ther suppress all tildes, and since there is no time evolution for the pressure we also suppress the time dependence. 4.5.1 Semi-norms and a decomposition of the pressure term The following semi-norms are useful when treating the pressure term. Namely, define hvi l; ; = l+j j3 k@ 1 1 @ 2 2 vk L 2 ( ) (4.57) for all 2N 2 0 ,l2Z withj j +l 3, and all 0< 0 . Also let hvi l;n = X jj=n; 3 =0 sup 0< 0 hvi l; 0 ; = X jj=n; 3 =0 sup 0< 0 l+n3 k@ 1 1 @ 2 2 vk L 2 ( ) (4.58) for alln 0 withn +l 3. Note that we have the inequalityhD 0 k vi l;n hvi lk;n+k , wherehD 0k vi l;n = P jj=n; 3 =0 sup 0< 0 l+n3 kD 0k vk L 2 ( ) , andkD 0k vk L 2 is defined above. Next, we shall estimate the pressure term arising in (4.19), i.e., P = 1 X m=3 0 @ X jj=m 3 sup 0< 0 m3 @ @ j k @ k p L 2 ( ) 1 A m3 (m 3)! s : (4.59) Similarly to (4.34)–(4.39), we let C; > 0 be such that P jj=j k@ Dk L 1 ( ) C(j 3)! s = j for allj 0. Assuming that < , it follows from the Leibniz rule 72 and the bound m j (mj 3)! s (j 3)! s (m 3)! s C that the pressure term is bounded as PCkDpk W 2;1 ( ) +C 1 X m=3 0 @ X jj=m 3 sup 0< 0 m3 k@ Dpk L 2 ( ) 1 A m3 (m 3)! s ; (4.60) for some positive constant C = C(D;), where = = < 1 by assumption. We decompose the upper bound on the pressure term as follows PCkDpk W 2;1 ( ) +C 1 X m=3 m X 3 =0 3 h@ 3 3 Dpi 3 ;m 3 ! m3 (m 3)! s Ckpk W 3;1 ( ) +C (1 +)P 0 +P 1 +P 2 ; (4.61) where we have denoted the term with at most one normal derivative by P 0 = 1 X m=3 hDpi 0;m m3 (m 3)! s ; (4.62) and the terms with at least two normal derivatives (according toD =@ 3 +D 0 ) by P 1 = 1 X m=3 m X 3 =1 3 h@ 3 +1 3 pi 3 ;m 3 m3 (m 3)! s (4.63) and P 2 = 1 X m=3 m X 3 =2 3 h@ 3 3 pi 3 1;m 3 +1 m3 (m 3)! s : (4.64) 73 4.5.2 The elliptic Neumann problem for the pressure Under the change of variables : x7!y, the elliptic Neumann problem for the pressure (cf. [55]) becomes (omitting tildes) p =A ij @ ij p +B j @ j p +D ijkl @ i u j @ k u l ; in (4.65) @ 3 p =C j @ j p + ij u i u j ; on@ ; (4.66) where we denoted A ij = 1 0 B B B B @ (@ 1 ) 2 (@ 2 ) 2 0 @ 1 0 (@ 1 ) 2 (@ 2 ) 2 @ 2 @ 1 @ 2 0 1 C C C C A ; (4.67) B j = 1 0 B B B B @ 0 0 @ 11 @ 22 1 C C C C A ; (4.68) C j = 1 + 1=2 (2 + 1=2 ) 1=2 0 B B B B @ @ 1 @ 2 0 1 C C C C A ; (4.69) D ijkl = 1 ik jl + 1 k3 0 B B B B @ @ l 0 0 0 @ l 0 (@ l ) 2 @ 1 @ 2 @ l 1 C C C C A ; (4.70) ij = 1 3=2 0 B B B B @ @ 11 @ 12 0 @ 12 @ 22 0 0 0 0 1 C C C C A ; (4.71) 74 with = 1 + (@ 1 ) 2 + (@ 2 ) 2 : (4.72) The precise form of the above matrices is not essential; what is important for the following arguments is thatA 33 =C 3 = 0, and that the coefficientsA ij ;C j are small. We also denote f =D ijkl @ i u j @ k u l ; (4.73) g = ij u i u j : (4.74) 4.5.3 The interiorH 2 -regularity estimate Letp be the smooth solution of the elliptic Neumann problem p =A ij @ ij p +B j @ j p +f; in ; (4.75) @ 3 p =C j @ j p +g; on@ ; (4.76) where all coefficients are of Gevrey-class s. We have the following interior H 2 - regularity estimate. 4.5.1 Lemma. There exists a sufficiently small positive dimensional constant " such that ifA 33 = C 3 = 0,kA ij k L 1 ( ) ", andkC j k L 1 ( ) ", the smooth solutionp of (4.75)–(4.76) satisfies kD 2 pk L 2 ( +r ) C 0 kfk L 2 ( ) +kDgk L 2 ( ) + 1 +K 2 r kDpk L 2 ( ) ; (4.77) 75 for all 2 (0; 0 ) and for all 0 < r << 1, where K is as defined in (4.9), and C 0 =C 0 (A ij ;B j ;C j ) is a positive constant depending on the domain. The proof is standard and thus omitted. It relies on the fact that the elliptic operator acting onp in (4.75) is a small/lower-order perturbation of the Laplacian, and on the the fact that by (4.12) we haveC 0 dist( +r ; c )r=(1 +K 2 ), for some sufficiently large constantC 0 . 4.5.4 The estimation of tangential derivatives Fixk 3, and let 0 = ( 1 ; 2 ; 0)2N 3 0 be such thatj 0 j = k. Consider the system (4.75)–(4.76). The function@ 0 p satisfies the elliptic Neumann problem (@ 0 p) =A ij @ ij @ 0 p +B j @ j @ 0 p +@ 0 f + [@ 0 ;A ij @ ij ]p + [@ 0 ;B j @ j ]p; in ; (4.78) @ 3 (@ 0 p) =C j @ j @ 0 p +@ 0 g + [@ 0 ;C j @ j ]p; on@ : (4.79) We apply theH 2 -estimate (4.77) to the solution of (4.78)–(4.79), and bound the com- mutators using the Leibniz rule as k[@ 0 ;A ij @ ij ]pk L 2 ( ) X 0< 0 0 0 0 k@ 0 A ij k L 1 ( ) k@ 0 0 @ ij pk L 2 ( ) : (4.80) 76 The terms involving [@ 0 ;B j @ j ] and [@ 0 ;C j @ j ] are estimated similarly. Letting r = =k=3, we obtain k@ 0 D 2 pk L 2 ( +=k ) C 0 k@ 0 fk L 2 ( ) +k@ 0 Dgk L 2 ( ) + (1 +K 2 ) k k@ 0 Dpk +C 0 k X j=1 k j X j 0 j=j; 0 0 maxfk@ 0 A ij k L 1 ( ) ;k@ 0 B j k L 1 ( ) ;k@ 0 C j k L 1 ( ) g k@ 0 0 DD 0 pk L 2 ( ) +k@ 0 0 Dpk L 2 ( ) ; (4.81) where we usedA 33 = 0. Denote 0 ; = j 0 j2 maxfk@ 0 A ij k L 1 ( ) ;k@ 0 B j k L 1 ( ) ;k@ 0 C j k L 1 ( ) g; (4.82) for allj 0 j 2, and 0 ; = maxfk@ 0 A ij k L 1 ( ) ;k@ 0 B j k L 1 ( ) ;k@ 0 C j k L 1 ( ) g; (4.83) ifj 0 j = 1. Also let j = X j 0 j=j sup 0< 0 0 ; ; (4.84) Note that since , and hence A ij ;B j ;C j , is of Gevrey-class s, there exist C; > 0 such that j C (j 2)! s j ; (4.85) 77 for allj 1, where recall (1)! = 1. Since the boundary is fixed,C and the Gevrey- classs radius of the boundary are not functions of time. Multiplying estimate (4.81) by ( +=k) k2 , and using (1 + 1=k) k2 e fork 3, we obtain hD 2 pi 1; 0 ;+=k Chfi 1; 0 ; +ChDgi 1; 0 ; +C 0 ; kDD 0 pk L 2 ( ) +kDpk L 2 ( ) +C(1 +K 2 )k X j 0 j=k1; 0 0 0 ; kDD 02 pk L 2 ( ) +kDD 0 pk L 2 ( ) +Ck(k 1) X j 0 j=k2; 0 0 0 ; kDD 02 pk L 2 ( ) +C k4 k2 X j=2 k j X j 0 j=j; 0 0 0 ; hDD 0 pi 3; 0 0 ; +C k5 k3 X j=2 k j X j 0 j=j; 0 0 0 ; hDpi 3; 0 0 ; +CkhDpi 0; 0 ; +Ck X j 0 j=1; 0 0 0 ; hDD 0 pi 2; 0 0 ; +hDpi 2; 0 0 ; : (4.86) By taking the supremum over 0 < 0 1 of the above estimate and summing over allj 0 j =k 3, cf. Lemma 3.4.2, we obtain h@ 3 Dpi 1;k +hDpi 0;k+1 C hfi 0;k +hDgi 0;k +C(1 +K 2 ) k kDpk L 2 ( ) + ( k +k k1 )kD 2 pk L 2 ( ) +C(1 +K 2 )(k k1 +k 2 k2 )kD 3 pk L 2 ( ) +C k2 X j=1 k j j hDpi 0;kj+1 +C k4 k3 X j=1 k j j hDpi 0;kj ; (4.87) 78 where as usual we write = S 0< 0 . Estimate (4.87) is used to bound the termP 0 in the decomposition (4.61) ofP. Furthermore, using the bound (4.85) on j , estimate (4.61) implies h@ 3 Dpi 1;k +hDpi 0;k+1 C hfi 0;k +hDgi 0;k +C(1 + + 2 )(1 +K 2 ) kDpk L 2 ( ) +kD 2 pk L 2 ( ) +kD 3 pk L 2 ( ) (k 2)! s k +C k2 X j=1 k j (j 2)! s j hDpi 0;kj+1 +C k4 k3 X j=1 k j (j 2)! s j hDpi 0;kj ; (4.88) for allk 3, whereC depends onC 0 and 0 1, while = ( ) is fixed. 4.5.5 The transfer of normal to tangential derivatives We use the special structure of the coefficientsA ij andB j to rewrite (4.75) as @ 33 p = (a 1 @ 1 +a 2 @ 2 )@ 3 p +b@ 3 p +c (@ 11 +@ 22 )p +f = (ar 0 )@ 3 p +b@ 3 p +c 0 p +f; (4.89) where, as above, (cf. (4.67),(4.68), and (4.72)) a i =2 @ i ; b = @ 11 +@ 22 ; c = 1 : (4.90) Since , and hencea;b, andc, is a function of (y 1 ;y 2 ) only, we obtain from (4.89) that fork 2 we have @ k 3 p = (ar 0 )@ k1 3 p +b@ k1 3 p +c 0 @ k2 3 p +@ k2 3 f: (4.91) 79 Note that in the case of the half-space (cf. Chapter 3 or [39]), identity (4.89) simplifies to@ 33 p = 0 p +f, which allows one to obtain an explicit formula for@ k 33 p in terms off and ( 0 ) k p. The combinatorial structure of this transfer of normal to tangential derivatives is encoded in the coefficientsM of Chapter 3. In the case of the present paper, it is highly inconvenient use the recursion formula (4.91) to explicitly calculate @ k 3 p in terms off and tangential derivatives ofp. Instead we use the fact that we may choose << 1 and recursively bootstrap to estimates on higher number of normal derivatives acting onp. By applying@ 0 , wherej 0 j = n, to (4.91), using the Leibniz rule, the H¨ older inequality, we obtain k@ k 3 @ 0 pk L 2 ( ) k@ k2 3 @ 0 fk L 2 ( ) +C n X j=0 n j X j 0 j=j; 0 0 0 ; k@ 0 0 D 0 @ k1 3 pk L 2 ( ) +k@ 0 0 @ k1 3 pk L 2 ( ) +k@ 0 0 0 @ k2 3 pk L 2 ( ) ; (4.92) where we have denoted 0 ; and j similarly to (4.82)–(4.84) (replaceA ij ;B j ;C j by a;b;c). Sincea;b, andc are of Gevrey-classs (they only depend on ), as in (4.85), there exist C; > 0 with j C(j 2)! s = j . Multiplying the bound (4.92) by n+k4 , it follows that h@ k 3 pi k1; 0 ; h@ k2 3 fi k1; 0 ; + n X j=0 n j X j 0 j=j; 0 0 0 ; h@ k1 3 D 0 pi k+1; 0 0 ; +h@ k2 3 0 pi k+1; 0 0 ; +h@ k1 3 pi k+1; 0 0 ; ; (4.93) 80 for alln +k 4, 0 0 , and2N 3 0 with 3 = 0, andj 0 j =n. Estimate (4.93) above will be used to bound the terms with high number of normal derivatives in the pressure estimate, namelyP 1 , andP 2 . 