JNL: NON PIPS: 476361 TYPE: PAP TS: NEWGEN DATE: 15/4/2014 EDITOR: TT SPELLING: UK

| London Mathematical Society Nonlinearity

Nonlinearity 27 (2014) 1–21 UNCORRECTED PROOF

Primitive equations with continuousinitial data

Igor Kukavica1, Yuan Pei1, Walter Rusin2 andMohammed Ziane1

1 Department of Mathematics, University of Southern California, Los Angeles, CA 90089, USA2 Department of Mathematics, Oklahoma State University, Stillwater, OK 74078, USA

E-mail: kukavica@usc.edu, ypei@usc.edu, walter.rusin@okstate.edu and ziane@usc.edu

Received 4 June 2013, revised 10 March 2014Accepted for publication 27 March 2014Published

Recommended by E S Titi

AbstractWe address the well-posedness of the primitive equations of the ocean withcontinuous initial data. We show that the splitting of the initial data into aregular finite energy part and a small bounded part is preserved by the equationsthus leading to existence and uniqueness of solutions.

Keywords: XXXMathematics Subject Classification: XXX

1. Introduction

AQ1

AQ2

In this paper, we analyse the structure of the nonlinearity of the primitive equations

∂tvk − ν�vk +2∑

j=1

∂j (vjvk) + ∂3(wvk) + ∂kp = 0, k = 1, 2

2∑k=1

∂kvk + ∂3w = 0 (1.1)

enabling us to establish a well-posedness theory for data with no differentiability properties.In particular, we obtain the uniqueness of weak solutions under a mild assumption that theinitial data are only continuous in the space variables.

The primitive equations of the atmosphere and the ocean are considered to be thefundamental model for meteorology and climate prediction [P]. Indeed, the full compressible

0951-7715/14/020001+21$33.00 © 2014 IOP Publishing Ltd & London Mathematical Society Printed in the UK 1

Nonlinearity 27 (2014) 000 I Kukavica et al

Navier–Stokes equations, which govern the dynamics of the atmosphere and the ocean, arevery complicated and contain phenomena which are not interesting from the geophysical pointof view, such as shocks and sound waves. The Boussinesq approximation along with thehydrostatic balance leads to the primitive equations. The main part of the system consistsof the momentum equations and the conservation of mass, a simplified version of which isgiven above. The primitive equations also contain the thermodynamic equations (diffusion oftemperature), as well as the diffusion of humidity for the atmosphere and diffusion of salinityfor the ocean, see [PTZ, TZ].

The mathematical theory started with the work of Lions, Temam and Wang [LTW1,LTW2, LTW3] who set the analytical foundation for the equations and established theglobal existence of weak solutions for square 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, whichwas established by Bresch et al [BGMR1, BGMR2, BGMR3, BGMR4] and independentlyby 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 works on the primitive equations, see[CT2,GH,GZ,H,J,K,KZ1,KZ2,KZ3,STT,SV] (see also [KTVZ,MW,R,RTT1,RTT2] for theinviscid case).

In conclusion, the global existence of weak solution without uniqueness is known for bothtwo and three space dimensions. Imposing the H 1 regularity for initial data leads to globalexistence and uniqueness of solutions (2D and 3D).

The state of analysis for the primitive equations may seem to be much better than theone for the Navier–Stokes equations. However, when considering the uniqueness of weaksolutions of the primitive equations in 2D, a classical and elementary fact for the 2D Navier–Stokes equations, we face the obstacle of a derivative loss in the nonlinearity. For this reason,the well-posedness in Lp for the primitive equations remains open in both the 2D and 3Dcases for any p. We note the derivative loss in the nonlinearity constitutes a primary reasonfor the ill-posedness and finite time blow-up for the inviscid primitive equations in Sobolevspaces [R, CINT].

The main goal of this paper is to obtain spaces with less differentiability than H 1 for well-posedness. In this spirit Bresch et al proved in [BKL] the uniqueness in 2D for weak solutionswith ∂x3v0 ∈ H 1/2. In this paper, we establish the well-posedness (existence and uniqueness)of solutions with only continuous initial data that require no differentiability. Our approachrelies on the splitting of the initial data into a smooth finite energy part and a small boundedpart. In our reasoning we exploit the fact that this splitting is preserved by the equation,the main difficulty being caused by the pressure and the derivative loss terms. We note thatthe main reason for the choice of the space in which we seek solutions is the possibility ofestablishing uniqueness. For the uniqueness statements in which a splitting method has beenused in different contexts, see [BK, LM].

The paper is organized as follows. In section 2, we set up the notation, and write asimple form of the primitive equations which contains the main mathematical difficulties.The results that we obtain in this paper easily carry over to the general systems ofequations. Our main existence result is contained in theorem 2.2, while theorem 2.3asserts uniqueness. The proof of theorem 2.2 relies on the H 2 regularity of solutions forwhich we give a sketch in appendix A. Section 3 contains the proof of the consistencyof splitting given continuous initial data decaying at infinity (or simply continuous inthe case of periodic data). Finally, section 4 provides the proof of the uniqueness ofsolutions.

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Nonlinearity 27 (2014) 000 I Kukavica et al

2. Main results

Let �0 = R2 × [−h, 0]. The primitive equations of the ocean read

∂tuk − ν�uk +3∑

j=1

∂j (ujuk) + ∂kp = 0, k = 1, 2

3∑k=1

∂kuk = 0. (2.1)

Let

u(x, t) = (u1(x, t), u2(x, t), u3(x, t)) = (v(x, t), w(x, t)), (2.2)

where v = (v1, v2) and w are the horizontal and the vertical components respectively. Theinitial data v0 = (v01, v02): �0 → R

2 satisfy

div2

∫ 0

−h

v0 dx3 = 0 (2.3)

in the sense of distributions. We emphasize that the pressure is two dimensional and thatthere is no equation for u3 other than divu = 0. As in [CT1], we equip system (2.1) with theNeumann boundary conditions

∂v

∂x3= 0 and w = 0 (2.4)

on x3 = −h and x3 = 0. Since our initial data are merely continuous and ∂v/∂x3 may notbe well defined, we take advantage of the fact that the above formulation is equivalent to theproblem on

� = R2 × [−h, h]

with periodic boundary conditions in the x3 direction (see [GKVZ, Pe]). By this extension,the u1 and u2 components of the velocity field u are even in the x3 variable, whereas the u3

component is odd.We denote by C0(�) the space of continuous functions on � which vanish at infinity. To

simplify the notation, we denote by

B∞ε (� × [0, T )) = {

u ∈ L∞t L∞

x (�): ‖u‖L∞t L∞

x (�) � ε}

the ball of radius ε in the space L∞x,t (�).

Let us define the spaces

H ={v ∈ L2 : div2

∫ h

−h

v dx3 = 0 on R2

}, (2.5)

and V = H ∩ {v ∈ H 1 : v(·, x3) = v(·, −x3), v(·, x3) = v(·, x3 + 2h)}.Definition 2.1. We say that v ∈ L∞([0, ∞), H) ∩ L2

loc([0, ∞), V ) is a weak solution of (2.1)with initial data v0 ∈ H if for every ϕ ∈ D(� × [0, ∞)) such that divϕ = 0 we have

−2∑

k=1

∫ ∞

0

∫�

uk∂tϕk + ν

2∑k=1

∫ ∞

0

∫�

∇uk · ∇ϕk −3∑

j=1

2∑k=1

∫ ∞

0

∫�

ujuk∂jϕk

−2∑

k=1

∫�

v0kϕk(·, 0) = 0. (2.6)

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Nonlinearity 27 (2014) 000 I Kukavica et al

We recall the following result regarding the existence of weak solutions to the primitiveequations.

