WEAK TIME-PERIODIC SOLUTIONS TO THE COMPRESSIBLE NAVIER-STOKES EQUATIONS∗

Hong CAI（蔡虹）S

chool of Mathematical Sciences，Xiamen University，Xiamen 361005，China

E-mail∶caihong19890418@163.com

Zhong TAN（谭忠）

School of Mathematical Sciences and Fujian Provincial Key Laboratory on Mathematical Modeling and Scientific Computing，Xiamen University，Xiamen 361005，China

E-mail∶tan85@xmu.edu.cn



WEAK TIME-PERIODIC SOLUTIONS TO THE COMPRESSIBLE NAVIER-STOKES EQUATIONS∗

Hong CAI（蔡虹）S

chool of Mathematical Sciences，Xiamen University，Xiamen 361005，China

E-mail∶caihong19890418@163.com

Zhong TAN（谭忠）

School of Mathematical Sciences and Fujian Provincial Key Laboratory on Mathematical Modeling and Scientific Computing，Xiamen University，Xiamen 361005，China

E-mail∶tan85@xmu.edu.cn

The compressible Navier-Stokes equations driven by a time-periodic external force are considered in this article.We establish the existence of weak time-periodic solutions and improve the result from［3］in the following sense∶we extend the class of pressure functions，that is，we consider lower exponent γ.

Compressible fluid；Navier-Stokes equations；weak time-periodic solutions

2010 MR Subject Classification35M10；35Q35；35B10.

1　Introduction

This article studies the existence of weak time-periodic solutions of the following compressible Navier-Stokes equations with the time-periodic external force in R3

with the boundary conditions

where ν is the outer normal vector and［v（t，x）］τdenotes the projection of a vector v（t，x）on the tangent plane to∂Ω at the point x（see［3］for more details）.And the unknown functions ρ= ρ（t，x）≥0 and u=（u1（t，x），u2（t，x），u3（t，x））denote the density and the velocity，respectively. Furthermore，the viscosity coefficientsµ，λ are assumed to be constants satisfyingµ＞0 and λ+≥0.The pressure P is a nondecreasing function of the density；more specifically，we assume that

a＞0 is a positive constant，and γ＞1 is the adiabatic constant.In addition，f（t，x）=（f1（t，x），f2（t，x），f3（t，x））is the external force with a period ω＞0；say

For simplicity，we assume Ω to be a cube，that is，

Therefore，the boundary conditions（1.2）read

Thus，throughout this article，a suitable function-space framework is provided by the spatially periodic functions，that is，functions is defined on the torus

and for any（t，x）∈Q，ρ，u，f satisfy the following geometrical conditions，

for any i，j=1，2，3，where Yisatisfies

Note that u satisfies（1.5），then the Poincar´e inequality is automatically satisfied，that is，

As this article is devoted to finding the existence of weak time-periodic solutions，it is convenient to consider the time t belonging to the one dimensional sphere

Moreover，we set

Time periodic or time almost periodic processes are frequently observed in many real world applications of fluid mechanics.They are represented by the time periodic solutions of their associated mathematical models.Concerning the weak time periodic problem for the compressible Navier-Stokes equations，Feireisl［3］first studied three dimensional compressible Navier-Stokes equations driven by a time-periodic external force.They imposed so-called no-stick boundary condition.For the three dimensional flat boundary case，this condition means that the vorticity is perpendicular（see［3］）.Using the Faedo-Galerkin method and the vanishing viscosity method，they obtained the existence of weak time-periodic solutions for three dimensionalcompressible Navier-Stokes equations under the restriction

And for ferrofluids driven by the time periodic external forces，using the ideas and techniques in［3］，Yan［13］showed that such system has the weak time-periodic solutions for γ＞95.Recently，Feireisl［2］showed the existence of at least one weak time periodic solution to the Navier-Stokes-Fourier problem under the basic hypothesis that the system is allowed to dissipate the thermal energy through the boundary.

In this article，inspired by［3-5］，we will investigate the existence of weak time-periodic solutions to the problem（1.1）-（1.3），that is

for the adiabatic constant satisfies

which is an improvement of Feireisl［3］.

