TIME PERIODIC SOLUTION TO THE

http://actams.wipm.ac.cn

TIME PERIODIC SOLUTION TO THE

COMPRESSIBLE NAVIER-STOKES EQUATIONS IN A PERIODIC DOMAIN

Chunhua JIN(^7Ss)

School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China E-mail:jinchhua@126.com

Tong YANG(^Õ)^†

Department of Mathematics, City University of Hong Kong, Hong Kong, China E-mail:matyang@cityu.edu.hk

Dedicated to Professor Boling Guo on the occasion of his 80th birthday

Abstract This article is concerned with the time periodic solution to the isentropic com- pressible Navier-Stokes equations in a periodic domain. Using an approach of parabolic regularization, we first obtain the existence of the time periodic solution to a regularized problem under some smallness and symmetry assumptions on the external force. The result for the original compressible Navier-Stokes equations is then obtained by a limiting process.

The uniqueness of the periodic solution is also given.

Key words Time periodic solution; compressible Navier-Stokes equation; topology degree;

energy method

2010 MR Subject Classification 35Q30; 35B10; 35M10

1 Introduction

Consider the compressible Navier-Stokes equations for isentropic fluid (ρt+∇ ·(ρu) = 0,

ρ(ut+ (u· ∇)u) +∇P(ρ) =µ∆u+ (µ+λ)∇divu+ρf(x, t), (1.1) where x∈(−L, L)³, ρ(x, t),u(x, t) = (u1, u2, u3)(x, t) are the density and velocity,P(ρ) is the pressure,f = (f1, f2, f3)(x, t) is a given external force, which is assumed to be periodic in both

∗Received April 16, 2015; revised January 19, 2016. C. Jin’s research is supported by the Program for New Century Excellent Talents in University of the Ministry of Education (NCET-13-0804), NSFC (11471127), Guangdong Natural Science Funds for Distinguished Young Scholar (2015A030306029), The Excellent Young Teachers Program of Guangdong Province (HS2015007), Pearl River S&T Nova Program of Guangzhou (2013J2200064), and Special support program of Guangdong Province; T. Yang is supported by the General Research Fund of Hong Kong, CityU 104511.

†Corresponding author

space and time with period 2L and T respectively, andµ and λare the viscosity coefficients satisfying

µ >0, λ+2µ 3 ≥0.

The purpose of this article is to study the existence of periodic solution around a constant state (¯ρ,0). Rewrite (1.1) in the perturbation form as

(σt+ ¯ρ∇ ·u=−∇ ·(σu),

ut−(1−h(σ))(¯µ∆u+ (¯µ+ ¯λ)∇divu) +γρ∇σ¯ =−(u· ∇)u−g(σ)∇σ+f(x, t), (1.2) whereσ=ρ−ρ, ¯¯ µ=µ/ρ, ¯¯ λ=λ/ρ,¯ γ=P^′(¯ρ)/ρ¯², and

h(σ) = σ

σ+ ¯ρ and g(σ) = P^′(σ+ ¯ρ)

σ+ ¯ρ −P^′(¯ρ)

¯ ρ .

Before stating the main result, we give some notations, which will be used throughout this article,

Ω = (−L, L)³, QT = Ω×(0, T), and for multi-indexα= (α1, α2, α3),

∂_x^α=∂_x^α1∂_x^α2∂_x^α3, |α|=

3

X

i=1

αi,

|u(·, t)|²_K= X

|α|=K

k∂^αu(·, t)k²_L²_(Ω), kuk_H^k =ku(·, t)k_H^k(Ω),

X ={(σ, u) :σ∈L^∞((0, T);H^K(Ω)), u∈L²((0, T);H^K+1(Ω))∩L^∞((0, T);H^K(Ω));

σ, usatisfies (a), (b), (c) of Theorem 1.1}.

Theorem 1.1 Assume that P(ρ) is a smooth function in a neighborhood of ¯ρ, K ≥4 is an integer, and f ∈L²((0, T);H^K−1(Ω)) withf(−x, t) = −f(x, t). If RT

0 kf(·, t)k²_HK−1dt is suitably small, then the problem (1.2) admits a solution (σ, u)∈ X ∩G^R_K such that

(a) (σ, u) is periodic with the space period 2Land time periodT; (b) R

Ωσ(x, t)dx≡0,R

Ωu(x, t)dx≡0;

(c) σ(x, t) =σ(−x, t),u(x, t) =−u(−x, t).

Moreover, the solution (σ, u) is unique within this class, provided that sup

t∈(0,T)

k(σ, u)(s)k_H^K is sufficiently small.

Remark 1.2 Theorem 1.1 shows that there is a periodic solution with constant total mass, momentum. The condition on symmetry of the external force f is to ensure that the Poinc´are inequality holds. In fact, when there is a given external force acting on the fluid, only the conservation of the total mass holds in general. It is worth noting that some structural condition on f seems to be necessary, because (1.1) implies that

d dt

Z

Ω

ρudx= Z

Ω

ρfdx.

If f ≥ 0 (or f ≤0) for any (x, t) ∈ QT with f 6≡0, then we have _dt^d R

Ωρudx > 0 (or <0) even when f is very small. This implies that the smallness on the external force itself cannot guarantee the existence of time periodic solution.