Theorem 2.1 ([LTW1, LTW2, TZ]). For every v0 ∈ H , there exists a weak solution for theprimitive equation (2.1) as in definition 2.1.

We now state our main results. For simplicity, we assume that the viscosity constant ν

equals 1. The first theorem provides the existence of solutions to equation (2.1).

Theorem 2.2. Assume v0 ∈ C0(�) ∩ L2(�). There exists a constant C > 0 and a weaksolution satisfying for any ε > 0 the decomposition

v ∈ C([0, ∞), H 2) ∩ L2loc([0, ∞), H 3) + B∞

Cε(� × [0, ∞))

to equation (2.1). Namely, v = v + V where v ∈ L∞([0, ∞), H 2) ∩ L2loc([0, ∞), H 3) and

V ∈ B∞Cε(� × [0, ∞)), where C is a constant independent of the solution.

Note that the obtained solution is continuous on R3 × (0, ∞). The second main result

addresses the uniqueness of solutions in the class C([0, T ], H 2) ∩ L2([0, T ], H 3) + B∞ε (� ×

[0, T )) for a sufficiently small ε > 0.

Theorem 2.3. Let T > 0. Suppose that v(1) and v(2) are two weak solutions of theprimitive equations (2.1) with the same initial data v0 ∈ C0(�) such that v(1), v(2) ∈L∞([0, T ], H 2) ∩ L2([0, T ], H 3) + B∞

ε (� × [0, T )) where ε > 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.

Remark 2.4. Theorems 2.2 and 2.3 are also valid in the case � = T2 × [−h, 0]. In this

situation we replace the requirement v0 ∈ C0(�) ∩ L2(�) with v0 ∈ C(�).

3. Proof of existence

The proof of theorem 2.2 uses the splitting approach introduced by Brezis and Kato in [BK]. Itwas also used by Calderon in [C] for the construction of weak solutions for the Navier–Stokesequations with Lp initial data (see also [GK] for construction of global solutions). Here weuse the splitting in the uniform norm.

In the proof of theorem 2.2 we use the following result concerning the higher regularity ofsolutions of primitive equations. The reader can find a more general statement in [PW]. Forcompleteness, we include a simple proof in appendix A.

Theorem 3.1. Assume v0 ∈ V and let v ∈ L∞([0, T ], V ) ∩ L2([0, T ], H 2) be the associatedsolution of equation (2.1). If in addition v0 ∈ H 2, then v ∈ L∞([0, T ], H 2)∩L2([0, T ], H 3).

Proof of theorem 2.2. Let v0 ∈ C0(�) ∩ L2(�) and fix a sufficiently small ε > 0, to bechosen below. Then let ϕδ be a standard mollifier which is in addition radially symmetric.More precisely, let ϕ be a smooth, compactly supported, non-negative, and radially symmetricfunction such that

∫�

ϕ = 1. For δ > 0, let ϕδ(x) = δ−3ϕ(x/δ). Given v0 = v0 ∗ ϕδ andfor δ sufficiently small, we have ‖v0 − v0‖L2 , ‖v0 − v0‖L∞ < ε. The function v0 is infinitelysmooth and satisfies v0 ∈ Hk for all k ∈ N . Due to radial symmetry of ϕ, it is also even inthe vertical variable (so that it satisfies the Neumann boundary condition) and it satisfies thedivergence-free condition (2.3). (Naturally, the size of the Hk norms of v0 depends on ε andgrows as ε → 0. However, this has no bearing as we do not need to consider the limit ε → 0.)

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Nonlinearity 27 (2014) 000 I Kukavica et al

By [CT1] (see also [KZ1]), we know that system (2.1) with the initial data v0 has a uniquestrong solution v ∈ C([0, T ], H 1) ∩ L2([0, T ], H 2) for any T > 0. Note that V0 = v0 − v0

satisfies ‖V0‖L2 , ‖V0‖L∞ < ε. Taking the difference of the equations for u = (v, w), andu = (v, w) we obtain that U = (V , W) = u − u = (v − v, w − w) satisfies

∂tVk − �Vk +2∑

j=1

Vj∂jVk + W∂3Vk +2∑

j=1

vj∂jVk + w∂3Vk

+2∑

j=1

Vj∂jvk + W∂3vk + ∂kP = 0, k = 1, 2

2∑k=1

∂kVk + ∂3W = 0, (3.1)

where P = p − p. Since V0 ∈ L∞(�) ∩ L2(�), we have by interpolation V0 ∈ Lq(�) for allq ∈ [2, ∞] with ‖V0‖Lq < ε. Now, we obtain a priori estimates for V in Lq(�). Let

φq =2∑

k=1

‖|Vk|q/2‖2L2 , φq =

2∑k=1

‖∇(|Vk|q/2)‖2L2 . (3.2)

We multiply the equations (3.1)1 by Vk|Vk|q−2, integrate over �, and sum for k = 1, 2. We get

1

q

d

dt

2∑k=1

∫�

|Vk|q +4(q − 1)

q2

3∑j=1

2∑k=1

∫�

∂j (|Vk|q/2)∂j (|Vk|q/2)

= −2∑

k=1

∫�

Vk|Vk|q−2∂kP −2∑

j,k=1

∫�

Vj∂jvkVk|Vk|q−2 −2∑

k=1

∫�

W∂3vkVk|Vk|q−2

= I1 + I2 + I3, (3.3)

where we integrated by parts. We proceed now to estimating the terms I1, I2 and I3.Regarding the term I1, we integrate by parts and obtain

−2∑

k=1

∫�

Vk|Vk|q−2∂kP =2∑

k=1

∫�

P∂k(Vk|Vk|q−2). (3.4)

Holder’s inequality then yields

I1 � C

2∑k=1

∫�

|P ||Vk|q/2−1|∂k(|Vk|q/2)| � C

2∑k=1

‖|P ||Vk|q/2−1‖L2‖∂k(|Vk|q/2)‖L2 . (3.5)

In order to estimate the first factor on the right-hand side of (3.5), we note that for k = 1, 2

‖|P ||Vk|q/2−1‖2L2 =

∫�

P 2|Vk|q−2 =∫

R2P 2

∫ h

−h

|Vk|q−2 dx3 dx1 dx2

� ‖P 2‖Lq/4(R2)

∥∥∥∥∫ h

−h

|Vk|q−2 dx3

∥∥∥∥Lq/(q−4)(R2)

� ‖P 2‖Lq/4(R2)

∫ h

−h

∥∥|Vk|q−2∥∥

Lq/(q−4)(R2)dx3, (3.6)

where we used Holder’s and Minkowski’s inequalities. By the Gagliardo–Nirenberg–Sobolevinequality, we then have∫ h

