Primitive equations with continuous initial data

Transcription

1 London Mathematical Society Nonlinearity Nonlinearity 27 ( UNCORRECTED PROOF Primitive euations with continuous initial data Igor Kuavica 1, Yuan Pei 1, Walter Rusin 2 and Mohammed Ziane 1 1 Department of Mathematics, University of Southern California, Los Angeles, CA 90089, USA 2 Department of Mathematics, Olahoma State University, Stillwater, OK 74078, USA uavica@usc.edu, ypei@usc.edu, walter.rusin@ostate.edu and ziane@usc.edu Received 4 June 2013, revised 10 March 2014 Accepted for publication 27 March 2014 Published Recommended by ESTiti Abstract We address the well-posedness of the primitive euations of the ocean with continuous initial data. We show that the splitting of the initial data into a regular finite energy part and a small bounded part is preserved by the euations thus leading to existence and uniueness of solutions. Keywords: XXX Mathematics Subect Classification: XXX AQ1 AQ2 1. Introduction In this paper, we analyse the structure of the nonlinearity of the primitive euations t v νv + (v v + 3 (wv + p = 0, = 1, 2 v + 3 w = 0 (1.1 enabling us to establish a well-posedness theory for data with no differentiability properties. In particular, we obtain the uniueness of wea solutions under a mild assumption that the initial data are only continuous in the space variables. The primitive euations of the atmosphere and the ocean are considered to be the fundamental model for meteorology and climate prediction [P]. Indeed, the full compressible /14/ $ IOP Publishing Ltd & London Mathematical Society Printed in the UK 1 JNL: NON PIPS: TYPE: PAP TS: NEWGEN DATE: 15/4/2014 EDITOR: TT SPELLING: UK

2 Navier Stoes euations, which govern the dynamics of the atmosphere and the ocean, are very complicated and contain phenomena which are not interesting from the geophysical point of view, such as shocs and sound waves. The Boussines approximation along with the hydrostatic balance leads to the primitive euations. The main part of the system consists of the momentum euations and the conservation of mass, a simplified version of which is given above. The primitive euations also contain the thermodynamic euations (diffusion of temperature, as well as the diffusion of humidity for the atmosphere and diffusion of salinity for the ocean, see [PTZ, TZ]. The mathematical theory started with the wor of Lions, Temam and Wang [LTW1, LTW2, LTW3] who set the analytical foundation for the euations and established the global existence of wea solutions for suare integrable initial data in the spirit of Leray. The H 2 regularity of the associated stationary linear problem was obtained in [Z1, Z2]. This result implied the local existence of strong solutions with initial data in H 1, which was established by Bresch et al [BGMR1, BGMR2, BGMR3, BGMR4] and independently by Hu et al [HTZ]. The global existence of strong solutions with initial data in H 1 was proven by Cao and Titi in [CT1]. For other wors on the primitive euations, see [CT2,GH,GZ,H,J,K,KZ1,KZ2,KZ3,STT,SV] (see also [KTVZ,MW,R,RTT1,RTT2] for the inviscid case. In conclusion, the global existence of wea solution without uniueness is nown for both two and three space dimensions. Imposing the H 1 regularity for initial data leads to global existence and uniueness of solutions (2D and 3D. The state of analysis for the primitive euations may seem to be much better than the one for the Navier Stoes euations. However, when considering the uniueness of wea solutions of the primitive euations in 2D, a classical and elementary fact for the 2D Navier Stoes euations, we face the obstacle of a derivative loss in the nonlinearity. For this reason, the well-posedness in L p for the primitive euations remains open in both the 2D and 3D cases for any p. We note the derivative loss in the nonlinearity constitutes a primary reason for the ill-posedness and finite time blow-up for the inviscid primitive euations in Sobolev spaces [R, CINT]. The main goal of this paper is to obtain spaces with less differentiability than H 1 for wellposedness. In this spirit Bresch et al provedin[bkl] the uniueness in 2D for wea solutions with x3 v 0 H 1/2. In this paper, we establish the well-posedness (existence and uniueness of solutions with only continuous initial data that reuire no differentiability. Our approach relies on the splitting of the initial data into a smooth finite energy part and a small bounded part. In our reasoning we exploit the fact that this splitting is preserved by the euation, the main difficulty being caused by the pressure and the derivative loss terms. We note that the main reason for the choice of the space in which we see solutions is the possibility of establishing uniueness. For the uniueness statements in which a splitting method has been used in different contexts, see [BK, LM]. The paper is organized as follows. In section 2, we set up the notation, and write a simple form of the primitive euations which contains the main mathematical difficulties. The results that we obtain in this paper easily carry over to the general systems of euations. Our main existence result is contained in theorem 2.2, while theorem 2.3 asserts uniueness. The proof of theorem 2.2 relies on the H 2 regularity of solutions for which we give a setch in appendix A. Section 3 contains the proof of the consistency of splitting given continuous initial data decaying at infinity (or simply continuous in the case of periodic data. Finally, section 4 provides the proof of the uniueness of solutions. 2