Following the strategy in［3，4，7］，we introduce the definition of finite energy weak solution（ρ，u）to the problem（1.1）-（1.3）in the following sense：

Definition 1.1We call（ρ，u）the finite energy weak solution of the problem（1.1）-（1.3）if the following is satisfied.

（1）ρ，u belong to the classes

（2）The energy E（t）is bounded a.e.t∈S1and satisfies the energy inequality

in D′（S1），where

（3）The equations of（1.1）hold in the sense of D′（Q）.

（5）For any（t，x）∈Q，there holds Z

with a given positive mass m and conditions（1.4），（1.5）hold a.e.on Q.

（6）The first equation（1.1）is satisfied in the sense of renormalized solutions；it means that

holds in D′（Q）for any function b∈C1（R+）such that b′（z）=0 if z is large.

The following theorem is the main result of this article.

We shall follow the scheme［3］to construct the above weak time-periodic solution.In more detail，the proof of this theorem will be carried on by means of a three-level approximation scheme based on a modified system

where ǫ，δ＞0 are small，β＞0 sufficiently large，and M（t）∈C∝（R1），

Thus，we vanish the artificial viscosity ǫ and then vanish the artificial pressure δ to complete the construction of the desired time-periodic solution to the original system.

The rest of this article is devoted to the proof of Theorem 1.1 is and organized as follows. In Section 2，following the method in［3］，we present the following results，the existence result of the weak time-periodic solutions to the approximate system（2.1）-（2.3），the result of passing to the limit for n→∞and finally the vanishing viscosity limit result.In Section 3，we pass to the limit in the artificial pressure term；unlike［3］，we follow the idea in［4，5］to prove the existence of the convex function Ψ and get the strong convergence of the density.

Now，we introduce some notations，which will be used throughout this article.

NotationsThroughout this article，for simplicity，we will omit the variables t，x of functions if it does not cause any confusion.C denotes a generic positive constant，which may vary in different estimates.The norm in the Lebesgue Space Lp（T3）is denoted by‖·‖pfor p≥1. Wk，p（T3）（1≤k≤∞，1≤p≤∞）isthe usual Sobolev spaces.C（S1；Xweak）is the space of function g：［0，ω］→X that is continuous with respect to the weak topology.

2　The Faedo-Galerkin Approximation and the Vanishing Viscosity Limit

In this section，we will present the Faedo-Galerkin approximation result and the vanishing viscosity limit result in Feireisl［3］.More specifically，we first replace the original system by the following approximative version，that is，we look for an approximate solution（ρn，un）of the following problem for any fixed n：

The equation of ρn：

with the initial data ρn（0）satisfying

The equation of un：

with the initial condition

Here，for any fixed constant n，

and

Moreover，Xnis the finite-dimensional space defined by

Here，and in what follows，the symbols ak，k∈Z3denote the Fourier coefficients.Observe that all ψ∈Xnsatisfy（1.5）.

At this stage，we shall solve the Cauchy problem for（2.1）-（2.3）.More precisely，（2.1），（2.2）is solved directly while（2.3）is then obtained by the Banach fixed-point theorem.So，the existence of a time-periodic solution is got by the standard topological arguments，that is，a fixed-point of the corresponding period map on a bounded invariant set is founded.Thus，we have the following proposition；for the proof of the proposition，we refer to［3］Section 2 for more details.

Proposition 2.1Suppose that ǫ，δ，and β are the given positive parameters.Then，for any fixed n，the system（2.1）-（2.3）has a time-periodic solution ρn，un.Moreover，ρn∈C1（S1；C2（T3））is a classical solution of（2.1）on S1，and there exists K depending on n such that

The energy inequality

holds on S1，where

and the constant C is independent of n，ǫ，δ.

Furthermore，there exits a constant E1independently of n，such that

Next，let ǫ，δ be fixed and take the limit as n→+∞in the sequence of the approximate solutions constructed in Proposition 2.1 to obtain a time-periodic solution of the problem（1.11）-（1.12）.See Section 3 in［3］for the proof of the following proposition.