Let us review some previous related works. The time periodic solution to the incompressible Navier-Stokes equations has been extensively studied; see, for example [1–5]. Precisely, [1] is about the existence of time periodic solutions in a bounded domain Ω when the boundary moves periodically in time. In this article, Serrin introduced a method to solve the periodic problem, that is, for a time-periodic force f and any initial value, the solution of the corresponding initial-value problem will converge (as t→ ∞) to some state, which is considered as an initial value , and this yields a time-periodic solution. Another approach was introduced by Yudovich [2]. He considered the Poincar´e map from an initial value u(x,0) to the state u(x, T), where T is the period of the given external forcef, and u(x, t) is the solution corresponding to the initial datau(x,0). A time-periodic solution is then identified as a fixed point of this Poincar´e map. This approach is also introduced in the same time by Prodi in another article [3]. In [4], the authors considered the reproductive property, which can be regarded as a generalization of time periodicity, that is, a solution with the same value as the initial data at some time T without the assumption of the periodicity on the external force f. If f is time periodic, then the reproductive solution is also time periodic. Recently, [5] extends these results to the inhomogeneous boundary conditions and gives the existence and uniqueness of small time periodic strong solutions under the assumption that the inhomogeneous boundary data and the external forcef are small; moreover, using the reproductive property, [5] yields the existence of time periodic weak solution without any smallness assumption on boundary value and the external forcef.

On the other hand, for the compressible Navier-Stokes equations, there are only a few works on time periodic solutions. Among them, in 1983, Valli [9] first studied the existence of periodic solutions in a three spatially bounded domain with non-slip boundary condition, and using Serrin’s method, he showed the existence of a small strong periodic solution when the external forcef is small enough. In [10], Matsumura and Nishida considered the periodic solution for a one dimensional bounded domain in the lagrangian mass coordinate with external force or a piston. In a recent work [11], Feireisl, Mucha et al considered a full Navier-Stokes- Fourier system confined to a smooth bounded domain with no-slip boundary conditions, and obtained the existence of periodic solutions. As the authors in [11] pointed out, such a condition is in fact necessary, as energetically closed fluid systems do not possess non-trivial (changing in time) periodic solutions due to the second law of thermodynamics.

In addition to these results in a bounded domain, there are also some results for unbounded domain problems; for example, in [6–8], the authors showed the existence of time periodic solutions for the incompressible Navier-Stokes equations in the whole three-dimensional space, the half space and exterior domains, respectively. It is worth noticing that most of these existence results depend closely on the decay rate estimate of the corresponding linearized equation, where the technique of L^q −L^r estimate is used. However, this technique seems ineffective for the compressible Navier-Stokes equations, due to the fact that divu6= 0, and the appearance of some other nonlinear terms. Hence, for the compressible Navier-Stokes equations, by using the spectral analysis method for the optimal decay estimates, in the previous work, the existence of periodic solutions is obtained only when the space dimension N ≥ 5 inR^N. In a recent work [12, 13], B˘rezina and Kagei considered the time-periodic parallel flow in an n-dimensional infinite layerR^N−1×(0, l). In these article, the authors constructed a periodic

solution using the solution of a one-dimensional heat type equation in a bounded domain (0, l) for a special time-periodic external force of the form f= (f1(xn, t),0,· · · ,0, fn(xn)).

In this article, we focus on the compressible Navier-Stokes equations with periodic bound- ary. The main idea is based on an approach of parabolic regularization, and this method allows us to overcome difficult problems in proving the compactness of the operator. Thus, using the energy estimates and the topological degree method, we obtain the existence of periodic solutions for the regularized problem, then the existence of periodic solutions for (1.2) is ob- tained by a limiting process. On the basis of this, we also show the uniqueness of small periodic solutions.

2 Existence of Periodic Solutions

We will prove the existence of time periodic solutions in this section. For this, we first consider the following regularized problem

(σt−ε∆σ+ ¯ρ∇ ·u=−∇ ·(σu),

ut−(1−h(σ))(¯µ∆u+ (¯µ+ ¯λ)∇divu) +γρ∇σ¯ =−(u· ∇)u−g(σ)∇σ+f(x, t). (2.1) Set

GK =

(ρ, ω)∈L²((0, T);H^K(Ω))∩L^∞((0, T);H^K−1(Ω));ρ, ωsatisfies (a), (b), (c) ; and

G_K^R =

(ρ, ω)∈ GK; sup

0<t<T

k(ρ, ω(·, t)k²_HK−1+ Z T

0

k(ρ, ω)(·, t)k²_HKdt < R²

.

Define an operator

F : G^R_K×[0,1]→ GK, ((ρ, ω), τ)→(σ, u), withK≥4, where (σ, u) is the solution of the problem

(σt−ε∆σ+ ¯ρ∇ ·u=−τ∇ ·(ρω),

ut−(1−τ h(ρ))(¯µ∆u+ (¯µ+ ¯λ)∇divu) +γρ∇σ¯ =−τ((ω· ∇)ω+g(ρ)∇ρ) +τ f(x, t). (2.2) The following lemma shows that the operatorF is well defined.

Lemma 2.1 Assume that R is suitably small. Then for any (ρ, ω)∈ G_K^R, τ ∈[0,1], the problem (2.2) admits a unique solution (σ, u) inGK.

Proof Noticing thatK≥4, we have kρkL^∞≤µ sup

0<t<T

kρ(·, t)k_H^K−1 ≤µR.

Then, when R is suitably small, we have |h(ρ)| ≤ 1/2. By the classical theory of parabolic equations, the existence of solutions can be obtained. To prove uniqueness, suppose to the contrary, there exists two solutions (σ1, u1), (σ2, u2) for some (ρ, ω) ∈ G_K^R, τ ∈ [0,1]. Let σ=σ1−σ2,u=u1−u2. Then, we have

(σt−ε∆σ+ ¯ρ∇ ·u= 0,

ut−(1−τ h(ρ))(¯µ∆u+ (¯µ+ ¯λ)∇divu) +γρ∇σ¯ = 0.