3 2. Main results Let 0 = R 2 [, 0]. The primitive euations of the ocean read 3 t u νu + (u u + p = 0, = 1, 2 3 u = 0. (2.1 Let u(x, t = (u 1 (x, t, u 2 (x, t, u 3 (x, t = (v(x, t, w(x, t, (2.2 where v = (v 1,v 2 and w are the horizontal and the vertical components respectively. The initial data v 0 = (v 01,v 02 : 0 R 2 satisfy 0 div 2 v 0 dx 3 = 0 (2.3 in the sense of distributions. We emphasize that the pressure is two dimensional and that there is no euation for u 3 other than divu = 0. As in [CT1], we euip system (2.1 with the Neumann boundary conditions v = 0 and w = 0 (2.4 x 3 on x 3 =and x 3 = 0. Since our initial data are merely continuous and v/ x 3 may not be well defined, we tae advantage of the fact that the above formulation is euivalent to the problem on = R 2 [, h] with periodic boundary conditions in the x 3 direction (see [GKVZ, Pe]. By this extension, the u 1 and u 2 components of the velocity field u are even in the x 3 variable, whereas the u 3 component is odd. We denote by C 0 ( the space of continuous functions on which vanish at infinity. To simplify the notation, we denote by Bɛ ( [0,T= { u L t L x (: u L t L x ( ɛ } the ball of radius ɛ in the space L x,t (. Let us define the spaces h H = {v :div 2 } v dx 3 = 0onR 2, (2.5 and V = H {v H 1 : v(,x 3 = v(, x 3, v(,x 3 = v(,x 3 +2h}. Definition 2.1. We say that v L ([0,, H loc ([0,, V is a wea solution of (2.1 with initial data v 0 H if for every ϕ D( [0, such that divϕ = 0 we have 3 u t ϕ + ν u ϕ u u ϕ 0 0 v 0 ϕ (, 0 = 0. (

4 We recall the following result regarding the existence of wea solutions to the primitive euations. Theorem 2.1 ([LTW1, LTW2, TZ]. For every v 0 H, there exists a wea solution for the primitive euation (2.1 as in definition 2.1. We now state our main results. For simplicity, we assume that the viscosity constant ν euals 1. The first theorem provides the existence of solutions to euation (2.1. Theorem 2.2. Assume v 0 C 0 ( (. There exists a constant C > 0 and a wea solution satisfying for any ɛ>0the decomposition v C([0,, H 2 loc ([0,, H 3 + BCɛ ( [0, to euation (2.1. Namely, v = v + V where v L ([0,, H 2 loc ([0,, H 3 and V BCɛ ( [0,, where C is a constant independent of the solution. Note that the obtained solution is continuous on R 3 (0,. The second main result addresses the uniueness of solutions in the class C([0,T],H 2 ([0,T],H 3 + Bɛ ( [0,Tfor a sufficiently small ɛ>0. Theorem 2.3. Let T > 0. Suppose that v (1 and v (2 are two wea solutions of the primitive euations (2.1 with the same initial data v 0 C 0 ( such that v (1,v (2 L ([0,T],H 2 ([0,T],H 3 + Bɛ ( [0,T where ɛ > 0 is sufficiently small. Then v (1 = v (2 for 0 t T. The proof of theorem 2.2 is given in section 3, while theorem 2.3 is proven in section 4. Remar 2.4. Theorems 2.2 and 2.3 are also valid in the case = T 2 [, 0]. In this situation we replace the reuirement v 0 C 0 ( ( with v 0 C(. 3. Proof of existence The proof of theorem 2.2 uses the splitting approach introduced by Brezis and Kato in [BK]. It was also used by Calderón in [C] for the construction of wea solutions for the Navier Stoes euations with L p initial data (see also [GK] for construction of global solutions. Here we use the splitting in the uniform norm. In the proof of theorem 2.2 we use the following result concerning the higher regularity of solutions of primitive euations. The reader can find a more general statement in [PW]. For completeness, we include a simple proof in appendix A. Theorem 3.1. Assume v 0 V and let v L ([0,T],V ([0,T],H 2 be the associated solution of euation (2.1. If in addition v 0 H 2, then v L ([0,T],H 2 ([0,T],H 3. Proof of theorem 2.2. Let v 0 C 0 ( ( and fix a sufficiently small ɛ>0, to be chosen below. Then let ϕ δ be a standard mollifier which is in addition radially symmetric. More precisely, let ϕ be a smooth, compactly supported, non-negative, and radially symmetric function such that ϕ = 1. For δ>0, let ϕ δ(x = δ 3 ϕ(x/δ. Given v 0 = v 0 ϕ δ and for δ sufficiently small, we have v 0 v 0, v 0 v 0 L <ɛ. The function v 0 is infinitely smooth and satisfies v 0 H for all N. Due to radial symmetry of ϕ, it is also even in the vertical variable (so that it satisfies the Neumann boundary condition and it satisfies the divergence-free condition (2.3. (Naturally, the size of the H norms of v 0 depends on ɛ and grows as ɛ 0. However, this has no bearing as we do not need to consider the limit ɛ 0. 4