The density ρ≥0 and satisfies（1.4），with

The fluid velocity u∈L2（S1；W1，2（T3））satisfies（1.5）a.e.on Q.The energy Eδ［ρ，u］∈L∝（S1）such that

holds in D′（S1），where

and the constant C is independent of ǫ and δ.

Now，we give the existence result derived from passing to the limit as ǫ go to 0 for the approximate problem（2.6），（2.7）while δ is kept fixed.The proof of the following proposition is fulfilled by Section 4 in［3］and we will not give the details here.

Proposition 2.3Given γ＞53，β＞5，δ＞0，then there exists a time-periodic solution（ρ，u）of the problem

Moreover，ρ，u satisfy（1.4）-（1.5），and

The equation（2.10）holds in the sense of renormalized solutions and the energy Eδ［ρ，u］satisfies

in D′（S1），where

and the constant C is independent of δ.

In next section，we will complete the proof of Theorem 1.1 by vanishing the artificial pressure.

3　Passing to the Limit in the Artificial Pressure Term

In this section，our ultimate goal is devoted to letting δ→0 in（2.10），（2.11）and complete the proof of Theorem 1.1.In order to prove the main theorem，we shall start with the following lemma on L1convergence（see［9］Lemma 1.1）.

Lemma 3.1If ψ：R→（−∞，+∞］is a proper，lower semi-continuous，and strictly convex function，D⊂Rmis a domain with bounded measure，and

with p＞1，then it holds that

Before starting the technical part of the proof of Theorem 1.1，we present a straightforward consequence of the energy estimate；the proof can be referred to Lemma 4.1 in［3］.

Lemma 3.2Let ρ≥0，u satisfy

Then，there holds

where C is a constant depending onµ，λ，‖f‖L∝（Q），P denotes a convex function such that

and

satisfying the energy inequality

According，the weak periodic solutions constructed in Proposition 2.3 will be denoted by（ρδ，uδ）.We now first derive the estimates of ρδ，uδindependent of δ＞0，where the technique is inspired by［3］.

Consider the operators

where∆−1stands for the inverse of the Laplacian on the space of spatially periodic functions with zero mean.We have

with the standard elliptic regularity results：

With the help of the above operators，we have the following assertion，which plays a crucial role in the proof of our main result.

Lemma 3.3Let（ρδ，uδ）be the sequence of weak time-periodic solutions of problem（2.10），（2.11）obtained in Proposition 2.3，then

are bounded independently of δ，

ProofIntegrating the energy inequality（2.12）over S1and by（1.7），we obtain

which implies

By the fact that ρδis a renormalized solution of（2.10），we see that for some ϑ＞0，

In view of（3.3）and the regularity results achieved in Proposition 2.3，we are allowed to take1，2，3，as a test function for（2.11）.Thus，we have

where the bilinear operator

and

At this moment，we will estimate the terms on the right-hand side of（3.6）steps by steps.Here，we only consider the last three terms，where the others are simple.The main tools used is thefact thatis bounded independently of δ，the Hüolder inequality，the Sobolev embedding theorems together with the estimates for Aipresented in（3.3）.Therefore，one has

where the constant C is independent of δ provided

where the constant C is independent of δ provided ϑ≤2γ−3 3γ.

where s＞3，so the constant C is independent of δ provided

Hence，it follows from（3.6）-（3.9）that

In view of（3.4）and the interpolation inequality，we have

Combining this with（3.10），it yields

Moreover，interpolating between the space L1and Lγ+ϑ，we deduce

Finally，by virtue of Lemma 3.2 and the above inequality，formula（3.12）reads

Consequently，（3.13）implies thatdxdt is bound independently of δ.And the boundedness offollows from Lemma 3.2 and（3.4）.Hence， we obtain the desired estimates.This completes the proof of Lemma 3.3.

Then，we are in a position to pass to the limit as δ→0 in the sequence of the approximate solutions obtained in Proposition 2.3.Indeed，by Lemma 3.3，we have

Similarly as in［3］Lemma 4.3，the other information obtained from the uniform energy estimates of Lemma 3.3 is

However，we must be careful about the case of the pressure；indeed，Lemma 3.3 guarantees

but not more.So，the limit δ→0+is quite clear，and the only last proof will consist in showing that the weak limitis in fact equal to ργ.