From the proof of (2.3) in the following Lemma 2.2, we have d

dt γ|σ(·, t)|²_K+|u(·, t)|²_K

+εγ|σ(·, t)|²_K+1+2

3µ|u(·, t)|¯ ²_K+1≤0.

Then, the Poinc´are inequality implies (σ, u) = (0,0), and the uniqueness of solutions is proved.

On the other hand, as both (σ(x, t), u(x, t)) and (σ(−x, t),−u(−x, t)) are the solution of (2.2), by uniqueness, σ(x, t) = σ(−x, t), u(x, t) = −u(−x, t). And this completes the proof of the

lemma.

Next, we show thatF is a compact operator.

Lemma 2.2 IfRis suitably small, then the operatorF is compact.

Proof Assume that (σ, u) is the solution of problem (2.2). For each multi-index αwith

|α| = K, by applying ∂_x^α to (2.2), multiplying the first and second equation by γ∂_x^ασ, ∂_x^αu respectively, and taking integration over Ω, we obtain

γ 2

d dt

Z

Ω

|∂^ασ|²dx+εγ Z

Ω

|∇∂^ασ|²dx−γρ¯ Z

Ω

∂^αu∂^α(∇σ)dx=−τ γ Z

Ω

∂^α(∇ ·(ρω))∂^ασdx and

1 2

d dt

Z

Ω

|∂^αu|²dx+ (1−τ h(ρ))

¯ µ Z

Ω

|∇∂^αu|²dx+ (¯µ+ ¯λ) Z

Ω

|∂^αdivu|²dx

=−γρ¯ Z

Ω

∂^αu∂^α(∇σ)dx+τ Z

Ω

¯

µh^′(ρ)∂^α∇u∂^αu∇ρ+ (¯µ+ ¯λ)h^′(ρ)∂^αu∂^α(divu)∇ρ dx +

Z

Ω

X

1≤l≤α

α l

∂^l(1−τ h(ρ))∂^α−l(¯µ∆u+ (¯µ+ ¯λ)∇divu)∂^αudx +τ

Z

Ω

∂^α−1((ω· ∇)ω+g(ρ)∇ρ)∂^α+1udx+τ Z

Ω

∂^α−1f ∂^α+1udx.

Summing up the above equalities yields 1

2 d dt

Z

Ω

γ|∂^ασ|²+|∂^αu|²

dx+εγ Z

Ω

|∇∂^ασ|²dx + (1−τ h(ρ))

¯ µ Z

Ω

|∇∂^αu|²dx+ (¯µ+ ¯λ) Z

Ω

|∂^αdivu|²dx

=−τ γ Z

Ω

∂^α(∇ ·(ρω))∂^ασdx +τ

Z

Ω

¯

µh^′(ρ)∂^α∇u∂^αu∇ρ+ (¯µ+ ¯λ)h^′(ρ)∂^αu∂^α(divu)∇ρ dx +

Z

Ω

X

1≤l≤α

α l

∂^l(1−τ h(ρ))∂^α−l(¯µ∆u+ (¯µ+ ¯λ)∇divu)∂^αudx +τ

Z

Ω

∂^α−1((ω· ∇)ω+g(ρ)∇ρ)∂^α+1udx+τ Z

Ω

∂^α−1f ∂^α+1udx

=I1+I2+I3+I4+I5. Firstly, note that

kρkL^∞(QT)≤C sup

0<t<T

kρ(·, t)k_H^K−1< CR.

Then for sufficiently smallR, we have

kρkL^∞(QT)≤ρ¯ 2,

which implies that

(1−τ h(ρ))≥2 3.

By the periodic boundary condition, we see thatk∂^β(ρ, u)kL²≤Ck∇∂^β(ρ, u)kL² for any multi- indexβ with|β| ≥0. Note that|α| ≥4, then we have

|I1| ≤Ck∇∂^ασk_L²(k∂^αρk_L²kωkL^∞+k∂^αωk_L²kρkL^∞)

≤ εγ

2 k∇∂^ασk²_L2+C1(|ω|²_Kkρk²_HK−1+|ρ|²_Kkωk²_HK−1),

|I2| ≤Ck∇∂^αuk²_L²k∇ρkL^∞≤C2Rk∇∂^αuk²_L²,

|I3| ≤Ck∂^αuk_L²(kρkL^∞k∇²uk_H^K−1+k∇²ukL^∞kρk_H^K−1)

≤Ckρk_H^K−1|u|²_K+1≤C3R|u|²_K+1,

|I4| ≤C(|ω|KkωkL^∞+|ω|K−1k∇ωkL^∞+|ρ|KkρkL^∞+|ρ|K−1k∇ρkL^∞)k∂^α+1ukL²

≤C4(|ω|Kkωk_H^K−1+|ρ|Kkρk_H^K−1)|u|K+1,

|I5| ≤ |f|K−1|u|K+1. Then for sufficiently smallR >0, we have

d

dt γ|σ(·, t)|²_K+|u(·, t)|²_K

+εγ|σ(·, t)|²_K+1+2

3µ|u(·, t)|¯ ²_K+1

≤C5(|ω(·, t)|²_K+|ρ(·, t)|²_K)(kω(·, t)k²_HK−1+kρ(·, t)k²_HK−1) +C6|f(·, t)|²_K−1. (2.3) Integrating this inequality from 0 toT yields

εγ Z T

0

|σ(·, t)|²_K+1dt+2 3µ¯

Z T 0

|u(·, t)|²_K+1dt

≤C5 sup

0<t<T

(kω(·, t)k²_HK−1+kρ(·, t)k²_HK−1) Z T

0

(|ω(·, t)|²_K+|ρ(·, t)|²_K)dt +C6

Z T 0

|f(·, t)|²_K−1dt

=M^∗. (2.4)

Then, there exists a timet^∗∈(0, T) such that εγT|σ(·, t^∗)|²_K+1+2

3µT¯ |u(·, t^∗)|²_K+1≤M^∗.