−h

∥∥|Vk|q−2∥∥

Lq/(q−4)(R2)dx3 =

∫ h

−h

∥∥|Vk|q/2∥∥(2q−4)/q

L(2q−4)/(q−4)(R2)dx3

� C

∫ h

−h

‖|Vk|q/2‖(2q−8)/q

L2(R2)‖∇2(|Vk|q/2)‖4/q

L2(R2)dx3. (3.7)

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Nonlinearity 27 (2014) 000 I Kukavica et al

We bound the right-hand side of the above expression using Holder’s inequality by

C‖|Vk|q/2‖(2q−8)/q

L2 ‖∇2(|Vk|q/2)‖4/q

L2 . (3.8)

From (3.6)–(3.8) we obtain

C

2∑k=1

‖|P ||Vk|q/2−1‖L2 � C

2∑k=1

‖P 2‖1/2Lq/4‖|Vk|q/2‖(q−4)/q

L2 ‖∇2(|Vk|q/2)‖2/q

L2

= C

2∑k=1

‖P ‖Lq/2‖|Vk|q/2‖(q−4)/q

L2 ‖∇2(|Vk|q/2)‖2/q

L2 . (3.9)

The Lq/2 norm of the pressure may be estimated using lemma 3.3 below. Therefore, we boundthe expression (3.9) by

Cq

2∑k=1

(‖Vk‖2Lq + ‖vk‖Lq ‖Vk‖Lq

) ‖|Vk|q/2‖(q−4)/q

L2 ‖∇2(|Vk|q/2)‖2/q

L2 . (3.10)

Thus we obtain

I1 � Cq

2∑k=1

(‖Vk‖2Lq + ‖vk‖Lq ‖Vk‖Lq

) ‖|Vk|q/2‖(q−4)/q

L2 ‖∇2(|Vk|q/2)‖(q+2)/q

L2

= Cq

2∑k=1

(‖|Vk|q/2‖4/q

L2 + ‖vk‖Lq ‖|Vk|q/2‖2/q

L2

)‖|Vk|q/2‖(q−4)/q

L2 ‖∇2(|Vk|q/2)‖(q+2)/q

L2

� Cq

2∑k=1

‖|Vk|q/2‖L2‖∇2(|Vk|q/2)‖(q+2)/q

L2 + Cq

×2∑

k=1

‖vk‖Lq ‖|Vk|q/2‖(q−2)/q

L2 ‖∇2(|Vk|q/2)‖(q+2)/q

L2

� Cqφ1/2q φ(q+2)/2q

q + Cq‖v‖Lq φ(q−2)/2qq φ(q+2)/2q

q . (3.11)

In order to estimate the term I2, we apply Holder’s inequality

I2 = −2∑

j,k=1

∫�

Vj∂jvkVk|Vk|q−2 �2∑

j,k=1

‖∂j vk‖L3‖VjVk|Vk|q/2−2‖L2‖|Vk|q/2‖L6 . (3.12)

We bound the right-hand side of (3.12) using the Gagliardo–Nirenberg–Sobolev inequality andobtain

I2 � C

2∑j,k=1

(‖∂j vk‖1/2

L2 ‖∇∂j vk‖1/2L2 + ‖∂j vk‖L2

)‖|V |q/2‖L2

(‖∇(|V |q/2)‖L2 + ‖|V |q/2‖L2

)� C

2∑j,k=1

‖∂j vk‖1/2L2 ‖∇∂j vk‖1/2

L2 φ1/2q φ1/2

q + C

2∑j,k=1

‖∂j vk‖L2φ1/2q φ1/2

q

+ C

2∑j,k=1

‖∂j vk‖1/2L2 ‖∇∂j vk‖1/2

L2 φq + C

2∑j,k=1

‖∂j vk‖L2φq. (3.13)

Regarding the term I3, we observe that since

W = −2∑

i=1

∫ x3

−h

∂iVi dx3, (3.14)

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Nonlinearity 27 (2014) 000 I Kukavica et al

we have after integration by parts

I3 = −2∑

k=1

∫�

W∂3vkVk|Vk|q−2 =2∑

i,k=1

∫�

∂i

( ∫ x3

−h

Vi dx3

)∂3vkVk|Vk|q−2

= −2∑

i,k=1

∫�

( ∫ x3

−h

Vi dx3

)∂i3vkVk|Vk|q−2 −

2∑i,k=1

∫�

( ∫ x3

−h

Vi dx3

)∂3vk∂i

(Vk|Vk|q−2

)= I31 + I32. (3.15)

Using Holder’s and Minkowski’s inequalities, we bound the term I31 as

I31 � C

2∑i,k=1

‖Vi‖Lq ‖∂i3vk‖L2‖|Vk|(q−2)/2‖L3q/(q−3)‖|Vk|q/2‖L6 , (3.16)

which we rewrite as

I31 � C

2∑i,k=1

‖Vi‖Lq ‖∂i3vk‖L2‖|Vk|q/2‖(q−2)/q

L(3q−6)/(q−3)‖|Vk|q/2‖L6 . (3.17)

Applying the Gagliardo–Nirenberg–Sobolev inequality, we bound the right-hand side of(3.17) by

C

2∑i,k=1

‖Vi‖Lq ‖∂i3vk‖L2

(‖|Vk|q/2‖(q−4)/(2q−4)

L2 ‖∇(|Vk|q/2)‖q/(2q−4)

L2 + ‖|Vk|q/2‖L2

)(q−2)/(q)

×(‖∇(|Vk|q/2)‖L2 + ‖|Vk|q/2‖L2

), (3.18)

whence

I31 � C

2∑i,k=1

‖∂i3vk‖L2φ1/4q φ3/4

q + C

2∑i,k=1

‖∂i3vk‖L2φ1/2q φ1/2

q

+ C

2∑i,k=1

‖∂i3vk‖L2φ3/4q φ1/4

q + C

2∑i,k=1

‖∂i3vk‖L2φq. (3.19)

In order to estimate the term I32, we use Holder’s and Minkowski’s inequalities and obtain

I32 = −1∑

i,k=2

∫�

( ∫ x3

−h

Vi dx3

)∂3vk∂i

(Vk|Vk|q−2

)� C

2∑i,k=1

‖Vi‖Lq ‖∂3vk‖L6‖∂i(Vk|Vk|q−2)‖L6q/(5q−6) . (3.20)

Since

∂i

(Vk|Vk|q−2

) = 2(q − 1)

qVk|Vk|q/2−2∂i

(|Vk|q/2)

(3.21)

we bound the right-hand side of (3.20) using Holder’s inequality by

C

2∑i,k=1

‖Vi‖Lq ‖∂3vk‖L6‖|Vk|q/2−1‖L3q/(q−3)

∥∥∂i

(|Vk|q/2)∥∥

L2 , (3.22)

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Nonlinearity 27 (2014) 000 I Kukavica et al

which we rewrite as

C

2∑i,k=1

‖Vi‖Lq ‖∂3vk‖L6‖|Vk|q/2‖(q−2)/(q)

L(3q−6)/(q−3)

∥∥∂i

(|Vk|q/2)∥∥

L2 . (3.23)

Then by the Gagliardo–Nirenberg–Sobolev inequality, we obtain

I32 � C

2∑i,k=1

‖Vi‖Lq (‖∇∂3vk‖L2 + ‖∂3vk‖L2)

