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Certain regularity problems in fluid dynamics

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Certain regularity problems in fluid dynamics

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Content CERTAIN REGULARITY PROBLEMS IN FLUID DYNAMICS by Yuan Pei A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulllment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (APPLIED MATHEMATICS) Aug 2014 Copyright 2014 Yuan Pei Dedication To my grandfather and my parents ii Acknowledgments I would truly like to thank my thesis advisor, Prof. Igor Kukavica, for his guid- ance, enthusiasm, encouragement, and enlightenment for all my years at USC. My research and dissertation are only possible due to his unreservedly shared wisdom and knowledge. I also deeply appreciate Prof. Sergey Lototsky, Paul Newton, Walter Rusin, and Mohammed Ziane for serving my qualifying examination and dissertation committee and for their insightful comments and advice on this thesis. Besides, many thanks must go to Prof. Francis Bonahon, Cymra Haskell, and Alan Schumitzky for their help on my graduate study and teaching skills. I have been lucky to be a member of the Department of Mathematics at USC with all the nice faculty and sta, who make my years here so enjoyable. I must also acknowledge and cherish all my friends whose friendship, hospitality, humor, and knowledge have always inspired and entertained me. Last but not least, I would like to thank all my family members, especially my parents Qingping Xu and Xinli Pei. They deserve special thanks for not only providing me the best environment to grow up but also caring me and educating me with so much patience, whole-hearted commitment and endless love that I will never forget. iii Table of Contents Dedication ii Acknowledgments iii Abstract vi Chapter 1: On the Partial Regularity of the Navier-Stokes Equations 1 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Notation and main theorems . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Fundamental lemmas and fractal estimates . . . . . . . . . . . . . . 8 1.4 On the -fractal dimension . . . . . . . . . . . . . . . . . . . . . . . 29 Chapter 2: On the Well-posedness of the Primitive Equations of the Ocean 40 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.2 Main results on the whole domain . . . . . . . . . . . . . . . . . . . 43 2.3 Proof of existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.4 Proof of uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 2.5 The results on the periodic domain . . . . . . . . . . . . . . . . . . . 74 2.6 Construction of solutions . . . . . . . . . . . . . . . . . . . . . . . . 76 iv Appendix A: The Detailed Proof of Regularity of Solutions for the Partial Regularity Results 84 Appendix B: Proof of the Higher Regularity of Solutions to the Primitive Equations of the Ocean 92 References 98 v Abstract In the rst chapter of this dissertation, we address the partial regularity for a suitable weak solutions of the Navier-Stokes system in a bounded space-time domain D. We show that the parabolic fractal dimension of the singular set is less than or equal to 45=29, which is an improvement of the earlier result from [K4]. Also, we introduce the new -fractal dimension and prove that the dimension of the singular set is bounded by 3=2 for a certain range of . The second chapter addresses the existence and uniqueness of the solutions for the primitive equations of the ocean with continuous initial datum. We split the initial data into a regular nite energy part and a small bounded part and show that the equations are preserved by the splitting, which enables us to prove the well-posedness of solutions. We provide a priori estimates and the construction of the solutions. Keywords: Navier-Stokes equations, weak solutions, fractal dimension, regularity, primitive equations. vi Chapter 1: On the Partial Regularity of the Navier-Stokes Equations 1.1 Introduction The classical results on the existence of weak solutions of the 3D Navier-Stokes equations @u @t u + 3 X j=1 @ j (u j u) +rp =f (1.1) ru = 0 have been proven by Leray and Hopf [CF,Le,T] for square integrable initial conditions and with proper assumptions on the forcing term. It has also been known since Leray's time that the 2D Navier-Stokes equations have a unique regular solution that exists globally in time. However, the global uniqueness and regularity of the weak solutions for the 3D Navier-Stokes equations still remain open. In the papers [S1{S3], Scheer addressed the partial regularity of solutions by estimating the size of the singular set for a given suitable weak solution. Namely, Scheer proved the following theorem. 1 Theorem 1.1. For f = 0, there exists a weak solution of the Navier-Stokes equations whose singular set S satises H 5=3 (S)<1; and H 1 (S\ ( ftg))<1; uniformly for all time t at which the solution does not blow up. HereH k denotes the k dimensional Hausdor measure. The partial regularity problem was further studied by Cafarrelli, Kohn, and Nirenberg in the classical paper [CKN], where they obtained the following result. Theorem 1.2. For any suitable weak solution of the Navier-Stokes system on an open set in space-time, the associated singular set S satisesP 1 (S) = 0: HereP 1 denotes the parabolic Hausdor measure where in the denition the coverings by balls are replaced with coverings by parabolic cylinders. In [RS1,RS2], Robinson and Sadowski initiated the study of the fractal dimension, also known as the box-counting dimension of the singular set by showing that the set of the space-time singular points have fractal dimension at most 5=3. Later, Kukavica proved in [K4] that the fractal dimension of the singular set is at most 135=82. In this thesis, we rst address the problem of the fractal dimension of the singular 2 set. By denition, the fractal dimension is dierent from the Hausdor dimension used by [CKN] since it requires covering the set in space-time with parabolic cylinders of the same radius rather than allowing radii of the cubes to vary. We show that the parabolic fractal dimension of the set of the space-time singular points has dimension at most 45=29. This improvement over [K4] is mainly from a better treatment of the pressure term. By considering the gradient of the pressure, we are able to improve the upper bound of the right side of the local energy inequality. In addition, due to the dierent treatment of the pressure, we no longer need the intermediate radius as used in [K4]. Also, we introduce a new test function replacing the backward heat kernel whose main feature is the constant spatial integral. This property is explored when estimating the linear part of the local energy inequality. In the proof, we apply a dyadic decomposition for each term in the local energy inequality which is dierent from [K4] where only non-linear terms were dyadically decomposed. Moreover, we introduce the new-fractal dimension. In the denition, rather than cover the singular set with parabolic cylinders of xed radius, we cover the bounded set inR 3 R with parabolic cubes of radii between and , where 0<< 1 and some 1. This new dimension is formally between the Hausdor dimension and the fractal dimension. Namely, by setting ! 1 we obtain the usual fractal dimension and by letting !1 the dimension nally converges to the Hausdor dimension. 3 However, conceptually, this dimension seems more close to the concept of the fractal dimension. Besides, the new-fractal dimension enables us to get a better dimension as low as 3=2, which is strictly smaller than our rst result 45=29. We point out that the constant we choose here is independent of the energy and the solution to the Navier-Stokes system so our result is general in this sense. 1.2 Notation and main theorems We start by recalling the denition of a suitable weak solution of the Navier-Stokes system @u @t u + 3 X j=1 @ j (u j u) +rp =f (1.2) ru = 0; where we have set the viscosity to 1. Let D be a bounded domain. We say that a pair (u;p) is a suitable weak solution of the Navier-Stokes equations if it satises (i) u2L 1 t L 2 x (D) andru2L 2 t L 2 x (D) with p2L 5=3 (D); (ii) f2L 5=3 (D) is divergence free; (iii) the Navier-Stokes system (1.2) holds inD 0 (D); 4 (iv) the local energy inequality holds in D, i.e., Z R 3 juj 2 j T dx + 2 ZZ R 3 R jr(u)j 2 dxdt ZZ R 3 R juj 2 ( t + ) + (juj 2 + 2p)ur + 2(uf) dxdt (1.3) for all 2C 1 0 (D) such that 0 in D and for almost all T2R. Now we recall from [K2] the denition of the parabolic fractal dimension. Let AR 3 R. Forr> 0, we denote byN(r) the minimal number of centered parabolic cylinders Q r (x;t) =B r (x) (tr 2 ;t +r 2 ) needed to cover the bounded subset A, i.e., N(r) = minfN2N 0 :9(x 1 ;t 1 ); ; (x N ;t N )2R 3 R; such that A[ N i=1 Q r (x i ;t i )g: Then the parabolic fractal dimension is dened by dim pf (A) = lim sup r!0 + logN(r) logr : 5 Note that the dimension does not change if instead of the centered parabolic cylinders we use the non-centered ones Q r (x;t) =B r (x) (tr 2 ; 0): Recall that a point (x;t)2 D is regular for a suitable weak solution (u;p) if the solution is bounded in some neighborhood of (x;t), and that a point in singular otherwise. We denote by S the set of singular points. The next statement, which provides an estimate of the fractal dimension of the singular set, is our rst main result (cf. [KP]). Theorem 1.3. For (u;p) as above, we have dim pf (S\K) 45=29; for any compact set KD. In order to prove Theorem 1.3, we provide the next statement about the regularity criterion for a point. Theorem 1.4. There exists a suciently small universal constant > 0, such that 6 if 0 1, and if ZZ Q juj 10=3 +jruj 2 +jpj 5=3 +jrpj 5=4 +jfj 5=3 dxdt 45=29 ; then (0; 0) is a regular point. Proof of Theorem 1.3. Fix a compact subsetKD as in the statement of Theorem 1.3, and let s2 (0; dist(K;D c )) be such that s< 1. For every (x;t)2S\K, we have by Theorem 1.4 ZZ Q s=3 (x;t) juj 10=3 +jruj 2 +jpj 5=3 +jrpj 5=4 +jfj 5=3 dxdt (s=3) 45=29 with Q s (x;t) D. Applying the Vitali covering argument to the above family of cylinders, we nd a disjoint nite subfamilyfQ s=3 (x j ;t j )g n j=1 , such that S\K[ n j=1 Q s (x j ;t j ): 7 Therefore, since the parabolic cylinders Q s=3 (x j ;t j ) are disjoint, we get I = ZZ D juj 10=3 +jruj 2 +jpj 5=3 +jrpj 5=4 +jfj 5=3 dxdt n X j=1 ZZ Q s=3 (x j ;t j ) juj 10=3 +jruj 2 +jpj 5=3 +jrpj 5=4 +jfj 5=3 dxdt n (s=3) 45=29 : Therefore, we conclude N(r)n I (s=3) 45=29 ; which implies dim pf (S\K) 45=29. 