Quansen JIU (酒全森)

School of Mathematical Sciences, Capital Normal University, Beijing 100048, China

E-mail : jiuqs@cnu.edu.cn

Fengchao WANG (王鳳超)?

Advanced Institute of Natural Sciences, Beijing Normal University at Zhuhai, Zhuhai 519087, China School of Mathematical Sciences, Beijing Normal University and Key Laboratory of Mathematics and Complex Systems, Ministry of Education Beijing 100875, China

E-mail : wfcwymm@163.com

Here (x,z) ∈ ? = M × (0,1), with M = (0,L1). For simplicity, we suppose that L1= 1. The boundary conditions are that

and the initial conditions are

The primitive equations for atmosphere dynamics are one of the fundamental models in geophysical flows (see [22]). Since the rigorous arguments by Lions, Temam and Wang in [20],mathematical theories regarding the incompressible primitive equations have been widely developed (see [1, 5–11, 18, 23, 26]and references therein). In recent years, there have appeared some mathematical analyses on compressible atmosphere models (see [12–14, 17]). Very recently, Jiu, Li and Wang[16]proved the uniqueness of the weak solutions obtained by Gatapov and Kazhikhov[14], and the global existence of weak solutions for three-dimensional compressible primitive equations was proved by Wang, Dou and Jiu [25]. However, up until now, there has been little study on the nonhomogeneous incompressible primitive equations.

In this article,we will prove the local existence and uniqueness of strong solutions to the twodimensional nonhomogeneous primitive equations (1.1)–(1.3) by a semi-Galerkin method as in[3]where the compressible Navier-Stokes equations in three-dimensional is considered. The main difference between the two models is the lack of a vertical momentum equation in the primitive equations. Thus there is a strongest nonlinear term in the horizontal momentum equation which enjoys nonlinearity of the quadratic in first order derivatives of horizontal velocity. This is why we only consider the two dimensional case. It seems difficult to achieve an appropriate estimate for utbecause utis multipled by ρ, possibly vanishing due to the initial vacuum.We will overcome this difficulty by estimatingby using the ideas in [3]. In addition,ρ0∈ W1,6(?) is needed to prove the uniqueness of strong solutions. Motivated by Choe and Kim [3, 4], we construct the approximate system (3.1)–(3.2) and prove that they have an approximate solution (ρn,un) satisfying

Then we prove that the quantityis bounded in a small time interval(0,T?), and derive the desired a priori estimates. Next,with the boundedness ofin a small time interval (0,T?)? (0,T?) in hand, by compactness arguments, we establish the existence and uniqueness of the strong solutions to (1.1)–(1.3).

The rest of the article is organized as follows: in Section 2, we present some preliminary lemmata which will be used frequently throughout the paper. In Section 3, we show the existence of global solutions to the approximate system(3.1)by using the semi-discrete-Galerkin approximation, and we derive some a priori estimates. In Section 4, we will give the proofs of Theorem 2.4 and Theorem 2.6.

2 Preliminaries and Main Results

In this section, we present some basic facts needed later, and then state our main results.First of all, we introduce some notations for function spaces. Throughout this article, for positive integer k and positive number q ∈ [1,∞], we use Lqand Wk,qto denote the standard Lebesgue and Sobolev spaces, respectively, on the domain ?. When q =2, we use Hk, instead of Wk,2. In addition,we define the following function spaces for primitive equations(1.1)–(1.3):

We have the following lemma about the regularity result of hydrostatic Stokes system:

Lemma 2.1(see [24]) Letwith L > 0,α = 1,2,··· ,n ? 1, and let

α? ={(x′,z)∈ O × R;0

possesses a unique solution(u,P)which belongs towhere the Sobolev spacedefined asis periodic on ?O}. Moreover,we have the continuous dependence,that is,there exists a constant C =C(?,νh,νv)>0 such that

The next lemma regarding the compactness of ρ due to Diperna-Lions, is standard.

Lemma 2.2(see [21]) For any fixed T ∈ (0,∞), we suppose thatsatisfies

where C denotes various positive constants independent of n.

Then, ρnconverges in C([0,T];Lp(?)) for all p ∈ [1,∞) to the unique solution ρ, bounded on ?×(0,T), of

Finally, we state a Gronwall type inequality which will be used to guarantee the existence of solutions.

Lemma 2.3(see [2, 19]) (Bihari-LaSalle inequality) Let f and g be non-negative continuous functions defined on [0,∞), and let h be a continuous non-decreasing function defined on[0,∞) and h(f)>0 on (0,∞). If f satisfies the integral inequality

where α is a non-negative constant, then

where the function G is defined by

and G?1is the inverse function of G and T is chosen so that

Our main results are stated as follows:

Theorem 2.4Assume that the initial data (ρ0,u0) satisfies the initial regularity

and the compatibility condition

for some P0∈ V and g(x,z) ∈ L2(?). Then there exists a small time T?∈ (0,T), such that the system(1.1)–(1.3)has a weak solution (ρ,u,w,P) which satisfies the following regularities:

Remark 2.5If the boundary conditions (1.2) are replaced by the boundary conditions

we can also obtain the result of Theorem 2.4.

