of the Three-Dimensional Primitive Equations with Only Horizontal Viscosity and Diffusion

of the Three-Dimensional Primitive Equations with Only Horizontal Viscosity and Diffusion

CHONGSHENG CAO Florida International University

JINKAI LI

Weizmann Institute of Science AND

EDRISS S. TITI Texas A&M University Weizmann Institute of Science

Abstract

In this paper, we consider the initial boundary value problem of the three-dimen- sional primitive equations for planetary oceanic and atmospheric dynamics with only horizontal eddy viscosity in the horizontal momentum equations and only horizontal diffusion in the temperature equation. Global well-posedness of the strong solution is established for anyH²initial data. AnN-dimensional loga- rithmic Sobolev embedding inequality, which bounds theL¹-norm in terms of theL^q-norms up to a logarithm of theL^p-norm forp > N of the first-order derivatives, and a system version of the classic Grönwall inequality are exploited to establish the required a prioriH²estimates for global regularity. © 2016 Wi- ley Periodicals, Inc.

1 Introduction

The primitive equations are derived from the Boussinesq system of incompress- ible flow and they form a fundamental block in models for planetary oceanic and atmospheric dynamics; see, e.g., Lewandowski [16], Majda [20], Pedlosky [21], Vallis [25], and Washington and Parkinson [26]. Due to their importance, the primitive equations has been studied analytically by many authors; see, e.g., [17, 18, 20, 22, 24] and the references therein.

In this paper, we consider the following version of primitive equations with only horizontal eddy viscosities and only horizontal diffusion due to strong dominant horizontal turbulence mixing:

@tvC.v rH/vCw@´vC rHp HvCf0kv D0;

(1.1)

@´pCT D0;

(1.2)

Communications on Pure and Applied Mathematics, Vol. LXIX, 1492–1531 (2016)

© 2016 Wiley Periodicals, Inc.

rH vC@´wD0;

(1.3)

@tT Cv rHT Cw@´T _HT D0;

(1.4)

where the horizontal velocityv D.v¹; v²/, the vertical velocityw, the temperature T, and the pressurep are the unknowns, andf0is the Coriolis parameter. In this paper, we use the notations r^H D .@x; @y/ and H D @²_x C@_y² to denote the horizontal gradient and the horizontal Laplacian, respectively.

For the primitive equations with full viscosities and full diffusion, the mathemat- ical analysis was initialed in 1990s by Lions, Temam, and Wang [17–19], where among other issues they established the global existence of weak solutions. The uniqueness of weak solutions for the two-dimensional case was later proved by Bresch, Guillén-González, Masmoudi, and Rodríguez-Bellido [1]; however, the uniqueness of weak solutions for the three-dimensional case is still unclear. Lo- cal well-posedness of strong solutions was obtained by Guillén-González, Mas- moudi, and Rodríguez-Bellido [12]. Global existence of strong solutions for the two-dimensional case was established by Bresch, Kazhikhov, and Lemoine in [2]

and Temam and Ziane in [24], while the three-dimensional case was established in [8]. Global strong solutions for the three-dimensional case were also obtained by Kobelkov [13] later by using a different approach; see also the subsequent arti- cles by Kukavica and Ziane [14, 15].

The systems considered in all the papers [1, 2, 8, 12–15, 17–19, 22, 24] are as- sumed to have full dissipation, i.e., with both full viscosities and full diffusion.

Both physically and mathematically, it is also important and interesting to study the system with partial dissipation, i.e., with only partial viscosities or only partial diffusion. The first result in this direction for the primitive equations was obtained in [9], where the authors considered the system with full viscosities but only ver- tical diffusion and proved that such a system has a unique global strong solution, provided the local-in-time one exists. As the complement and a generalization of [9], the local and global well-posedness of strong solutions were recently estab- lished in [6] withH²initial data. As the counterpart of [6], global well-posedness of strong solutions to the primitive equations with full viscosities but only hori- zontal diffusion was later obtained in [5], still forH²initial data. Notably, smooth solutions to the inviscid primitive equation, with or without coupling to the temper- ature equation, has been shown [4] to blow up in finite time (see also Wong [27]).

Note that in all the papers [1,2,5,6,8,9,12–15,17–19,22,24], no matter whether the systems are considered to have full or partial diffusion in the temperature equa- tion, they are assumed to have full viscosities in the horizontal momentum equa- tions. Physically, in the oceanic and atmospheric dynamics, the horizontal scales are much lager than the vertical one with dominant strong horizontal turbulence mixing that induces horizontal viscosities, i.e., system (1.1)–(1.4).