By Poinc´are inequality, we haveγ|σ(·, t^∗)|²_K+|u(·, t^∗)|²_K≤CM^∗. Then, integrating (2.3) from t^∗to tfor anyt∈(t^∗, T], we have

γ|σ(·, t)|²_K+|u(·, t)|²_K ≤CM^∗. By the time periodicity, we further have

γ|σ(·,0)|²_K+|u(·,0)|²_K ≤CM^∗. Repeating the above process yields

sup

0<t<T{γkσ(·, t)k²_HK+ku(·, t)k²_HK} ≤CM^∗. (2.5)

Integrating (2.3) fromttot+h, then integrating it from 0 to T over t, we have Z T

0

(γ|σ(·, t+h)|²_K+|u(·, t+h)|²_K)−(γ|σ(·, t)|²_K+|u(·, t)|²_K)dt

≤2C5h sup

0<t<T

(kω(·, t)k²_HK−1 +kρ(·, t)k²_HK−1) Z T

0

(|ω(·, s)|²_K+|ρ(·, s)|²_K)ds + 2C6h

Z T 0

|f(·, s)|²_K−1ds. (2.6)

For each multi-indexβ with|β|=K−1, by applying∂_x^β to (2.2), and multiplying the first and second equation by (∂_x^βσ)t, (∂_x^βu)trespectively, then integrating overQT, we have

Z T 0

(k(∂^βσ)tk²_L2+k(∂^βu)tk²_L2dt

≤C Z T

0

|u(·, t)|²_K+|ρ(·, t)|²_K−1|ω(·, t)|²_K+|ρ(·, t)|²_K|ω(·, t)|²_K−1+|ρ(·, t)|²_K−1

|u(·, t)|²_K+1+|σ(·, t)|²_K+|u(·, t)|⁴_K+|ρ(·, t)|²_K−1|ρ(·, t)|²_K+|f(·, s)|²_K−1 dt

≤C sup

0<t<T

|ρ(·, t)|²_K−1 Z T

0

(1 +|ω(·, t)|²_K+|ρ(·, t)|²_K+|u(·, t)|²_K+1)dt+C Z T

0

|f(·, s)|²_K−1dt +C sup

0<t<T

|ω(·, t)|²_K−1 Z T

0

|ρ(·, t)|²_Kdt+CT sup

0<t<T

(|u(·, t)|²_K+|u(·, t)|⁴_K). (2.7) From (2.4), (2.5), (2.6), and (2.7), we see thatF is a compact operator and this completes the

proof of the lemma.

The next lemma is about the continuity of the operatorF.

Lemma 2.3 WhenR is suitably small, the operatorF is continuous.

Proof Assume that (ρn, ωn)⊂ G_K^R,τn∈[0,1], (ρ, ω)⊂ G_K^R,τ ∈[0,1], and

n→∞lim sup

t∈(0,T)

k(ρn−ρ, ωn−ω)(·, t)k²_HK−1+ Z T

0

k(ρn−ρ, ωn−ω)(·, t)k²_HKdt= 0 as lim

n→∞τn=τ. Let (σn, un) =F((ρn, ωn), τn), (σ, u) =F((ρ, ω), τ). Then, (σn−σ, un−u) is a periodic solution of following equation



























σet−ε∆eσ+ ¯ρ∇ ·ue= (τ−τn)∇ ·(ρω)−τn∇ ·((ρn−ρ)ω+ρn(ωn−ω)), uet−(1−τ h(ρn))(¯µ∆ue+ (¯µ+ ¯λ)∇diveu) +γρ∇¯ eσ

= (τ−τn)h(ρ)(¯µ∆u+ (¯µ+ ¯λ)∇divu) +τn(h(ρ)−h(ρn))(¯µ∆u+ (¯µ+ ¯λ)∇divu)

−(τn−τ)((ωn· ∇)ωn+g(ρn)∇ρn)−τ (ωn−ω)∇ωn+ω∇(ωn−ω)

−τ (g(ρn)−g(ρ))∇ρn+g(ρ)∇(ρn−ρ)

+ (τn−τ)f(x, t),

(2.8)

with periodic boundary condition. Similar to the proof for the compactness of the operatorF, and noticing that

Z T 0

ku(·, t)k²_HK+1dt < C, we obtain

n→∞lim sup

t∈(0,T)

k(σn−σ, un−u)(·, t)k²_HK−1+ Z T

0

k(σn−σ, un−u)(·, t)k²_HKdt= 0,

which gives the continuity of the operatorF.

From Lemmas 2.2 and 2.3, we see that the operatorF is completely continuous.

Proposition 2.4 Assume that

f(x, t) =−f(−x, t), f ∈L²((0, T);H^K−1) andRT

0 kf(·, t)k²_HKdtis sufficiently small, then problem (2.1) admits a periodic solutionu∈ G_K^R. Proof Firstly, we see that solving problem (2.1) inGKis equivalent to solving the equation

U−F(U,1) = 0, U= (σ, u)∈ GK.

In what follows, we apply the topological degree theory. We first chooseR >0 such that (I−F(·, τ))(∂BˆR(0))6= 0, for anyτ ∈[0,1], (2.9) where ˆBR(0) is the ball of radiusR centered at the origin inGK. If (2.9) holds, then to show the existence of solution, we only need to show that

deg(I−F(·,1),BˆR(0),0)6= 0. (2.10) For this, we will show that there existsR >0 such that (2.9) holds. We prove it by contradiction.