Theorem 2.6Assume that (2.1)–(2.2) in Theorem 2.4 holds. If ρ0∈ H1(?)∩ W1,6(?),then there exists a small time T?∈ (0,T?)such that the system(1.1)–(1.3)has a unique strong solution (ρ,u,w,P) satisfying

Remark 2.7The H3estimate of horizontal velocity u can be obtained because of the higher regularity estimates of the hydrostatic Stokes system with a Dirichlet-Periodic boundary condition (see Lemma 2.1).

Remark 2.8The local well-posedness of the three-dimensional nonhomogeneous incompressible primitive equations remains unsolved. The main difficulty lies in obtaining the higher regularity estimate ofWe will consider this problem in a later work.

3 Proof of Main Results

3.1 Semi-Discrete-Galerkin approximation

In this section, we will prove Theorem 2.4. First, we construct the approximate solutions by the semi-discrete-Galerkin method. To do this, we define a finite-dimensional space Xn=span{ψ1,··· ,ψn}, n ∈ N, where the function ψnis the n-th eigenfunction to the following eigenvalue problem of the Dirichlet-Periodic problem to the hydrostatic Stokes equations in ?:

Here λnis the n-th eigenvalue of the hydrostatic Stokes operator (see[15]). The sequencecan be renormalized in such a way that it is an orthonormal basis of L2, which is also an orthogonal basic ofand a basis inLetting(ρ0,u0)satisfy the hypotheses of Theorem 2.4, for the initial density, we assume that

Now we are going to consider the approximate system

with the initial data

where X = (x,z). If we assume thatthe standard theory of ordinary differential equations gives us the particle trajectoryWhat is more, Y depends continuously on the velocity Vn; that is,

where Y1,Y2are two particle paths of two different velocity fieldsrespectively,which pass through the same point X at the time 0. Here C is a positive constant only depending upon the domain ?. The divergence-free vector field Vnimplies that the mapping Mt:X → Y(t,X)is a C2-diffeomorphism ofonto itself, and then X(t,Y) ∈If we denote ρn(Y,t)= ρn0(X(t,Y)), it is easy to check that ρnis the unique solution of

naturally,it is continuous with respect to Vn∈ C(0,T;Xn). Noting that wn=we have that ρn=Sn[un]is continuous with respect to un∈ C(0,T;Xn).

Next,we are looking for the approximate solution un∈C(0,T;Xn)to the following system:

In order to obtain the solution of the above system, we denote

Hence, the system (3.5) can be rewritten as

Next we will extend the above local solution to be a global one. To this end,taking φ=unin (3.1)2and integrating by parts, we obtain

Furthermore, the density ρnis bounded, and bounded away from the blow with a positive constant, which means that

It follows from the above uniform estimates that Tn= T. In other words, we get the global approximate solution (ρn,un) to systems (3.1) and (3.2), satisfying

3.2 A priori estimates

In the section, we will establish some uniform higher estimates with respect to n for the approximate solution (ρn,un,wn). To this end, for any t ∈ (0,T) and T >0, we denote

First,we estimate the density ρn. From the continuity equation(3.1)1and the incompressibility condition ?xun+ ?zwn=0, we immediately derive that

As we have done for (3.7), we have the basic energy estimate

for t ∈(0,T).

Now we estimate the first termof Ψ(t) as the following lemma:

Lemma 3.1The following inequality holds true:

Here C is a positive constant depending only on the initial data and the terminal time T, but is independent of n.

ProofTaking φ=untin (3.1)2and integrating by parts, we obtain

Applying Young’s inequality, one gets

We estimate the first term of (3.11) as follows:

Here we used the Gagliardo-Nirenberg interpolation equality for the direction

Similarly, the second term can be bounded as

Substituting the above estimates into (3.11), we have

In order to estimate the termin (3.12), we consider the following hydrostatic Stokes system (see [24]):

By using Lemma 2.1, we have

Substituting (3.12) into (3.14), we have

With (3.12)and (3.15)in hand, we can prove that there exists a parameter ?0>0 so small that the following inequality holds:

Integrating (3.16) over (0,t), we complete the proof of (3.10).

Next, we estimate the second term of Ψ(t) and write it as the following lemma:

Lemma 3.2For any t>0, the following inequality holds:

Here C is a positive constant andwill be defined as below.

ProofDifferentiating (3.1)2with respect to t and using (3.1)1yields

Taking φ=untin the equation obtained above, it follows from integrating by parts and using(3.1)1that

Next, we will estimate the terms of the left hand side in (3.18) one by one:

Substituting the above estimates K1–K15into (3.18), we obtain

Then, integrating (3.19) with respect to time t over (τ,t)? (0,T), we have

Taking the limits τ → 0 on the both sides of (3.22), we conclude that

Letting τ → 0 in (3.20), we obtain the inequality (3.17).