×(‖|Vk|q/2‖(q−4)/(2q)

L2 ‖∇(|Vk|q/2)‖1/2L2 + ‖|Vk|q/2‖(q−2)/q

L2

) ∥∥∂i

(|Vk|q/2)∥∥

L2

� C

2∑i,k=1

(‖∇∂3vk‖L2 + ‖∂3vk‖L2)(‖|V |q/2‖1/2L2 ‖∇(|Vk|q/2)‖1/2

L2 + ‖|V |q/2‖L2)‖∂i(|Vk|q/2)‖L2

� C

2∑k=1

‖∇∂3vk‖L2φ1/4q φ3/4

q + C

2∑k=1

‖∇∂3vk‖L2φ1/2q φ1/2

q

+ C

2∑k=1

‖∂3vk‖L2φ1/4q φ3/4

q + C

2∑k=1

‖∂3vk‖L2φ1/2q φ1/2

q . (3.24)

From (3.3), (3.11), (3.13), (3.19) and (3.24), we conclude

1

q

d

dtφq +

4(q − 1)

q2φq � Cqφ1/2

q φ(q+2)/2qq + Cq‖v‖Lq φ(q−2)/2q

q φ(q+2)/2qq

+ C

2∑j,k=1

‖∂j vk‖1/2L2 ‖∇∂j vk‖1/2

L2 φ1/2q φ1/2

q + C

2∑j,k=1

‖∂j vk‖L2φ1/2q φ1/2

q

+ C

2∑j,k=1

‖∂j vk‖1/2L2 ‖∇∂j vk‖1/2

L2 φq + C

2∑j,k=1

‖∂j vk‖L2φq

+ C

2∑i,k=1

‖∂i3vk‖L2φ3/4q φ1/4

q + C

2∑i,k=1

‖∂i3vk‖L2φq

+ C

2∑k=1

‖∇∂3vk‖L2φ1/4q φ3/4

q + C

2∑k=1

‖∇∂3vk‖L2φ1/2q φ1/2

q

+ C

2∑k=1

‖∂3vk‖L2φ1/4q φ3/4

q + C

2∑k=1

‖∂3vk‖L2φ1/2q φ1/2

q . (3.25)

Applying Young’s inequality, we obtain

1

q

d

dtφq +

4(q − 1)

q2φq � Cq(3q+2)/(q−2)φq/(q−2)

q +1

9qφq + Cq(3q+2)/(q−2)‖v‖2q/(q−2)

Lq φq +1

9qφq

+Cq

2∑j,k=1

‖∂j vk‖L2‖∇∂j vk‖L2φq +1

9qφq + Cq

2∑j,k=1

‖∂j vk‖2L2φq +

1

9qφq

+C

2∑j,k=1

‖∂j vk‖1/2L2 ‖∇∂j vk‖1/2

L2 φq + C

2∑j,k=1

‖∂j vk‖L2φq

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Nonlinearity 27 (2014) 000 I Kukavica et al

+Cq1/32∑

i,k=1

‖∂i3vk‖4/3L2 φq +

1

9qφq + C

2∑i,k=1

‖∂i3vk‖L2φq

+Cq32∑

k=1

‖∇∂3vk‖4L2φq +

1

9qφq + Cq

2∑k=1

‖∇∂3vk‖2L2φq +

1

9qφq

+Cq32∑

k=1

‖∂3vk‖4L2φq +

1

9qφq + Cq

2∑k=1

‖∂3vk‖2L2φq +

1

9qφq, (3.26)

which leads to

d

dtφq + φq � Cq4q/(q−2)φq/(q−2)

q + Cq4q/(q−2)‖v‖2q/(q−2)

Lq φq + Cq22∑

j,k=1

‖∂j vk‖L2‖∇∂j vk‖L2φq

+ Cq22∑

j,k=1

‖∂j vk‖2L2φq + Cq

2∑j,k=1

‖∂j vk‖1/2L2 ‖∇∂j vk‖1/2

L2 φq + Cq

2∑j,k=1

‖∂j vk‖L2φq

+ Cq4/32∑

i,k=1

‖∂i3vk‖4/3L2 φq + Cq

2∑i,k=1

‖∂i3vk‖L2φq + Cq42∑

k=1

‖∇∂3vk‖4L2φq

+ Cq22∑

k=1

‖∇∂3vk‖2L2φq + Cq4

2∑k=1

‖∂3vk‖4L2φq + Cq2

2∑k=1

‖∂3vk‖2L2φq. (3.27)

Following [Ku], we note that

φq �φ

5/3q − Cφ

10/3q/2

Cφ4/3q/2

. (3.28)

Inequality (3.27) then yields

d

dtφq +

φ5/3q

Cφ4/3q/2

� Cφ2q/2 + Cq4q/(q−2)φq/(q−2)

q + Cq4q/(q−2)‖v‖2q/(q−2)

Lq φq

+ Cq22∑

j,k=1

‖∂j vk‖L2‖∇∂j vk‖L2φq + Cq22∑

j,k=1

‖∂j vk‖2L2φq

+ Cq

2∑j,k=1

‖∂j vk‖1/2L2 ‖∇∂j vk‖1/2

L2 φq + Cq

2∑j,k=1

‖∂j vk‖L2φq

+ Cq4/32∑

i,k=1

‖∂i3vk‖4/3L2 φq + Cq

2∑i,k=1

‖∂i3vk‖L2φq + Cq42∑

k=1

‖∇∂3vk‖4L2φq

+ Cq22∑

k=1

‖∇∂3vk‖2L2φq + Cq4

2∑k=1

‖∂3vk‖4L2φq + Cq2

2∑k=1

‖∂3vk‖2L2φq,

(3.29)

which by writing φq = C(φ5/3q /Cφ

4/3q/2)

3/5φ4/5q/2 for each factor φq on the right-hand side and

using Young’s inequality in order to absorb the φ5/3q /Cφ

4/3q/2 terms leads to

d

dtφq +

φ5/3q

Cφ4/3q/2

� Cφ2q/2 + Cq10φ

2q/(q−1)

q/2 + Cq10‖v‖5q/(q−2)

Lq φ2q/2

+Cq52∑

j,k=1

‖∂j vk‖5/2L2 ‖∇∂j vk‖5/2

L2 φ2q/2

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Nonlinearity 27 (2014) 000 I Kukavica et al

+Cq52∑

j,k=1

‖∂j vk‖5L2φ

2q/2 + Cq5/2

2∑j,k=1

‖∂j vk‖5/4L2 ‖∇∂j vk‖5/4

L2 φ2q/2

+Cq5/22∑

j,k=1

‖∂j vk‖5/2L2 φ2

q/2 + Cq10/32∑

i,k=1

‖∂i3vk‖10/3L2 φ2

q/2

+Cq5/22∑

i,k=1

‖∂i3vk‖5/2L2 φ2

q/2 + Cq102∑

k=1

‖∇∂3vk‖10L2φ

2q/2

+Cq52∑

k=1

‖∇∂3vk‖5L2φ

2q/2 + Cq10

2∑k=1

‖∂3vk‖10L2φ

2q/2 + Cq5

2∑k=1

‖∂3vk‖5L2φ

2q/2.