1.3 Fundamental lemmas and fractal estimates We start with the description of the test function used in the proofs below. First, x a function 2C 1 0 (R 3 ) such that = 1 on B 3=4 =B 3=4 (0; 0), and suppB 1 , with Z R 3 dx = 1: 8 Let 0 < r =2 where 2 (0; 1), and denote throughout = r=. Also let 2C 1 (R) be a function such that = 1 on [1;1) and supp [0;1). We shall use the test function (x;t) =r 2 G(x;r 2 t) t + 2 2 t + (r=2) 2 (r=2) 2 x where G is the backward heat kernel, i.e., G(x;t) = 1 (4(r 2 t)) 3=2 exp jxj 2 4(r 2 t) : From [K1,K2], we recall the fundamental property j t (x;t) + (x;t)jC r 2 5 : We also have the bounds j(x;t)j 1 Cr ; (x;t)2Q r j(x;t)j Cr 2 R 3 ; (x;t)2Q 2R nQ R (1.4) 9 and jr(x;t)j Cr 2 R 4 ; (x;t)2Q 2R nQ R where r < R. In order to obtain these estimates, we need the following dyadic decomposition of Q . Let 0 (t) = t + 4r 2 3r 2 ; and m (t) = t + 4(2 m r) 2 3(2 m r) 2 t + 4(2 m1 r) 2 3(2 m1 r) 2 ; for m 1. Similarly to the bounds on , we have that on D m = (x;t) :jxj 2 m r;(2 m+1 r) 2 t(2 m1 r) 2 ; for m 1,j m j 1 andjj C=2 3m r. For time derivatives, we havej@ t m j C=(2 m r) 2 andj@ t j C=2 5m r 3 . As for the spatial derivatives we havej@ j C=2 (3+jj)m r jj+1 . Also, denote D 0 =f(x;t) :jxjr;(2r) 2 t 0g, and g denotes the spatial average over the specied cylinders of any function g. 10 In the following lemmas, we estimate the three terms on the right side of the local energy inequality. First, we introduce the notations (x;t) () = 1 1=2 ku(;t)k L 2 x (B(x)) ; (x;t) () = 1 1=2 kruk L 2 x;t (Q(x;t)) ; (x;t) () = 1 1=2 kuk L 10=3 x;t (Q(x;t)) : Also, for simplicity, we write () and () for (0;0) () and (0;0) (), respectively. Lemma 1.5. We have ZZ tt 0 juj 2 ( t + )dxdt C r ()() +C r 2 () 2 ; forr 2 t 0. 11 Proof. Denoting by I 1 the left side of the above inequality, we have I 1 = ZZ tt 0 juj 2 ( t + )dxdt = Z t 0 2 =4 Z R 3 (juj 2 juj 2 )( t + )dxdt + Z t 0 2 =4 Z R 3 juj 2 ( t + )dxdt + Z t 0 2 =4 Z R 3 (juj 2 )( t + )dxdt =I 11 +I 12 +I 13 Denote by m 0 the largest integer such that (2 m r) 2 < 2 so that we have 1 = m 0 X m=0 m (t) for t2 [ 2 ; 0]. Then we apply the above equality to I 11 in order to get I 11 = m 0 X m=0 ZZ Dm (juj 2 juj 2 )(( m ) t + ( m ))dxdt C m 0 X m=0 juj 2 juj 2 L 1 x;t (Dm) 1 2 5m r 3 C m 0 X m=0 juj 2 juj 2 L 1 x;t (Q 2 m r ) 1 2 5m r 3 (1.5) 12 Then by the H older's inequality and the Gagliardo-Nirenberg inequalities, with (2 m r) 2 < 2 ; the right side of (1.5) is less than or equal to C m 0 X m=0 (2 m r) 2 juj 2 juj 2 L 5=4 t L 15=7 x (Q) 1 2 5m r 3 C m 0 X m=0 1 2 3m r kuk L 10=3 x;t (Q) kruk L 2 x;t (Q) : Summing the above series gives jI 11 j r ()(): Denote = K(t) , where K(t) = Cr 2 =(r 2 t) 3=2 , and recall thatjuj 2 is the spatial average ofjuj 2 over B , which is a function of time variable only. Also, for 2 =4tt 0 , we have 4(t + 2 ) 3 2 = 1: 13 Thus, we deduce I 12 = Z t 0 2 =4 Z R 3 juj 2 ( t + )dxdt = Z t 0 2 =4 K(t)juj 2 dt Z R 3 ( t + )dx = Z t 0 2 =4 K(t)juj 2 dt (@ t Z R 3 dx + Z R 3 dx) = 0; and the last equality holds due to the constant spatial integral of . Next, for I 13 , by the H older's inequality, we have I 13 Cr 2 5 ku 2 k L 2 x;t (Q) Cr 2 3 kuk 2 L 10=3 x;t (Q) : Therefore, the lemma follows. Lemma 1.6. We have I 2 = ZZ tt 0 juj 2 u j @ j dxdt C r 3=2 () 1=2 () 5=2 ; forr 2 t 0. 14 Proof. Using the same dyadic decomposition as in Lemma 1.5, we have I 2 = m 0 X m=0 ZZ ftt 0 g\Dm juj 2 u j @ j ( m ) C m 0 X m=0 r 2 (2 m r) 4 juj 2 juj 2 L 5=3 t L 15=8 x (Q 2 m r ) kuk L 10=3 t L 15=7 x (Q 2 m r ) : We apply the Gagliardo-Nirenberg inequality to deduce that the right side of the above inequality is less than or equal to Cr 2 m 0 X m=0 1 (2 m r) 7=2 kuk 3=2 L 10=3 x;t (Q 2 m r ) kruk 1=2 L 2 x;t (Q 2 m r ) kuk L 10=3 x;t (Q 2 m r ) C m 0 X m=0 1 2 7m=2 r 3=2 kuk 5=2 L 10=3 x;t (Q 2 m r ) kruk 1=2 L 2 x;t (Q 2 m r ) (1.6) C m 0 X m=0 1 2 7m=2 r 3=2 kuk 5=2 L 10=3 x;t (Q) kruk 1=2 L 2 x;t (Q) : Then the lemma follows by summing the above series in m. For the next lemma, we rst denote, (x;t)() = 1 1=2 kpk 1=2 L 5=3 x;t (Q(x;t)) ; (x;t)() = 1 1=2 krpk 1=2 L 5=4 x;t (Q(x;t)) : 15 Also we write () and () for (0;0) () and (0;0) (), respectively. Lemma 1.7. We have jI 3 j = ZZ tt 0 pu j @ j dxdt C r 3=2 ()()(); forr 2 t 0. Proof. Similarly to the previous proof, we write I 3 = m 0 X m=0 ZZ ftt 0 g\Dm pu j @ j ( m ) C m 0 X m=0 1 2 4m r 2 kppk L 5=3 x;t (Q 2 m r ) kuk L 10=3 x;t (Q 2 m r ) C m 0 X m=0 1 2 4m r 2 kppk 1=2 L 5=3 x;t (Q 2 m r ) kppk 1=2 L 5=4 x;t (Q 2 m r ) kuk L 10=3 x;t (Q 2 m r ) C m 0 X m=0 1 2 4m r 2 kppk 1=2 L 5=3 x;t (Q 2 m r ) (2 m r) 1=2 kppk 1=2 L 5=4 t L 15=7 x (Q 2 m ) kuk L 10=3 x;t (Q 2 m r ) : Then by the Gagliardo-Nirenberg inequality, the far right side of the above inequality 16 is less than or equal to C m 0 X m=0 1 2 7m=2 r 3=2 kpk 1=2 L 5=3 x;t (Q 2 m r ) krpk 1=2 L 5=4 x;t (Q 2 m r ) kuk L 10=3 x;t (Q 2 m r ) (1.7) C m 0 X m=0 1 2 7m=2 r 3=2 kpk 1=2 L 5=3 x;t (Q) krpk 1=2 L 5=4 x;t (Q) kuk L 10=3 x;t (Q) : Thus Lemma 1.7 follows by summing the above series in m. The next lemma provides the pressure estimates. Lemma 1.8. For 0<r, we have krpk L 5=4 x;t (Qr ) C()() +C r 12=5 7=5 () 2 : Proof. For i;j 2f1; 2; 3g, denote U ij =u i u j . Let 0 : R 3 ! [0; 1] be a C 1 function such that = 1 on B 3=5 and supp()B 4=5 and let (x) = 0 (x=). Using p =@ i (u j @ j u i ) from taking the divergence of the Navier-Stokes equation, we write (rp) =@ ij (rU ij ) + (@ ij )rU ij @ j (rU ij @ i ) @ i (rU ij @ j )rp + 2@ j ((@ j )rp) 17 Using the Newtonian potential N(x) = 1 4jxj we may invert the Laplacian as rp =R i R j (rU ij ) +N ((@ ij )rU ij )@ j N (rU ij @ i ) @ i N (rU ij @ j )N (rp) + 2@ j N ((@ j )rp) =q 1 +q 2 +q 3 +q 4 +q 5 +q 6 ; where R i is the i-th Riesz transform. Now we estimate each of the above six terms. For q 1 , we have by the Calder on-Zygmund theorem and the H older's inequality kq 1 k L 5=4 x (Br ) kq 1 k L 5=4 x (R 3 ) C X i;j krU ij k L 5=4 (B) C X i;j ku i ru j k L 5=4 x (B) Ckuk L 10=3 x (B) kruk L 2 x (B) : 18 We apply H older's inequality and integrate in time fromr 2 to 0 in order to get kq 1 k L 5=4 x (Qr ) C kuk L 10=3 x (B) kruk L 2 x (B) L 5=4 t (r 2 ;0) Ckuk L 10=3 x;t (Q) kruk L 2 x;t (Q) : For q 2 , we usej@ ij j 1= 2 and the Sobolev inequalities to obtain kq 2 k L 5=4 x (Br ) Cr 12=5 kq 2 k L 1 x (Br ) Cr 12=5 X i;j krU ij @ ij k L 1 x (B) r 12=5 3 X i;j ku i ru j k L 1 x (B) Cr 12=5 3 kuk L 2 x (B) kruk L 2 x (B) C r 12=5 kuk L 10=3 x (B) kruk L 2 x (B) : forr 2 < t < 0. Therefore by integrating in time fromr 2 to 0 and using the H older's inequality, we obtain kq 2 k L 5=4 x (Qr ) C r 12=5 kuk L 10=3 x (B) kruk L 2 x (B) L 5=4 t (r 2 ;0) C r 12=5 kuk L 10=3 x;t (Q) kruk L 2 x;t (Q) : 19 For q 3 , usingj@ i j 1=, we get kq 3 k L 5=4 x (Br ) Cr 12=5 kq 3 k L 1 x (Br ) Cr 12=5 2 X i;j krU ij @ i k L 1 x (B) r 12=5 3 X i;j ku i ru j k L 1 x (B) Cr 12=5 3 kuk L 2 x (B) kruk L 2 x (B) C r 12=5 kuk L 10=3 x (B) kruk L 2 x (B) : forr 2 <t< 0. Then by integrating in time fromr 2 to 0 we obtain kq 3 k L 5=4 x (Qr ) C r 12=5 kuk L 10=3 x (B) kruk L 2 x (B) L 5=4 t (r 2 ;0) C r 12=5 kuk L 10=3 x;t (Q) kruk L 2 x;t (Q) : For q 4 , interchanging the sub-indices i and j in the forgoing estimates for q 3 , we get kq 4 k L 5=4 x (Qr ) Cr 12=5 2 k X i;j krU ij @ j k L 1 x (B) k L 5=4 t (r 2 ;0) C r 12=5 kuk L 10=3 x (B) kruk L 2 x (B) L 5=4 t (r 2 ;0) C r 12=5 kuk L 10=3 x;t (Q) kruk L 2 x;t (Q) : 20 Regarding q 5 we have kq 5 k L 5=4 x;t (Qr ) Cr 12=5 kq 5 k L 5=4 t L 1 x (Qr ) C r 12=5 3 krpk L 5=4 t L 1 x (Qr ) C r 12=5 krpk L 5=4 x;t (Q) : As for q 6 , we have kq 6 k L 5=4 x;t (Qr ) Cr 12=5 kq 6 k L 5=4 t L 1 x (Qr ) C r 12=5 2 k(@ i )rpk L 5=4 t L 1 x (Qr ) C r 12=5 3 krpk L 5=4 t L 1 x (Q) C r 12=5 krpk L 5=4 x;t (Q) : Thus the lemma follows by combining the bounds of q 1 through q 6 . Lemma 1.9. For 0<r, we have kpk L 5=3 x;t (Qr ) Cr 1=5 9=5 ()() +Cr 2 ()() +C r 9=5 4=5 () 2 Proof. Following the same notation and inverting the Laplacian as in Lemma 1.8, we 21 obtain p =R i R j (U ij ) +N ((@ ij )U ij )@ j N (U ij @ i ) @ i N (U ij @ j )N (p) + 2@ j N ((@ j )p) =p 1 +p 2 +p 3 +p 4 +p 5 +p 6 ; where R i is the i-th Riesz transform. Next we estimate each of the above six terms. For p 1 , we have by the Calder on-Zygmund theorem and the H older's inequality kp 1 k L 5=3 (Br ) kp 1 k L 5=3 (R 3 ) C X i;j kU ij k L 5=3 (Br ) C X i;j ku i k L 30=13 (B) ku j u j k L 6 (B) Ckuk L 30=13 (B) kruk L 2 (B) C 2=5 kuk L 10=3 (B) kruk L 2 (B) : Then we integrate in time fromr 2 to 0 in order to get kp 1 k L 5=3 (Qr ) C 2=5 kuk L 10=3 (B) ruk L 2 (B) k L 5=3 (r 2 ;0) Cr 1=5 2=5 kuk L 10=3 (B) ruk L 2 (B) k L 2 (r 2 ;0) : 22 By applying the H older's inequality we obtain kp 1 k L 5=3 (Qr ) Cr 1=5 2=5 kuk L 2 t L 10=3 x (Q) kruk L 2 x;t (Q) Cr 1=5 4=5 kuk L 10=3 x;t (Q) kruk L 2 x;t (Q) : For p 2 , we usej@ ij j 1= 2 and the Sobolev inequalities, forr 2 <t< 0, to get kp 2 k L 5=3 (Br ) Cr 9=5 kp 2 k L 1 (Br ) C r 9=5 kU ij @ ij k L 1 (B) C r 9=5 3 kU ij k L 1 (B) Cr 9=5 2 X i;j ku i k L 2 (B) ku j u j k L 6 (B) C r 9=5 7=5 kuk L 10=3 (B) kruk L 2 (B) : Then we integrate in time fromr 2 to 0 to obtain kp 2 k L 5=3 (Qr ) C r 9=5 7=5 kuk L 10=3 (B) ruk L 2 (B) k L 5=3 (r 2 ;0) C r 2 7=5 kuk L 10=3 (B) ruk L 2 (B) k L 2 (r 2 ;0) C r 2 kuk L 10=3 x;t (Q) kruk L 2 x;t (Q) : 23 For p 3 , usingj@ i j 1= we have kp 3 k L 5=3 (Br ) Cr 9=5 kp 3 k L 1 (Br ) C r 9=5 2 kU ij @ i k L 1 (B) C r 9=5 3 kU ij k L 1 (B) Cr 9=5 2 X i;j ku i k L 2 (B) ku j u j k L 6 (B) C r 9=5 7=5 kuk L 10=3 (B) kruk L 2 (B) : Then we integrate in time fromr 2 to 0 and apply the H older's inequality to get kp 3 k L 5=3 (Qr ) C r 9=5 7=5 kuk L 10=3 (B) ruk L 2 (B) k L 5=3 (r 2 ;0) C r 2 7=5 kuk L 10=3 (B) ruk L 2 (B) k L 2 (r 2 ;0) C r 2 kuk L 10=3 x;t (Q) kruk L 2 x;t (Q) : For p 4 , we estimate analogously as for p 3 by swapping the sub-indices i and j as kp 4 k L 5=3 (Br ) Cr 9=5 kp 4 k L 1 (Br ) C r 9=5 2 kU ij @ j k L 1 (B) C r 9=5 3 kU ij k L 1 (B) Cr 9=5 2 X i;j ku i k L 2 (B) ku j u j k L 6 (B) C r 9=5 7=5 kuk L 10=3 (B) kruk L 2 (B) : 24 Thus, we bound p 4 by integrating in time fromr 2 to 0 as kp 4 k L 5=3 (Qr ) C r 9=5 7=5 kuk L 10=3 (B) ruk L 2 (B) k L 5=3 (r 2 ;0) C r 2 kuk L 10=3 x;t (Q) kruk L 2 x;t (Q) : Regarding p 5 we have kp 5 k L 5=3 x;t (Qr ) Cr 9=5 kp 5 k L 5=3 t L 1 x (Br(r 2 ;0)) C r 9=5 kpk L 5=3 t L 1 x (R 3 (r 2 ;0)) C r 9=5 3 kpk L 5=3 t L 1 x (Q) C r 9=5 3 6=5 kpk L 5=3 x;t (Q) C r 9=5 kpk L 5=3 x;t (Q) : As for p 6 , we proceed similarly as kp 6 k L 5=3 x;t (Qr ) Cr 9=5 kp 6 k L 5=3 t L 1 x (Br(r 2 ;0)) C r 9=5 2 kp@ j k L 5=3 t L 1 x (R 3 (r 2 ;0)) C r 9=5 3 kpk L 5=3 t L 1 x (Q) C r 9=5 3 6=5 kpk L 5=3 x;t (Q) C r 9=5 kpk L 5=3 x;t (Q) : Thus, the lemma follows by combining the bounds of p 1 through p 6 . 