From the mathematical point of view, there are two obvious difficulties in study- ing system (1.1)–(1.4). One is that the strongest nonlinear term, i.e.,.R´

hrH v d /@´v, is quadratic in the first derivatives of the unknowns. This is caused by

the lack of the dynamical equation for the vertical component of the velocity. The other one is that, due to the lack of the vertical viscosity in the horizontal momen- tum equations, one cannot expect any smoothing effect in the vertical direction.

The aim of this paper is to show that strong solutions exist globally for system (1.1)–(1.4), subject to some initial and boundary conditions, for any H² initial data. More precisely, we consider the problem in the domain0 DM . h; 0/, withM D .0; 1/.0; 1/, and supplement system (1.1)–(1.4) with the following boundary and initial conditions:

v; wandT are periodic inxandy;

(1.5)

.@´v; w/j´D h;0D.0; 0/; Tj´D hD1; Tj^´D0D0;

(1.6)

.v; T /j^tD0D.v0; T0/:

(1.7)

By replacingT andpbyTC^´_handp ^´_2h², respectively, then system (1.1)–(1.4) with (1.5)–(1.7) is reduced to

@tvC.v r^H/vCw@´vC r^Hp HvCf0kv D0;

(1.8)

@´pCT D0;

(1.9)

r^H vC@´wD0;

(1.10)

@tT Cv rHT Cw

@´T C1 h

HT D0;

(1.11)

subject to the boundary and initial conditions

v; w; T are periodic inxandy;

(1.12)

.@´v; w/j´D h;0 D0; Tj´D h;0D0;

(1.13)

.v; T /j^tD0D.v0; T0/:

(1.14)

Here, for simplicity, we still use T0 to denote the initial temperature in (1.14), though it is obtained by replacing theT0in (1.7) byT0 ´

h.

For the same reasons used to in [5, 6], system (1.8)–(1.14) defined on 0 is equivalent to the following system defined onWDM . h; h/:

@tvC.v rH/vCw@´vC rHp HvCf0kv D0;

(1.15)

@´pCT D0;

(1.16)

r^H vC@´wD0;

(1.17)

@tT Cv rHT Cw

@´T C1 h

_HT D0;

(1.18)

subject to the boundary and initial conditions

v; w; p; andT are periodic inx; y; ´;

(1.19)

vandpare even in´; andwandT are odd in´;

(1.20)

.v; T /j^tD0D.v0; T0/:

(1.21)

Note that the restriction on the subdomain0of a solution.v; w; p; T /to system (1.15)–(1.21) is a solution to the original system (1.8)–(1.14). Because of this, throughout this paper, we are mainly concerned with the study of system (1.15)–

(1.21) defined on, while the well-posedness results for system (1.8)–(1.14) de- fined on₀follow as a corollary of those for system (1.15)–(1.21).

One can check that system (1.15)–(1.21) is equivalent to (see [9] for example)

@tv _HvC.v rH/v

Z ´

hrH v.x; y; ; t /d

@´v Cf0kvC rH

ps.x; y; t / Z ´

h

T .x; y; ; t /d

D0;

(1.22)

r^H Z h

h

v.x; y; ´; t /d´D0;

(1.23)

@tT _HT Cv rHT Z ´

hr^H v.x; y; ; t /d

@´T C1 h

D0;

(1.24)

subject to the following boundary and initial conditions:

vandT are periodic inx; y; ´;

(1.25)

vandT are even and odd in´; respectively;

(1.26)

.v; T /j^tD0D.v0; T0/:

(1.27)

Before stating our main results, let’s introduce some necessary notation and give the definitions of strong solutions. Throughout this paper, for1q 1, we useL^q./; L^q.M /, and W^m;q./; W^m;q.M / to denote the standard Lebesgue and Sobolev spaces, respectively. Forq D 2, we useH^m instead ofW^m;2. We use W_per^m;q./and H_per^m to denote the spaces of periodic functions in W^m;q./

andH^m./, respectively. For simplicity, we still use the notation L^p andH^m to denote the N product spaces .L^p/^N and .H^m/^N, respectively. We always usekukp to denote theL^p-norm ofu. Moreover, for convenience, we often use k.f1; : : : ; fn/k²2to denote the summationPn

iD1kfik²2.

DEFINITION 1.1. Given a positive time T, letv0 2 H²./andT0 2 H²./

be two periodic functions such that they are even and odd in ´, respectively. A couple.v; T /is called astrong solutionto system (1.22)–(1.27) (or equivalently (1.15)–(1.21)) on.0;T/if

(i) vandT are periodic inx; y; ´, and they are even and odd in´, respectively;

(ii) vandT have the regularities

.v; T /2L¹.0;TIH².//\C.Œ0;TIH¹.//;

.r^Hv;r^HT /2L².0;TIH².//; .@tv; @tT /2L².0;TIH¹.//I (iii) v andT satisfy equations (1.22)–(1.24) a.e. in.0;T/and the initial

condition (1.27).