4.5.6 Bounds forP 0 ;P 1 , andP 2 For the termP 0 with a low number of tangential derivatives we have the bound P 0 C 1 P 0 +C 1 (1 +K 2 )kpk H 4 ( ) +C 1 1 X m=4 hfi 0;m1 +hDgi 0;m1 m3 (m 3)! s ; (4.94) where = = < 1, is the Gevrey-class radius of the boundary, C = C( ) and C 1 = C 1 ( ;;) are sufficiently large constant positive constants. As usual, = S 0< 0 . Note that the condition < 1 is natural, as the flow may not have arbitrarily large radius of Gevrey-class regularity close to the boundary. Under the assumption< 1, we also have the bounds P 1 C 1 (P 0 +P 1 ) +C 1 kpk H 3 ( ) +C 1 1 X m=3 m X 3 =1 3 h@ 3 1 3 fi 3 ;m 3 m3 (m 3)! s ; (4.95) and P 2 C 1 (P 0 +P 1 +P 2 ) +C 1 kpk H 3 ( ) +C 1 1 X m=3 m X 3 =2 3 h@ 3 2 3 fi 3 1;m 3 +1 m3 (m 3)! s ; (4.96) 81 where C = C( ) is a fixed sufficiently large constant, while C 1 = C 1 ( ;;) has additional dependence on the Gevrey-class norm and radius of cf. (4.85) and the parameter. First we prove the bound forP 0 . Proof of (4.94). Lettingk =m 1 in (4.88), and recalling the definition (4.62) ofP 0 , we obtain P 0 hDpi 0;3 +C 1 X m=4 hfi 0;m1 +hDgi 0;m1 m3 (m 3)! s +C(1 +K 2 )(1 + 2 )kpk H 3 ( ) 1 X m=4 (m 3)! s m3 m3 (m 3)! s +C 1 X m=4 m3 X j=1 m 1 j (j 2)! s j hDpi 0;mj m3 (m 3)! s +C 1 X m=5 m4 X j=1 m 1 j (j 2)! s j hDpi 0;mj1 m3 (m 3)! s : Using the fact that for alls 1,m 4, and 1jm 3 we have m 1 j (j 2)! s (mj 3)! s (m 3)! s C; (4.97) and recalling that we have = = < 1, we estimate the discrete convolution and obtain P 0 C 1 P 0 +C 1 (1 +K 2 )kpk H 4 ( ) +C 1 1 X m=4 hfi 0;m1 +hDgi 0;m1 m3 (m 3)! s ; (4.98) whereC is a dimensional constant andC 1 =C 1 ( ; 0 ;), concluding the proof. The estimates forP 1 andP 2 are symmetric, and so to avoid redundancy we only give the proof of (4.95). 82 Proof of (4.95). Letk = 3 + 1 andn = m 3 (so thatn +k 4) in (4.93), to obtain that for allj 0 j =m 3 we have h@ 3 +1 3 pi 3 ; 0 ; h@ 3 1 3 fi 3 ; 0 ; + m 3 X j=0 m 3 j X j 0 j=j; 0 0 0 ; h@ 3 3 D 0 pi 3 +2; 0 0 ; +h@ 3 1 3 D 02 pi 3 +2; 0 0 ; +h@ 3 3 pi 3 +2; 0 0 ; : Taking the supremum over 0< 0 < 1, and summing over allj 0 j =m 3 , the above estimate implies P 1 C 1 X m=3 m X 3 =1 3 h@ 3 1 3 fi 3 ;m 3 m3 (m 3)! s +C 1 X m=3 m X 3 =1 m 3 X j=0 m 3 j j 3 m3 (m 3)! s h@ 3 3 pi 3 +1;mj 3 +1 +h@ 3 1 3 pi 3 ;mj 3 +2 +h@ 3 3 pi 3 +2;mj 3 : Using the bound (4.85) on j and the combinatorial estimate m 3 j (j 2)! s (mj 3)! s (m 3)! s C; (4.99) 83 which holds for allm 3, 1 3 m, and 0jm 3 , we obtain P 1 C 1 X m=3 m X 3 =1 3 h@ 3 1 3 fi 3 ;m 3 m3 (m 3)! s +C 1 X m=3 m X 3 =1 m 3 X j=0 j 3 1 h@ 3 3 pi 3 +1;mj 3 +1 mj3 (mj 3)! s +C 1 X m=3 m X 3 =1 m 3 X j=0 j+1 3 1 h@ 3 3 pi 3 +2;mj 3 mj4 (mj 4)! s + 2 C 1 X m=3 m X 3 =1 m 3 X j=0 j 3 2 h@ 3 1 3 pi 3 ;mj 3 +2 mj3 (mj 3)! s : (4.100) Here, as before we denoted = = < 1. It is convenient to reverse the summation order in the above estimate and write P 1 C 1 X m=3 m X 3 =1 3 h@ 3 1 3 fi 3 ;m 3 m3 (m 3)! s +C 1 X m=3 m1 X j=0 j mj1 X 3 =0 3 h@ 3 +1 3 pi 3 +2;mj 3 mj3 (mj 3)! s +C 1 X m=3 m X j=1 j mj X 3 +3; 3 =0 3 h@ 3 +1 3 pi mj 3 mj3 (mj 3)! s + 2 C 1 X m=3 m2 X j=0 j mj2 X 3 =0 3 h@ 3 +1 3 pi 3 +2;mj 3 mj3 (mj 3)! s +C 1 X m=3 m1 X j=0 j hpi 1;mj+1 mj3 (mj 3)! s C 1 X m=3 m X 3 =1 3 h@ 3 1 3 fi 3 ;m 3 m3 (m 3)! s +T 1 +T 2 +T 3 +T 4 : (4.101) 84 The terms T 1 ;T 2 ;T 3 , and T 4 are bounded by estimating the discrete convolution P m P j x j y mj , and using the fact that since < 1 we have P j0 j = 1=(1). We have the following estimate T 1 C 1 P 0 + C 1 P 1 +C 1 kpk H 3 ( ) ; (4.102) whereC =C( ) is a positive constant, andC 1 has additional dependance on, and. Similarly we obtain T 2 C 1 P 0 + C 1 P 1 +C 1 kpk H 3 ( ) ; (4.103) T 3 2 C 1 P 0 + 2 C 1 P 1 +C 1 kpk H 3 ( ) ; (4.104) and T 4 C 1 P 0 +C 1 kpk H 3 ( ) ; (4.105) withC =C( )> 0, andC 1 =C 1 ( ;;)> 0. The proof is concluded by combining (4.101)–(4.105). 85 4.5.7 Gevrey-class estimates for the pressure 4.5.2 Lemma. There exists a sufficiently small constant > 0 depending only on , such that if , then we have PC 1 X m=3 hDgi 1;m1 m3 (m 3)! s +C(1 +K 2 )kpk H 4 ( ) +Ckpk W 3;1 ( ) +C 1 X m=3 m X 3 =2 3 h@ 3 2 3 fi 3 1;m 3 +1 m3 (m 3)! s +C 1 X m=3 m X 3 =1 3 h@ 3 1 3 fi 3 ;m 3 m3 (m 3)! s (4.106) whereC =C( ) is a fixed positive constant. Proof of Lemma 4.5.2. By combining estimates (4.94)–(4.96) we obtain that for< 1 P 0 +P 1 +P 2 ( +)C 1 P 0 + C 1 P 1 + C 1 P 2 +C 1 (1 +K 2 )kpk H 4 ( ) +C 1 1 X m=4 hfi 0;m1 +hDgi 0;m1 m3 (m 3)! s +C 1 1 X m=3 m X 3 =2 3 hD@ 3 2 3 fi 3 ;m 3 m3 (m 3)! s ; (4.107) for a sufficiently large, fixed constant C = C ( ) > 0, and C 1 = C 1 ( ;;) > 0. Define =( ) by = 1 1 + 4C : (4.108) It is clear that may be fixed for all time, as it only depends on the boundary of the domain. Whenever , we have == , and therefore ( +)=(1) 2=(1) 1=(2C ), by the choice of (4.108). Thus the terms involvingP 0 ,P 1 , 86 andP 2 on the right side of (4.107), may be absorbed on the left side of (4.107) and the proof of the lemma is completed. 