25 The following lemma provides an estimate for the forcing term. Lemma 1.10. For 0<r, we have jI 4 j = 2 ZZ tt 0 (uf)dxdt C r 2 5=2 kuk L 10=3 x;t (Q(x;t)) kfk L 5=3 x;t (Q(x;t)) : Proof. This follows immediately after applying the H older's inequality and the esti- mate (1.4). Proof of Theorem 1.4: By assuming ZZ Q juj 10=3 +jruj 2 +jpj 5=3 +jrpj 5=4 +jfj 5=3 dxdt 45=29 ; we get kuk L 10=3 x;t (Q) 3=10 27=58 ; (1.8) kruk L 2 x;t (Q) 1=2 45=58 ; (1.9) 26 kpk L 5=3 x;t (Q) 3=5 27=29 ; (1.10) krpk L 5=4 x;t (Q) 4=5 36=29 ; (1.11) kfk L 5=3 x;t (Q) 3=5 27=29 : (1.12) Together with r = 30=29 M ; and by Lemmas 1.5, 1.6, and 1.7, we have for the right side of the local energy 27 inequality (1.3) ZZ R 3 R juj 2 ( t + ) + (juj 2 + 2p)ur + 2(uf) dxdt C r kuk L 10=3 x;t (Q) kruk L 2 x;t (Q) + Cr 2 3 kuk 2 L 10=3 x;t (Q) + C r 3=2 kuk 5=2 L 10=3 x;t (Q) kruk 1=2 L 2 x;t (Q) + C r 3=2 kpk 1=2 L 5=3 x;t (Q) krpk 1=2 L 5=4 x;t (Q) kuk L 10=3 x;t (Q) + Cr 2 5=2 kuk L 10=3 x;t (Q) kfk L 5=3 x;t (Q) : Then we substitute (1.8) through (1.11) to bound the far right side of the above inequality by C 4=5 6=29 +C 3=5 +C +C +C 9=10 28=29 C( 4=5 + ): On the other hand, by the lower bound of , the left side of the energy inequality (1.3) satises Z R 3 juj 2 j t 0 dx + 2 Z t 0 1 Z R 3 jr(u)j 2 dxdt C r ku(;t 0 )k 2 L 2 x (Br ) + C r kruk 2 L 2 x (Qr ) = 2 (r) + 2 (r): 28 Therefore we get maxf(r);(r)gC . If we also have (r)C , then it follows that (0; 0) is a regular point, by [CKN]. Indeed, by Lemma 1.8, we obtain krpk L 5=4 x;t (Qr ) C( 30=29 ) 3=5 2=5 1=2 45=58 3=10 27=58 +C 30=29 12=5 4=5 36=29 =C 4=5 328=145 +C 4=5 192=145 2C 4=5 192=145 : Therefore, (r) = 1 r 1=2 krpk 1=2 L 5=4 x;t (Qr ) C r 1=2 2=5 96=145 C 2=5 21=145 C 2=5 : Combining the above upper bounds, in particular, using the proof in the Appendix, the theorem is proved. 1.4 On the -fractal dimension In this section we prove that for a certain range of , the -fractal dimension of the singular set of solutions to the Navier-Stokes equations is at most 3=2. We take the advantage of our test function as well as its dyadic decomposition to bound the 29 energy on each dyadic cube Q 2 m r rather than simply on Q . First we introduce the -fractal dimension, where 2 [1;1). Let AR 3 R be bounded. For r 0 2 (0; 1) and d 0, let F d; r 0 (A) = inf X i R d i :9(x 1 ;t 1 ); ; (x N ;t N )2R 3 R such that A N [ i=1 Q R i (x i ;t i ) with r 0 R 1 ;:::;R N r 0 and F d; (A) = lim r 0 !0 F d; r 0 (A): Then the -fractal dimension is dened by dim -pf (A) = inffd 0 :F d; (A) = 0g: Note that when = 1 the-fractal dimension agrees with the fractal dimension, while for =1 it coincides with the parabolic Hausdor dimension; in the latter case, the condition r 0 R 1 ;:::;R N r 0 is interpreted as 0R 1 ;:::;R N r 0 (which is the reason why we required r 0 2 (0; 1) above). Actually, by [F], the denition of the 30 -fractal dimension is generalized as follows, for r 0 > 0, let H h r 0 ;d; (A) = inf X i h(r) :A [ i Q r (x;t) ; where the function h(t) =h(t;r 0 ;d;);t 0 is dened as h(t;r 0 ;d;) = 8 > > > > > > < > > > > > > : 0 if 0<t<r 0 t d if r 0 tr 0 0 if t>r 0 and H h d; (A) = lim r 0 !0 H h r 0 ;d; (A): Then the denition of the -fractal dimension is dim -pf (A) = inffd 0 :H h d; (A) = 0g: By substituting the function h(t;r 0 ;d;) by ~ h(t) = ~ h(t;r 0 ;d), which is dened as ~ h(t;r 0 ;d) =t d fr=r 0 g ; 31 where is the characteristic function, we recover the denition of the fractal dimension in the previous section. The next statement contains our second main result: it provides the bound for the -fractal dimension. Theorem 1.11. For 21=20, the parabolic -fractal dimension of the singular set is less than or equal to 3=2. Namely, dim -pf (S\K) = 3=2; for any compact set KD. We need the next theorem to prove Theorem 1.11 and we debrief the proof after the statement of the theorem. Theorem 1.12. There exists a suciently small universal constant > 0, such that if 0 1, and ZZ Q R juj 10=3 +jruj 2 +jpj 5=3 +jrpj 5=4 +jfj 5=3 dxdt R 3=2 (1.13) for R, 21=20 and =M with M suciently large, then (0; 0) is a regular point. 32 Proof of Theorem 1.11 assuming Theorem 1.12. Fix a compact subset K D as in the statement of Theorem 1.11, and let s2 (3 ; 3) be such that s < 1. Note if Q s (x;t)\D c 6=; then we replace Q s (x;t) by Q s (x;t)\D. Thus, for every (x;t)2S\K, we have by Theorem 1.12 ZZ Q s=3 (x;t) juj 10=3 +jruj 2 +jpj 5=3 +jrpj 5=4 +jfj 5=3 dxdt (s=3) 3=2 : By applying the Vitali covering lemma to the above family of parabolic cylinders, we nd a disjoint nite subfamilyfQ s=3 (x j ;t j )g n j=1 , such that S\K[ n j=1 Q s (x j ;t j ): Therefore, since the parabolic cylinders Q s=3 (x j ;t j ) are disjoint, we get I = ZZ D juj 10=3 +jruj 2 +jpj 5=3 +jrpj 5=4 +jfj 5=3 dxdt n X j=1 ZZ Q s=3 (x j ;t j ) juj 10=3 +jruj 2 +jpj 5=3 +jrpj 5=4 +jfj 5=3 dxdt n (s=3) 3=2 : 33 Therefore, we obtain N(r)n I (s=3) 3=2 ; which implies dim -pf (S\K) 3=2. Proof of Theorem 1.12. By the assumption of the theorem, we have kuk L 10=3 x;t (Q 2 m r ) 3=10 (2 m r) 9=20 ; (1.14) kruk L 2 x;t (Q 2 m r ) 1=2 (2 m r) 3=4 ; (1.15) kpk L 5=3 x;t (Q 2 m r ) 3=5 (2 m r) 9=10 ; (1.16) krpk L 5=4 x;t (Q 2 m r ) 4=5 (2 m r) 6=5 ; (1.17) kfk L 5=3 x;t (Q 2 m r ) 3=5 (2 m r) 9=10 ; (1.18) 34 for 2 m r, 0mm 0 . Then we modify the proof of the previous lemmas in Section since we now have the above better estimates on each level of the dyadic decomposition. From (1.5) in the proof of Lemma 1.5, we proceed to estimate I 11 as follows I 11 m 0 X m=0 juj 2 juj 2 L 1 x;t (Q 2 m r ) 1 2 5m r 3 C m 0 X m=0 (2 m r) 2 kjuj 2 juj 2 k L 5=4 t L 15=7 x (Q 2 m r ) 1 2 5m r 3 C m 0 X m=0 1 2 3m r kuk L 10=3 x;t (Q 2 m r ) kruk L 2 x;t (Q 2 m r ) C m 0 X m=0 1 2 3m r 3=10 (2 m r) 9=20 1=2 (2 m r) 3=4 : Using the same estimates on I 13 , and by substituting the analogues of equations (1.8) and (1.9) by the condition of the theorem, we obtain that the far right side of the above inequality is bounded by C 4=5 r 1=5 m 0 X m=0 1 2 9m=5 C 4=5 r 1=5 : (1.19) Here and in the sequel, we choose r = . From (1.6) in the proof of Lemma 1.6, we 35 bound the expression in (1.19) by C m 0 X m=0 1 2 7m=2 r 3=2 kuk 5=2 L 10=3 x;t (Q 2 m r ) kruk 1=2 L 2 x;t (Q 2 m r ) C m 0 X m=0 1 2 7m=2 r 3=2 ( 3=10 ) 5=2 ((2 m r) 9=20 ) 5=2 ((2 m r) 3=4 ) 1=2 C m 0 X m=0 1 2 7m=2 r 3=2 (2 m r) 3=2 C : Next we estimate (1.7) of Lemma 1.7 using the -fractal condition as C m 0 X m=0 1 2 7m=2 r 3=2 kpk 1=2 L 5=3 x;t (Q 2 m r ) krpk 1=2 L 5=4 x;t (Q 2 m r ) kuk L 10=3 x;t (Q 2 m r ) C m 0 X m=0 1 2 7m=2 r 3=2 3=10 ((2 m r) 9=10 ) 1=2 4=10 ((2 m r) 6=5 ) 1=2 3=10 (2 m r) 9=20 C m 0 X m=0 1 2 7m=2 r 3=2 (2 m r) 3=2 C : Summerizing, we bound the right side of the energy inequality by C 4=5 =5 +C 3=5 23+9=10 + 2C +C 9=10 223=20 C 36 since 21=20. Therefore, by the same notation as in the proof of Theorem 1.4 and using the local energy inequality, we get 2 (r) + 2 (r)C . This implies that (0; 0) is a regular point and the proof is completed. In order to complete our estimate on the size of the singular set, we need to ll in the gap for 2 [1; 21=20] and calculate the -fractal dimension directly as an expression of . Thus, instead of (1.13), we assume for R2 [ ;] ZZ Q juj 10=3 +jruj 2 +jpj 5=3 +jrpj 5=4 +jfj 5=3 dxdt R ! ; (1.20) where !2 [3=2; 45=29] is to be determined. First, (1.20) implies kuk L 10=3 (Q R ) 3=10 R 3!=10 ; (1.21) kruk L 2 (Q R ) 1=2 R !=2 ; (1.22) kpk L 5=3 (Q R ) 3=5 R 3!=5 ; (1.23) krpk L 4=5 (Q R ) 4=5 R 4!=5 ; (1.24) kfk L 5=3 (Q R ) 3=5 R 3!=5 ; (1.25) for all R2 [ ;]. With r = =2 where is to be determined, we estimate the 37 integrals I 1 , I 2 and I 3 as in the previous lemmas as follows. First, we have I 1 Cr 2 3 kuk 2 L 10=3 (Q) C 3=5 r 2 33!=5 =C 3=5 r 2+3!=53= : Let m 0 be as in the setup of the dyadic decomposition, and let m 1 be the largest integer such that (2 m1 ) 2 2 : Then we have I 2 = m 1 X m=0 + m 0 X m=m 1 +1 ZZ Dm (juj 2 m (t))u j @ j ( m )dxdt The term in the rst sum are estimated as in the proof of Theorem 1.4 while the terms in the second sum are treated as in Theorem 1.12. We get I 2 C m 1 X m=0 1 2 7m=2 r 3=2 ( 3=10 3!=10 ) 5=2 ( 1=2 !=2 ) 1=2 +C m 0 X m=m 1 +1 1 2 7m=2 r 3=2 ( 3=10 (2 m r) 3!=10 ) 5=2 ( 1=2 (2 m r) !=2 ) 1=2 : 38 Summing up both geometric series and using 2 m 1 C , we get I 2 C !3=2 + C !+27=2 r (1)(7=2!) : Choosing = 30=(9 + 20) and ! = 45=(9 + 20), we get I 2 C . Similarly, we obtain the smallness of I 3 and (r). We thus conclude that for 2 [1; 21=20], the -fractal dimension of the singular set is bounded from above by 45=(9 + 20). 