5 By [CT1] (see also [KZ1], we now that system (2.1 with the initial data v 0 has a uniue strong solution v C([0,T],H 1 ([0,T],H 2 for any T > 0. Note that V 0 = v 0 v 0 satisfies V 0, V 0 L <ɛ. Taing the difference of the euations for u = (v, w, and u = (v, w we obtain that U = (V, W = u u = (v v, w w satisfies t V V + V V + W 3 V + v V + w 3 V + V v + W 3 v + P = 0, = 1, 2 V + 3 W = 0, (3.1 where P = p p. Since V 0 L ( (, we have by interpolation V 0 L ( for all [2, ] with V 0 L <ɛ. Now, we obtain a priori estimates for V in L (. Let φ = V /2 2 L, φ 2 = ( V /2 2 L. (3.2 2 We multiply the euations (3.1 1 by V V 2, integrate over, and sum for = 1, 2. We get V 4( dt 2 ( V /2 ( V /2 = V V 2 P V v V V 2 W 3 v V V 2, = I 1 + I 2 + I 3, (3.3 where we integrated by parts. We proceed now to estimating the terms I 1, I 2 and I 3. Regarding the term I 1, we integrate by parts and obtain V V 2 P = P (V V 2. (3.4 Hölder s ineuality then yields I 1 C P V /2 1 ( V /2 C P V /2 1 ( V /2. (3.5 In order to estimate the first factor on the right-hand side of (3.5, we note that for = 1, 2 h P V /2 1 2 L = P 2 V 2 2 = P 2 V 2 dx 3 dx x 2 R 2 h L P 2 L /4 (R 2 V 2 dx 3 /( 4 (R 2 h P 2 L /4 (R 2 V 2 L /( 4 (R 2 dx 3, (3.6 where we used Hölder s and Minowsi s ineualities. By the Gagliardo Nirenberg Sobolev ineuality, we then have h V 2 h L /( 4 (R 2 dx 3 = V /2 (2 4/ L (2 4/( 4 (R 2 dx 3 h C V /2 (2 8/ (R 2 2 ( V /2 4/ (R 2 dx 3. (3.7 5

6 We bound the right-hand side of the above expression using Hölder s ineuality by C V /2 (2 8/ 2 ( V /2 4/. (3.8 From (3.6 (3.8 we obtain C P V /2 1 C P 2 1/2 V L /4 /2 ( 4/ 2 ( V /2 2/ = C P L /2 V /2 ( 4/ 2 ( V /2 2/. (3.9 The L /2 norm of the pressure may be estimated using lemma 3.3 below. Therefore, we bound the expression (3.9by ( C V 2 L + v L V L V /2 ( 4/ 2 ( V /2 2/. (3.10 Thus we obtain ( I 1 C V 2 L + v L V L V /2 ( 4/ 2 ( V /2 (+2/ = C C ( V /2 4/ + v L V /2 2/ V /2 ( 4/ 2 ( V /2 (+2/ V /2 2 ( V /2 (+2/ v L V /2 ( 2/ 2 ( V /2 (+2/ φ (+2/2 Cφ 1/2 v L φ ( 2/2 In order to estimate the term I 2, we apply Hölder s ineuality I 2 =, V v V V 2 φ (+2/2. (3.11 v L 3 V V V /2 2 V /2 L 6. (3.12, We bound the right-hand side of (3.12 using the Gagliardo Nirenberg Sobolev ineuality and obtain ( I 2 C v 1/2 v 1/2 + v V /2 ( ( V /2 + V /2 C,, v 1/2 v 1/2 φ 1/2 φ 1/2, v 1/2 v 1/2 φ, v φ 1/2 φ 1/2 v φ. (3.13 Regarding the term I 3, we observe that since x3 W = i V i d x 3, (3.14 i=1 6,

7 we have after integration by parts I 3 = W 3 v V V 2 = = i, i, i ( x 3 ( x 3 V i d x 3 i3 v V V 2 V i d x 3 3 v V V 2 i, ( x 3 V i d x 3 3 v i ( V V 2 = I 31 + I 32. (3.15 Using Hölder s and Minowsi s ineualities, we bound the term I 31 as I 31 C which we rewrite as I 31 C V i L i3 v V ( 2/2 L 3/( 3 V /2 L 6, (3.16 i, i, V i L i3 v V /2 ( 2/ L (3 6/( 3 V /2 L 6. (3.17 Applying the Gagliardo Nirenberg Sobolev ineuality, we bound the right-hand side of (3.17 by ( ( 2/( C V i L i3 v V /2 ( 4/(2 4 ( V /2 /(2 4 + V /2 i, whence ( ( V /2 + V /2, (3.18 I 31 C i, i3 v φ 1/4 i, φ 3/4 i3 v φ 3/4 φ 1/4 i, i3 v φ 1/2 φ 1/2 i3 v φ. (3.19 In order to estimate the term I 32, we use Hölder s and Minowsi s ineualities and obtain 1 ( x 3 ( I 32 = V i d x 3 3 v i V V 2 Since C i,=2 i, V i L 3 v L 6 i (V V 2 L 6/(5 6. (3.20 i, ( i V V 2 2( 1 = V V /2 2 ( i V /2 (3.21 we bound the right-hand side of (3.20 using Hölder s ineuality by C V i L 3 v L 6 V /2 1 ( L 3/( 3 i V /2, (3.22 i, 7