(3.30)

Let

M = max{‖v‖L∞([0,T ],Lq (�)), ‖∇v‖L∞([0,T ],L2(�)), ‖∇2v‖L∞([0,T ],L2(�))}. (3.31)

The estimate (3.30) can be rewritten asd

dtφq � Cq10φ

2q/(q−1)

q/2 + Cq10M5q/(q−2)φ2q/2 + Cq5M5φ2

q/2 + Cq5/2M5/2φ2q/2

+ Cq10/3M10/3φ2q/2 + Cq10M10φ2

q/2. (3.32)

For simplicity assume M � C (i.e. we let constants depend on M). Inequality (3.32) leads tod

dtφq � Cq10φ

2q/(q−1)

q/2 + Cq10φ2q/2. (3.33)

Let Rq(t) = ‖V ‖L∞([0,t],Lq (�)). Integrating (3.33) over (0, t) and raising both sides to power1/q, and using ‖V0‖Lq � ε we obtain

Rq(t) �(Cq10(R

q

q/2(t))q/(q−1) + Cq10R

q

q/2(t) + εq)1/q

. (3.34)

In order to pass to the limit q → ∞ we use lemma 3.2 below. We start with the estimate forq = 8. In this case, inequality (3.29) yields

d

dtφ8 � Cφ

4/38 + C‖v‖8/3

L8 φ8 + C

2∑j,k=1

‖∂j vk‖L2‖∇∂j vk‖L2φ8

+ C

2∑j,k=1

‖∂j vk‖2L2φ8 + C

2∑j,k=1

‖∂j vk‖1/2L2 ‖∇∂j vk‖1/2

L2 φ8 + C

2∑j,k=1

‖∂j vk‖L2φ8

+ C

2∑i,k=1

‖∂i3vk‖4/3L2 φ8 + C

2∑i,k=1

‖∂i3vk‖L2φ8 + C

2∑k=1

‖∇∂3vk‖4L2φ8

+ C

2∑k=1

‖∇∂3vk‖2L2φ8 + C

2∑k=1

‖∂3vk‖4L2φ8 + C

2∑k=1

‖∂3vk‖2L2φ8. (3.35)

We note that initially, by interpolation φ8 � ε8. Furthermore, there exists a sufficiently smallTε > 0 such that by Gronwall’s inequality we obtain the bound

R8(Tε) � 2ε. (3.36)

Furthermore, for ε sufficiently small, using lemma 3.2 with Sn = R2n (Tε) and n0 = 3 we get

supn

R2n (Tε) � Cε, (3.37)

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Nonlinearity 27 (2014) 000 I Kukavica et al

whence we have Vk ∈ L∞([0, Tε], L∞(�)) and

‖Vk‖L∞([0,Tε ],L∞(�)) � Cε (3.38)

for k = 1, 2. Moreover, note that since v0 ∈ H , by theorem 2.1 there exists 0 < T1 < Tε

such that v(T1) ∈ V . Therefore, there exists T1 < T2 < Tε such that v(T2) ∈ H 2. Thus, weobserve that the splitting

v ∈ C([0, T ], H 2) ∩ L2([0, T ], H 3) + B∞Cε(� × [0, T ))

holds on [0, T ] by theorem 3.1, where T > 0 is arbitrary. The theorem is thus proven. �

Lemma 3.2. Let Sn0 , Sn0+1, Sn0+2, . . . be a sequence satisfying the recurrence relation

Sn �(A2Kn(S2n

n−1)2n/(2n−4) + A2KnS2n

n−1 + ε2n)1/2n

, (3.39)

where A � 1, K > 0, and ε ∈ (0, 1]. Then there exists a constant C > 0 such that

supn

Sn � Cε (3.40)

provided Sn0 � ε and ε is sufficiently small.

Proof of lemma 3.2. Consider the sequence defined by αn0 = 1 and

αn+1 = 41/2n

A1/2n

2Kn/2n

αn (3.41)

for n = n0, n0 + 1, . . .. Note that by A � 1 we have

αn � 1, n = n0, n0 + 1, . . . (3.42)

as well as

α = supn

αn = limn

αn < ∞,

which follows from∞∏

n=n0

41/2n

A1/2n

2Kn/2n

< ∞.

Assuming that ε > 0 is so small that αε � 1, we obtain

αnε � 1, n = n0, n0 + 1, . . . . (3.43)

We shall prove by induction that

Sn � αnε, n = n0, n0 + 1, . . . . (3.44)

Clearly, the claim holds for n = n0. Assuming that (3.44) holds up to n − 1, we have

Sn �(A2Kn(S2n

n−1)2n/(2n−4) + A2KnS2n

n−1 + ε2n)1/2n

�(2A2KnS2n

n−1 + ε2n)1/2n

�(2A2Knα2n

n−1 + 1)1/2n

ε

�(4A2Knα2n

n−1

)1/2n

ε, (3.45)

where we used Sn−1 � αn−1ε � 1 in the first and (3.42) in the last inequality. We thus obtainSn � 41/2n

A1/2n

2Kn/2n

αn−1ε = αnε by (3.41). The induction step is thus established and theproof is complete. �

Lemma 3.3. The pressure P = P(x1, x2) in equation (3.1) satisfies the estimate

‖P ‖Lq/2 = ‖p − p‖Lq/2 � Cq

2∑k=1

(‖Vk‖2Lq + ‖vk‖Lq ‖Vk‖Lq

). (3.46)

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Nonlinearity 27 (2014) 000 I Kukavica et al

Proof of lemma 3.3. Recall that u satisfies the equation

∂tvk − �vk +2∑

j=1

∂j (vjvk) + ∂3(wvk) + ∂kp = 0, k = 1, 2 (3.47)

and u satisfies the equations

∂tvk − �vk +2∑

j=1

∂j (vjvk) + ∂3(wvk) + ∂kp = 0, k = 1, 2 (3.48)

Applying the averaging operator

M[·] = 1

2h

∫ h

−h

· dx3 (3.49)

to equations (3.47)–(3.48) and taking their difference, we obtain

∂tM[Vk] − M[�Vk] +2∑

j=1

∂j

(M[vjvk] − M[vjvk]

)+ ∂k(p − p) = 0, k = 1, 2. (3.50)

Using Vk = vk − vk , and applying div2 to equation (3.50), we obtain

− �2(p − p) =2∑

j,k=1

∂j ∂k(M[VkVj ] + M[vkVj ] + M[Vkvj ]) −2∑

k=1

∂kM[�Vk]

=2∑

j,k=1

∂j ∂k(M[VkVj ] + M[vkVj ] + M[Vkvj ]) −2∑

k=1

∂kM[�2Vk]

−2∑

k=1

∂kM[∂33Vk]

=2∑

j,k=1

∂j ∂k(M[VkVj ] + M[vkVj ] + M[Vkvj ]) −2∑

k=1

(∂3kVk(·, h)

− ∂3kVk(·, −h))

=2∑

j,k=1

∂j ∂k(M[VkVj ] + M[vkVj ] + M[Vkvj ]), (3.51)

where we used the periodicity in the x3 direction. Therefore, we have P =(−�2)

−1∂j ∂k(VkVj + vkVj + Vkvj ). Using the Calderon–Zygmund theorem, we get

‖P ‖Lq/2 � Cq

2∑k=1

‖M[VkVj ] + M[vkVj ] + M[Vkvj ]‖Lq/2

� Cq

2∑k=1

(‖Vk‖2Lq + ‖vk‖Lq ‖Vk‖Lq

), (3.52)

completing the proof of the lemma. �

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Nonlinearity 27 (2014) 000 I Kukavica et al

4. Proof of uniqueness

In this section we take advantage of the splitting to show uniqueness of solutions.