39 Chapter 2: On the Well-posedness of the Primitive Equations of the Ocean 2.1 Introduction In the second chapter of the dissertation, we analyze the primitive equations of the ocean and the atmosphere. We address the global existence and uniqueness of the solutions for the primitive equations without assuming any dierentiability on the initial datum. The primitive equations govern the motion of the atmosphere and the ocean and thus are the fundamental model for meteorology, climate prediction, and geophysical uid dynamics. They are derived from the full compressible Navier-Stokes equations which are complicated and contain phenomena that are not interesting from the geophysical point of view, including shocks and sound waves. The core of the system consists of the momentum equations and the conservation of mass, as is given below, @v k @t v k + 2 X j=1 @ j (v j v k ) +@ z (wv k ) +@ k p = 0; k = 1; 2 (2.26) 2 X j=1 @ k v k +@ z w = 0: 40 The full primitive equations also include the dynamic equations for the diusion of temperature, humidity, and the salinity of the ocean. The mathematical framework for the primitive equations was established by Lions, Temam, and Wang. They obtained a series of results including the global existence of a class of weak solutions, as well as the numerical schemes used to compute solutions (cf. [LTW1, LTW2, LTW3]). Subsequently, in [TZ], Temam and Ziane proved the existence and uniqueness results for H 1 initial data. They also obtained the global existence of weak solutions for a squared integrable initial data. In 2005, Cao and Titi obtained in [CT] the global existence of strong solutions by solving the Neumann boundary problem on the top and the bottom. Later, Kukavica and Ziane proved in [KZ1] the existence and uniqueness of global strong solutions in the case of the Dirichlet boundary conditions on the bottom of the ocean with the general bottom topography. They also obtained the uniform gradient bounds in [KZ2]. In summary, the H 1 regularity for initial data leads to the global well-posedness of solution in both two and three dimensions. However, the uniqueness of the weak solutions remain unknown for both 2D and 3D. Unlike the Navier-Stokes equations, whose uniqueness of weak solutions in 2D is a classical and fundamental result, the primitive equations present the obstacle as derivative loss in the nonlinear terms, leading to an outstanding open problem. 41 Furthermore, the well-posedness of the primitive equations with L p initial datum is rather dicult that it still remains open (cf. [BGMR]). In this dissertation, our goal is to prove the well-posedness in spaces that are larger than H 1 . In particular, we establish existence and uniqueness of solutions with only continuous initial data that requires no dierentiability. Our approach is the splitting of the initial data into a smooth nite energy part and a small bounded part, and consider the equations satised by the two parts separately. Since the splitting is preserved by the equation, we already have the existence and uniqueness of solutions with smooth initial datum proved in [KZ1]. Therefore, we focus on the equations with the small initial velocity that are uniformly bounded and small. The main diculty comes from the pressure and the derivative loss terms. The rest of the chapter is organized as follows. We rst set up the notation and introduce the functional spaces. Next we obtain a priori estimates of the solutions in L p spaces so that we can prove our main theorems on the existence. We also state and prove the higher regularity results. Uniqueness is obtained in the subsequent section, followed by the results on the periodic domain and datum. 42 2.2 Main results on the whole domain Let 0 =R 2 [h; 0]. The primitive equations of the ocean read @u k @t u k + 2 X j=1 @ j (u j u k ) +@ z (wu k ) +@ k p = 0; k = 1; 2 (2.27) 3 X k=1 @ k u k = 0; where u(x;t) = (u 1 (x;t);u 2 (x;t);u 3 (x;t)) = (v(x;t);w(x;t)): Here v = (v 1 ;v 2 ) and w are the horizontal and the vertical components respectively. The initial data v 0 = (v 01 ;v 02 ): !R 2 satises div 2 Z 0 h v 0 dz = 0 in the sense of distributions, where we identify z and x 3 as the vertical component of the coordinates. We point out that the pressure is only two dimensional and we do not have equation for the vertical component of the velocity w other than the divergence free conditionru = 0. Next we describe the Neumann boundary conditions that we equip to the system 43 (2.27). On the top and the bottom of the domain, where x 3 =h and x 3 = 0, respectively, we have @v @x 3 = 0 and w = 0: Due to the fact that our initial datum are merely continuous and @v=@x 3 may not be well-dened, we take advantage of the fact that the above formulation is equivalent to the same problem but on the extended domain =R 2 [h;h] where we impose the periodic boundary condition in the x 3 direction. By such extension, the horizontal components (u 1 ;u 2 ) of the velocity eld u become even in the vertical x 3 variable whereas the vertical component u 3 is odd. We point out that this extension helps us in the pressure estimates in the proof of our main theorem. Next we introduce the functional spaces that we will be working with. Denote by C 0 ( ) the space of continuous functions on that vanish at innity, and for simplicity, we denote by B 1 ;T ( [0;T )) = u2L 1 t L 1 x ( ):kuk L 1 t L 1 x ( ) the ball of radius in the space L 1 x;t ( ). 44 As in [TZ], we dene the spaces H = v2 (L 2 ( )) 2 : div 2 Z h h vdz = 0 onR 2 and V =H\ v2 (H 1 ) 2 : v(;x 3 ) =v(;x 3 );v(;x 3 ) =v(;x 3 + 2h) : We now state our rst main results. For simplicity, assume that the external force f vanishes and that the viscosity equals 1. First we recall the denition of the weak solution to (2.27). Denition 2.13. We say that v2L 1 ([0;1);H)\L 2 loc ([0;1);V ) is a weak solution of the equation (2.27) with initial data v 0 2H if for every 2D( [0;1)) such that div = 0 we have 2 X k=1 Z 1 0 Z u k @ t k dxdt + 2 X k=1 Z 1 0 Z ru k r k dxdt 3 X j=1 2 X k=1 Z 1 0 Z u j u k @ j k dxdt 2 X k=1 Z v 0k k (; 0)dxdt = 0: We also recall the following result concerning the existence of weak solutions to the primitive equations. 45 Theorem 2.14. For every v 0 2 H, there exists a weak solution for the primitive equation (2.27) as in the above denition. The next theorem provides the existence of solutions to the equation (2.29) below. Theorem 2.15. Assume v 0 2C 0 ( )\L 2 ( ), and let T 0 be arbitrary. For any positive constant > 0, there exists a constant C 0 and a weak solution satisfying the decomposition v2C([0;T ];H 2 )\L 2 loc ([0;1);H 3 ) +B 1 C;T ( [0;1)) to the equation (2.27). Namely, v =v +V where v2L 1 ([0;T ];H 2 )\L 2 loc ([0;1);H 3 ) and V2B 1 C;T ( [0;1)): Here C is a constant independent of the solution. Note that due to the embedding of H 2 ( ) into the space of continuous functions that vanish at innity and the fact that > 0 in the above theorem is arbitrarily small, the obtained solution is continuous onR 3 [0;T ]. The next theorem provides the uniqueness of the above solution in the space C([0;T];H 2 )\L 2 ([0;T];H 3 ) + B 1 C;T ( [0;T )). 46 Theorem 2.16. Supposev (1) andv (2) are two weak solutions of the primitive equation (2.27) with the same initial data v 0 2C 0 ( ) such that v (1) , v (2) 2L 1 ([0;T];H 2 )\ L 2 ([0;T ];H 3 ) +B 1 ( [0;T )) where > 0 is suciently small. Then v (1) =v (2) for 0tT . 2.3 Proof of existence In order to prove Theorem 2.15, we use the splitting method introduced by Brezis and Kato in [BK], and here we use the splitting in the uniform norm. In the proof of the existence result, we recall the following theorem concerning the higher regularity of solutions of primitive equations and we provide its proof in the last section. Theorem 2.17. Assume v 0 2V and suppose that v2L 1 ([0;T ];V )\L 2 ([0;T ];H 2 ) are the associated solutions of the system 2.27. If in addition v 0 2 H 2 , then v2 L 1 ([0;T ];H 2 )\L 2 ([0;T ];H 3 ). Proof of Theorem 2.15. Let v 0 2 C 0 ( )\L 2 ( ) and x 0. Denote by the standard smooth mollier that is also radially symmetric. More specically, let be a non-negative, smooth, and radially symmetric function such that R = 1. For positive constant , let (x) = 3 (x=). Denote by v the convolution v 0 and 47 for suciently small , we have kv 0 v 0 k L 2; kv 0 v 0 k L 1: The function v 0 is innitely smooth and we have v 0 2 H k for all k 2 N. The divergence-free condition and the Neumann boundary condition are satised since v 0 is even in the vertical variable, due to the radial symmetry of . We point out that the H k -norm of v 0 indeed depends on and might blow up as ! 0. However, we will not consider the limit as tends to 0 and instead, we x arbitrarily small in our following proof. From [KZ1] and [CT], we know that the system (2.27) has a unique strong solution v2C([0;T ];H 1 )\L 2 ([0;T ];H 2 ): Note that V 0 = v 0 v 0 satiseskV 0 k L 2,kV 0 k L 1 . Next we take the dierence of the equations for u = (v;w), and for u = (v;w) to obtain that the dierence 48 U = (V;W ) =uu = (vv;ww) satises @V k @t V k + 2 X j=1 V j @ j V k +W@ z V k + 2 X j=1 v j @ j V k +w@ z V k + 2 X j=1 V j @ j v k +W@ z v k +@ k P = 0; k = 1; 2 (2.28) rU = 0 where P =pp. By interpolation we have V 0 2L q ( ) for all q2 [2;1] andkV 0 k L q sinceV 0 2L 1 ( )\L 2 ( ) and both norms are bounded by. Next we obtain a priori estimates for V in L q ( ). Let q = 2 X k=1 kjV k j q=2 k 2 L 2; = 2 X k=1 kr(jV k j q=2 )k 2 L 2: 49 We multiply the equations (2.28) by V k jV k j q2 , integrate over , and sum for k = 1; 2. We have for q = 2; 4; 8; ; 2 l ; , 1 q d dt 2 X k=1 Z jV k j q + 4(q 1) q 2 3 X j=1 2 X k=1 Z @ j (jV k j q=2 )@ j (jV k j q=2 ) = 2 X k=1 Z V k jV k j q2 @ k P 2 X j;k=1 Z V j @ j v k V k jV k j q2 2 X k=1 Z W@ z v k V k jV k j q2 =I 1 +I 2 +I 3 ; (2.29) where we used the boundary condition when integrating by parts. We proceed to estimate the terms I 1 , I 2 and I 3 in order to show that P 2 k=1 kV k k L 2q is uniformly bounded in q using an iterative argument. First we estimate the term I 1 . Since V 0 also vanishes at innity and on the extended domain , V 1 and V 2 are also even in the vertical z variable, we integrate by parts to obtain I 1 = 2 X k=1 Z V k jV k j q2 @ k P = 2 X k=1 Z P@ k (V k jV k j q2 ): 50 Then H older's inequality implies I 1 C 2 X k=1 Z jPjjV k j q=21 j@ k (jV