DEFINITION 1.2. A couple .v; T / is called a global strong solution to system (1.22)–(1.27) if it is a strong solution on.0;T/for anyT 2.0;1/.

The main result of this paper is the following global well-posedness result.

THEOREM1.3. Suppose that the periodic functionsv0; T02H²./are even and odd in´, respectively. Then system(1.22)-(1.27) (or equivalently(1.15)–(1.21)) has a unique global strong solution.v; T /that is continuously dependent on the initial data.

The key issue in proving Theorem 1.3 is establishing the a priori L¹.0;TI H².//estimates on the strong solutions. Our analysis shows that once theL¹.0;

TIH².//estimate onuD@´vis obtained, all the required estimates of the other derivatives can be successfully achieved. Unfortunately, due to the lack of the ver- tical viscosity in the horizontal momentum equations, such anL¹.0;TIH².//

estimate cannot be obtained solely without the contribution of the other derivative of the velocity. We observe that in all the arguments of existing articles, the full viscosities play an essential role in obtaining theL¹.0;TIH².//estimate on uD@´v, and thus the existing arguments cannot be applied in our case. Still be- cause of the lack of vertical viscosity, one will encounterkvk²₁, which appears as the coefficients in the higher-order energy inequalities; in other words, the energy inequalities we arrive to are all of the form

(1.28) d

dtf Ckvk²₁f Cother termsI see Section 4 for details.

Since we do not know whetherkvk²₁is integrable in time, we cannot obtain the required estimate forf directly from this kind of energy inequalities. Besides, re- calling thatf will represent quantities involving theH²-norm ofv, thoughkvk²₁ can be bounded byf, by the Sobolev embedding inequality, one will still be un- able to obtain the global-in-timeH²estimate forvby the simple application of the standard embedding inequality. Observe that ifkvk²₁andf have the relationship

kvk²₁ C logf;

then the energy inequality (1.28) implies the global-in-time estimate for f. To guarantee this relationship, thanks to the logarithmic Sobolev embedding inequal- ity (Lemma 2.2 below), it suffices to prove that theL^q-norms ofv grow no faster thanCpq. By taking advantage of the property that the pressurepdepends essen- tially only on the horizontal spatial variables, and using the Ladyzhenskaya-type

inequalities (Lemma 2.1 below) for a class of integrals in three dimensions, one can successfully prove the desired growth of theL^q-norms ofv, and thus obtain the a prioriH²estimates, and hence the global regularity.

It should be pointed out that, due to the anisotropic structure of the momen- tum equation (1.22) (the advection term.v rH/vand the vertical advection term

R´

hr^H vd

@´v play different roles), the treatments for different derivatives of the same order will vary. More precisely, when dealing with the derivatives of the same order, the treatment of the vertical derivatives precedes that of the hor- izontal ones, because the estimates of the horizontal derivatives may depend on those of the vertical ones, see Proposition 4.1 and Proposition 4.2, below, for the details. Accordingly, a system version of the classic Grönwall inequality, Lemma 2.3 below, is exploited to derive the a priori bounds from the energy inequalities.

We believe that this system version of the Grönwall inequality is interesting on its own, and in fact it can benefit us when using the energy approach; see Remark 2.4 below.

As a corollary of Theorem 1.3, we have the following theorem, which states the global well-posedness of strong solutions to system (1.8)–(1.14). Strong solutions to system (1.8)–(1.14) are defined similarly as before.

THEOREM 1.4. Letv0 andT0 be two functions defined on0 such that they are both periodic in x andy. Denote by v₀^ext andT₀^ext the even and odd extensions in´ofv0 andT0, respectively. Suppose thatv^ext₀ ; T₀^ext 2 H_per² ./. Then system (1.8)–(1.14)has a unique global strong solution.v; T /.

Theorem 1.4 follows directly by applying Theorem 1.3 with initial data.v₀^ext; T₀^ext/and restricting the solution to the subdomain0.

The rest of this paper is arranged as follows: in Section 2 we collect some pre- liminary results that will be used in subsequent sections; in Section 3 we establish the a priori low-order energy estimates, which are independent of the regularization parameter"for strong solutions to a regularized system, while the" independent higher-order energy inequalities are given in Section 4. In Section 5, with the aid of the a priori estimates and the higher-order energy inequalities achieved in the previous two sections, we first establish the a prioriH² estimates for the strong solutions to the regularized system, and then obtain the global well-posedness of strong solutions to system (1.22)–(1.27) (or, equivalently, system (1.15)–(1.21)) by a standard approach. Finally, in the appendix, anN-dimensional logarithmic Sobolev embedding inequality is established.

Throughout this paper,C denotes a general constant that may be different from line to line.