4.5.1 Remark. The condition < is not restrictive; it is a manifestation of the fact that the velocity field cannot have arbitrarily large Gevrey-class radius close to the boundary, it must be bounded from above by the Gevrey-class radius of the boundary. In the following lemma we use the definitions of f and g (cf. (4.73), (4.74)) to bound the right side of (4.106) in terms of the velocity. 4.5.3 Lemma. For =( )> 0 as in Lemma 4.5.2, if < , then we have PCkDuDuk H 1 ( ) +Ckuuk H 2 ( ) +C(1 +K 2 )kpk H 4 ( ) +Ckpk W 3;1 ( ) +C 1 X m=3 hD(uu)i 1;m1 m3 (m 3)! s +C 1 X m=3 m1 X 3 =0 3 +1 h@ 3 3 (DuDu)i 3 +3;m1 3 m3 (m 3)! s ; (4.109) whereC =C( )> 0 is a sufficiently large constant. Proof of Lemma 4.5.3. Denote the right side of (4.106) by T 1 f +T 2 f +T g +C(1 + K 2 )kpk H 4 ( ) +Ckpk W 2;1 ( ) . First we estimate the term T g =C 1 X m=3 hDgi 1;m1 m3 (m 3)! s : (4.110) 87 Recall thatg = u i u j ij (cf. (4.74)). As in the proof of (4.94) and (4.95), we denote ; = jj k@ ij k L 1 ( ) , and j = P jj=j sup 0< 0 ; . Since ij is of Gevrey- classs (cf. (4.71)) there existC; such that j C(j 3)! s = 0 , for allj 0 (recall that we writen! = 1 ifn 0). By the Leibniz rule, we have T g C 1 X m=3 m X j=0 X jj=m; 3 1 X jj=j; sup 0< 0 m3 k@ k L 1 ( ) k@ (uu)k L 2 ( ) m3 (m 3)! s : (4.111) We split this sum into four pieces according toj = m,j = m 1,j = m 2, and 0jm 3. We obtain T g C 1 X m=3 m3 X j=0 m j j hD(uu)i 1;mj1 m3 (m 3)! s +C 1 X m=3 m kuuk L 2 ( ) +m m1 kuuk H 1 ( ) +m 2 m2 kuuk H 2 ( ) m3 (m 3)! s (4.112) Using the bound j C(j 3)! s = j , the combinatorial estimate m j (j 3)! s (mj 3)! s =(m 3)! s C; and == < 1, we obtain T g C 1 X m=3 m3 X j=0 j hD(uu)i 1;mj1 mj3 (mj 3)! s +Ckuuk H 2 ( ) C 1 X m=3 hD(uu)i 1;m1 m3 (m 3)! s +Ckuuk H 2 ( ) ; (4.113) 88 for some sufficiently large constantC =C( ). We now estimate the termsT 1 f andT 2 f . We have T 1 f =C 1 X m=3 m X 3 =1 3 h@ 3 1 3 fi 3 ;m 3 m3 (m 3)! s C 1 X m=3 X jj=m1 3 +1 sup 0< 0 m3 k@ fk L 2 ( ) m3 (m 3)! s ; and similarly T 2 f =C 1 X m=3 m X 3 =2 3 h@ 3 2 3 fi 3 1;m 3 +1 m3 (m 3)! s C 1 X m=3 X jj=m1 3 +2 sup 0< 0 m3 k@ fk L 2 ( ) m3 (m 3)! s : Recall that cf. (4.73) we have f = @ i u j @ k u l D ijkl , where D ijkl is of Gevrey-class s (cf. (4.70)), and therefore we have j C(j 2)! s = j , for allj 0. Here we have denoted ; = maxfjj2;0g k@ D ijkl k L 1 ( ) , and also j = P jj=j sup 0< 0 ; . From the above estimates and the Leibniz rule we obtain that T 1 f +T 2 f C 1 X m=3 m1 X j=0 m 1 j X jj=m1 X jj=j; 3 +1 sup 0< 0 m3 k@ D ijkl k L 1 ( ) k@ (DuDu)k L 2 ( ) m3 (m 3)! s C 1 X m=3 m3 X j=0 m 1 j j mj1 X 3 =0 3 +1 h@ 3 3 (DuDu)i 3 +3;mj1 3 m3 (m 3)! s +C 1 X m=3 ( m1 +m m2 )kDuDuk H 1 ( ) m3 (m 3)! s : (4.114) 89 Using the bound j C(j 2)! s = j , the combinatorial estimate m 1 j (j 2)! s (mj 3)! s (m 3)! s C; (4.115) and the fact that == < 1, from (4.114) we obtain T 1 f +T 2 f C 1 X m=3 m3 X j=0 j mj1 X 3 =0 3 +1 h@ 3 3 (DuDu)i 3 +3;mj1 3 mj3 (mj 3)! s ! +CkDuDuk H 1 ( ) C 1 X m=3 m1 X 3 =0 3 +1 h@ 3 3 (DuDu)i 3 +3;m1 3 m3 (m 3)! s +CkDuDuk H 1 ( ) ; (4.116) for some sufficiently largeC =C( )> 0. This concludes the proof of the lemma. 4.5.8 Proof of Lemma 4.3.3 Here we use the estimate obtained in Lemma 4.5.3 to boundP in terms of the Gevrey- class norm of the velocity, and prove the estimate (4.23). In view of Lemma 4.5.3, we need to estimate the terms 1 X m=3 m1 X 3 =0 3 +1 h@ 3 3 (DuDu)i 3 +3;m1 3 m3 (m 3)! s C 1 X m=3 m1 X j=0 X jj=m1 X jj=j; 3 +1 sup 0< 0 m3 k@ Du@ Duk L 2 ( ) m3 (m 3)! s =R; (4.117) 90 and the lower order term 1 X m=3 hD(uu)i 1;m1 m3 (m 3)! s C 1 X m=3 m X j=0 X jj=m; 3 1 X jj=j; sup 0< 0 m3 k@ u@ uk L 2 ( ) m3 (m 3)! s =S: (4.118) Similarly to the estimate for the the commutator termC (cf. Proof of Lemma 4.3.2), boundingR andS is achieved by splitting the above sums according to the relative sizes ofj andmj. This idea was introduced in Chapter 3 (see also [39]). Namely, we write the right side of (4.117) asR 1 +R 2 +R 3 +R low +R high +R 4 +R 5 , according toj = 0; 1; 2, 3j [(m 1)=2], [(m 1)=2] + 1jm 3,j =m 