8 which we rewrite as C i, V i L 3 v L 6 V /2 ( 2/( L (3 6/( 3 i ( V /2. (3.23 Then by the Gagliardo Nirenberg Sobolev ineuality, we obtain I 32 C C C V i L ( 3 v + 3 v i, ( V /2 ( 4/(2 ( V /2 1/2 + V /2 ( 2/ ( i V /2 ( 3 v + 3 v ( V /2 1/2 ( V /2 1/2 + V /2 i ( V /2 i, 3 v φ 1/4 φ 3/4 3 v φ 1/4 φ 3/4 3 v φ 1/2 3 v φ 1/2 φ 1/2 From (3.3, (3.11, (3.13, (3.19 and (3.24, we conclude dt φ 4( 1 + φ 2 Cφ 1/2 φ (+2/2 v L φ ( 2/2 φ (+2/2,, i, v 1/2 v 1/2 φ 1/2 φ 1/2 v 1/2 v 1/2 φ i3 v φ 3/4 3 v φ 1/4 3 v φ 1/4 φ 1/4 φ 3/4 φ 3/4 φ 1/2. (3.24, v φ, i3 v φ i, 3 v φ 1/2 3 v φ 1/2 v φ 1/2 φ 1/2 φ 1/2 φ 1/2. (3.25 Applying Young s ineuality, we obtain dt φ 4( 1 + φ 2 C (3+2/( 2 φ /( φ (3+2/( 2 v 2/( 2 L φ φ +C +C,, v v φ φ v 1/2 v 1/2 φ 8, v φ, v 2 φ φ

9 +C 1/3 +C 3 +C 3 i, i3 v 4/3 φ φ 3 v 4 φ φ 3 v 4 φ φ i3 v φ i, which leads to d dt φ + φ C 4/( 2 φ /( 2 4/( 2 v 2/( 2 L φ 2 2 v 2 L φ 2, 4/3 i, 3 v 2 φ φ 3 v 2 φ φ, (3.26 v v φ, v 1/2 v 1/2 φ, i3 v 4/3 φ 2 3 v 2 L φ 2 4 i3 v φ 4 i, 3 v 4 L φ 2 2 v φ, 3 v 4 L φ 2 3 v 2 L φ 2. (3.27 Following [Ku], we note that φ φ5/3 Cφ 10/3 /2. (3.28 Cφ 4/3 /2 Ineuality (3.27 then yields d dt φ + φ5/3 Cφ/2 2 4/( 2 φ /( 2 Cφ 4/3 /2 2 v v φ 2,, 4/3 i, v 1/2 v 1/2 φ i3 v 4/3 φ 2 3 v 2 L φ 2 4 4/( 2 v 2/( 2 L φ v 2 L φ 2, v φ, i3 v φ 4 i, 3 v 4 L φ v 4 L φ 2 3 v 2 L φ 2, (3.29 which by writing φ = C(φ 5/3 /Cφ 4/3 /2 3/5 φ 4/5 /2 for each factor φ on the right-hand side and using Young s ineuality in order to absorb the φ 5/3 /Cφ 4/3 /2 terms leads to d dt φ + φ5/3 Cφ 4/3 Cφ/ φ 2/( 1 /2 10 v 5/( 2 L φ2 /2 /2 +C 5 v 5/2 v 5/2 φ 2 /2, 9

10 +C 5 +C 5/2 +C 5/2 +C 5 v 5 L φ 2 2 /2 5/2,, i,, v 5/2 φ 2 /2 10/3 i3 v 5/2 φ 2 / v 5 L φ 2 2 /2 10 v 5/4 v 5/4 φ 2 /2 i, i3 v 10/3 φ 2 /2 3 v 10 φ 2 /2 3 v 10 φ 2 /2 5 3 v 5 L φ 2 2 /2. (3.30 Let M = max{ v L ([0,T ],L (, v L ([0,T ], (, 2 v L ([0,T ], (}. (3.31 The estimate (3.30 can be rewritten as d dt φ C 10 φ 2/( 1 /2 10 M 5/( 2 φ/2 2 5 M 5 φ/2 2 5/2 M 5/2 φ/2 2 10/3 M 10/3 φ/ M 10 φ/2 2. (3.32 For simplicity assume M C (i.e. we let constants depend on M. Ineuality (3.32 leads to d dt φ C 10 φ 2/( 1 /2 10 φ/2 2. (3.33 Let R (t = V L ([0,t],L (. Integrating (3.33 over (0,tand raising both sides to power 1/, and using V 0 L ɛ we obtain ( R (t C 10 (R /2 (t/( 1 10 R /2 (t + ɛ 1/. (3.34 In order to pass to the limit we use lemma 3.2 below. We start with the estimate for = 8. In this case, ineuality (3.29 yields d dt φ 8 Cφ 4/3 8 v 8/3 φ L 8 8 v v φ 8, v 2 L φ 2 8, i, i3 v 4/3 φ 8 3 v 2 L φ 2 8, v 1/2 v 1/2 φ 8 i3 v φ 8 i, 3 v 4 L φ 2 8 v φ 8, 3 v 4 L φ v 2 L φ 2 8. (3.35 We note that initially, by interpolation φ 8 ɛ 8. Furthermore, there exists a sufficiently small T ɛ > 0 such that by Grönwall s ineuality we obtain the bound R 8 (T ɛ 2ɛ. (3.36 Furthermore, for ɛ sufficiently small, using lemma 3.2 with S n = R 2 n(t ɛ and n 0 = 3weget sup R 2 n(t ɛ Cɛ, (3.37 n 10