2 Preliminaries

In this section we collect some preliminary results that will be used in the rest of this paper, and we start with the following Ladyzhenskaya-type inequality in three

dimensions for a class of integrals, which will be frequently used throughout the paper.

LEMMA2.1. The following inequalities hold true:

Z

M

Z h

hj.x; y; ´/jd´

Z h

hj'.x; y; ´/ .x; y; ´/jd´

dx dy

Cmin n

kk¹⁼²2 kk¹⁼²2 C krHk¹⁼²2

k'k2k k¹⁼²2 k k¹⁼²2 C krH k¹⁼²2

;

kk2k'k¹⁼²₂ k'k¹⁼²₂ C krH'k¹⁼²₂

k k¹⁼²₂ k k¹⁼²₂ C krH k¹⁼²₂ o

;

and Z

M

Z h

hj.x; y; ´/jd´

Z h

hj'.x; y; ´/rH‰.x; y; ´/jd´

dx dy Cmin

n

kk¹⁼²2 kk¹⁼²2 C kr^Hk¹⁼²2

k'k²k‰k¹⁼²₁ rH²‰

1=2 2 ; kk²k'k¹⁼²2 k'k¹⁼²2 C kr^H'k¹⁼²2

k‰k¹⁼²₁ rH²‰

1=2 2

o

; for every; '; ; ‰such that the right-hand sides make sense and are finite, where C is a positive constant depending only onh. Moreover, if has the form D r^Hf for some functionf, then by the Poincáre inequality, the lower-order term kk¹⁼²2 in the parentheses can be dropped in the above inequalities, and the same can also be said for'and .

PROOF. Similar inequalities have been established in [7, 9]. However, for com- pleteness, we sketch the proofs here, and the ideas used here are the same as in those papers. By the Hölder and Minkowski inequalities, we deduce

Z

M

Z h

hj.x; y; ´/jd´

Z h

hj'.x; y; ´/ .x; y; ´/jd´

dx dy

Z

M

Z h

hj.x; y; ´/jd´

Z h

hj'.x; y; ´/j²d´

1=2

Z h

hj .x; y; ´/j²d´

1=2

dx dy

min Z

M

ˇ ˇ ˇ ˇ

Z h hjjd´

ˇ ˇ ˇ ˇ

2

dx dy

1=2Z

M

ˇ ˇ ˇ ˇ

Z h

hj'j²d´

ˇ ˇ ˇ ˇ

2

dx dy 1=4

; Z

M

ˇ ˇ ˇ ˇ

Z h hjjd´

ˇ ˇ ˇ ˇ

4

dx dy

1=4Z

M

Z h

hj'j²d´ dx dy 1=2

Z

M

ˇ ˇ ˇ ˇ

Z h

hj j²d´

ˇ ˇ ˇ ˇ

2

dx dy 1=4

Cmin

kk² Z h

hk'k²4;M d´

1=2

; Z h

hkk4;Md´k'k2

Z h

hk k²4;Md´

1=2

: Similarly,

Z

M

Z h

hj.x; y; ´/jd´

Z h

hj'.x; y; ´/rH‰.x; y; ´/jd´

dx dy

C min

kk2

Z h

hk'k²4;Md´

1=2

; Z h

hkk4;Md´k'k² Z h

hkrH‰k²4;Md´

1=2

:

It follows from the two-dimensional Ladyzhenskaya and Gagliardo-Nirenberg in- equalities that

Z h

hkk4;M d´C Z h

hkk¹⁼²_2;Mkk¹⁼²_H1.M /d´

Ckk¹⁼²2 kk¹⁼²2 C kr^Hk¹⁼²2

; Z h

hk k²4;M d´C Z h

hk k^2;Mk kH¹.M /d´

Ck k².k k²C kr^H k²/;

Z h

hjrH‰k4;Md´C Z h

hk‰k1;Mk‰kH².M /d´Ck‰k1krH²‰k²: The conclusions follow from combining the previous five inequalities.

The following logarithmic Sobolev inequality, which bounds theL¹-norm in terms of theL^q-norms up to the logarithm of the norms of the high-order deriva- tives will play an important role in establishing the a priori H² estimates later.

Some relevant inequalities can be found in [3, 10, 11], where the two-dimensional case is considered.

LEMMA 2.2. LetF 2 W^1;p./, withp > 3, be a periodic function. Then the following inequality holds true:

kFk1 C_p;max (

1;sup

r2

kFkr

r )

log.kFkW^1;p./Ce/

for any > 0.