2, and respectivelyj =m 1. Note that by symmetry (replacej bymj) the termsR 1 and R 5 ,R 2 andR 4 , and alsoR low andR high , have the same upper bounds. We have the estimates R 1 +R 5 CkDuk L 1 ( ) kuk H 3 ( ) +CkDuk L 1 ( ) kuk Y (4.119) R 2 +R 4 CkD 2 uk L 1 ( ) kuk H 2 ( ) +CkD 2 uk L 1 ( ) kuk H 3 ( ) +C 2 kD 2 uk L 1 ( ) kuk Y (4.120) R 3 C 2 kuk H 5 ( ) kuk Y ; (4.121) and also R low +R high C( 3=2 + (1 +K 3 ) 3 )kuk X kuk Y : (4.122) 91 The proofs of (4.119)–(4.122) are similar to those in Section 3.5 and those in Sec- tion 4.4 of the chapter paper, and are thus omitted. Combined they give the desired estimate onP. To estimateS one proceeds similarly. Note though that this is a lower order term. We have the following bound SC kuk L 1 ( ) + 2 kDuk L 1 ( ) + 3 kD 2 uk L 1 ( ) kuk Y +C( 5=2 + (1 +K 3 ) 4 )kuk X kuk Y +C(1 + 2 ) kuk 2 W 2;1 ( ) +kuk 2 H 3 ( ) ; (4.123) whereC > 0 is a constant that may depend on . The proof of (4.123) is omitted (see Section 3.5 for details). By collecting the above estimates, and the lower order terms from (4.109), we conclude the proof of the pressure estimate. 4.6 Global Gevrey-class persistence In this section we prove that the local, short time estimates of Section 4.3 may be combined together to obtain global (in space) Gevrey-class a priori estimates that are valid for allt<T , the maximal time of existence of the Sobolev solution. Let T < T be fixed. We shall prove that the solution u(t) is of Gevrey-class s on [0;T ] and give a lower bound on the radius of Gevrey-class regularity. For this purpose letfx g N =1 be points on@D determined as follows. In a small neighborhood ofx the boundary ofD is the graph of a Gevrey-class function , i.e., there exists r > 0 sufficiently small such thatD =D\B r (x ) =fx2 B r (x ) : x 3 > (x 1 ;x 2 )g. Moreover, we can pickr small enough so that after composing with a rigid body rotation about x we havek@ 1 k L 1 +k@ 2 k L 1 ", where " > 0 is the fixed universal constant of Lemma 4.5.1. For all 2f1;:::;Ng we let = 92 D\B r =2 (x ). We takeN large enough so that there exists a compactly embedded open set D with analytic boundary, such that [ S 1N =D. To obtain Gevrey-class regularity in the interior ofD, we cover with finitely many, sufficiently small, analytic chartsfD g N+N 0 N+1 , chosen as follows. Denote by a ball insideD , and letr = dist( ; (D ) c ), where2fN + 1;:::;N +N 0 g. We letN 0 be large enough so that 1 N X =1 (x) + N+N 0 X =N+1 (x)C (4.124) for allx2D, whereC 1 is a sufficiently large constant. For s 0 fixed, define by t;s (a) the particle trajectory with initial condition s;s (a) =a, i.e., the unique smooth solution to d dt X(t) =u(X(t);t) X(s) =a: Note that t;0 (a) = t (a), where t is as defined in (4.5)–(4.6). Since the flow map t;s :D7!D is a bijection, cf. (4.124), we also have 1 P N+N 0 =1 t;s( ) (x)C for all 0st and allx2D. LetT 0 = 0, and defineT 1 as the maximal time 0 = T 0 < T 1 T such that for all T 0 t T 1 we have that t;T 0 ( )D for all 2f1;:::;N +N 0 g. Note that ifT 1 < T , then by the maximality ofT 1 , there exists2f1;:::;N +N 0 g with T 1 ;T 0 ( )\ (D ) c 6=;. Thus there exists andx 0 2 such thatj T 1 ;T 0 (x 0 )x 0 j 93 r =2 r , where r = min 1N+N 0fr =2g is a fixed constant. We obtain that if T 1 <T , thenT 1 may be estimated from below via Z T 1 T 0 ku(;t)k W 1;1 (D) dtr : (4.125) For each 2 f1;:::;N + N 0 g, let (x 1 ;x 2 ;x 3 ) = (x 1 ;x 2 ;x 3 (x 1 ;x 2 )) = (y 1 ;y 2 ;y 3 ) be a boundary straightening map and define e = ( ). Note that this is exactly the setup from Section 4.2. Let u (x;t) = u(x;t) D (x) and for y = (x)2 (D ) definee u (y;t) =u (x;t). Let 0 = (T 0 ) be the uniform radius of Gevrey-class regularity of the initial data u 0 . By possibly decreasing 0 by a factor, we may assume that 0 , where = (D) > 0 is as in Lemma 4.5.2, and is the uniform radius of Gevrey-class regularity of@D. Sinceu (T 0 ) has Gevrey-class radius 0 , we have that ke u (T 0 ;y)k Xa 0 <1 for all 2f1;:::;N +N 0 g, where 0 < a 1 measures the possible decrease in the Gevrey-class radius after composing with the boundary straightening map (cf. Remark 4.2.1). Therefore, on [T 0 ;T 1 ] we can apply Theo- rem 4.3.1 for each chartf g N =1 , respectively Remark 4.3.1 forf g N+N 0 =N+1 , to obtain that for all2f1;:::;N +N 0 g we have (cf. (4.30)) ke u (;t)k X (t) Q 0 +C Z t T 0 1 +K 2 (s) M 2 (s)ds (4.126) whereQ 0 = max 2f1;:::;N+N 0 g ke u (;T 0 )k Xa 0 ,C =C(D) is a positive constant, and the radius of Gevrey-class regularity(t) is bounded from below (cf. (4.32)) by (t)a 0 1 +CtQ 0 +Ct 2 M 2 (T 0 ) 2 exp CK(T 0 )CK(t) (4.127) 94 for allT 0 t T 1 . Here we recall thatK(t) = R t 0 ku(;s)k W 1;1 (D) ds, andM(t) = ku(;t)k H r (D) . Therefore, modulo composing with ( ) 1 we obtain that the localized velocityu (x;t) is of Gevrey-classs on [T 0 ;T 1 ] for each2f1;:::;N +N 0 g. By (4.124) we obtain thatu(;t) is of Gevrey-classs on [T 0 ;T 1 ], with uniform radius of Gevrey-class regularity bounded from below bya times the right side of (4.127). We proceed inductively. Letk 1 be fixed. Since T k ;T k1 (D) =D , as above for t = 0 we coverD with local chartsf g N+N 0 =1 and defineT k+1 as the maximal time T k+1 T such that t;T k ( )D for all2f1;:::;N +N 0 g. Similarly to (4.125) we obtain that ifT k+1 <T , thenT k+1 may be estimated from Z T k+1 T k ku(;t)k W 1;1 (D) dtr : (4.128) The induction assumption is thatu(x;T k ) if of Gevrey-classs, the uniform (over x2D) radius of Gevrey-class regularity ofu(x;T k ) is bounded from below by k =a 2 k1 1 +C(T k T k1 )Q k1 +C(T k T k1 ) 2 M 2 (T k1 ) 2 exp CK(T k1 )CK(T k ) ; (4.129) and that the Gevrey-class norm at t = T k , given by Q k = max 2f1;:::;N+N 0 g ke u (;T k )k Xa k , is bounded as Q k Q k1 +C Z T k T k1 1 +K 2 (s) M 2 (s)ds: (4.130) 95 We apply Theorem 4.3.1, respectively Remark 4.3.1, on each local chart , and con- clude thate u (y;t) is of Gevrey-classs on2 [T k ;T k+1 ] for all2f1;:::;N +N 0 g, with Gevrey-class norm bounded as ke u (;t)k X (t) Q k +C Z t T k 1 +K 2 (s) M 2 (s)ds; (4.131) and radius of Gevrey-class regularity(t) bounded from below by a k 1 +C(tT k )Q k +C(tT k ) 2 M 2 (T k ) 2 exp CK(T k )CK(t) : (4.132) Modulo composing with the inverse map of , if follows from (4.124) and (4.131) thatu(x;t) is of Gevrey-classs for allt2 [T k ;T k+1 ] with radius bounded from below bya times the quantity in (4.132). Moreover, lettingt =T k+1 in (4.131)–(4.132) we obtain that the induction assumptions (4.129)–(4.130) hold for the next iteration step. T T * k ~ T k ~ -1 T k+1 T k T 0 T 1 T 2 T 3 (τ ,Q ) 0 0 (τ ,Q ) 1 1 (τ ,Q ) 2 2 (τ ,Q ) 3 3 (τ ,Q ) k k (τ ,Q ) k+1 k+1 (τ ,Q ) k k ~ ~ Figure 4.2: The lower bound on theG s -radius obtained by patching local in time bounds. 96 We claim that for each fixedT <T the inductive argument described above stops after finitely many steps, i.e., there exists ak 1 such thatT k =T . To see this, note that ifT k <T , then from (4.125) and (4.128) we obtain kr Z T k 0 ku(;t)k W 1;1 (D) dt Z T 0 ku(;t)k W 1;1 (D) dt<1; (4.133) which cannot hold for allk 1, proving the claim. Moreover, we proved that it takes at most [K(T )=r ] + 1 applications of Theorem 4.3 to show thatu(;T ) is uniformly of Gevrey-classs, where [] denotes the integer part, andK(t) is as usual defined by (4.9). It is left to prove that the uniform radius of Gevrey-class regularity(T ) ofu(;T ) depends explicitly on the initial data and K(T ). Let k = [K(T )=r ] + 1 and hence T = T k . It follows from the above paragraph that (T ) k . By the induction assumptions (4.129)–(4.130) we bound k from below as k a 2k 0 k Y j=1 exp CK(T j1 )CK(T j ) 1 +C(T j T j1 )Q j1 +C(T j T j1 ) 2 M 2 (T j1 ) 2 : (4.134) Sincea 2k exp(2k log(1=a )) exp(2K(T ) log(1=a )=r ) we obtain that k 0 exp CK(T ) k Y j=1 1 +C(T j T j1 )Q j1 +C(T j T j1 ) 2 M 2 (T j1 ) 2 (4.135) for a sufficiently large constant C depending only on the domain. To estimate the product term in the above inequality we note that by (4.130) we have that Q j1 97 Q 0 +CM 2 (0) exp(CK(T j1 )), while from the Sobolev energy estimate we obtain M 2 (T j1 )M 2 (0) exp(CK(T j1 )). Therefore we have k 0 exp CK(T ) k Y j=1 1 +C(T j T j1 )Q 0 +C(T j T j1 )(1 +T )M 2 (0) exp CK(T j1 ) 2 0 exp CK(T ) exp C k X j=1 K(T j1 ) k Y j=1 1 +C(T j T j1 )Q 0 +C(T j T j1 )(1 +T )M 2 (0) 2 : By using the inequality between the arithmetic and the geometric mean, and the fact thatk =CK(T ), we obtain (T ) 0 exp C k X j=1 K(T j ) 1 + CTQ 0 +CT 2 M 2 (0) k 2k 0 exp CK 2 (T ) exp CTQ 0 CT 2 M 2 (0) : (4.136) Therefore we have proven the following statement, which is the main theorem of this paper. 