k j q=2 )jC 2 X k=1 kjPjjV k j q=21 k L 2k@ k (jV k j q=2 )k L 2: (2.30) We estimate the two factors on the right side of (2.30) separately. Regarding the rst factor, we have for k = 1; 2 kjPjjV k j q=21 k L 2 = Z P 2 jV k j q2 = Z R 2 P 2 Z h h jV k j q2 dzdx 1 dx 2 kP 2 k L q=4 (R 2 ) Z h h jV k j q2 dz L q=(q4) (R 2 ) kP 2 k L q=4 (R 2 ) Z h h kjV k j q2 k L q=(q4) (R 2 ) dz; where we used the H older's inequality and the Minkowski's Inequality. Then we rewrite the last factor in the above inequality and use the two dimensional Gagliardo- Nirenberg-Sobolev inequality to obtain Z h h kjV k j q2 k L q=(q4) (R 2 ) dz = Z h h kjV k j q=2 k (2q4)=q L (2q4)=(q4) (R 2 ) dz C Z h h kjV k j q=2 k (2q8)=q L 2 (R 2 ) kr 2 (jV k j q=2 )k q=4 L 2 (R 2 ) dz: 51 Therefore, we bound the right side of the above expression using H older's inequality by CkjV k j q=2 k (2q8)=q L 2 kr 2 (jV k j q=2 )k q=4 L 2 : Combining the above estimates, we bound I 1 by C 2 X k=1 kjPjjV k j q=21 k L 2k@ k (jV k j q=2 )k L 2 C 2 X k=1 kP 2 k 1=2 L q=4 kjV k j q=2 k (q4)=q L 2 kr 2 (jV k j q=2 )k q=2 L 2 =C 2 X k=1 kPk q=2 kjV k j q=2 k (q4)=q L 2 kr 2 (jV k j q=2 )k q=2 L 2 : Taking into account the pressure estimates given in Lemma 2.19 below, we further bound the above expression by Cq 2 X k=1 (kV k k 2 L q +kv k k L qkV k k L q)kjV k j q=2 k (q4)=q L 2 kr 2 (jV k j q=2 )k q=2 L 2 : 52 Therefore, we obtain I 1 Cq 2 X k=1 kV k k L q +kv k k L qkV k k L q kjV k j q=2 k (q4)=q L 2 kr 2 (jV k j q=2 )k q=2+1 L 2 =Cq 2 X k=1 kjV k j q=2 k q=4 L 2 +kv k k L qkjV k j q=2 k 2=q L 2 kjV k j q=2 k (q4)=q L 2 kr 2 (jV k j q=2 )k (q+2)=q L 2 Cq 2 X k=1 kjV k j q=2 k L 2kr 2 (jV k j q=2 )k (q+2)=q L 2 +Cq 2 X k=1 kv k k L qkjV k j q=2 k (q2)=q L 2 kr 2 (jV k j q=2 )k (q+2)=q L 2 Cq 1=2 q (q+2)=2q q +Cqkvk L q (q2)=2q q (q+2)=2q q : Regarding the estimates of I 2 , we rst apply the H older's inequality to obtain I 2 = 2 X j;k=1 Z V j @ j v k V k jV k j q2 2 X j;k=1 k@ j v k k L 3kV j V k jV k j q=22 k L 2kjV k j q=2 k L 6; (2.31) where we imposedL 3 norm onv due to its higher regularity. We bound the right side 53 of (2.31) by using the Gagliardo-Nirenberg-Sobolev inequality and get I 2 C 2 X j;k=1 k@ j v k k 1=2 L 2 kr@ j v k k 1=2 L 2 +k@ j v k k L 2 kjV k j q=2 k L 2 kr(jV k j q=2 )k L 2 +kjV k j q=2 k L 2 C 2 X j;k=1 k@ j v k k 1=2 L 2 kr@ j v k k 1=2 L 2 1=2 q 1=2 q +C 2 X j;k=1 k@ j v k k L 2 1=2 q 1=2 q +C 2 X j;k=1 k@ j v k k 1=2 L 2 kr@ j v k k 1=2 L 2 q +C 2 X j;k=1 k@ j v k k L 2 q : Again, we take the advantage of the fact thatv is the regular solution to the primitive system (2.27) in the above estimates. In order to estimate the term I 3 , note that due to the incompressibility condition W = 2 X i=1 Z x 3 h @ i V i dz; 54 and integration by parts, we have I 3 = 2 X k=1 Z W@ z v k V k jV k j q2 = 2 X i;k=1 Z @ i Z x 3 h V i dz @ z v k V k jV k j q2 = 2 X i;k=1 Z Z x 3 h V i dz @ iz v k V k jV k j q2 2 X i;k=1 Z Z x 3 h V i dz @ z v k @ i V k jV k j q2 =I 31 +I 32 : Then we use the H older's inequality and the Minkowski's inequality to bound the term I 31 as I 31 C 2 X i;k=1 kV i k L qk@ iz v k k L 2kjV k j (q2)=2 k L 3q=(q3)kjV k j q=2 k L 6; which by rewriting the third factor, is equivalent to I 31 C 2 X i;k=1 kV i k L qk@ iz v k k L 2kjV k j q=2 k (q2)=q L (3q6)=(q3) kjV k j q=2 k L 6: (2.32) Thus, applying the Gagliardo-Nirenberg-Sobolev inequality, we bound the right side 55 of (2.32) by C 2 X i;k=1 kjV k j q=2 k (q4)=(2q4) L 2 kr(jV k j q=2 )k q=(2q4) L 2 +kjV k j q=2 k L 2 (q2)=q kV i k L qk@ iz v k k L 2 kr(jV k j q=2 )k L 2 +kjV k j q=2 k L 2 ; whence I 31 C 2 X i;k=1 k@ iz v k k L 2 1=4 q 3=4 q +C 2 X i;k=1 k@ iz v k k L 2 1=2 q 1=2 q +C 2 X i;k=1 k@ iz v k k L 2 3=4 q 1=4 q +C 2 X i;k=1 k@ iz v k k L 2 q : Regarding the term I 32 , by using H older's inequality and Minkowski's inequality we have I 32 = 2 X i;k=1 Z Z x 3 h V i dz @ z v k @ i V k jV k j q2 C 2 X i;k=1 kV i k L qk@ z v k k L 6k@ i V k jV k j q2 k 6q=(5q6) : (2.33) By using the product rule to the last factor in the above inequality, we get @ i V k jV k j q2 = 2(q 2) q V k jV k j q=22 @ i jV k j q=2 : 56 Therefore, using the H older's inequality, we bound the right side of (2.33) by C 2 X i;k=1 kV i k L qk@ z v k k L 6kjV k j q=21 k L 3q=(q3) @ i jV k j q=2 L 2 ; and we rewrite it as C 2 X i;k=1 kV i k L qk@ z v k k L 6kjV k j q=2 k (q2)=q L (3q6)=(q3) @ i jV k j q=2 L 2 : Then we use the Gagliardo-Nirenberg-Sobolev inequality to obtain I 32 C 2 X i;k=1 kjV k j q=2 k (q4)=(2q) L 2 kr jV k j q=2 k 1=2 L 2 +kjV k j q=2 k (q2)=q L 2 kV i k L q kr@ z v k k L 2 +k@ z v k k L 2 @ i jV k j q=2 L 2 C 2 X i;k=1 kjV k j q=2 k 1=2 L 2 kr jV k j q=2 k 1=2 L 2 +kjVj q=2 k L 2 kr@ z v k k L 2 +k@ z v k k L 2 @ i jV k j q=2 L 2 C 2 X k=1 kr@ z v k k L 2 1=4 q 3=4 q +C 2 X k=1 kr@ z v k k L 2 1=2 q 1=2 q +C 2 X k=1 k@ z v k k L 2 1=4 q 3=4 q +C 2 X k=1 k@ z v k k L 2 1=2 q 1=2 q : 57 Combining the above estimates for I 1 , I 2 , and I 3 , we conclude 1 q d q dt + 4(q 1) q 2 q Cq 1=2 q (q+2)=2q q (q+2)=2q q +Cqkvk L q (q2)=2q q (q+2)=2q q +C 2 X j;k=1 k@ j v k k 1=2 L 2 kr@ j v k k 1=2 L 2 1=2 q 1=2 q +C 2 X j;k=1 k@ j v k k L 2 1=2 q 1=2 q +C 2 X j;k=1 k@ j v k k 1=2 L 2 kr@ j v k k 1=2 L 2 q +C 2 X j;k=1 k@ j v k k L 2 q +C 2 X i;k=1 k@ iz v k k L 2 3=4 q 1=4 q +C 2 X i;k=1 k@ iz v k k L 2 q +C 2 X k=1 kr@ z v k k L 2 1=4 q 3=4 q +C 2 X k=1 kr@ z v k k L 2 1=2 q 1=2 q +C 2 X k=1 k@ z v k k L 2 1=4 q 3=4 q +C 2 X k=1 k@ z v k k L 2 1=2 q 1=2 q : 58 Thus, we apply the Young's inequality in order to get 1 q d q dt + 4(q 1) q 2 q 1 q q +Cq (3q+2)=(q2) q=(q2) q +Cq (3q+2)=(q2) kvk 2q=(q2) L q q +Cq 2 X j;k=1 k@ j v k k L 2kr@ j v k k L 2 q +Cq 2 X j;k=1 k@ j v k k 2 L 2 q +C 2 X j;k=1 k@ j v k k 1=2 L 2 kr@ j v k k 1=2 L 2 q +C 2 X j;k=1 k@ j v k k L 2 q +Cq 1=3 2 X i;k=1 k@ iz v k k 4=3 L 2 q +C 2 X i;k=1 k@ iz v k k L 2 q +Cq 3 2 X k=1 kr@ z v k k 4 L 2 q +Cq 2 X k=1 kr@ z v k k 2 L 2 q +Cq 3 2 X k=1 k@ z v k k 4 L 2 q +Cq 2 X k=1 k@ z v k k 2 L 2 q 59 where we combined all the terms q =9q. Then we rewrite the above inequality as d q dt + q Cq (4q)=(q2) q=(q2) q +Cq (4q)=(q2) kvk 2q=(q2) L q q +Cq 2 2 X j;k=1 k@ j v k k L 2kr@ j v k k L 2 q +Cq 2 2 X j;k=1 k@ j v k k 2 L 2 q +Cq 2 X j;k=1 k@ j v k k 1=2 L 2 kr@ j v k k 1=2 L 2 q +Cq 2 X j;k=1 k@ j v k k L 2 q +Cq 4=3 2 X i;k=1 k@ iz v k k 4=3 L 2 q +Cq 2 X i;k=1 k@ iz v k k L 2 q +Cq 4 2 X k=1 kr@ z v k k 4 L 2 q +Cq 2 2 X k=1 kr@ z v k k 2 L 2 q +Cq 4 2 X k=1 k@ z v k k 4 L 2 q +Cq 2 2 X k=1 k@ z v k k 2 L 2 q : (2.34) Note that by using the Gagliardo-Nirenberg-Sobolev inequality, we have q 5=3 q C 10=3 q=2 C 4=3 q=2 ; which yields d q dt + 5=3 q C 4=3 q=2 C 2 q=2 +Cq 4q=(q2) q=(q2) q +Cq 4q=(q2) kvk 2q=(q2) L q q +Cq 2 2 X j;k=1 k@ j v k k L 2kr@ j v k k L 2 q +Cq 4 2 X k=1 kr@ z v k k 4 L 2 q : 60 Then in order to absorb the 5=3 q =C 4=3 q=2 term, we write q =C( 5=3 q =C 4=3 q=2 ) 3=5 4=5 q=2 for each factor q on the right side of the above inequality and use the Young's inequality in order to get d q dt + 5=3 q C 4=3 q=2 C 2 q=2 +Cq 10 2q=(q1) q=2 +Cq 10 kvk 5q=(q2) L q 2 q=2 +Cq 5 2 X j;k=1 k@ j v k k 5=2 L 2 kr@ j v k k 5=2 L 2 2 q=2 +Cq 5 2 X j;k=1 k@ j v k k 5 L 2 2 q=2 +Cq 5=2 2 X j;k=1 k@ j v k k 5=4 L 2 kr@ j v k k 5=4 L 2 2 q=2 +Cq 5=2 2 X j;k=1 k@ j v k k 5=2 L 2 2 q=2 +Cq 10=3 2 X i;k=1 k@ iz v k k 10=3 L 2 2 q=2 +Cq 5=2 2 X i;k=1 k@ iz v k k 5=2 L 2 2 q=2 +Cq 10 2 X k=1 kr@ z v k k 10 L 2 2 q=2 +Cq 5 2 X k=1 kr@ z v k k 5 L 2 2 q=2 +Cq 10 2 X k=1 k@ z v k k 10 L 2 2 q=2 +Cq 5 2 X k=1 k@ z v k k 5 L 2 2 q=2 : (2.35) Next let M = max kvk L 1 ([0;T ];L q ( )) ;krvk L 1 ([0;T ];L 2 ( )) ;kr 2 vk L 1 ([0;T ];L 2 ( )) : 61 Then we bound the right side of (2.35) as d q dt Cq 10 2q=(q1) q=2 +Cq 10 M 5q=(q2) 2 q=2 +Cq 5 M 5 2 q=2 +Cq 5=2 M 5=2 2 q=2 +Cq 10=3 M 10=3 2 q=2 +Cq 10 M 10 2 q=2 : (2.36) Now suppose M is bounded by some constant C, i.e., C depends on M. We further reduce inequality (2.36) as d q dt Cq 10 2q=(q1) q=2 +Cq 10 2 q=2 : (2.37) By denoting R q (t) =kVk L 1 ([0;t];L q ( )) and intergating (2.37) over time (0;t), we obtain R q (t) q Cq 10 R q q=2 (t) q=(q1) +Cq 10 R q q=2 (t) + q ; where we used the fact thatkV 0 k L q is suciently small. In order to pass to the limit q!1, we need the following lemma. Lemma 2.18. Let S n 0 , S n 0 +1 , S n 0 +2 , be a sequence that satises the following 62 recurrence inequality S n A2 Kn S 2 n n1 2 n =(2 n 4) +A2 Kn S 2 n n1 + 2 n 1=2 n ; where A 1, K 0, and 2 (0; 1]. Then there exists a constant C 0 such that sup n S n C provided S n 0 and is suciently small. Proof of Lemma 2.18. With n 0 = 1, we dene the following sequence n+1 = 4 1=2 n A 1=2 n 2 Kn=2 n n ; (2.38) for n =n 0 ;n 0 + 1; . Note that since A 1, we have n 1; n =n 0 ;n 0 + 1; and = sup n n = lim n n 1; 63 which follows from the convergence 1 Y n=n 0 4 1=2 n A 1=2 n 2 Kn=2 n 1: If we assume that 0 is small enough that 1, we get n 1; ;n =n 0 ;n 0 + 1; Next we prove by induction that S n n ; n =n 0 ;n 0 + 1; Since n 0 1, we have S n 0 n 0 . Namely, the claim holds for n 0 . Now assume that the claim is true for up to n 1, i.e., S n1 n1 , we have S n A2 Kn S 2 n n1 2 n =(2 n 4) +A2 Kn S 2 n n1 + 2 n 1=2 n 2A2 Kn S 2 n n1 + 2 n 1=2 n 2A2 Kn 2 n n1 1=2 n 4A2 Kn 2 n n1 1=2 n : where we used the fact that n1 1 in the last inequality. Therefore we obtain 64 S n 4 1=2 n A 1=2 n n1 = n by the recurrence relation (2.38). Now with the induction step established, the proof of lemma is complete. Back to the proof of Theorem 2.15. We note that by interpolation we have 8 8 , and there exists a small T > 0 such that R 8 (T ) 2 by the Gr onwall's inequality. Moreover, for suciently small , by the above Lemma 2.18 with n 0 = 3 and S n =R 2 n(T ) we obtain sup n R 2 n(T )C: Therefore, combining (2.35) with q = 8, we have V k 2L 1 ([0;T ];L 1 ( )) and kV k k L 1 ([0;T];L 1 ( )) C for k = 1; 2. Note that since v 0 2H, by Theorem 2.14 there exists 0<T 1 <T such that v(T 1 )2 V . Thus, there exists T 1 < T 2 < T such that v(T 2 )2 H 2 . Then by Theorem 2.15 we have the splitting v2C([0;T ];H 2 )\L 2 ([0;T ];H 3 ) +B 1 C ( [0;T )) 65 on [0;T ]. Since T > 0 is arbitrary, the theorem is proven. Next we prove the lemma regarding the pressure estimates. Lemma 2.19. For the pressure term P =P(x 1 ;x 2 ) in the primitive system (2.28), we have kPk L q=2 =kppk L q=2Cq 2 X k=1 kV k k 2 L q +kv k k L qkV k k L q : Proof of Lemma 2.19. First we rewrite equations satised byu = (v;w) andu = (v;w) as @v k @t v k + 2 X j=1 @ j (v j v k ) +@ z (wv k ) +@ k p = 0; k = 1; 2; (2.39) and @v k @t v k + 2 X j=1 @ j (v j v k ) +@ z (wv k ) +@ k p = 0; k = 1; 2; (2.40) where we used the divergence-free conditionsru = 0 andru = 0. Next, we apply 66 the vertical direction average operator M() = 1 h Z h h dz; to the equations (2.39) and (2.40) and take their dierence in order to obtain @M[V k ] @t M[V k ] + 2 X j=1 @ j (M[V j V k ]M[v j v k ]) +@ k P = 0; k = 1; 2; (2.41) where V k = v k v k . Then by taking the two-dimensional divergence of (2.41), we obtain 2 P = 2 X j;k=1 @ j @ k (M[V k V j ] +M[v k V j ] +M[V k v j ]) 2 X k=1 @ k M[V k ] = 2 X j;k=1 @ j @ k (M[V k V j ] +M[v k V j ] +M[V k v j ]) 2 X k=1 @ k M[ 2 V k ] 2 X k=1 @ k M[@ zz V k ] = 2 X j;k=1 @ j @ k (M[V k V j ] +M[v k V j ] +M[V k v j ]) 2 X k=1 (@ zk V k (;h)@ zk V k (;h)) = 2 X j;k=1 2 X j;k=1 @ j @ k (M[V k V j ] +M[v k V j ] +M[V k v j ]); 67 where we used the boundary condition, the divergence-free condition, and the periodic extension in the z direction. Thus, we have P = 2 X j;k=1 ( 2 ) 1 @ j @ k (M[V k V j ] +M[v k V j ] +M[V k v j ]) = 2 X j;k=1 R j R k (V k V j +v k V j +V k v j ); where R i is the i-th Riesz transform. Applying the Calder on-Zygmund theorem, we deduce kPk L q=2Cq 2 X k=1 kM[V k V j ] +M[v k V j ] +M[V k v j ]k L q=2 Cq 2 X k=1 kV k k 2 L q +kv q k L qkV k k L q : Therefore, the proof of lemma is complete. 2.4 Proof of uniqueness In this section we take advantage of the splitting method in order to prove the uniqueness of solutions. 68 Proof of Theorem 2.16. Let (u (1) ;p (1) ) = (v (1) ;w (1) ;p (1) ) and (u (2) ;p (2) ) = (v (2) ;w (2) ;p (2) ) be two weak solutions of (2.27) with the same initial data as stated in Theorem 2.15. Let u =u (1) u (2) . Then u = (v;w) satises @v k @t v k + 2 X j=1 v 2 j @ j v k +w 2 @ z v k + 2 X j=1 v j @ j v (1) k +w@ z v (1) k +@ k (p (1) p (2) ) = 0; k = 1; 2 (2.42) 2 X k=1 @ k v k +@ z w = 0; with initial data v(0) =v 0 = 0. Let V (t) = 2 X k=1 kv k k 2 L 2 ! 1=2 and V (t) = 2 X k=1 krv k k 2 L 2 ! 1=2 : 69 We multiply equations (2.42) for k = 1; 2 by v k , and integrate over . Summing for k = 1; 2 and integrating by parts, we obtain 1 2 d dt 2 X k=1 Z jv k j 2 + 2 X k=1 Z jrv k j 2 = 2 X j;k=1 Z v j @ j v (1) k v k 2 X k=1 Z w@ z v (1) k v k = 2 X j;k=1 Z v j v (1) k @ j v k + 2 X k=1 Z wv (1) k @ z v k =I 1 +I 2 ; where we used the incompressibility condition on u. Then we estimate the terms I 1 andI 2 . RegardingI 1 , we splitv (1) into a regular part and a small essentially bounded part as I 1 = 2 X j;k=1 Z v j v (1) k @ j v k = 2 X j;k=1 Z v j v (1) k @ j v k + 2 X j;k=1 Z v j ~ v (1) k @ j v k =I 11 +I 12 ; where v (1) k 2L 1 ([0;T ];H 2 )\L 2 ([0;T ];H 3 ) and ~ v (1) k 2B C 70 as in Theorem 2.15. In order to estimate the termI 11 , we apply the H older's inequality and the Gagliardo-Nirenberg-Sobolev inequality to get 2 X j;k=1 Z Om v j v (1) k @ j v k 2 X j;k=1 kv j k L 3kv (1) k k L 6k@ j v k k L 2 C 2 X j;k=1 kv j k 1=2 L 2 krv j k 1=2 L 2 krv (1) k k L 2k@ j v k k L 2: Therefore I 11 Ckrv (1) k L 2V (t) 1=2 V (t) 3=2 : For the term I 12 , we have I 12 = 2 X j;k=1 Z v j v (1) k @ j v k 2 X j;k=1 kv j k L 2kv (1) k k L 1k@ j v k k L 2 Thus, I 12 CV (t)V (t): 71 Now we estimate the term I 2 . We take advantage of the fact that the essentially bounded part is small after we split it as I 2 = 2 X k=1 Z wv (1) k @ z v k = 2 X k=1 Z wv (1) k @ z v k + 2 X k=1 Z w~ v (1) k @ z v k =I 21 +I 22 : First we estimate I 21 and integration by parts implies 2 X k=1 Z wv (1) k @ z v k = 2 X k=1 Z @ z wv (1) k v k 2 X k=1 Z w@ z v (1) k v k : Applying the H older's inequality to the right side of the above equality, we bound it by 2 X k=1 k@ z wk L 2kv (1) k k L 6kv k k L 3 + 2 X k=1 kwk L 2k@ z v (1) k k L 6kv k k L 3: Thus, by the Gagliardo-Nirenberg-Sobolev inequality and w = 2 X i=1 Z x 3 h @ i v i dz; (2.43) 72 we bound I 21 as I 21 C 2 X k=1 krvk L 2krv (1) k k L 2kv k k 1=2 L 2 krv k k 1=2 L 2 +C 2 X k=1 krvk L 2kr@ z v (1) k k L 2kv k k 1=2 L 2 krv k k 1=2 L 2 : Hence I 21 Ckrv (1) k L 2V (t) 1=2 V (t) 3=2 +Ckr@ z v (1) k L 2V (t) 1=2 V (t) 3=2 : In order to estimate the term I 22 , we apply the H older's inequality to obtain I 22 = 2 X k=1 Z w~ v (1) k @ z v k 2 X k=1 kwk L 2k~ v (1) k k L 1k@ z v k k L 2 krvk L 2k~ v (1) k L 1krvk L 2CV (t) 2 ; where we used (2.43). Combining the above estimates on I 1 and I 2 yields 1 2 dV 2 dt +V 2 Ckrv (1) k L 2V (t) 1=2 V (t) 3=2 +CV (t)V (t) +Ckr@ z v (1) k L 2V (t) 1=2 V (t) 3=2 +CV 2 : 73 By Young's inequality we get 1 2 dV 2 dt +V 2 Ckrv (1) k 4 L 2V (t) 2 +CV (t) 2 +Ckr@ z v (1) k 4 L 2V (t) 2 +CV (t) 2 : Assuming that is suciently small, we get 1 2 dV 2 dt Ckrv (1) k 4 L 2 +CV (t) 2 +Ckr@ z v (1) k 4 L 2V (t) 2 : Since the initial data vanishes, i.e., V (0) = 0, by the Gr onwall's inequality we have V (t) = 0: Therefore, u (1) =u (2) , and the uniqueness of solutions is proven. 2.5 The results on the periodic domain In this section we state and prove the theorem on the existence and uniqueness of the primitive equations with continuous initial data on the domain =T 2 [h; 0]. We employ the Neumann boundary condition as in Section . 74 Theorem 2.20. Assume v 0 2 C( ) and let T 0 be arbitrary. For any posi- tive constant , there exists a constant C 0 and a weak solution satisfying the decomposition v2C([0;T ];H 2 )\L 2 ([0;T ];H 3 ) +B 1 C;T ( ) to the equations (2.27). Namely, v =v +V where v2L 1 ([0;T ];H 2 )\L 2 ([0;T ];H 3 ) and V2B 1 C;T ( ), where the constant C is independent of the solution. Furthermore, suppose v (1) and v (2) are two weak solutions of the primitive equations (2.27) with the same initial data v 0 2 C( ), such that v (1) , v (2) inL 1 ([0;T];H 2 )\L 2 ([0;T];H 3 ) + B 1 ( ) where 0 is suciently small. Then v (1) =v (2) for 0tT . Proof of Theorem 2.20. Note that on the periodic domainT 2 [h; 0], we no longer need the assumption v 0 2L 2 ( ). Evenly extending the domain in the vertical direc- tion, approximating the initial datav 0 with smoothv as in the beginning of the proof of Theorem 2.15, we obtain similarly the estimates and results parallel to those of Theorem 2.15 and Theorem 2.16. 75 2.6 Construction of solutions In Sections and Section we obtained a priori estimates that leads to the existence and uniqueness of the solutions to the primitive equations with only continuous initial data. In this section, we provide the construction of such solutions. We adopt the approach known as retarded mollication from the classic paper [CKN]. Namely, we build up the solution by using Aubin's Lemma in order to pass to the limit for the sequence of solutions of the regularized equations, following the results in [LTW1,LTW2,TZ]. Since our results hold true similarly in the whole domainR 2 [h;h] as well as the periodic domainT 3 , we proceed our argument as follows overT 3 . In our following reasoning, we denote by v the nite energy part of the solution and we have w = Z x 3 h div 2 vdz: For V 0 as in the proof of Theorem 2.15 and the standard mollier , consider the sequence V () 0 = V 0 . For any xed , by the earlier results of Lions, Temam, 76 Wang, and Ziane, we readily have the existence of a weak solution V () to the system @V () k @t V () k + 2 X j=1 V () j @ j V () k +W () @ z V () k + 2 X j=1 v j @V () k +w@ z V () k + 2 X j=1 V () j @ j v k +W () @ z v k +@ k P () = 0; k = 1; 2 (2.44) 2 X k=1 @ k V () k +@ z W () = 0; with initial data V () 0 . By multiplying equations (2.44) by V () k , integrating over and summing for k = 1; 2 we get 1 2 dkV () k 2 L 2 dt +krV () k 2 L 2 = 2 X j;k=1 Z V () j @ j v k V () k 2 X k=1 Z W () @ z v k V () k : 77 Next, using the boundary conditions and the divergence free condition on the solution, we integrate by parts and obtain 1 2 dkV () k 2 L 2 dt +krV () k 2 L 2 = 2 X j;k=1 Z V () j v k @ j V () k + 2 X k=1 Z W () v k @ z V () k = 2 X j;k=1 Z V () j v k @ j V () k 2 X k=1 Z @ z W () v k V () k 2 X k=1 Z W () @ z v k V () k Ckvk H 2 2 X j;k=1 Z jV () j jj@ j V () k j + 2 X k=1 Z j@ z W () jjV () k j ! + 2 X k=1 kW () k L 2k@ z v k k L 6kV () k k L 3; where we used the H older's inequality in the last step. By the Gagliardo-Nirenberg- Sobolev inequality we have 1 2 kV () k 2 L 2 dt +krV () k 2 L 2Ckvk H 2 2 X j;k=1 Z jV () j jj@ j V () k j + 2 X i;k=1 Z j@ i V () i jjV () k j ! +C 2 X k=1 kW () k L 2kr@ z v k k L 2kV () k k 1=2 L 2 krV () k k 1=2 L 2 : Now by using W () = 2 X i=1 Z x 3 0 @ i V () i dz (2.45) 78 we obtain 1 2 kV () k 2 L 2 dt +krV () k 2 L 2Ckvk H 2 2 X j;k=1 Z jV () j jj@ j V () k j + 2 X i;k=1 Z j@ i V () i jjV () k j ! +Ckvk H 2 2 X i;k=1 k@ i V () i k L 2kV () k k 1=2 L 2 krV () k k 1=2 L 2 ; where the Minkowski's inequality was applied. Thus, using the H older's inequality and Young's inequality implies 1 2 kV () k 2 L 2 dt + 1 2 krV () k 2 L 2Ckvk 2 H 2kV () k 2 L 2 +Ckvk 4 H 2kV () k 2 L 2: Therefore, by applying the Gr onwall's inequality to the above expression, we obtain a uniform in estimate for V () in L 1 ([0;T];L 2 )\L 2 ([0;T];H 1 ). Next we need a uniform bound on @ t V () k and thus an estimate on @ k P () . Similar to the beginning of the proof of Theorem 2.16, we apply the divergence-free condition and rewrite equations (2.44) as @V () k @t V () k + 2 X j=1 @ j (V () j V () k ) +@ z (W () V () k ) + 2 X j=1 @ j (v j V () k ) +@ z (wV () k ) + 2 X j=1 @ j (V () j v k ) +@ z (W () v k ) +@ k P () = 0; k = 1; 2: (2.46) 79 By applying the averaging operator as in the proof of Lemma 2.19 to the above equations and since the resulting expression is actually two-dimensional, i.e., M@ z 0; we have M@V () k @ t MV () k + 2 X j=1 M@ j (V () j V () k ) + 2 X j=1 M@ j (v j V () k ) + 2 X