Thanks to Lemma 3.1 and Lemma 3.2, we will prove that Ψ(t) is locally bounded for some T?>0. To this end, Combining the estimates (3.10) and (3.17), we have

In other words,

If we denote Υ(t)=ln(Ψ(t)/C), the inequality (3.24) reduces to

Using Lemma 2.3, there exists a small time T?such that

that is,

Recall the classical regularity estimates on the hydrostatic Stokes system, so that by

we have

Noticing the fact that

for any t ∈ (0,T?), we can show that

Next we will show the higher regularity estimates of ρnas the following lemma:

Lemma 3.3There exists a time T?∈ (0,T?) such that the following inequality holds:

where C is some constant.

ProofDifferentiating the equation (3.1)1with respect to x and z, respectively, yields

Multiplying (3.28)by ?zρnand ?zρn, respectively,and integrating(by parts)over ?,we obtain

Similarly, we multiply ?zρn|?zρn|4and ?zρn|?zρn|4on (3.28)1and (3.28)2, respectively. Using the same method for |?ρn|2, we obtain

From (3.29) and (3.30), we have

Therefore, Lemma 2.1 yields

By direct estimates, we have

We control the term I3as follows:

Substituting I1–I3into (3.33), we obtain

Combining (3.34) and (3.31), we deduce that

Integrating (3.36) with respect to time t, it holds that

Using the Lemma 2.3, we complete the proof of this lemma.

It follows by (3.34) and Lemma 3.3 that

4 Proof of Theorem 2.4 and Theorem 2.6

In this section, we prove the local existence and uniqueness of strong solutions to the nonhomogeneous incompressible primitive equations (1.1), subject to (1.2)–(1.3). First, we consider the existence of weak solutions, in other words, we will prove the Theorem 2.4.

4.1 Proof of Theorem 2.4

In this subsection, we will prove the existence of the weak solutions to (1.1)–(1.3).

ProofFrom the above estimates (3.8), (3.9), (3.25), (3.26) and (3.27), we obtain the following uniform regularities:

By the regularities (4.1), we can prove, up to an extracted subsequence, that

Adapting the standard argument, we can easily show that the limit (ρ,u,w) is a weak solution to the system (1.1)–(1.3) with the initial dataThanks to the lower semi-continuity of the convex function, we have

Now we prove the Theorem 2.4. Recalling the regularity (2.1) for the initial data (ρ0,u0), we construct a new initial densitywhich satisfies

By the previous proof,we know that the system(1.1)–(1.3)with the initial data(ρ0?,u0)exists as a local weak solution(ρ?,u?),and that the weak solution satisfies the uniform estimates(4.2)and (4.3) withreplaced by ρ0?.

Taking the limits ? → 0, we obtain a local weak solution (ρ,u,w) for (1.1)–(1.3) with the initial data (ρ0,u0). Then the proof of Theorem 2.4 is complete.

4.2 Proof of Theorem 2.6

In the next subsection, we will prove the existence and uniqueness of strong solutions to the system (1.1)–(1.3).

4.2.1 Existence of strong solutions

ProofFor T?∈ (0,T?), from the above estimates (3.8)–(3.9), (3.25)–(3.27), as well as Lemma 3.3 and (3.38), we obtain the following uniform regularities:

Thanks to the above regularities (4.5), up to an extracted subsequence, we can easily obtain the following weak convergence results:

Furthermore, noticing that ρnt= ?un?xρn? wn?zρn∈ L2(0,T?;L6)∩ L∞(0,T?;Lr), 1

Therefore,by the Aubin-Lions compactness lemma, we have the following strong convergences:

Using the previous compactness results, (4.6) and (4.8), one can check that

With the above convergences in hand, by taking the limit n → ∞, we can easily prove the existence of strong solutions for the system (1.1)–(1.2) with the initial dataNow we are ready to prove the existence of strong solutions to Theorem 2.6. As we have done in the proof of Theorem 2.4, we also construct a new initial densitywhich satisfies

Then for the system (1.1)–(1.2) with the initial datathere exists a strong solution(ρ?,u?,w?).

Letting ? → 0, the limit (ρ,u,w) is a local strong solution for (1.1)–(1.2) with the initial data (ρ0,u0), which satisfies

Then we complete the proof of the existence of strong solutions.

4.2.2 Uniqueness of strong solutions

In this part,we will prove the continuous dependence upon the initial data and the uniqueness of the local strong solutions.

ProofAssume that (ρ,u,w) andare two solutions of (1.1) with corresponding pressures P andand initial data (ρ0,u0) andrespectively. It is clear that

Multiplying the above equation byand integrating (by parts) over ?, we obtain

From the horizontal momentum equation (1.1)2, we have

Subtracting the above two equations,

For J1, we have the following estimate:

Direct estimates give

Substituting J1–J3into (4.13), we obtain

Combining (4.11) with (4.14), we get

The above inequality proves the continuous dependence of the solutions on the initial data. In particular, whenfor any 0 ≤ t ≤ T?. Thus we have completed the proof of uniqueness.

AcknowledgementsThe authors would like to thank Professor Li Jinkai not only for posing the problem, but also for making many valuable suggestions.