4.6.1 Theorem. Let u 0 be divergence-free and of Gevrey-class s, with s 1, on a Gevrey-classs, open, bounded domainDR 3 , andr 5. Then the unique solution u(;t)2C([0;T );H r (D)) to the initial value problem (4.1)–(E.4) is of Gevrey-classs 98 for allt<T , whereT 2 (0;1] is the maximal time of existence inH r (D). Moreover, the radius(t) of Gevrey-class regularity of the solutionu(;t) satisfies (t)C 0 exp C Z t 0 ku(s)k W 1;1ds 2 ! exp Ctku 0 k X 0 Ct 2 ku 0 k 2 H r ; (4.137) for allt < T , whereC is a sufficiently large constant depending only on the domain D, 0 is the radius of Gevrey-class regularity of the initial datau 0 , andku 0 k X 0 is its Gevrey-class norm. 4.6.1 Remark. Theorem 4.6.1 also holds in the case of a two-dimensional Gevrey- class domain. In 2D it is known thatku(s)k W 1;1C exp(Ct) for some positive con- stantC = C(D;u 0 ), and therefore estimate (4.137) shows that the radius of Gevrey- class regularity of the solution is bounded from below byC exp(C exp(Ct)) for some C > 0, depending on the domain and on the initial data. We note that such a lower bound on(t) was obtained in the 2D analytic cases = 1 by Bardos, Benachour, and Zerner [7], whereas in the non-analytic Gevrey-class case on domains with generic boundary, Theorem 4.6.1 is the first such result (see also Chapter 2 or [38] for the periodic domain, and Chapter 3 [39] for the half-plane). 4.7 Appendix I: Combinatorial identities 4.7.1 Lemma. Letfa g, andfb ; g be sequences of positive numbers, where;2 N 3 0 . The identity X jj=m X jj=j; X j j=k; a b ; = 0 @ X j j=k a 1 A 0 @ X jj=mk X jj=jk; b ; 1 A 99 holds for positive integersj;k;m such thatkjm. Proof. X jj=m X jj=j; X j j=k; a b ; = X jj=m X j j=k; a X jj=j; b ( )( ); = X jj=m X j j=k; a X jj=jk; b ( ); = X jj=m X j j=k; a d ; (4.138) where we let d = X jj=jk; b ( ); : By Lemma 3.4.2 the far right side of (4.138) may be written as 0 @ X j j=k a 1 A 0 @ X jj=mk d 1 A = 0 @ X j j=k a 1 A 0 @ X jj=mk X jj=jk; b ; 1 A ; which concludes the proof of the lemma. 4.7.2 Lemma. Let 0 < < 1 andfa m;j g m0;j0 be a sequence of positive numbers. Then we have 1 X m=3 m X j=1 j X k=0 k a mk;jk = 3 1 a 0;0 + 2 1 (a 1;0 +a 1;1 ) + 1 (a 2;0 +a 2;1 +a 2;2 ) + 1 1 X m=3 a m;0 + 1 1 1 X m=3 m X j=1 a m;j : (4.139) 100 Proof. By re-indexing we have 1 X m=3 m X j=1 j X k=1 k a mk;jk = 1 X m=3 m X k=1 k m X j=k a mk;jk ! = 1 X m=3 m X k=1 k mk X j=0 a mk;j ! = 1 X m=3 m X k=1 k b mk ; where we denotedb l = P l j=0 = a l;j . By summing the geometric series in the far right side of the above equality may be re-written as ( 3 b 0 + 2 b 1 + P 1 j=2 b l )=(1). Therefore we obtain 1 X m=3 m X j=1 j X k=0 k a mk;jk = 1 X m=3 m X j=1 a m;k + 3 1 a 0;0 + 2 1 (a 1;0 +a 1;1 ) + 1 1 X m=2 m X j=0 a m;j ; and (4.139) follows by grouping appropriate terms. 4.8 Appendix II: Bounding the derivative of a supre- mum 4.8.1 Lemma. LetfF (t)g 2[0; 0 ] be a family of nonnegativeC 1 functions, where 0 1 is a fixed constant. Assume that 1.f _ F (t)g 2[0; 0 ] is a uniformly equicontinuous family, 2. for every fixedt, the functionsF (t) and _ F (t) depend continuously on. 101 Then for every fixedt2 (0;1) we have d + dt sup 2[0; 0 ] F (t) = lim sup h!0 + 1 h sup 2[0; 0 ] F (t +h) sup 2[0; 0 ] F (t) ! sup 2[0; 0 ] _ F (t): (4.140) Proof. Fixt2 (0;1). For a fixed2 [0; 0 ], andh> 0, we have F (t +h) = Z t+h t _ F (s)ds +F (t) Z t+h t sup 2[0; 0 ] _ F (s) ! ds + sup 2[0; 0 ] F (t): Therefore 1 h sup 2[0; 0 ] F (t +h) sup 2[0; 0 ] F (t) ! 1 h Z t+h t sup 2[0; 0 ] _ F (s) ! ds; and if we can prove that sup 2[0; 0 ] _ F (t) is a continuous function of t, then (4.140) holds, concluding the proof of the lemma. The fact that sup 2[0; 0 ] _ F (t) is a continuous function of t follows directly from the definition of uniform equicontinuity and the inequality sup 2[0; 0 ] a() sup 2[0; 0 ] b() sup 2[0; 0 ] ja()b()j; which holds for all functionsa;b : [0; 0 ]!R. 4.8.2 Lemma. Lete v =@ e u, for some2N 3 0 , ande u as in Lemma 4.3.1. Let f (t) = jj ke v(t;)k L 2 ( e ;t ) ; (4.141) andF (t) = f 2 (t). Then the familyfF (t)g 2[0; 0 ] satisfies the conditions (i) and (ii) of Lemma 4.8.1. 