j=1 M@ j (V () j v k ) +@ k P () = 0; where we took advantage of the fact that the pressure is merely two-dimensional, not depending on the z variable. By dierentiating in the horizontal variables x 1 and x 2 , summing up for k = 1; 2, and using the divergence-free condition, we obtain 2 P () = 2 X j;k=1 M@ j @ k (V () l V () k ) + 2 X j;k=1 M@ j @ k (v j V () k ) + 2 X j;k=1 M@ j @ k (V () j v k ): 80 Therefore, we have P () = 2 X j;k=1 ( 2 ) 1 @ j @ k M[V () j V () k ] +M[v j V () k ] +M[V () j v k ] = 2 X j;k=1 R j R k M[V () j V () k ] +M[v j V () k ] +M[V () j v k ] ; where R i is the i-th Riesz transform. Thus, by the Calder on-Zygmund theorem we get kP () k L 3=2 (T 2 ) C 2 X j;k=1 kM[V () j V () k ]k L 3=2 (T 2 ) +kM[v j V () k ]k L 3=2 (T 2 ) +kM[V () j v k ]k L 3=2 (T 2 ) : Hence H older's inequality and Minkowski's inequality imply that kP () k L 3=2C 2 X j;k=1 kV () j k L 3kV () k k L 3 +kv j k L 3kV () k k L 3 +kV d j k L 3kv k k L 3 ; which leads to kP () k L 3=2CkV () k L 2krV () k L 2 +Ckvk 1=2 L 2 krvk 1=2 L 2 kV () k 1=2 L 2 krV () k 1=2 L 2 ; 81 where we used the Gagliardo-Nirenberg-Sobolev inequality. Therefore, we have obtained a uniform bound on P () in L 2 ([0;T];L 3=2 ), and thus @ k P () is uniformly bounded in L 2 ([0;T ];H 3=2 ). Next we rewrite the equation (2.46) as @V () k @t = V () k 2 X j=1 @ j (V () j V () k )@ z (W () V () k ) 2 X j=1 @ j (v j V () k )@ z (wV () k ) 2 X j=1 @ j (V () j v k )@ z (W () v k )@ k P () ; k = 1; 2: Note that V () k is uniformly bounded in L 2 ([0;T ];H 2 ), and the terms 2 X j=1 @ j (V () j V () k ); 2 X j=1 @ j (v j V () ); and 2 X j=1 @ j (V () j v k ) are all uniformly bounded in L 2 ([0;T ];H 3=2 ). Regarding the @ z (W () V () k ) term, for 2H 2 we have Z T 3 @ z (W () V () k ) = Z T 3 W () V () k @ z = Z T 3 2 X i=1 @ i Z x 3 0 V () i dz V () k @ z = 2 X i=1 Z T 3 Z x 3 0 V () i dz @ i V () k @ z + 2 X i=1 Z T 3 Z T 3 V () i dz V () k @ iz : Applying the Minkowski's inequality and combining the estimates for V () and rV () imply that the terms V () i @ i V () k and V () i V () k are uniformly bounded in 82 L 4=3 ([0;T ];H 2 ), and thus, @ t V () k is uniformly bounded in L 4=3 ([0;T ];H 2 ). Finally, we apply the Aubin-Lions Lemma to conclude the existence of a sequence of V () converging strongly in L 2 ([0;T];L 2 ) to V , which also satises the same energy estimates as V () . Moreover, by passing to a subsequence, we ensure that the conver- gence is almost everywhere pointwise, which implies that such limit inherits the a priori estimates as was obtained in Section with uniqueness proven in Theorem 2.16. 83 Appendix A: The Detailed Proof of Regularity of Solutions for the Partial Regularity Results In the appendix we present the proof of the regularity of the point (x;t) of the pair of solution (u;p) to the Navier-Stokes system provided that the quantity maxf (x;t) (r); (x;t) (r); (x;t) (r)g is suciently small near (x;t). We state the main theorem next. Theorem A.21. There exists a suciently small universal constant > 0 such that if (x 0 ;t 0 )2 D and (x 0 ;t 0 ) (r) < for suciently small r, then (x 0 ;t 0 ) is a regular point. Next we denote (x;t) (r) = (x;t) (r) + (x;t) (r) + 3 (x;t) (r) where =r= for 0<r=2 and 1. We start with the following lemma, Lemma A.22. For 0<r=2 and Q (x;t)D, we have (x;t) (r)C 1=2 () +C 17=5 ()() 84 and (x;t) (r)C 1=2 (x;t) () +C 17=5 (x;t) () 2 Proof. By the denition of (x;t) (r), we have (x;t) (r) (x;t) (r) so it is sucient to prove the rst inequality. Without the loss of generality, we assume (x;t) = (0; 0). From the generalized energy inequality (1.3) and using the upper bounds of the test function , we get (r) 2 +(r) 2 Z Br juj 2 j t 0 dx + 2 ZZ Qr jruj 2 dxdt C r 2 () 2 +C r 3=2 () 1=2 () 5=2 + C r 2 kppk L 5=4 t L 15=7 x (Q) kuk L 5 t L 15=8 x (Q) =I 1 +I 2 +I 3 : For I 2 , we have by the Gagliardo-Nirenberg inequalities that kuk L 10=3 x;t (Q) Ckuk 2=5 L 1 t L 2 x (Q) kruk 3=5 L 2 x;t (Q) (A.47) 85 from which we obtain I 2 C r 3=2 () 1=2 1 1=2 5=2 kuk L 1 t L 2 x (Q) kruk 3=2 L 2 x;t (Q) C r 3=2 ()() 2 : For I 3 , we have rst kppk L 5=4 t L 15=7 x (Q) krpk L 5=4 x;t (Q) by Poincar e's inequality. Then by the interpolation inequality we get kuk L 5 t L 15=8 x (Q) kuk 11=15 L 1 t L 3=2 x (Q) kruk 4=15 L 4=3 t L 2 x (Q) : Combining the last two inequalities, we obtain I 3 C r 3=2 () 11=15 () 4=15 () 2 86 Therefore, we have (r) +(r)C() +C 3=4 () 1=2 () +C 3=4 () 11=30 () 2=15 (): On the other hand, from our pressure estimates Lemma (1.8), together with the interpolation relation (A.47), we obtain krpk L 5=4 x;t (Qr ) Cr 3=5 2=5 kruk 8=5 L 2 x;t (Q) kuk 2=5 L 1 t L 2 x (Q) +C r 12=5 krpk L 5=4 x;t (Q) which implies (r) 2 C r 2=5 () 2=5 () 8=5 +C r 7=5 () 2 : Thus by collecting the upper bounds of (r) +(r) and (r) 2 we get (r)C() +C 3=4 () 1=2 () +C 3=4 () 11=30 () 2=15 () +C 17=5 () 2=5 () 8=5 +C 8=5 () 2 : (A.48) 87 Applying Cauchy-Schwarz inequality and using ();()() and 3 ()(); , we obtain (r)C() +C( 1=4 () 1=2 () 1=2 )( 1 () 1=2 ) +C( 1=4 () 11=30 () 2=15 () 1=2 )(() 1=2 ) +C 17=5 ()(() 2=5 () 3=5 ) +C 7=5 () C() +C 1=2 ()() +C 2 () +C 1=2 () 11=15 () 4=15 +C 2 () 2 +C 17=5 ()() +C 7=5 () C 1=2 () +C 17=5 ()(): Thus, the lemma is proven. Since we have shown in the proof of Theorem 1.4 that (r) and (r) are su- ciently small for small r, in particular, (r) is suciently small, then by a standard iterative argument, we have the next lemma 88 Lemma A.23. For suciently small (r), and every 2 (0; 0 ) where 0 2 (0; 1), there exist r 1 ;r 2 > 0 and M > 0 such that maxf (x;t) (r); (x;t) (r); (x;t) (r) 2 gMr ; for (x;t)2B (x 0 ;t 0 ) (r 1 ) and r2 (0;r 2 ). In order to prove that the point (x;t) is regular, we need the following result from [O]. Lemma A.24. Let DR 3 R be a bounded domain. Assume that (i) sup (x;t)2D sup >0 RR D\B(x;t) jg(y;s)j q dyds<1 and (ii) g2L m (D) for some mq> 1, and 0< 5. For > 0, dene h(x;t) = ZZ D g(y;s) (jxyj + p ts) 5 dyds: Then for all m2 (m;1) such that 1 m > 1 m 1 q 5 we have h2L m (D). 89 Therefore we are able to show that u is in Morrey space. In order to complete the proof of Theorem A.21, we use the argument from [K1]. Namely, we iterate the solution (u;p) into higherL p -spaces. Since the rest of the proof is the same as in [K1], we omit the details. Hence, we obtain the regularity at the point (x;t). For the purpose of completeness, we state another regularity criterion from [K2], which is also compatible with our approach. Theorem A.25. There exists a suciently small universal constant > 0 with the following property. If (x 0 ;t 0 )2D, and if there exists r 0 > 0 such that 1 r 1=2 kfk L 10=7 x;t (Qr (x 0 ;t 0 )) ; for (x;t)2B r (x 0 ;t 0 ) and r2 (0;r 0 ], and if lim sup r!0+ (x 0 ;t 0 ) (r); then (x 0 ;t 0 ) is a regular point. We also point out that the following theorem, combined with the calculation above, also implies the regularity of a point (x;t). 90 Theorem A.26. Let (u;p) be a suitable weak solution of the Navier-Stokes equations in a domain D. Assume that Q D. Then there exists a suciently small constant 2 (0; 1] such that if 2 (0; 1) and if 1 2=3 kuk L 3 (Q) + 1 11=10 kpk L 5=4 t L 2 x (Q) ; then (x 0 ;t 0 ) is a regular point. For the proof, c.f. Theorem 1 from [V] or Remark 6.2.5 from [K1]. 91 Appendix B: Proof of the Higher Regularity of Solutions to the Primitive Equations of the Ocean In this section we prove Theorem 2.17 that we used to obtain the a priori estimates of the solutions to the primitive equations (2.27). Proof of Theorem 2.17 Without loss of generality, we set = 1. Let A(t) = 2 X k=1 kv k (;t)k 2 L 2 ! 1=2 and A(t) = 2 X k=1 krv k (;t)k 2 L 2 ! 1=2 : Applying the Laplacian operator to the primitive equations (2.27) k for k = 1; 2, multiplying by v k and integrating over the domain , we get 1 2 d dt 2 X k=1 Z jv k j 2 + 2 X k=1 Z jrv k j 2 = 2 X j;k=1 Z (v j @ j v k )v k 2 X k=1 Z (w@ z v k )v k =I 1 +I 2 ; (B.49) 92 where we used the divergence-free condition and the fact that the pressure is indepen- dent of the z variable. Next we estimate the terms I 1 and I 2 individually. Regarding I 1 , rst we integrate by parts and obtain I 1 = 3 X i=1 2 X j;k=1 Z @ i v j @ j v k @ i v k + 3 X i=1 2 X j;k=1 Z v j @ ij v k @ i v k =I 11 +I 12 : In order to estimate the termI 11 , we apply the H older's inequality and the Gagliardo- Nirenberg-Sobolev inequality and get I 11 3 X i=1 2 X j;k=1 k@ i v j k L 6k@ j v k k L 3k@ i v k k L 2 C 3 X i=1 2 X j;k=1 kr@ i v j k L 2k@ j v k k 1=2 L 2 kr@ j v k k 1=2 L 2 k@ i v k k L 2: Letting H(t) = 2 X k=1 krv k k 2 L 2 ! 1=2 and H(t) = 2 X k=1 krrv k k 2 L 2 ! 