Proof of theorem 2.3. Let (u(1), p(1)) and (u(2), p(2)) be two weak solutions of (1.1) withinitial data v0 as in theorem 2.2. Let u = u(1) − u(2). Note that u = (v, w) satisfies

∂tvk − �vk +2∑

j=1

v(2)j ∂j vk + w(2)∂3vk +

2∑j=1

vj∂jv(1)k + w∂3v

(1)k + ∂k(p

(1) − p(2)) = 0,

k = 1, 2,

2∑k=1

∂kvk + ∂3w = 0, (4.1)

with the initial data v0 = 0. Let

V (t) =(

2∑k=1

‖vk‖2L2

)1/2

, V (t) =(

2∑k=1

‖∇vk‖2L2

)1/2

. (4.2)

Multiplying (4.1)k for k = 1, 2 by vk , integrating over �, summing over k, and integrating byparts we obtain

1

2

d

dt

2∑k=1

∫�

|vk|2 +2∑

k=1

∫�

|∇vk|2 = −2∑

j,k=1

∫�

vj∂jv(1)k vk −

2∑k=1

∫�

w∂3v(1)k vk

=2∑

j,k=1

∫�

vjv(1)k ∂j vk +

2∑k=1

∫�

wv(1)k ∂3vk

= I1 + I2, (4.3)

where we used the fact that u is divergence-free.In order to estimate the term I1 we split it into two parts

I1 =2∑

j,k=1

∫�

vjv(1)k ∂j vk =

2∑j,k=1

∫�

vjv(1)k ∂j vk +

2∑j,k=1

∫�

vj v(1)k ∂j vk = I11 + I12, (4.4)

where v(1)k ∈ L∞([0, T ], H 2)∩L2([0, T ], H 3) and v

(1)k ∈ B∞

Cε(�× [0, T )) as in theorem 2.2.Regarding the term I11, we apply Holder’s inequality and obtain

2∑j,k=1

∫�

vjv(1)k ∂j vk �

2∑j,k=1

‖vj‖L3‖v(1)k ‖L6‖∂jvk‖L2

� C

2∑j,k=1

‖vj‖1/2L2 ‖∇vj‖1/2

L2 ‖∇v(1)k ‖L2‖∂jvk‖L2 , (4.5)

where we used the Gagliardo–Nirenberg–Sobolev inequality. Therefore

I11 � C‖∇v(1)‖L2V (t)1/2V (t)3/2. (4.6)

In order to estimate the term I12, we note that

I12 =2∑

j,k=1

∫�

vj v(1)k ∂j vk �

2∑j,k=1

‖vj‖L2 ‖v(1)k ‖L∞‖∂jvk‖L2 , (4.7)

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Nonlinearity 27 (2014) 000 I Kukavica et al

whence

I12 � CεV (t)V (t). (4.8)

Regarding the term I2 on the right-hand side of (4.3), we split it into

I2 =2∑

k=1

∫�

wv(1)k ∂3vk =

2∑k=1

∫�

wv(1)k ∂3vk +

2∑k=1

∫�

wv(1)k ∂3vk = I21 + I22. (4.9)

In order to estimate the term I21, we note that2∑

k=1

∫�

wv(1)k ∂3vk = −

2∑k=1

∫�

∂3wv(1)k vk −

2∑k=1

∫�

w∂3v(1)k vk. (4.10)

Using Holder’s inequality, we may bound the right-hand side of the above expression by

2∑k=1

‖∂3w‖L2‖v(1)k ‖L6‖vk‖L3 +

2∑k=1

‖w‖L2‖∂3v(1)‖L6‖vk‖L3 . (4.11)

Therefore, by the Gagliardo–Nirenberg–Sobolev inequality and (3.14) we obtain

I21 � C

2∑k=1

‖∇v‖L2‖∇v(1)k ‖L2‖vk‖1/2

L2 ‖∇vk‖1/2L2

+ C

2∑k=1

‖∇v‖L2‖∇∂3v(1)‖L2‖vk‖1/2

L2 ‖∇vk‖1/2L2 , (4.12)

whence

I21 � C‖∇v(1)‖L2V (t)1/2V (t)3/2 + C‖∇∂3v(1)‖L2V (t)1/2V (t)3/2. (4.13)

Regarding the term I22, by Holder’s inequality we obtain

I22 =2∑

k=1

∫�

wv(1)k ∂3vk �

2∑k=1

‖w‖L2 ‖v(1)k ‖L∞‖∂3vk‖L2

� ‖∇v‖L2 ‖v(1)‖L∞‖∇v‖L2 � CεV (t)2, (4.14)

where we used (3.14). Estimates (4.3)–(4.14) yield

1

2

d

dtV 2 + V 2 � C‖∇v(1)‖L2V (t)1/2V (t)3/2 + CεV (t)V (t)

+ C‖∇∂3v(1)‖L2V (t)1/2V (t)3/2 + CεV (t)2. (4.15)

Using Young’s inequality we get

1

2

d

dtV 2 + V 2 � C‖∇v(1)‖4

L2V (t)2 + CεV (t)2 + C‖∇∂3v(1)‖4

L2V (t)2 + CεV (t)2, (4.16)

and assuming that ε is small enough, we obtain

1

2

d

dtV 2 � C‖∇v(1)‖4

L2V (t)2 + CεV (t)2 + C‖∇∂3v(1)‖4

L2V (t)2. (4.17)

Since V (0) = 0, by Gronwall’s inequality

V (t) = 0, (4.18)

and thus u(1) = u(2). This concludes the proof of the theorem. �

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Nonlinearity 27 (2014) 000 I Kukavica et al

5. Construction of solutions

In section 3 we presented a priori estimates which lead to the existence of solutions in theconsidered class. At this point we briefly outline a construction of such solutions. To avoidunessential technical complications, we restrict ourselves to the case � = T

3. The case� = R

2 × [−h, h] is handled similarly.Fix V0 as in the proof of theorem 2.2. Let v be the solution to the corresponding finite

energy part and denote w = ∫ x3

−hdiv2v dx3. For a standard mollifier ϕδ , consider the sequence

V(δ)

0 = V0 ∗ ϕδ . For any fixed δ, by the results in [LTW1, LTW2, TZ] we obtain the existenceof a weak solution V (δ) to the system

∂tV(δ)k − �V

(δ)k +

2∑j=1

V(δ)j ∂jV

(δ)k + W(δ)∂3V

(δ)k +

2∑j=1

vj∂jV(δ)k + w∂3V

(δ)k

+2∑

j=1

V(δ)j ∂j vk + W(δ)∂3vk + ∂kP

(δ) = 0, k = 1, 2,

2∑k=1

∂kV(δ)k + ∂3W

(δ) = 0, (5.1)

with the initial data V(δ)