Letting δ→0 in（2.10），（2.11），we obtain the weak time-periodic solutions ρ，u satisfying

where i=1，2，3.

In the rest of this section，we will proveTo show the strong convergence of the density will be the most difficult task in our limit passage.Unlike［3］，we follow the idea of［4，9］that consist in using cut-off functions to control the density.More precisely，let T∈C∝（R）be concave and satisfy

then build a sequence of functions Tkdefined as follows

Recall that ρδ，uδis a renormalized solution to（2.10）impliesPassing to the limit for δ→0+，we deduce that

where

and

Next，we introduce some lemmas，which will be used in the proof of the strong convergence of the density.For the proof of these lemmas，we refer to［4］for details.

Lemma 3.4Let ρδ，uδbe the solutions obtained in Proposition 2.3，then it holds that

Lemma 3.5There exists a constant C independent of k such that

The proof of the following lemma can be proved in a similar way as［4］；the details are omitted.

Lemma 3.6The limit functions ρ，u obtained in this section is a renormalized solution to（3.18），that is，

holds in D′（Q）for any b∈C1（R+）with b′（z）=0，for z is large.

To complete the proof of Theorem 1.1，we now introduce the functions

It is not difficult to see that Lkcan be written as

Additionally，by virtue of（3.20），we have

whereand the bound in（3.23）depends solely on α；in particular，it is independent of k（see［11］Lemma 6.15 and 7.57 for details），and next，

for 1≤α＜γ by approximating z logz≈Lk（z）.In particularso taking φ∈D（T3），φ=1 for x∈T3as the test function for the difference of（3.20）and（3.21），then，for any τ1＜τ2，integrating with respect to t，we have

Passing to the limit for δ→0 and by（3.22），we have

In view of Lemma 3.4，we can estimate the right-hand side of the above inequality as

which，together with（3.25），implies

In the sequel，we shall need the following crucial lemmas，which can be proved in a similar way as Lemma 6.1，Lemma 5.3 in［5］.

Lemma 3.7There exists a constant d＞0，such that

where

and Ψ is the convex function from Lemma 3.8.

Then，Ψ is convex，strictly increasing for z≥0，Ψ（0）=0.

At this stage，it remains to see what happen when k→∞in（3.26）.First，we have

for certain p1，p2＞0 independent of k.

In addition，by the definition of Lk（ρδ），we also have

where ε is a sufficiently small constant，such that γ＞1+ε.Hence，passing to the limit for δ→0 in（3.28）and by（3.22），（3.24），we obtain

Seeing that，in accordance with Lemma 3.7，（3.27），and（3.29），one can pass to the limit in（3.26）for k→∞to conclude

where the term on the right-hand side can be estimated as follows，

Next，observe that

and hence it follows that when 1≤p≤γ，

On the other hand，we have for 1≤p＜γ，

Noting that Tkis concave，and therefore，also，by（3.32），（3.33）and Tk（ρ）≤ρ，we have

This result，in combination with Lemma 3.5，yields that the term on the right-hand side of（3.30）tends to zero for k→∞.

In view of Lemma 3.6，ρ is the renormalized solution to（3.18），and due to［11］Lemma 6.15 and 7.57，we see that

in particular，

Therefore，it follows from（3.23），（3.29），（3.35），and the convexity of ρlogρ，we know that the function

is continuous，bounded，and nonegative on S1.Furthermore，（3.30）implies that for any τ1＜τ2，

Consequently，D≡0，then，by Lemma 3.1，we have the strong convergence of the sequence ρδ，that is，

which means that

This completes the proof of Theorem 1.1.

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September 26，2014；revised April 25，2015.The first author is supported by National Natural Science Foundation of China-NSAF（11271305，11531010）and the Fundamental Research Funds for Xiamen University（201412G004）.The second author is supported by National Natural Science Foundation of China-NSAF（11271305，11531010）.