Let ((σ, u), τ) be a solution of (2.9) for some smallR >0 by replacing (ρ, ω) with (σ, u) such that ((σ, u), τ) satisfies

(σt−ε∆σ+ ¯ρ∇ ·u=−τ∇ ·(σu),

ut−(1−τ h(σ))(¯µ∆u+ (¯µ+ ¯λ)∇divu) +γρ∇σ¯ =−τ(u· ∇u+g(σ)∇σ) +τ f(x, t). (2.11) For each multi-indexαwith|α|=K, by applying∂_x^αto (2.11), and multiplying the equations byγ∂_x^ασ,∂_x^αurespectively, then the integrating over Ω, we obtain

1 2

d dt

X

|α|=K

Z

Ω

γ|∂^ασ|²+|∂^αu|²

dx+εγ X

|α|=K

Z

Ω

|∇∂^ασ|²dx

+ (1−τ h(σ)) X

|α|=K

¯ µ Z

Ω

|∇∂^αu|²dx+ (¯µ+ ¯λ) Z

Ω

|∂^αdivu|²dx

=−τ γ X

|α|=K

Z

Ω

∂^α(∇ ·(σu))∂^ασdx

+τ X

|α|=K

Z

Ω

¯

µh^′(σ)∂^α∇u∂^αu∇σ+ (¯µ+ ¯λ)h^′(σ)∂^αu∂^α(divu)∇σ dx

+ X

|α|=K

Z

Ω

X

1≤l≤α

α l

∂^l(1−τ h(σ))∂^α−l(¯µ∆u+ (¯µ+ ¯λ)∇divu)∂^αudx

+τ X

|α|=K

Z

Ω

∂^α−1((u· ∇)u+g(σ)∇σ)∂^α+1udx+τ X

|α|=K

Z

Ω

∂^α−1f ∂^α+1udx

=J1+J2+J3+J4+J5. (2.12)

Next, we estimate each term on the right hand side of (2.12). ForJ1, we have

|J1| ≤τ γ 2

X

|α|=K

k∇ukL^∞k∂^ασk²_L²

+τ γ Z

Ω



 X

0≤l≤α−1

α l

∂^l(∇σ)∂^α−lu∂^ασ+∂^lσ∂^α−l(∇ ·u)∂^ασ



dx

≤ X

|α|=K

τ γ

2 k∇ukL^∞k∂^ασk²_L2+ X

|α|=K

k∂^ασk_L²(k∇ukL^∞k∇σk_H^K−1 +k∇uk_H^K−1k∇σkL^∞+kσkL^∞k∇²uk_H^K−1+k∇²ukL^∞kσk_H^K−1

≤C˜1kukH^K−1kσk²_HK+ ˆC1|σ|²_Kkσk²_HK−1+η|u|²_K+1. For other terms, we have

|J2| ≤C X

|α|=K

k∇σkL^∞k∂^α∇ukL²k∂^αukL²≤C˜2|σ|²_K−1|u|²_K+η|u|²_K+1,

|J3| ≤C X

|α|=K

k∂^αuk_L²(kσkL^∞k∇²uk_HK−1+k∇²ukL^∞kσk_HK−1)

≤C|σ|K−1|u|K+1|u|K ≤C˜3|σ|²_K−1|u|²_K+η|u|²_K+1,

|J4| ≤C(|u|KkukL^∞+|u|K−1k∇ukL^∞+|σ|KkσkL^∞+|σ|K−1k∇σkL^∞)|u|K+1

≤C(|u|˜ K|u|K−1+|σ|K|σ|K−1)|u|K+1

≤C˜4(|u|²_K|u|²_K−1+|σ|²_K|σ|²_K−1) +η|u|²_K+1,

|J5| ≤ |f|K−1|u|K+1≤C˜5|f|²_K−1+η|u|²_K+1. Taking 5η <¹₃µ¯ to be sufficiently small, we obtain

1 2

d

dt γ|σ(·, t)|²_K+|u(·, t)|²_K

+εγ|σ(·, t)|²K+1+1

3µ|u(·, t)|¯ ²K+1

≤Cb |u(·, t)|²_K−1+|σ(·, t)|²_K−1

|u(·, t)|²_K+|σ(·, t)|²_K

+δ|σ(·, t)|²_K + ˜C5|f(·, t)|²_K−1, (2.13) whereδ is a small constant to be determined later.

We now turn to estimate |σ|K. For|β| =K−1, applying ∂^β_x to (2.11)₂, and multiplying the resulting equation by∂_x^β∇σ, then integrating them over Ω, we obtain

γρ¯ Z

Ω

|∇∂^βσ|²dx+ Z

Ω

∂^βut∇∂^βσdx

= Z

Ω

∂^β{(1−τ h(σ))(¯µ∆u+ (¯µ+ ¯λ)∇divu)}∇∂^βσdx

−τ Z

Ω

∂^β(u· ∇u+g(σ)∇σ)∇∂^βσdx+τ Z

Ω

∂^βf(x, t)∇∂^βσdx.

On the other hand, applying ∇∂_x^β to (2.11)₁, and multiplying the resulting identity by ∂_x^βu, and then integrating over Ω, it yields

Z

Ω

∂^βu∇∂^βσtdx−ε Z

Ω

∂^β∇σ∂^β∆udx−τρ¯ Z

Ω

|∂^β(divu)|²dx=τ Z

Ω

∂^β(divσu)∂^β(divu)dx.