0 . Multiplying (5.1)1 by V(δ)k , integrating over � and summing over k

we obtain

1

2

d

dt‖V (δ)‖2

L2 + ‖∇V (δ)‖2L2 = −

2∑j,k=1

∫�

V(δ)j ∂j vkV

(δ)k −

2∑k=1

∫�

W(δ)∂3vkV(δ)k . (5.2)

Integrating by parts in both terms on the right-hand side of the above equation we get

1

2

d

dt‖V (δ)‖2

L2 + ‖∇V (δ)‖2L2

=2∑

j,k=1

∫�

V(δ)j vk∂jV

(δ)k +

2∑k=1

∫�

W(δ)vk∂3V(δ)k

=2∑

j,k=1

∫�

V(δ)j vk∂jV

(δ)k −

2∑k=1

∫�

∂3W(δ)vkV

(δ)k −

2∑k=1

∫�

W(δ)∂3vkV(δ)k

� C‖v‖H 2

( 2∑j,k=1

∫�

|V (δ)j ||∂jV

(δ)k | +

2∑k=1

∫�

|∂3W(δ)||V (δ)

k |)

+2∑

k=1

‖W(δ)‖L2‖∂3vk‖L6‖V (δ)k ‖L3 , (5.3)

where we used Hoder’s inequality in the last term. We use the divergence-free condition andthe Gagliardo–Nirenberg–Sobolev inequality to obtain

1

2

d

dt‖V (δ)‖2

L2 + ‖∇V (δ)‖2L2 � C‖v‖H 2

( 2∑j,k=1

∫�

|V (δ)j ||∂jV

(δ)k | +

2∑i,k=1

∫�

|∂iV(δ)i ||V (δ)

k |)

+ C

2∑k=1

‖W(δ)‖L2‖∇∂3vk‖L2‖V (δ)k ‖1/2

L2 ‖∇V(δ)k ‖1/2

L2 , (5.4)

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Nonlinearity 27 (2014) 000 I Kukavica et al

which using

W(δ) = −2∑

i=1

∫ x3

0∂iV

(δ)i dx3 (5.5)

leads to

1

2

d

dt‖V (δ)‖2

L2 + ‖∇V (δ)‖2L2 � C‖v‖H 2

( 2∑j,k=1

∫�

|V (δ)j ||∂jV

(δ)k | +

2∑i,k=1

∫�

|∂iV(δ)i ||V (δ)

k |)

+ C‖v‖H 2

2∑i,k=1

‖∂iV(δ)i ‖L2‖V (δ)

k ‖1/2L2 ‖∇V

(δ)k ‖1/2

L2 , (5.6)

where we used Minkowski’s inequality. Therefore by Holder’s and Young’s inequalities,

1

2

d

dt‖V (δ)‖2

L2 +1

2‖∇V (δ)‖2

L2 � C‖v‖2H 2‖V (δ)‖2

L2 + C‖v‖4H 2‖V (δ)‖2

L2 . (5.7)

By Gronwall’s inequality applied to the above expression, we obtain an estimate for V (δ) inL∞([0, T ], L2) ∩ L2([0, T ], H 1) which is uniform in δ. In order to apply the Aubin–Lionslemma we also need a uniform bound on ∂tV

(δ)k and thus an estimate on ∂kP

(δ). Note that dueto the divergence-free condition, equation (5.1) may be rewritten as

∂tV(δ)k − �V

(δ)k +

2∑j=1

∂j (V(δ)j V

(δ)k ) + ∂3(W

(δ)V(δ)k ) +

2∑j=1

∂j (vjV(δ)k ) + ∂3(wV

(δ)k )

+2∑

j=1

∂j (V(δ)j vk) + ∂3(W

(δ)vk) + ∂kP(δ) = 0, k = 1, 2. (5.8)

Averaging in the x3 direction, and using M∂3 ≡ 0, for k = 1, 2 we get

∂tMV(δ)k − M�V

(δ)k +

2∑j=1

M∂j(V(δ)j V

(δ)k ) +

2∑j=1

M∂j(vjV(δ)k )

+2∑

j=1

M∂j(V(δ)j vk) + ∂kP

(δ) = 0, (5.9)

where we also took advantage of the fact that the pressure does not depend on x3. Applying∂k , summing over k and using the divergence-free condition leads to

− �2P(δ) =

2∑j,k=1

M∂j∂k(V(δ)j V

(δ)k ) +

2∑j,k=1

M∂j∂k(vjV(δ)k ) +

2∑j,k=1

M∂j∂k(V(δ)j vk). (5.10)

Proceeding as in the proof of lemma 3.3, we obtain

P (δ) =2∑

j,k=1

RjRk

(M(V

(δ)j V

(δ)k ) + M(vjV

(δ)k ) + M(V

(δ)j vk)

), (5.11)

where Ri denotes the Riesz transform. Therefore by the Calderon-Zygmund theorem

‖P (δ)‖L3/2(T2) � C

2∑j,k=1

(‖M(V(δ)j V

(δ)k )‖L3/2(T2) + ‖M(vjV

(δ)k )‖L3/2(T2) + ‖M(V

(δ)j vk)‖L3/2(T2)).

(5.12)

16

Nonlinearity 27 (2014) 000 I Kukavica et al

Applying Holder’s and Minkowski’s inequalities we get

‖P (δ)‖L3/2 � C

2∑j,k=1

(‖V (δ)j ‖L3‖V (δ)

k ‖L3 + ‖vj‖L3‖V (δ)k ‖L3 + ‖V (δ)

j ‖L3‖vk‖L3), (5.13)

which in turn by the Gagliardo–Nirenberg–Sobolev inequality leads to

‖P (δ)‖L3/2 � C‖V (δ)‖L2‖∇V (δ)‖L2 + C‖v‖1/2L2 ‖∇v‖1/2

L2 ‖V (δ)‖1/2L2 ‖∇V (δ)‖1/2

L2 . (5.14)

Hence, we obtain a uniform bound on P (δ) in L2([0, T ], L3/2), and thus ∂kP(δ) is uniformly

bounded in L2([0, T ], H−3/2). We now rewrite equation (5.8) in the form

∂tV(δ)k = �V

(δ)k −

2∑j=1

∂j (V(δ)j V

(δ)k ) − ∂3(W

(δ)V(δ)k ) −

2∑j=1

∂j (vjV(δ)k ) − ∂3(wV

(δ)k )

−2∑

j=1

∂j (V(δ)j vk) − ∂3(W

(δ)vk) − ∂kP(δ), k = 1, 2. (5.15)

Observe that �V(δ)k is uniformly bounded in L2([0, T ], H−2), whereas

∑2j=1 ∂j (V

(δ)j V

(δ)k ),∑2

j=1 ∂j (vjV(δ)k ) and

∑2j=1 ∂j (V

(δ)j vk) are uniformly bounded in L2([0, T ], H−3/2).