1=2 ; 93 we get I 11 CA(t)H(t) 1=2 H(t) 1=2 A(t): Similar argument implies I 12 3 X i=1 2 X j;k=1 kv j k L 6k@ ij v k k L 3k@ i v k k L 2 C 3 X i=1 2 X j;k=1 krv j k L 2k@ ij v k k 1=2 L 2 kr@ ij v k k 1=2 L 2 k@ i v k k L 2: Therefore, we have I 12 CA(t) 1=2 A(t) 3=2 H(t): Next we start the estimates of the term I 2 with integration by parts, and obtain I 2 = 3 X i=1 2 X k=1 Z @ i w@ z v k @ i v k + 3 X i=1 2 X k=1 Z w@ iz v k @ i v k =I 21 +I 22 : 94 In order to estimate the term I 21 , we apply the H older's inequality and get I 21 3 X i=1 2 X k=1 Z R 2 k@ i wk L 1 z k@ z v k k L 2 z k@ i v k k L 2 z dx 1 dx 2 C 3 X i=1 2 X k=1 Z R 2 k@ i wk 1=2 L 2 z k@ iz wk 1=2 L 2 z +k@ i wk L 2 z k@ z v k k L 2 z k@ i v k k L 2 z dx 1 dx 2 C 3 X i=1 2 X k=1 k@ i wk 1=2 L 2 k@ iz wk 1=2 L 2 z L 4 x 1 x 2 k@ z v k k L 2 z L 8 x 1 x 2 k@ i v k k L 2 +C 3 X i=1 2 X k=1 k@ i wk L 2 z L 4 x 1 x 2 k@ z v k k L 2 z L 4 x 1 x 2 k@ i v k k L 2; (B.50) where we used the Agmon's inequality in the z direction for the second inequality above and the H older's inequality in (B.50). Using the Gagliardo-Nirenberg-Sobolev inequality, the incompressibility, as well as w = 2 X j=1 Z x 3 h @ j v j dz; (B.51) the far right side of inequality (B.50) is bounded by C 3 X i=1 2 X k=1 kr 2 @ i vk 3=4 L 2 kr 2 r 2 @ i vk 1=4 L 2 k@ z v k k 1=4 L 2 kr 2 @ z v k k 3=4 L 2 k@ i v k k L 2 +C 3 X i=1 2 X k=1 kr 2 @ i vk 1=2 L 2 krr 2 @ i vk 1=2 L 2 k@ z v k k 1=2 L 2 kr 2 @ z v k k 1=2 L 2 k@ i v k k L 2; 95 where we used the Minkowski's inequality. Hence, I 21 CA(t) 3=4 A(t) 5=4 H(t) 1=4 H(t) 3=4 +CA(t) 1=2 A(t) 3=2 H(t) 1=2 H(t) 1=2 : Proceeding similarly, we estimate the term I 22 as I 22 C 3 X i=1 2 X k=1 Z R 2 kwk L 1 z k@ iz v k k L 2 z k@ i v k k L 2 z dx 1 dx 2 C 3 X i=1 2 X k=1 Z R 2 kwk 1=2 L 2 z k@ z wk 1=2 L 2 z +kwk L 2 z k@ iz v k k L 2 z k@ i v k k L 2 z dx 1 dx 2 ; (B.52) where we used the Agmon's inequality in the z direction. Then we further bound the right side of (B.52) by C 3 X i=1 2 X k=1 kwk 1=2 L 2 k@ z wk 1=2 L 2 z L 4 x 1 x 2 k@ iz v k k L 2 z L 8 x 1 x 2 k@ i v k k L 2 +C 3 X i=1 2 X k=1 kwk L 2 z L 4 x 1 x 2 k@ iz v k k L 2 z L 4 x 1 x 2 k@ i v k k L 2: 96 By using the divergence-free condition, the Gagliardo-Nirenberg-Sobolev inequality, and (B.51), we bound the above expression by C 3 X i=1 2 X k=1 kr 2 vk 3=4 L 2 kr 2 r 2 vk 1=4 L 2 k@ iz v k k 1=4 L 2 kr 2 @ iz v k k 3=4 L 2 k@ i v k k L 2 +C 3 X i=1 2 X k=1 C 3 X i=1 2 X k=1 kr 2 vk 1=2 L 2 krr 2 vk 1=2 L 2 k@ iz v k k 1=2 L 2 kr 2 @ iz v k k 1=2 L 2 k@ i v k k L 2; where we used the Minkowski's inequality. Therefore, for I 22 we have I 22 CA(t) 1=4 A(t) 7=4 H(t) 4=3 H(t) 1=4 +A(t) 1=2 A(t) 3=2 H(t) 1=2 H(t) 1=2 : Finally, combining all the above estimates leads to d dt A 2 +A 2 CH(t)H(t)A(t) 2 +CH(t) 4 A(t) 2 +CH(t) 2=3 H(t) 2 A(t) 2 +H(t) 2 H(t) 2 A(t) 2 +CH(t) 6 H(t) 2 A(t) 2 : (B.53) Due to the results in [KZ1, KZ2], we have H(t)2 L 1 (0;T) and H(t)2 L 2 (0;T). Thus, we conclude from (B.53) and the Gr onwall's inequality that A(t) is bounded in L 1 (0;T ) and A(t)is bounded in L 2 (0;T ). Theorem 2.17 is proven. 97 References [B] Y. Brenier, Remarks on the derivation of the hydrostatic Euler equations, Bull. Sci. Math. 127 (2003), no. 7, 585-595. [BGMR] D. Bresch, F. Guill en-Gonz alez, N. Masmoudi, and M. A. Rodr guiz-Bellido, In the uniqueness of weak solutions of the two-dimensional primitive equations, Dierential Integral Equations, 16, (2003), 77{94. [BK] H. Br egis and T. Kato, Remarks on the Schrodinger operator with singular complex potentials, J. Math. Pures Appl. (9) 58 (1979), no. 2, 137-151. [CF] P. Constantin and C. Foias: Navier-Stokes equations, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1988. [CKN] L. Caarelli, R.Kohn, and L. Nirenberg: Partial regularity of suitable weak solutions of the Navier-Stokes equations, Comm. Pure Appl. Math. 35(1982), no. 6, 771-831. [CT] C. Cao and E.S. Titi, Global well-posedness of the three-dimensional viscous primitive equations of large scale ocean and atmosphere dynamics, Annals of Math. 166, (2007), 245{267. 98 [EFNT] A. Eden, C. Foias, B. Nicolaenko, and R. Temam, \Exponential attractors for dissipative evolution equations," RAM: Research in Applied Mathematics, vol. 37, Masson, Paris, 1994. [F] K. Falconer, \Fractal geometry{Mathematical foundations and applications," second ed., John Wiley & Sons Inc., Hoboken, NJ, 2003, Mathematical foundations and applications. [FT] C. Foia s and R. Temam, Some analytic and geometric properties of the solutions of the evolution Navier-Stokes equations, J. Math. Pures Appl. (9) 58 (1979), no. 3, 339{368. [G] M. Giaquinta: Introduction to regularity theory for nonlinear elliptic systems, Lectures in Mathematics, Birkh auser Verlag, Berlin (1993), Chapter 6. [H] C. Hu, Asymptotic analysis of the primitive equations under the small depth assumption, Nonlinear Anal. 61 (2005), no. 3, 425-460. [Ho] E. Hopf, Uber die Anfangswertaufgabe f ur die hydrodynamischen Grundgle- ichungen, Math. Nachr. 4 (1951), 213{231. [K1] I. Kukavica: On partial regularity for the Navier-Stokes equations, Discrete Contin. Dynam. Systems 21(2008), 717-728. 99 [K2] I. Kukavica: The partial regularity results for the Navier-Stokes equations, Proceedings of the workshop on "Partial dierential equations and uid mechanics," Warwick, U.K., 2008. [K3] I. Kukavica: Regularity for the Navier-Stokes equations with a solution in a Morrey space, Indiana Univ. Math. J. 57(2008), 2843-2860. [K4] I. Kukavica: The fractal dimension of the singular set for solutions of the Navier-Stokes system, Nonlinearity, 2009. [KP] I. Kukavica and Y. Pei: An estimate on the parabolic fractal dimension of the singular set for solutions of the Navier-Stokes system, Nonlinearity. 25 (2012), 1{9. [Kp1] I. Kukavica, On the dissipative scale for the Navier-Stokes equation, Indiana Univ. Math. J. 48 (1999), no. 3, 1057{1081. [Kp2] I. Kukavica, Existence of a solution 'in the large' for the 3D large-scale ocean dynamics equations, C. R. Math. Acad. Sci. Paris, 343, (2006), 283{286. [KPRZ] , I. kukavica, Y. Pei, W. Rusin, and M. Ziane, Primitive equations with continuous initial data, Nonlinearity. 27 (2014) 1-21. 100 [KZ1] I. Kukavica and M. Ziane, On the regularity of the primitive equations of the ocean, Nonlinearity. 20 (2007), 2739{2753. [KZ2] I. Kukavica and M. Ziane, The regularity of solutions of the primitive equations of the ocean in space dimension three, C. R. Math. Acad. Sci. Paris. 345 (2007), 257{260. [L] P.G. Lemari e-Rieusset, Recent developments in the Navier-Stokes problem, Chapman & Hall/CRC Research Notes in Mathematics, vol. 431, Chapman & Hall/CRC, Boca Raton, FL, 2002. [Le] J. Leray: Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math. 63(1934), no.1, 193-248. [Li] F. Lin, A new proof of the Caarelli-Kohn-Nirenberg theorem, Comm. Pure Appl. Math. 51 (1998), no. 3, 241{257. [LS] O.A. Ladyzhenskaya and G.A. Seregin, On partial regularity of suitable weak solutions to the three-dimensional Navier-Stokes equations, J. Math. Fluid Mech. 1 (1999), no. 4, 356{387. [LTW1] J.L. Lions, R. Temam and S. Wang, New formulation of the primitive equations of the atmosphere and applications, Nonlinearity, 5, (1992), 237{ 188. 101 [LTW2] J.L. Lions, R. Temam and S. Wang, On the equations of the laege-scale ocean, Nonlinearity, 5, (1992), 1007{1053. [LTW3] J.L. Lions, R. Temam and S. Wang, Mathematical study of the coupled models of atmosphere and ocean (CAO III), J. Math. Pures Appl. 74, (1995), 105{163. [O] M. O'Leary: Conditions for the local boundedness of solutionsof the Navier- Stokes system in three dimensions, Comm. Partial Dierential Equations 28(2003), no. 3-4, 617-636. [P] J. Pedlosky, Geophysical Fluid Dynamics, 2nd Edition, Spring-Verlag, New York (1987). [Pe] M. Petcu, Gervey class regularity for the primitive equations in space dimen- sion 2, Asymptot. Anal. 39 (2004), no 1, 1-13. [RS1] J.C. Robinson and W. Sadowski: Decay of weak solutions and the singular set of the three-dimensional Navier-Stokes equations, Nonlinearity 20(2007), no.5, 1185-1191. [RS2] J.C. Robinson and W. Sadowski: Almost everywhere uniqueness of Lagrangian trajectories for suitable weak solutions of the three-dimensional Navier-Stokes equations, preprint. 102 [RS3] J.C. Robinson and W. Sadowski, Almost-everywhere uniqueness of Lagrangian trajectories for suitable weak solutions of the three-dimensional Navier-Stokes equations, Nonlinearity 22 (2009), no. 9, 2093{2099. [S1] V. Scheer: Partial regularity of solutions to the Navier-Stokes equations, Pacic J. Math. 66(1976), no.2, 535-552. [S2] V. Scheer: Turbulence and Hausdor dimension, Turbulence and Navier- Stokes equations (Proc. Conf., Univ. Paris-Sud, Orsay, 1975), Springer, Berlin, 1976, pp.174-183. Lecture Notes in Math., Vol.565. [S3] V. Scheer: Hausdor measure and the Navier-Stokes equations, Comm. Math. Phys. 55(1977), no.2, 97-112. [T] R. Temam: Navier-Stokes equations, AMS Chelsea Publishing, Providence, RI, 2001, Theory and Numerical Analysis, Reprint of the 1984 edition. [TZ] R. Temam and M. Ziane, Some mathematical problems in geophysical uid dynamics, Handbook of mathematical uid dynamics. Vol. (iii), 535{657, North-Holland, Amsterdam, 2004. [V] A.F. Vasseur, A new proof of partial regularity of solutions to Navier-Stokes equations, NoDEA Nonlinear Dierential Equations Appl. 14 (2007), no. 5-6, 753{785. 103 [W] J. Wolf, A direct proof of the Caarelli-Kohn-Nirenberg theorem, Parabolic and Navier-Stokes equations. Part 2, Banach Center Publ., vol. 81, Polish Acad. Sci. Inst. Math., Warsaw, 2008, pp. 533{552. [Z1] M. Ziane, Regularity results for Stokes type systems related to climatology, Appl. Math. Lett. 8 (1995), 53{58. [Z2] M. Ziane, Regularity results for the stationary primitive equations of the atmosphere and the ocean, Nonlinear Anal. 28 (1997), 289{313. 104

Abstract (if available)

Abstract In the first chapter of this dissertation, we address the partial regularity for a suitable weak solutions of the Navier-Stokes system in a bounded space‐time domain D. We show that the parabolic fractal dimension of the singular set is less than or equal to 45/29, which is an improvement of the earlier result from [K4]. Also, we introduce the new λ-fractal dimension and prove that the dimension of the singular set is bounded by 3/2 for a certain range of λ. The second chapter addresses the existence and uniqueness of the solutions for the primitive equations of the ocean with continuous initial datum. We split the initial data into a regular finite energy part and a small bounded part and show that the equations are preserved by the splitting, which enables us to prove the well‐posedness of solutions. We provide a priori estimates and the construction of the solutions.

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Creator Pei, Yuan (author)

Core Title Certain regularity problems in fluid dynamics

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Applied Mathematics

Publication Date 07/07/2014

Defense Date 04/30/2014

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

Tag fractal dimension,Navier-Stokes equations,OAI-PMH Harvest,primitive equations,regularity,weak solutions

Format application/pdf (imt)

Language English

Contributor Electronically uploaded by the author (provenance)

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

Creator Email peimathatpku@gmail.com,ypei@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c3-432069

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