PROOF. ExtendF periodically to the whole space. Take 2C₀¹.R³/, a func- tion such that 1onand0 1onR³. Setf D F . By Lemma A.1 (chooseRD1there) in the appendix, it holds that

kfkL¹.R³/C_p;max (

1;sup

r2

kfkL^r.R³/

r )

log.kfkW^1;p.R³/Ce/:

Noticing that

kFk1 kfkL¹.R³/; kfkL^r.R³/CkFkr; kfkW^1;p.R³/CkFkW^1;p./; we deduce

kFk₁ kfkL¹.R³/Cp;max (

1;sup

r2

kfkL^r.R³/

r )

log.kfkW^1;p.R³/Ce/

Cp;max (

1;sup

r2

kFkr

r )

log.kFkW^1;p./Ce/;

proving the conclusion.

The following lemma is a system version of the classic Grönwall inequality.

LEMMA2.3. Letm.t /,K.t /,A_i.t /, andB_i.t /be nonnegative functions such that Ai e;are absolutely continuous fori D1; : : : ; n,K2L¹_loc.Œ0;1//, and

m.t /K.t /log

n

X

iD1

Ai.t /:

Given a positive timeT, suppose that d

dtA1.t /CB1.t /m.t /A1.t /;

(2.1)

d

dtAi.t /CBi.t /m.t /Ai.t /CA^˛_{i 1}.t /Bi 1.t /;

(2.2)

fori D2; : : : ; nand for anyt 2.0;T/where˛1and 1are two constants.

Then it holds that

n

X

iD1

Ai.t /C

n

X

iD1

Z t 0

Bi.s/ds Q.t / 8t 2Œ0;T/;

where Q is a continuous function on Œ0;1/ that is determined byAi.0/, i D 1; : : : ; n;andK, given explicitly in equation(2.4)below.

PROOF. Multiplying inequality (2.1) by.˛C1/A^˛₁ yields d

dtA^˛₁^C1C.˛C1/A^˛₁B1 .˛C1/mA^˛₁^C1: Summing this with (2.2) fori D2leads to

d

dt A2CA^˛₁^C1

C˛A^˛₁B1CB2.˛C1/m.A2CA^˛₁^C1/:

SetA1DA1,A2 DA2CA^˛₁^C1,B1DB1, andB2 DB1CB2; then the above inequality gives

d

dtA₂CB₂.˛C1/mA₂: Continuing the previous procedure inductively, we obtain

A_i DAi CA^˛_{i 1}^C1; B_i DBiCB_{i 1}; i D2; : : : ; N;

d

dtA_iCB_i .˛C1/^{i 1}mA_i; i D1; : : : ; N:

In particular, it holds that

(2.3) d

dtAN CBN .˛C1/^{N 1}mAN: By the assumption onm.t /, the above inequality implies

d

dtAN.t /.˛C1/^{N 1}K.t /AN.t /logAN.t /:

Therefore

d

dt logA_N.t /.˛C1/^{N 1}K.t /logA_N; from which we obtain

logA_N e^{.˛C1/N 1}

Rt 0K.s/ds

logA_N.0/DWq0.t / and

A_N.t /e^{q0.t /}DWq1.t /

fort 2 Œ0;T/. Note thatq1 is an increasing function on Œ0;1/. Thanks to this estimate, it follows from integrating inequality (2.3) with respect totthat

A_N.t /C Z t

0

B_N.s/ds .˛C1/^{N 1} Z t

0

m.s/A_N.s/ds .˛C1/^{N 1}

Z t 0

AN.s/K.s/logAN.s/ds .˛C1/^{N 1}

Z t 0

K.s/dsq1.t /q0.t /DQ.t /;

for allt 2Œ0;T/, where

(2.4) Q.t /D.˛C1/^{N 1}

Z t 0

K.s/dsq1.t /q0.t /:

From this we obtain the conclusion.

Remark2.4. Lemma 2.3 indicates that when doing the energy estimates, step by step, the quantities appearing on the left-hand side in the previous steps can be treated freely as if they were a priori bounded, provided the coefficient term does not grow too fast compared to the quantities under consideration (no faster than the logarithm of their summation). This gives us room to handle some hard terms in the current step.

We also need the following Aubin-Lions lemma.

LEMMA 2.5 (Aubin-Lions Lemma; see Simon [23, cor. 4]). Assume that X, B, andY are three Banach spaces, withX ,!,!B ,!Y:Then it holds that

(i) IfF is a bounded subset ofL^p.0; TIX /, where1 p < 1, and ^@F_@t D f^@f_@t jf 2Fgis bounded inL¹.0; TIY /, thenF is relatively compact in L^p.0; TIB/.

(ii) IfF is bounded inL¹.0; TIX /and^@F_@t is bounded inL^r.0; TIY /, where r > 1, thenF is relatively compact inC.Œ0; T IB/.

Finally, we will use the following global existence result for a regularized sys- tem.