Regarding the third term on the right-hand side of (5.15), for ϕ ∈ H 2 we have

−∫

T3∂3(W

(δ)V(δ)k )ϕ =

∫T3

W(δ)V(δ)k ∂3ϕ = −

∫T3

2∑i=1

∂i

( ∫ x3

0V

(δ)i dx3

)V

(δ)k ∂3ϕ

=2∑

i=1

∫T3

( ∫ x3

0V

(δ)i dx3

)∂iV

(δ)k ∂3ϕ +

2∑i=1

∫T3

( ∫ x3

0V

(δ)i dx3

)V

(δ)k ∂i3ϕ. (5.16)

Minkowski’s inequality and the above obtained bounds for V (δ) and ∇V (δ) imply uniformbounds of V

(δ)i ∂iV

(δ)k and V

(δ)i V

(δ)k in L4/3([0, T ], H−2). Therefore, ∂tV

(δ)k is uniformly

bounded in L4/3([0, T ], H−2).Application of the Aubin–Lions lemma yields the existence of a sequence of V (δ)

converging strongly in L2([0, T ], L2) to V , which in addition satisfies the same energyestimate as V (δ). Furthermore, passing to a subsequence we can ensure that the convergence isa.e. pointwise. This implies that such limit inherits the a priori estimates presented in section 3,while theorem 2.3 guarantees uniqueness of the limit.

Acknowledgments

IK was supported in part by the NSF grant DMS-1311943, YP was supported in part by theNSF grants DMS-1009769, DMS-1109562 and DMS-1311943, WR was supported in partby the NSF grant DMS-1311964, while MZ was supported in part by the NSF grant DMS-1109562. We would like to express our gratitude to the referees for useful suggestions on theearlier versions of the paper.

Appendix A.

Proof of theorem 3.1. Without loss of generality, we may set ν = 1. Let A(t) =(∑2

k=1 ‖�vk(·, t)‖2L2)

1/2, and A(t) = (∑2

k=1 ‖∇�vk(·, t)‖2L2)

1/2. Applying � to (1.1)k for

17

Nonlinearity 27 (2014) 000 I Kukavica et al

k = 1, 2, multiplying by �vk , and integrating over �, we obtain

1

2

d

dt

2∑k=1

∫�

|�vk|2 +2∑

k=1

∫�

|∇�vk|2 = −2∑

i,k=1

∫�

�(vi∂ivk)�vk

−2∑

k=1

∫�

�(w∂3vk)�vk = I1 + I2, (A.1)

which follows from the divergence-free condition and the fact that the pressure is independentof the x3 variable. Regarding the first term on the right-hand side of (A.1), we integrate byparts and obtain

I1 =3∑

i=1

2∑j,k=1

∫�

∂ivj ∂j vk∂i�vk +3∑

i=1

2∑j,k=1

∫�

vj∂ij vk∂i�vk = I11 + I12. (A.2)

Using the Holder and the Gagliardo–Nirenberg–Sobolev inequalities we estimate the termI11 by

I11 �3∑

i=1

2∑j,k=1

‖∂ivj‖L6‖∂jvk‖L3‖∂i�vk‖L2

� C

3∑i=1

2∑j,k=1

‖∇∂ivj‖L2‖∂jvk‖1/2L2 ‖∇∂jvk‖1/2

L2 ‖∂i�vk‖L2 . (A.3)

Therefore, we have I11 � CA(t)H(t)1/2H (t)1/2A(t), where H(t) = (∑2

k=1 ‖∇vk‖2L2)

1/2 and

H (t) = (∑2

k=1 ‖∇∇vk‖2L2)

1/2. Similarly, we obtain

I12 �3∑

i=1

2∑j,k=1

‖vj‖L6‖∂ij vk‖L3‖∂j�vk‖L2

� C

3∑i=1

2∑j,k=1

‖∇vj‖L2‖∂ij vk‖1/2L2 ‖∇∂ij vk‖1/2

L2 ‖∂i�vk‖L2 , (A.4)

whence I12 � CH(t)A(t)1/2A(t)3/2. Integrating by parts in the term I2, we get

I2 =3∑

i=1

2∑k=1

∫�

∂iw∂3vk∂i�vk +3∑

i=1

2∑k=1

∫�

w∂i3vk∂i�vk = I21 + I22. (A.5)

Regarding the term I21, Holder’s inequality yields

I21 �3∑

i=1

2∑k=1

∫R2

‖∂iw‖L∞x3‖∂3vk‖L2

x3‖∂i�vk‖L2

x3dx1 dx2

� C

3∑i=1

2∑k=1

∫R2

(‖∂iw‖1/2

L2x3

‖∂i3w‖1/2L2

x3

+ ‖∂iw‖L2x3

)‖∂3vk‖L2

x3‖∂i�vk‖L2

x3dx1 dx2, (A.6)

where we used Agmon’s inequality in the x3 direction. The right-hand side of (A.6) is estimatedusing Holder’s inequality by

C

3∑i=1

2∑k=1

‖∂iw‖1/2L2 ‖∂i3w‖1/2

L2x3

L4x1x2

‖∂3vk‖L2x3

L8x1x2

‖∂i�vk‖L2

+ C

3∑i=1

2∑k=1

‖∂iw‖L2x3

L4x1x2

‖∂3vk‖L2x3

L4x1x2

‖∂i�vk‖L2 . (A.7)

18

Nonlinearity 27 (2014) 000 I Kukavica et al

Using the Gagliardo–Nirenberg–Sobolev inequality, the divergence-free condition and therelationship w = − ∑2

j=1

∫ x3

−h∂jvj dx3, the expression (A.7) may be estimated by

C

3∑i=1

2∑k=1

‖∇2∂iv‖1/2+1/4L2 ‖∇2∇2∂iv‖1/4

L2 ‖∂3vk‖1/4L2 ‖∇2∂3vk‖3/4

L2 ‖∂i�vk‖L2

+C

3∑i=1

2∑k=1

‖∇2∂iv‖1/2L2 ‖∇2∇2∂iv‖1/2

L2 ‖∂3vk‖1/2L2 ‖∇2∂3vk‖1/2

L2 ‖∂i�vk‖L2 , (A.8)

where we used Minkowski’s inequality. Therefore,

I21 � CA(t)3/4H(t)1/4H (t)3/4A(t)5/4 + CA(t)1/2H(t)1/2H (t)1/2A(t)3/2.(A.9)

Regarding the term I22, we proceed similarly and obtain

I22 � C

3∑i=1

2∑k=1

‖∇2v‖1/2+1/4L2 ‖∇2∇2v‖1/4

L2 ‖∂i3vk‖1/4L2 ‖∇2∂i3vk‖3/4

L2 ‖∂i�vk‖L2

+ C

3∑i=1

2∑k=1

‖∇2v‖1/2L2 ‖∇2∇2v‖1/2

L2 ‖∂i3vk‖1/2L2 ‖∇2∂i3vk‖1/2

L2 ‖∂i�vk‖L2

� CH(t)3/4A(t)1/4H (t)1/4A(t)7/4 + CH(t)1/2A(t)1/2H (t)1/2A(t)3/2. (A.10)

In conclusion, estimates (A.1)–(A.10) lead to

d

dtA2 + A2 � CH(t)H (t)A(t)2 + CH(t)4A(t)2 + CH(t)2/3H (t)2A(t)2

+ H(t)2H (t)2A(t)2 + CH(t)6H (t)2A(t)2. (A.11)

By the results in [KZ1, KZ2] we have H(t) ∈ L∞(0, T ) and H (t) ∈ L2loc(0, T ). Therefore,

from (A.11) and Gronwall’s inequality we obtain a bound on A(t) in L∞(0, T ) and A(t) inL2

loc(0, T ). �

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