11 whence we have V L ([0,T ɛ ],L ( and V L ([0,T ɛ ],L ( Cɛ (3.38 for = 1, 2. Moreover, note that since v 0 H, by theorem 2.1 there exists 0 <T 1 <T ɛ such that v(t 1 V. Therefore, there exists T 1 <T 2 <T ɛ such that v(t 2 H 2. Thus, we observe that the splitting v C([0,T],H 2 ([0,T],H 3 + BCɛ ( [0,T holds on [0,T] by theorem 3.1, where T>0is arbitrary. The theorem is thus proven. Lemma 3.2. Let S n0,s n0 +1,S n0 +2,...be a seuence satisfying the recurrence relation S n ( A2 Kn (Sn 1 2n /(2 n 4 2n + A2 Kn Sn 1 2n + 1/2 n ɛ2n, (3.39 where A 1, K>0, and ɛ (0, 1]. Then there exists a constant C>0such that sup S n Cɛ (3.40 n provided S n0 ɛ and ɛ is sufficiently small. Proof of lemma 3.2. Consider the seuence defined by α n0 = 1 and α n+1 = 4 1/2n A 1/2n 2 Kn/2n α n (3.41 for n = n 0,n 0 +1,... Note that by A 1wehave α n 1, n = n 0,n 0 +1,... (3.42 as well as α = sup α n = lim α n <, n n which follows from A 1/2n 2 Kn/2n <. n=n 0 4 1/2n Assuming that ɛ>0is so small that αɛ 1, we obtain α n ɛ 1, n = n 0,n 0 +1,... (3.43 We shall prove by induction that S n α n ɛ, n = n 0,n 0 +1,... (3.44 Clearly, the claim holds for n = n 0. Assuming that (3.44 holds up to n 1, we have S n ( A2 Kn (Sn 1 2n /(2 n 4 + A2 Kn S 2n n 1 2n + 1/2 n ɛ2n ( 2A2 Kn Sn 1 2n + 1/2 n ( ɛ2n 2A2 Kn αn 1 2n +1 1/2 n ɛ ( 4A2 Kn 1/2 αn 1 2n n ɛ, (3.45 where we used S n 1 α n 1 ɛ 1 in the first and (3.42 in the last ineuality. We thus obtain S n 4 1/2n A 1/2n 2 Kn/2n α n 1 ɛ = α n ɛ by (3.41. The induction step is thus established and the proof is complete. Lemma 3.3. The pressure P = P(x 1,x 2 in euation (3.1 satisfies the estimate ( P L /2 = p p L /2 C V 2 L + v L V L. (

12 Proof of lemma 3.3. Recall that u satisfies the euation t v v + and u satisfies the euations t v v + (v v + 3 (wv + p = 0, = 1, 2 (3.47 (v v + 3 ( wv + p = 0, = 1, 2 (3.48 Applying the averaging operator M[ ] = 1 h dx 3 (3.49 2h to euations (3.47 (3.48 and taing their difference, we obtain t M[V ] M[V ]+ ( M[v v ] M[v v ] + (p p = 0, = 1, 2. (3.50 Using V = v v, and applying div 2 to euation (3.50, we obtain 2 (p p = = = = (M[V V ]+M[v V ]+M[V v ], (M[V V ]+M[v V ]+M[V v ], M[ 33 V ] (M[V V ]+M[v V ]+M[V v ], 3 V (, M[V ] M[ 2 V ] ( 3 V (,h (M[V V ]+M[v V ]+M[V v ], (3.51, where we used the periodicity in the x 3 direction. Therefore, we have P = ( 2 1 (V V + v V + V v. Using the Calderón Zygmund theorem, we get P L /2 C C M[V V ]+M[v V ]+M[V v ] L /2 completing the proof of the lemma. ( V 2 L + v L V L, (