Summing up the above inequalities, we obtain γρ|σ|¯ ²_K+ X

|β|=K−1

d dt

Z

Ω

∂^βu∇∂^βσdx

≤ε|σ|K|u|K+1+τρ|u|¯ ²_K+|σu|K|u|K+4

3|u|K+1|σ|K+|u|K−1|u|K|σ|K

+|σ|K−1|σ|²_K+|f|K−1|σ|K

≤γ¯ρ

4 |σ|²_K+Cb2|u|²_K+1+Cb3 |u(·, t)|²_K−1+|σ(·, t)|²_K−1

|u(·, t)|²_K+|σ(·, t)|²_K

+C|f|²_K−1. By multiplying inequality (2.13) by A with ¹₆µA >¯ Cb2, and taking δ sufficiently small with δA < ^γ₄^ρ¯, then summing up the above inequalities, we obtain

d dt



 A

2(γ|σ(·, t)|²_K+|u(·, t)|²_K) + X

|β|=K−1

Z

Ω

∂^βu∇∂^βσdx



+Aεγ|σ(·, t)|²_K+1 +1

6µA|u(·, t)|¯ ²_K+1+γρ¯ 4 |σ|²_K

≤M1 |u(·, t)|²_K−1+|σ(·, t)|²_K−1

|u(·, t)|²_K+|σ(·, t)|²_K

+M2|f|²_K−1. (2.14) Let

ξ(t) = A

2(γ|σ(·, t)|²_K+|u(·, t)|²_K) + X

|β|=K−1

Z

Ω

∂^βu∇∂^βσdx.

It is easy to see that

C(|σ(·, t)|²_K+|u(·, t)|²_K)≤ξ(t)≤C(|σ(·, t)|²_K+|u(·, t)|²_K).

Note that we also have 1

6µA|u(·, t)|¯ ²_K+1+γρ¯

4 |σ|²_K ≥M(|σ(·, t)|²_K+|u(·, t)|²_K), for some positive constantM. Integrating (2.14) from 0 to T yields

M Z T

0

(|σ(·, t)|²_K +|u(·, t)|²_K)dt

≤M1 sup

0<t<T

|u(·, t)|²_K−1+|σ(·, t)|²_K−1 Z T

0

|u(·, t)|²_K+|σ(·, t)|²_K

dt+M2

Z T 0

|f|²_K−1dt

≤M1R²+M2

Z T 0

|f|²_K−1dt.

Using the Mean Value Theorem, there existsτ ∈(0, T) such that M T(|σ(·, τ)|²_K+|u(·, τ)|²_K)≤M1R²+M2

Z T 0

|f|²_K−1dt.

Integrating (2.14) fromτ tot for anyt∈(τ, T], we have ξ(t)≤ξ(τ) +M1R²+M2

Z T 0

|f|²_K−1dt≤Mf1R²+Mf2

Z T 0

|f|²_K−1dt.

Noticing thatξ(0) =ξ(T) and integrating (2.14) from 0 totfor anyt∈[0, T], we have sup

0<t<T

ξ(t)≤ξ(0) +M1R²+M2

Z T 0

|f|²_K−1dt

≤(M1+Mf1)R²+ (M2+Mf2) Z T

0

|f|²_K−1dt. (2.15) Then, we obtain

sup

0<t<Tk(σ, u)(·, t)k²_HK−1+ Z T

0

k(σ, u)(·, t)k²_HKdt

≤ sup

0<t<T

k(σ, u)(·, t)k²_HK+ Z T

0

k(σ, u)(·, t)k²_HKdt

≤M^∗R²+C^∗ Z T

0

|f|²_K−1dt, which implies

R≤M^∗R²+C^∗ Z T

0

|f|²_K−1dt.

Take R < _2M^1∗, and let C^∗RT

0 |f|²_K−1dt < ¹₂R, then the above inequality is a contradiction.

Hence, (2.9) holds. On the other hand, when τ= 0, (σ, u) = 0, that is,F(·,0) = 0. Therefore, for the chosenR >0, we have

deg(I−F(·,1),BˆR(0),0) =deg(I−F(·,0),BˆR(0),0) =deg(I,BˆR(0),0) = 1.

Thus, (2.10) holds. That is the problem (2.1) admits a time periodic solutionU ∈ G_K^R. The

proof is completed.

Proof of Theorem 1.1 (Existence) To avoid any confusion, we denote the solution of the regularized problem (2.1) by (σε, uε). From the proof of Proposition 2.4, we see that

sup

0<t<T

k(σε, uε)(·, t)k²_HK+ Z T

0

(kσε(·, t)k²_HK+kuε(·, t)k²_HK+1)dt≤CR,

where R is independent ofε. On the other hand, integrating from t to t+hof (2.13), then integrating it from 0 toT, we have

Z T 0

γ|σ(·, t+h)|²_K+|u(·, t+h)|²_K

− γ|σ(·, t)|²_K+|u(·, t)|²_K

dt≤Ch. (2.16) Thus, there exists a subsequence of (σε, uε), denoted by (σεn, uεn), such that

(σεn, uεn)⇀^∗ (σ, u) inL^∞((0, T);H^K(Ω));

uεn⇀ uinL²((0, T);H^K+1(Ω));

σεn →σinL²((0, T);H^K−1(Ω));

uεn→uin L²((0, T);H^K(Ω)).