102 Proof. Let e t = S 2[0; 0 ] e ;t , and t (x) be the particle trajectory with initial datax. Without loss of generality assume that e t 2 e D for allt> 0, and thatke vk L 2 ( e ) 6= 0. The fact that for a fixedt the familyF (t) depends continuously on, follows from the continuity of the integral with respect to the Lebesgue measure, and the fact that e v2L 1 ( e D). Also, from (4.2) and the fact that det(@=@x) = 1, we have d dt ke v(t;)k 2 L 2 ( e ;t ) = 2 Z g(t; t (x))v(t; t (x))dx; (4.142) and sincegv2 L 1 (D), the continuity of the integral implies that _ F (t) depends con- tinuously on, so that condition (ii) holds. To show that the familyf _ F (t)g is uniformly equicontinuous, let > 0. We need to show that there exists =()> 0 such thatj _ F (t) _ F (s)j< for alljtsj and all2 [0; 0 ]. By (4.142) and the mean value theorem we have d dt ke v(t;)k 2 L 2 ( e ;t ) d dt ke v(s;)k 2 L 2 ( e ;s ) = 2 Z g(t; t (x))v(t; t (x))g(s; s (x))v(s; s (x)) dx Cjtsj sup z2(t;s) sup x2 j@ t (gv)(z; z (x))j +ju j (z; z (x))@ j (gv)(z; z (x))j Cjtsj k@ t (gv)k L 1 (D) +kuk L 1 (D) kgvk W 1;1 (D) ; since e t 2 e D for allt > 0. Recall thate g = [@ ;e u j @ j k @ k ]e u +@ (@ j k @ k e p), and so the right side of the above is bounded by Cjtsjkuk 3 H r+jj (D) for some sufficiently larger. To conclude the proof of the lemma one follows standard arguments. 103 4.8.3 Lemma. Lete v =@ e u, for some2N 3 0 , ande u as in Lemma 4.3.1. LetM (t) 0 be an upper bound jj d dt ke v(t;)k L 2 ( e ;t ) M (t) (4.143) which holds for allt 0 and all2 [0; 0 ] such thatke v(t;)k L 2 ( e ;t ) > 0. Further- more, assume that sup 2[0; 0 ] M (t) is continuous int. Then we have d + dt sup 2[0; 0 ] jj ke v(t;)k L 2 ( e ;t ) sup 2[0; 0 ] M (t); (4.144) where we denote byd + a(t)=dt = lim sup h!0+ (a(t +h)a(t))=h the right derivative of a function. Proof. Letf = jj ke v(t;)k L 2 ( e ;t ) and letF =f 2 . Note that by assumption we have _ f (t) M (t) for all2 [0; 0 ] andt > 0. It follows from Lemmas 4.8.1 and 4.8.2 that d + dt sup 2[0; 0 ] F (t) sup 2[0; 0 ] _ F (t): Due to the continuity in of the familyf , sup 2[0; 0 ] F = sup 2[0; 0 ] f ! 2 : Therefore, d + dt sup 2[0; 0 ] f (t) = d + dt sup 2[0; 0 ] F (t) 2 sup 2[0; 0 ] f (t) sup 2[0; 0 ] f (t)M (t) sup 2[0; 0 ] f (t) sup 2[0; 0 ] M (t); (4.145) 104 for all t such that sup 2[0; 0 ] f (t) > 0. This concludes the proof of the lemma by noting that if g(t) is a nonnegative function such that (d + =dt)g(t) G(t) for all t such that g(t) > 0, with G(t) continuous, then g(t) g(t 0 ) + R t t 0 G(s) ds, for all 0t 0 <t. 105 References [1] S. Alinhac and G. 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Abstract (if available)
Abstract
The Euler equations are the classical model for the motion of an incompressible inviscid homogeneous fluid. This thesis addresses geometric qualitative properties of smooth solutions to the Euler equations, namely the persistence of analyticity and Gevrey-class regularity on domains with smooth boundary.
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University of Southern California Dissertations and Theses
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Asset Metadata
Creator
Vicol, Vlad Cristian
(author)
Core Title
Analyticity and Gevrey-class regularity for the Euler equations
School
College of Letters, Arts and Sciences
Degree
Doctor of Philosophy
Degree Program
Mathematics
Publication Date
07/23/2010
Defense Date
05/24/2010
Publisher
University of Southern California
(original),
University of Southern California. Libraries
(digital)
Tag
analyticity radius,Euler equations,Gevrey-class,OAI-PMH Harvest
Language
English
Contributor
Electronically uploaded by the author
(provenance)
Advisor
Kukavica, Igor (
committee chair
), Friedlander, Susan (
committee member
), Lototsky, Sergey Vladimir (
committee member
), Newton, Paul K. (
committee member
), Ziane, Mohammed (
committee member
)
Creator Email
vicol@usc.edu,vlad.vicol@gmail.com
Permanent Link (DOI)
https://doi.org/10.25549/usctheses-m3214
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UC1321916
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etd-Vicol-3909 (filename),usctheses-m40 (legacy collection record id),usctheses-c127-351483 (legacy record id),usctheses-m3214 (legacy record id)
Legacy Identifier
etd-Vicol-3909.pdf
Dmrecord
351483
Document Type
Dissertation
Rights
Vicol, Vlad Cristian
Type
texts
Source
University of Southern California
(contributing entity),
University of Southern California Dissertations and Theses
(collection)
Repository Name
Libraries, University of Southern California
Repository Location
Los Angeles, California
Repository Email
cisadmin@lib.usc.edu
Tags
analyticity radius
Euler equations
Gevrey-class