13 4. Proof of uniueness In this section we tae advantage of the splitting to show uniueness of solutions. Proof of theorem 2.3. Let (u (1,p (1 and (u (2,p (2 be two wea solutions of (1.1 with initial data v 0 as in theorem 2.2. Let u = u (1 u (2. Note that u = (v, w satisfies t v v + v (2 v + w (2 3 v + v v (1 + w 3 v (1 + (p (1 p (2 = 0, = 1, 2, v + 3 w = 0, (4.1 with the initial data v 0 = 0. Let ( 1/2 ( 1/2 V(t= v 2 L, V(t= v 2 2 L. (4.2 2 Multiplying (4.1 for = 1, 2byv, integrating over, summing over, and integrating by parts we obtain v 2 + v 2 = v v (1 2 dt v w 3 v (1 v, = v v (1 v + wv (1 3v = I 1 + I 2, (4.3 where we used the fact that u is divergence-free. In order to estimate the term I 1 we split it into two parts I 1 = v v (1 v = v v (1 v + v ṽ (1 v = I 11 + I 12, (4.4,,, where v (1 L ([0,T],H 2 ([0,T],H 3 and ṽ (1 BCɛ ( [0,Tas in theorem 2.2. Regarding the term I 11, we apply Hölder s ineuality and obtain v v (1 v v L 3 v (1 L 6 v C,,,, v 1/2 v 1/2 v (1 v, (4.5 where we used the Gagliardo Nirenberg Sobolev ineuality. Therefore I 11 C v (1 V(t 1/2 V(t 3/2. (4.6 In order to estimate the term I 12, we note that I 12 = v ṽ (1 v v ṽ (1 L v, (4.7,, 13

14 whence I 12 CɛV(t V(t. (4.8 Regarding the term I 2 on the right-hand side of (4.3, we split it into I 2 = wv (1 3v = wv (1 3v + wṽ (1 3v = I 21 + I 22. (4.9 In order to estimate the term I 21, we note that wv (1 3v = 3 wv (1 v w 3 v (1 v. (4.10 Using Hölder s ineuality, we may bound the right-hand side of the above expression by 3 w v (1 L 6 v L 3 + w 3 v (1 L 6 v L 3. (4.11 Therefore, by the Gagliardo Nirenberg Sobolev ineuality and (3.14 we obtain I 21 C v v (1 v 1/2 v 1/2 v 3 v (1 v 1/2 v 1/2, (4.12 whence I 21 C v (1 V(t 1/2 V(t 3/2 3 v (1 V(t 1/2 V(t 3/2. (4.13 Regarding the term I 22,byHölder s ineuality we obtain I 22 = wṽ (1 3v w ṽ (1 L 3v v ṽ (1 L v Cɛ V(t 2, (4.14 where we used (3.14. Estimates (4.3 (4.14 yield 2 dt V 2 + V 2 C v (1 V(t 1/2 V(t 3/2 ɛv(t V(t 3 v (1 V(t 1/2 V(t 3/2 ɛ V(t 2. (4.15 Using Young s ineuality we get 2 dt V 2 + V 2 C v (1 4 L V(t 2 ɛv(t v (1 4 L V(t 2 ɛ V(t 2, ( and assuming that ɛ is small enough, we obtain 2 dt V 2 C v (1 4 L V(t 2 ɛv(t v (1 4 L V(t 2. ( Since V(0 = 0, by Grönwall s ineuality V(t= 0, (4.18 and thus u (1 = u (2. This concludes the proof of the theorem. 14

15 5. Construction of solutions In section 3 we presented a priori estimates which lead to the existence of solutions in the considered class. At this point we briefly outline a construction of such solutions. To avoid unessential technical complications, we restrict ourselves to the case = T 3. The case = R 2 [, h] is handled similarly. Fix V 0 as in the proof of theorem 2.2. Let v be the solution to the corresponding finite energy part and denote w = x 3 div 2v d x 3. For a standard mollifier ϕ δ, consider the seuence V (δ 0 = V 0 ϕ δ. For any fixed δ, by the results in [LTW1, LTW2, TZ] we obtain the existence of a wea solution V (δ to the system t V (δ + V (δ + V (δ V (δ + W (δ 3 V (δ + v V (δ V (δ v + W (δ 3 v + P (δ = 0, = 1, 2, + w 3 V (δ V (δ + 3 W (δ = 0, (5.1 with the initial data V (δ 0. Multiplying (5.1 1 by V (δ, integrating over and summing over we obtain 2 dt V (δ 2 L + V (δ 2 2 L = V (δ 2 v V (δ W (δ 3 v V (δ. (5.2, Integrating by parts in both terms on the right-hand side of the above euation we get 2 dt V (δ 2 L + V (δ 2 2 = V (δ v V (δ + W (δ v 3 V (δ =,, V (δ v V (δ ( C v H 2 +, V (δ V (δ + 3 W (δ v V (δ 3 W (δ V (δ W (δ 3 v V (δ W (δ 3 v L 6 V (δ L 3, (5.3 where we used Höder s ineuality in the last term. We use the divergence-free condition and the Gagliardo Nirenberg Sobolev ineuality to obtain ( 2 dt V (δ 2 L + V (δ 2 2 L C v 2 H 2 V (δ V (δ + i V (δ i V (δ, i, W (δ 3 v V (δ 1/2 V (δ 1/2, (5.4 15