In what follows, we show thatσε∈ C^α,β(Ω×(0, T))). Clearly, we have σε(x, t)∈C^α(Ω) for α∈ (0,1) for anyt because σε ∈L^∞((0, T);H^K(Ω)) with K ≥4. So, we only need to show that there exists a constantβ∈(0,1) such that

|σε(x, t1)−σε(x, t2)| ≤C|t1−t2|^β (2.17) holds for anyt1, t2∈(0, T),x∈Ω. Without loss of generality, assume thatt2≤t1. Take a ball Br of radiusrcentered atx, withr=|t1−t2|^η,η= _2α+3¹ . Recalling (2.7) and using Poinc´are inequality, we have

Z

Br

|σε(y, t1)−σε(y, t2)|dy= Z

Br

Z t1

t2

∂σε(y, t)

∂t dt dy

≤C Z t1

t2

Z

Br

∂σε(y, t)

∂t

2

dydt

!1/2

|t1−t2|^1/2r^3/2

≤C|t1−t2|^1/2r^3/2.

By Mean Value Theorem, there existsx^∗∈Br such that

|σε(x^∗, t1)−σε(x^∗, t2)| ≤C|t1−t2|^1/2r^−3/2=C|t1−t2|^(1−3η)/2. Then, we have

|σε(x, t1)−σε(x, t2)|

≤ |σε(x, t1)−σε(x^∗, t1)|+|σε(x^∗, t1)−σε(x^∗, t2)|+|σε(x^∗, t2)−σε(x, t2)|

≤C(|t1−t2|^ηα+|t1−t2|^(1−3η)/2)

≤C|t1−t2|^α/(2α+3). Similarly, we also have

|uε(x1, t1)−uε(x2, t2)| ≤C|x1−x2|^α˜+|t1−t2|^β˜, (2.18) for ˜α,β˜∈(0,1), whereC is independent ofε. Thus, by Arzela-Ascoli Theorem, we have

(σεn, uεn)→(σ, u) uniformly.

Thus, (σ, u)∈ X is a time periodic solution of (2.1). And this completes the proof of Theorem

1.1.

3 The Uniqueness of Periodic Solutions

In this section, we consider the uniqueness of periodic solutions. Let (σ1, u1),(σ2, u2) ∈ G_K^R∩ X be the time periodic solutions of the problem. Setσ =σ1−σ2, u=u1−u2. Then, (σ, u) is a periodic solution of following equation

σt+ ¯ρ∇ ·u=−∇ ·(σu1)− ∇ ·(σ2u), (3.1) ut−(¯µ∆u+ (¯µ+ ¯λ)∇divu) +γρ∇σ¯

= (h(σ1)−h(σ2))(¯µ∆u2+ (¯µ+ ¯λ)∇divu2)−h(σ1)(¯µ∆u+ (¯µ+ ¯λ)∇divu)

−u∇u1−u2∇u−(g(σ1)−g(σ2))∇σ1−g(σ2)∇σ, (3.2) with periodic boundary condition. We will show the uniqueness if sup

t∈(0,T)

k(σi, ui)(s)kH^K is sufficiently small.

Proof of Theorem 1.1 (Uniqueness) We assume that sup

t∈(0,T)

k(σi, ui)(s)k_HK ≤δ

for some sufficiently smallδ >0. Note thatK≥4. For each multi-indexαwith|α|= 3≤K−1, by applying∂_x^αto (3.1), (3.2), and multiplying them byγ∂_x^ασ,∂_x^αurespectively, then integrating over Ω, we obtain

1 2

d dt

Z

Ω

X

|α|=3

(γ|∂^ασ|²+|∂^αu|²)dx+ ¯µ X

|α|=3

Z

Ω

|∇∂^αu|²dx+ (¯µ+ ¯λ) X

|α|=3

Z

Ω

|∂^αdivu|²dx

=−γ X

|α|=3

Z

Ω

∂^α(∇ ·(σu1))∂^ασdx−γ X

|α|=3

Z

Ω

∂^α(∇ ·(σ2u))∂^ασdx

+ X

|α|=3

Z

Ω

∂^α−1 (h(σ1)−h(σ2))(¯µ∆u2+ (¯µ+ ¯λ)∇divu2)

∂^α+1udx

+ X

|α|=3

Z

Ω

∂^α−1 h(σ1)(¯µ∆u+ (¯µ+ ¯λ)∇divu)

∂^α+1udx− X

|α|=3

Z

Ω

∂^α(u∇u1)∂^αudx

− X

|α|=3

Z

Ω

∂^α(u2∇u)∂^αudx− X

|α|=3

Z

Ω

∂^α((g(σ1)−g(σ2))∇σ1)∂^αudx

− X

|α|=3

Z

Ω

∂^α(g(σ2)∇σ)∂^αudx

=I1+I2+· · ·+I8.

Next, we estimate each term on the right hand side of the above equation. Firstly, for I1, we have

|I1| ≤γ 2

X

|α|=3

k∇u1kL^∞k∂^ασk²_L²

+γ Z

Ω



 X

0≤l≤2

α l

∂^l(∇σ)∂^α−lu1∂^ασ+∂^lσ∂^α−l(∇ ·u1)∂^ασ



dx

≤ X

|α|=3

τ γ

2 k∇u1kL^∞k∂^ασk²_L²+ X

|α|=3

k∂^ασkL²(k∇u1kL^∞k∇σkH²

+k∇u1k_H²k∇σkL^∞+kσkL^∞k∇²u1k_H²+k∇²u1kL^∞kσk_H²

≤Cδ|σ|²₃. ForI2, we have

|I2| ≤γ|σ|3(kσ2kL^∞|u|4+kukL^∞|σ2|4)≤Cδ(|σ|²₃+|u|²₄).