PROPOSITION2.6. Suppose that the periodic functionsv0; T0 2H²./are even and odd in´, respectively. Then for any " > 0, there is a unique global strong solution.v; T /to the following system:

@tvC.v r^H/v

Z ´

hr^H v.x; y; ; t /d

@´v Hv "@²_´v Cf0kvC rH

ps.x; y; t / Z ´

h

T .x; y; ; t /d

D0;

(2.5)

Z h

hrH v.x; y; ´; t /d´D0;

(2.6)

@tT Cv r^HT

Z ´

hr^H v.x; y; ; t /d

@´T C1 h

HT "@²_´T D0;

(2.7)

subject to the boundary and initial conditions(1.25)–(1.27), such that

.v; T /2L¹_loc.Œ0;1/IH².//\C.Œ0;1/IH¹.//\L²_loc.Œ0;1/IH³.//;

.@tv; @tT /2L²_loc.Œ0;1/IH¹.//:

PROOF. The proof can be given in the same way as in [6] (see proposition 2.1

there), and thus we omit it here.

3 Low-Order Energy Estimates

In this section, we work on the low-order energy estimates on the strong solution to system (2.5)–(2.7), subject to the boundary and initial conditions (1.25)–(1.27).

In particular, we prove that the growth of theL^q-norms ofvis not faster thanCpq for a constantC independent ofq.

PROPOSITION3.1. Let.v; T /be the global strong solution to system(2.5)–(2.7), subject to the boundary and initial conditions(1.25)–(1.27). Then for any T 2 .0;1/, we have the following:

(i) Basic energy estimates:

sup

0tT k.v; T /k²2C Z T

0 k.rHv;rHT;p

"@´v;p

"@´T /k²2dt K0.T/;

whereK0.T/DC e^T.kv0k²2CkT0k²2/for a positive constantC depending only onh.

(ii) L^q estimate onT, with2q 1: sup

0tT kTk^q kT₀k^q; whereTDT C^´_handT₀DT0C^´_h. (iii) L^q estimate onv, withq 2Œ4;1/:

sup

0tTkvk^qK1.T/e^CkT0k2qT.1C kv0k^q/p q;

where K1, a continuously increasing function, is determined by kv0k², kT0k2,kv0k4, andkT0k4.

PROOF.

(i) If we multiply equations (2.5) and (2.7) byvandT, respectively, sum the resulting equations, and integrate over, then it follows from integrating by parts and using (2.6) that

1 2

d dt

Z



.jvj²C jTj²/dx dy d´

C Z



jrHvj²C"j@´vj²C jrHTj²C"j@´Tj²

dx dy d´

D Z



Z ´ h

T d

r^H v 1 h

Z ´

hr^H v d

T

dx dy d´

CkTk²kr^Hvk² 1

2kr^Hvk²2CCkTk²2:

Thus d

dtk.v; T /k²2C k.r^Hv;r^HT;p

"@´v;p

"@´T /k²2CkTk²2; from which, by the Grönwall inequality, one obtains (i).

(ii) Recalling the definition ofT, using equation (2.7), one can easily check thatTsatisfies

@tTCv rHT

Z ´

hrH v.x; y; ; t /d

@´T _HT "@²_´TD0:

By multiplying the above equation by jTj^{q 2}T, with 2 q < 1and inte- grating over, it follows from integration by parts and using the divergence-free condition (2.6) that

1 q

d

dtkTk^qq0;

which implies the conclusion for2 q <1. The case thatq D 1follows by takingq! 1and using the fact thatkTk^q ! kTk1asq! 1.

(iii) Let 4 q < 1. TheL¹.0;TIL^q.//estimate onv is proved in two steps: a rough estimate and then a more refined estimate. As we shall see below, the latter is based on the former.

Step1. The rough L¹.0;TIL^q.//estimate onv. By multiplying equation (2.5) by jvj^{q 2}v and integrating the resulting equation over , then it follows from integrating by parts that

(3.1) 1 q

d dt

Z



jvj^qdx dy d´C Z



jvj^{q 2} jrHvj²C.q 2/ˇ ˇrHjvjˇ

ˇ

2

C"j@´vj²C.q 2/"ˇ ˇ@´jvjˇ

ˇ

2

dx dy d´DI1CI2; where

I1WD Z



r^H Z ´

h

T d

jvj^{q 2}v dx dy d´;

I2WD Z



r^Hps.x; y; t / jvj^{q 2}v dx dy d´:

An estimate forI1 is given as follows. By the Hölder and Young inequalities and using (ii), we deduce

I1D Z



rH

Z ´ h

T d

jvj^{q 2}v dx dy d´

D Z



Z ´ h

T d

.jvj^{q 2}rH vC.q 2/jvj^{q 3}v rHjvj/dx dy d´

Z



ˇ ˇ ˇ ˇ

Z ´ h

T d ˇ ˇ ˇ

ˇjvj^{q 2}.jrHvj C.q 2/ˇ ˇrHjvjˇ

ˇ/dx dy d´

CkTkqkvk

q 2 1 q

jvj^{q2 1}rHv

₂C.q 2/

jvj^{q2 1}rHjvj ₂

1 8

jvj^{q2 1}rHv

2

2C.q 2/

jvj^{q2 1}rHjvj

2 2

CC qkTk²qkvk^{q 2}q

1 8

jvj^{q2 1}rHv

2

2C.q 2/

jvj^{q2 1}rHjvj

2 2

CC q.1C kT0k²q/kvk^{q 2}q ; (3.2)

where the constantC is independent ofq 2Œ4;1/.