16 which using W (δ = i=1 x3 leads to ( 2 dt V (δ 2 L + V (δ 2 2 L C v 2 H 2 0 i V (δ i d x 3 (5.5 v H 2, i, V (δ V (δ + i, i V (δ i V (δ i V (δ i V (δ 1/2 V (δ 1/2, (5.6 where we used Minowsi s ineuality. Therefore by Hölder s and Young s ineualities, 2 dt V (δ 2 L V (δ 2 L C v 2 2 H V (δ 2 2 L v 4 2 H V (δ 2 2 L. (5.7 2 By Grönwall s ineuality applied to the above expression, we obtain an estimate for V (δ in L ([0,T], ([0,T],H 1 which is uniform in δ. In order to apply the Aubin Lions lemma we also need a uniform bound on t V (δ and thus an estimate on P (δ. Note that due to the divergence-free condition, euation (5.1 may be rewritten as t V (δ V (δ + (V (δ V (δ + 3 (W (δ V (δ + (v V (δ + 3 (wv (δ + (V (δ v + 3 (W (δ v + P (δ = 0, = 1, 2. (5.8 Averaging in the x 3 direction, and using M 3 0, for = 1, 2 we get t MV (δ MV (δ + M (V (δ V (δ + M (v V (δ + M (V (δ v + P (δ = 0, (5.9 where we also too advantage of the fact that the pressure does not depend on x 3. Applying, summing over and using the divergence-free condition leads to 2 P (δ =, M (V (δ V (δ +, M (v V (δ +, M (V (δ v. (5.10 Proceeding as in the proof of lemma 3.3, we obtain ( P (δ = R R M(V (δ V (δ + M(v V (δ + M(V (δ v, (5.11, where R i denotes the Riesz transform. Therefore by the Calderón-Zygmund theorem P (δ L 3/2 (T 2 C ( M(V (δ V (δ L 3/2 (T 2 + M(v V (δ L 3/2 (T 2 + M(V (δ v L 3/2 (T 2., 16 (5.12

17 Applying Hölder s and Minowsi s ineualities we get P (δ L 3/2 C ( V (δ, L 3 V (δ L 3 + v L 3 V (δ L 3 + V (δ L 3 v L 3, (5.13 which in turn by the Gagliardo Nirenberg Sobolev ineuality leads to P (δ L 3/2 C V (δ V (δ v 1/2 v 1/2 V (δ 1/2 V (δ 1/2. (5.14 Hence, we obtain a uniform bound on P (δ in ([0,T],L 3/2, and thus P (δ is uniformly bounded in ([0,T],H 3/2. We now rewrite euation (5.8 in the form t V (δ = V (δ (V (δ V (δ 3 (W (δ V (δ (v V (δ 3 (wv (δ (V (δ v 3 (W (δ v P (δ, = 1, 2. (5.15 Observe that V (δ is uniformly bounded in ([0,T],H 2, whereas 2 (V (δ V (δ, 2 (v V (δ and 2 (V (δ v are uniformly bounded in ([0,T],H 3/2. Regarding the third term on the right-hand side of (5.15, for ϕ H 2 we have 3 (W (δ V (δ ϕ = W (δ V (δ 3 ϕ = T 3 T 3 T 3 ( x 3 = V (δ i d x 3 i V (δ 3 ϕ + i=1 T 3 0 i=1 ( x 3 i i=1 T 3 ( x V (δ i d x 3 V (δ 3 ϕ V (δ i d x 3 V (δ i3 ϕ. (5.16 Minowsi s ineuality and the above obtained bounds for V (δ and V (δ imply uniform bounds of V (δ i i V (δ and V (δ i V (δ in L 4/3 ([0,T],H 2. Therefore, t V (δ is uniformly bounded in L 4/3 ([0,T],H 2. Application of the Aubin Lions lemma yields the existence of a seuence of V (δ converging strongly in ([0,T], to V, which in addition satisfies the same energy estimate as V (δ. Furthermore, passing to a subseuence we can ensure that the convergence is a.e. pointwise. This implies that such limit inherits the a priori estimates presented in section 3, while theorem 2.3 guarantees uniueness of the limit. Acnowledgments IK was supported in part by the NSF grant DMS , YP was supported in part by the NSF grants DMS , DMS and DMS , WR was supported in part by the NSF grant DMS , while MZ was supported in part by the NSF grant DMS We would lie to express our gratitude to the referees for useful suggestions on the earlier versions of the paper. Appendix A. Proof of theorem 3.1. Without loss of generality, we may set ν = 1. Let A(t = ( 2 v (,t 2 1/2, and Ā(t = ( 2 v (,t 2 1/2. Applying to (1.1 for 17