Moreover,

|I3| ≤ |u|4|(h(σ1)−h(σ2))(¯µ∆u2+ (¯µ+ ¯λ)∇divu2)|2

≤C|u|4(kσkL^∞|u2|4+|σ|2k∆u2kL^∞)

≤Cδ(|σ|²₃+|u|²₄), and

|I4| ≤C|u|²₄|σ2|2≤Cδ|u|²₄.

Furthermore,

|I5| ≤C|u|²₃|u1|4≤Cδ|u|²₃,

|I6| ≤Ck∇u2kL^∞|u|²₃+|u|3k∇ukL^∞|u2|3≤Cδ|u|²₃,

|I7| ≤ |u|4|(g(σ1)−g(σ2))∇σ1|2≤Cδ(|u|²₄+|σ|²₃),

|I8| ≤ |u|4|g(σ2)∇σ|2≤Cδ(|u|²₄+|σ|²₃).

Here, we use the fact that

|g(σ1)−g(σ2)|k ≤C|σ|max{2,k}, for smooth functiong. In fact, note that

g(σ1)−g(σ2) =g(σ1)−g(σ2)

σ1−σ2 (σ1−σ2) =f(σ1, σ2)σ.

Then,

|g(σ1)−g(σ2)|k ≤ |f|kkσkL^∞+|σ|k|f|L^∞ ≤C|σ|max{2,k}. Hence, we have

d dt

Z

Ω

(γ|σ(·, t)|²₃+|u(·, t)|²₃)dx+ 2¯µ|u(·, t)|²₄≤Ce1δ(|u(·, t)|²₄+|σ(·, t)|²₃). (3.3) For each multi-indexβ with |β|= 2, by applying∂_x^β,∇∂_x^β to (3.2), (3.1), and multiplying the resulting equations by∂_x^β∇σ, ∂_x^βurespectively, summing up the two resulting equations, then integrating over Ω, we have

γρ¯ X

|β|=2

Z

Ω

|∇∂^βσ|²dx+ d dt

X

|β|=2

Z

Ω

∂^βu∇∂^βσdx

= X

|β|=2

Z

Ω

∂^β(¯µ∆u+ (¯µ+ ¯λ)∇divu)∇∂^βσdx

+ X

|β|=2

Z

Ω

∂^β((h(σ1)−h(σ2))(¯µ∆u2+ (¯µ+ ¯λ)∇divu2))∇∂^βσdx

− X

|β|=2

Z

Ω

∂^β(h(σ2)(¯µ∆u+ (¯µ+ ¯λ)∇divu))∇∂^βσdx

− X

|β|=2

Z

Ω

∂^β(u∇u1+u2∇u)∇∂^βσdx

− X

|β|=2

Z

Ω

∂^β((g(σ1)−g(σ2))∇σ1)∇∂^βσdx− X

|β|=2

Z

Ω

∂^β(g(σ2)∇σ)∇∂^βσdx + ¯ρ X

|β|=2

Z

Ω

|∂^βdivu|²dx+ X

|β|=2

Z

Ω

∂^βdivu·∂^βdiv(σu1)dx

+ X

|β|=2

Z

Ω

∂^βdivu·∂^βdiv(σ2u)dx

=J1+J2+· · ·+J9.

In what follows, we estimate each term on the right hand side of the above equation as follows:

|J1| ≤ |σ|3|u|4, |J2| ≤C|σ|²₃|u2|4≤Cδ|σ|²₃,

|J3| ≤C|σ|3|u|4|σ2|2≤Cδ(|σ|²₃+|u|²₄),

|J4| ≤C(|u|2|u1|3+|u|3|u2|2)|σ|3≤Cδ(|σ|²₃+|u|²₄),

|J5| ≤C|σ|2|σ1|3|σ|3≤Cδ|σ|²₃, |J6| ≤Cδ|σ|²₃,

|J7| ≤ρ|u|¯ ²₃, |J8| ≤C|u|3|σ|3|u1|3≤Cδ(|σ|²₃+|u|²₄), |J9| ≤Cδ|u|²₄. Thus, we obtain

γρ|σ(·, t)|¯ ²₃+ 2d dt

X

|β|=2

Z

Ω

∂^βu∇∂^βσdx≤Ce2δ(|σ|²₃+|u|²₄) +Ce3|u|²₄. (3.4) Combining with (3.3), we obtain

d dt



A Z

Ω

(γ|σ(·, t)|²₃+|u(·, t)|²₃)dx+ 2 X

|β|=2

Z

Ω

∂^βu∇∂^βσdx





+ (2¯µA−Ce3)|u(·, t)|²₄+γρ|σ(·, t)|¯ ²₃

≤(Ce1A+Ce2)δ(|u(·, t)|²₄+|σ(·, t)|²₃).

TakingA= ^Ce_µ¯³, we obtain d

dt



A Z

Ω

(γ|σ(·, t)|²₃+|u(·, t)|²₃)dx+ 2 X

|β|=2

Z

Ω

∂^βu∇∂^βσdx





+ Ce3−(Ce1Ce3

¯

µ +Ce2)δ

!

|u(·, t)|²₄+ γρ¯−(Ce1Ce3

¯

µ +Ce2)δ

!

|σ(·, t)|²₃≤0.

Integrating the above inequality from 0 toT, then choosingδ suitably small, we have Z T

0

|u(·, t)|²₄dt+ Z T

0

|σ(·, t)|²₃dt≤0,

which implies thatu=σ= 0 a.e. inQT. The uniqueness is then proved.

Remark 3.1 From the proof of uniqueness, it is not difficult to see that for the compress- ible Navier-Stokes equations without external force, that is (1.2) withf ≡0, there is no small non-trivial time periodic solution.

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