By applying the operator _2h¹ Rh

hdivH./d´to equation (2.5) and using (2.6), it follows from integrating by parts that

Hps.x; y; t /DdivHdivH

1 2h

Z h h

v˝v d´

CdivH

1 2h

Z h h

f0kv d´

H

1 2h

Z h h

Z ´ h

T d Z

M

Z ´ h

T d dx dy

d´

:

Note that by assuming thatR

Mps.x; y; t /dx dyD0,ps.x; y; t /can be chosen in a unique way. Set

p_s⁰D 1 2h

Z h h

Z ´ h

T d Z

M

Z ´ h

T d dx dy

d´;

and decomposepsasps Dp_s⁰Cp¹_s Cp_s², with ( _Hp_s¹DdivHdivH 1

2h

Rh

h.v˝v/d´

inM;

R

Mp¹_sdx dyD0; p¹_s is periodic,

and (

Hp_s²DdivH 1 2h

Rh

hf0kv d´

inM;

R

Mp²_sdx dyD0; p²_s is periodic.

Then by the elliptic estimates, we have kp¹_sk^q;M Cq

Z h h

.v˝v/d´

_q;M Cq

Z h

hkvk²2q;Md´

(3.3)

for allq 2.1;1/, and krHp_s¹k2;M C

divH

Z h h

v˝v d´

_2;M CkjvjjrHvjk²; (3.4)

krHp_s²k2;M C

Z h h

kv d´

_2;M Ckvk2: (3.5)

By setting I2i WD

Z



r^Hp_sⁱ.x; y; t / jvj^{q 2}v dx dy d´; i D0; 1; 2;

then it is obvious that I2 D I20CI21 CI22. We estimateI2i, i D 0; 1; 2, as follows. ForI20, a similar argument similar to that for (3.2) yields

(3.6) I20 1

8

jvj^{q2 1}rHv

2

2C.q 2/

jvj^{q2 1}rHjvj

2 2

CC q.1C kT0k²q/kvk^{q 2}q ;

with constantC independent ofq. ForI22, by the Hölder, Minkowski, Ladyzhen- skaya, and Young inequalities, we deduce

I22D Z



rHp_s².x; y; t / jvj^{q 2}v dx dy d´

Z

M

jrHp_s².x; y; t /j²dx dy 1=2Z

M

Z h h

jvj^{q 1}d´

2

dx dy 1=2

CkrHp_s²k2;M

Z h h

kvk^{q 1}_{2.q 1/;M}d´

CkrHp_s²k2;M

Z h hkvk

1 q 1 2;Mkvk

q.q 2/

q 1 2q;M d´

CkrHp_s²k2;M

Z h hkvk

1 q 1 2;M

jvj^q2

q 2 q 1 2;M

jvj^q2

q 2 q 1 H¹.M /d´

CkrHp_s²k2;Mkvk

1 q 1 2

kvk

q.q 2/

q 1

q C kjvj^q2k

q 2 q 1 2

q 2

jvj^{q2 1}rHjvj ₂

^qq ²1

q 2 8

jvj^{q2 1}rHjvj

2

2CCkrHp_s²k

2.q 1/

q 2;M kvk

2 q 2

jvj^q2

2.q 2/

q 2 q^qq2 CCkrHp_s²k2;Mkvk

1 q 1 2 kvk

q.q 2/

q 1 q

q 2 8

jvj^{q2 1}rHjvj

2

2CC q.1C kvk²2/.1C krHp_s²k²2;M/kvk^{q 2}q

CC.1C kvk²2/.1C krHp_s²k²2;M/kvk

q.q 2/

q 1

q ;

C.1C kvk²2/.1C krHp_s²k²2;M/ qkvk^{q 2}q C kvk

q.q 2/

q 1

q

(3.7)

C q 2 8

jvj^{q2 1}rHjvj

2 2;

with the constant C independent of q 2 Œ4;1/. Recalling the elliptic estimate (3.5), the above inequality gives

I22 q 2 8

jvj^{q2 1}r^Hjvj

2

2CC 1C kvk²2

2

qkvk^{q 2}q C kvk

q.q 2/

q 1

q

: (3.8)