18 = 1, 2, multiplying by v, and integrating over, we obtain v 2 + v 2 = (v i i v v 2 dt i, (w 3 v v = I 1 + I 2, (A.1 which follows from the divergence-free condition and the fact that the pressure is independent of the x 3 variable. Regarding the first term on the right-hand side of (A.1, we integrate by parts and obtain 3 3 I 1 = i v v i v + v i v i v = I 11 + I 12. (A.2 i=1, i=1, Using the Hölder and the Gagliardo Nirenberg Sobolev ineualities we estimate the term I 11 by 3 I 11 i v L 6 v L 3 i v i=1, C 3 i=1, i v v 1/2 v 1/2 i v. (A.3 Therefore, we have I 11 CA(tH (t 1/2 H(t 1/2 Ā(t, where H(t = ( 2 v 2 1/2 and H(t = ( 2 v 2 1/2. Similarly, we obtain 3 I 12 v L 6 i v L 3 v i=1, C 3 i=1, v i v 1/2 i v 1/2 i v, whence I 12 CH (ta(t 1/2 Ā(t 3/2. Integrating by parts in the term I 2,weget 3 3 I 2 = i w 3 v i v + w i3 v i v = I 21 + I 22. i=1 i=1 Regarding the term I 21,Hölder s ineuality yields 3 I 21 i w L x3 3 v x3 i v x3 dx x 2 R 2 C i=1 3 i=1 R 2 (A.4 (A.5 ( i w 1/2 x 3 i3 w 1/2 x 3 + i w x3 3 v x3 i v x3 dx x 2, (A.6 where we used Agmon s ineuality in the x 3 direction. The right-hand side of (A.6 is estimated using Hölder s ineuality by 3 C i w 1/2 i3 w 1/2 x 3 L 4 3 v x 1 x x3 L 8 x 2 1 x i v 2 i=1 3 i=1 i w x3 L 4 x 1 x 3 v 2 x3 L 4 x 1 x i v (A.7

19 Using the Gagliardo Nirenberg Sobolev ineuality, the divergence-free condition and the relationship w = 2 x3 v d x 3, the expression (A.7 may be estimated by C 3 i=1 +C 2 i v 1/2+1/4 2 2 i v 1/4 3 v 1/4 2 3 v 3/4 i v 3 i=1 2 i v 1/2 2 2 i v 1/2 3 v 1/2 2 3 v 1/2 i v, (A.8 where we used Minowsi s ineuality. Therefore, I 21 CA(t 3/4 H(t 1/4 H(t 3/4 Ā(t 5/4 A(t 1/2 H(t 1/2 H(t 1/2 Ā(t 3/2.(A.9 Regarding the term I 22, we proceed similarly and obtain I 22 C 3 i=1 3 i=1 2 v 1/2+1/4 2 2 v 1/4 i3 v 1/4 2 i3 v 3/4 i v 2 v 1/2 2 2 v 1/2 i3 v 1/2 2 i3 v 1/2 i v CH (t 3/4 A(t 1/4 H(t 1/4 Ā(t 7/4 H (t 1/2 A(t 1/2 H(t 1/2 Ā(t 3/2. (A.10 In conclusion, estimates (A.1 (A.10 lead to d dt A2 + Ā 2 CH (t H (ta(t 2 H (t 4 A(t 2 H (t 2/3 H(t 2 A(t 2 + H(t 2 H(t 2 A(t 2 H (t 6 H(t 2 A(t 2. (A.11 By the results in [KZ1, KZ2] wehaveh(t L (0,Tand H(t loc (0,T. Therefore, from (A.11 and Grönwall s ineuality we obtain a bound on A(t in L (0,T and Ā(t in loc (0,T. References AQ3 [BK] [BGMR1] [BGMR2] [BGMR3] [BGMR4] [BKL] [C] [CINT] Brézis H and Kato T 1979 Remars on the Schrödinger operator with singular complex potentials J. Math. Pures Appl. ( Bresch D, Guillén-González F, Masmoudi N and Rodríguez-Bellido M A 2003 Uniueness of solution for the 2D primitive euations with friction condition on the bottom 7th Zaragoza-Pau Conf. on Applied Mathematics and Statistics (Jaca, Spain, 2001 (in Spanish (Monogr. Semin. Mat. García Galdeano vol 27 (Zaragoza: Univ. Zaragoza pp Bresch D, Guillén-González F, Masmoudi N and Rodríguez-Bellido M A 2003 Asymptotic derivation of a Navier condition for the primitive euations Asymptot. Anal Bresch D, Guillén-González F, Masmoudi N and Rodríguez-Bellido M A 2003 On the uniueness of wea solutions of the two-dimensional primitive euations Diff. Integral Ens Bresch D, Guillén-González F, Masmoudi N and Rodríguiz-Bellido M A 2003 In the uniueness of wea solutions of the two-dimensional primitive euations Diff. Integral Ens Bresch D, Kazhihov A and Lemoine J 2004/05 On the two-dimensional hydrostatic Navier Stoes euations SIAM J. Math. Anal (electronic Calderón C P 1990 Existence of wea solutions for the Navier Stoes euations with initial data in L p Trans. Am. Math. Soc Cao C, Ibrahim S, Naanishi K and Titi E S 2012 Finite-time blowup for the inviscid primitive euations of oceanic and atmospheric dynamics arxiv:

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22 QUERIES Page 1 AQ1 Please supply a minimum of three (and a maximum of seven eywords relevant to your article. Page 1 AQ2 Please provide MSC codes. Page 19 AQ3 Please chec the details for any ournal references that do not have a blue lin as they may contain some incorrect information. Pale purple lins are used for references to arxiv e-prints. Page 20 AQ4 Please provide complete publication details of ref [GKVZ].