As forI21, recalling the elliptic estimate (3.3), and using Lemma 2.6 and the Hölder, Ladyzhenskaya, and Young inequalities, we deduce

I21D Z



p_s¹.x; y; t /rH .jvj^{q 2}v/dx dy d´

Cq

Z

M

jp¹_s.x; y; t /j Z h

hjrHvj²jvj^{q 2}d´

1=2Z h

hjvj^{q 2}d´

1=2

dx dy

Cqkp¹_sk ^4q

qC2;M

jvj^{q2 1}rHv ₂

Z

M

Z h

hjvj^{q 2}d´

_q^2q₂ dx dy

^q_4q²

Cqkp¹_sk ^4q

qC2;M

jvj^{q2 1}r^Hv ₂

Z h h

Z

M

jvj^2qdx dy ^q_2q²

d´

1=2

Cq

Z h hkvk^28q

qC2;Md´

jvj^{q2 1}r^Hv ₂

Z h h

jvj^q2

2.q 2/

q

4;M d´

1=2

Cq

Z h

hkvk^4;Mkvk^2q;Md´

jvj^{q2 1}r^Hv ₂

Z h h

jvj^q2

2.q 2/

q

4;M d´

1=2

Cq

Z h

hkvk¹⁼²_2;Mkvk¹⁼²_H1.M /

jvj^q2

1 q

2;M

jvj^q2

1 q

H¹.M /d´

jvj^{q2 1}rHv ₂

Z h h

jvj^q2

q 2 q

2;M

jvj^q2

q 2 q

H¹.M /d´

1=2

Cqkvk¹⁼²2 kvk¹⁼²2 C krHvk¹⁼²2

jvj^{q2 1}rHv ₂

jvj^q2

1=2 2

jvj^q2

1=2 2 C

r^Hjvj^q2

1=2 2

1 8

jvj^{q2 1}r^Hv

2

2CCqŒkvk²2.kvk²2C kr^Hvk²2/C1kvk^qq

1 8

jvj^{q2 1}rHv

2

2CCq.1C kvk²2/.1C kvk²2C krHvk²2/kvk^qq:

With the aid of the above estimate, as well as (3.2), (3.6), and (3.8), it follows from the Young inequality that

I1CI2 DI1CI20CI21CI22

3

8 .q 2/

jvj^{q2 1}r^Hjvj

2 2C

jvj^{q2 1}r^Hv

2 2

CC q 1C kT0k²q

kvk^{q 2}q CC q 1C kvk²2

2

1C kvk^qq

CCq 1C kvk²2

1C kvk²2C kr^Hvk²2

kvk^qq

3

8 .q 2/

jvj^{q2 1}rHjvj

2 2C

jvj^{q2 1}rHv

2 2

CCq 1C kvk²2

1C kvk²2C kr^Hvk²2C kT0k²q

1C kvk^qq

: Substituting this into (3.1), one obtains

d

dtkvk^qqC5q 8

jvj^{q2 1}r^Hv

2 2C"

jvj^{q2 1}@´v

2 2

Cq 1C kvk²2

1C kvk²2C kr^Hvk²2C kT0k²q

1C kvk^qq

; from which, using (i), one obtains

sup

0tT kvk^qqC Z T

0

jvj^{q2 1}r^Hv

2 2C"

jvj^{q2 1}@´v

2 2

d´

exp

Cq

Z T

0

1C kvk²2

1C kvk²2C kr^Hvk²2C kT0k²q

ds

1C kv0k^qq

exp˚

Cqe^2T.T C1/ 1C kv0k²2C kT0k²2C kT0k²q

2

1C kv0k^qq

; in particular,

(3.9) sup

0tT kvk⁴4C Z T

0

jvjr^Hv

2

2d´K₁⁰.T/;

where

K₁⁰.T/De^{C e2T.TC1/.1Ckv0k22CkT0k22CkT0k24/2} 1C kv0k⁴4

for a positive constantC depending only onh.

Step2. The refinedL¹.0;TIL^q.//estimate onv. Noticing that all the con- stants C in the estimates forI1, I20, andI22 are independent ofq 2 Œ4;1/, it suffices to give a refined estimate forI21. Recalling the elliptic estimate (3.4), a similar argument to that for (3.7) yields

I21 Z

M

jr^Hp_s¹.x; y; t /j Z h

hjvj^{q 1}d´

dx dy

q 2 8

jvj^{q2 1}rHjvj

2 2

CC 1C kvk²2

1C krHp¹_sk²2;M

qkvk^{q 2}q C kvk

q.q 2/

q 1

q

C 1C kvk²2

1C kjvjjrHvjk²2

qkvk^{q 2}q C kvk

q.q 2/

q 1

q

Cq 2 8

jvj^{q2 1}r^Hjvj

2 2;