Weak time-periodic solutions to the compressible navier-stokes equations

Acta Mathematica Scientia 2016,36B(2):499–513 http://actams.wipm.ac.cn

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

é)

Hong CAI (

School of Mathematical Sciences, Xiamen University, Xiamen 361005, China E-mail : [email protected]

Zhong TAN (

§)

School of Mathematical Sciences and Fujian Provincial Key Laboratory on Mathematical Modeling and Scientific Computing, Xiamen University, Xiamen 361005, China E-mail : [email protected] Abstract 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. Key words

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

2010 MR Subject Classification

1

35M10; 35Q35; 35B10.

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  ∂ ρ + div(ρu) = 0, t (1.1) ∂t (ρu) + div(ρu ⊗ u) + ∇P (ρ) − µ∆u − (µ + λ)∇ div u = ρf, with the boundary conditions

u · ν = 0,

[Du · ν]τ = 0

for all t ∈ R1 , x ∈ ∂Ω,

(1.2)

where ν is the outer normal vector and [v(t, x)]τ denotes 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 coefficients µ, λ are assumed to be constants satisfying µ > 0 and λ + 32 µ ≥ 0. The pressure P is a nondecreasing function of the density; more specifically, we assume that P (ρ) = aργ , (1.3) ∗ Received

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 ChinaNSAF (11271305, 11531010).

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a > 0 is a positive constant, and γ > 1 is the adiabatic constant. In addition, f (t, x) = (f 1 (t, x), f 2 (t, x), f 3 (t, x)) is the external force with a period ω > 0; say f (t + ω, x) = f (t, x) for all t, x. For simplicity, we assume Ω to be a cube, that is, Ω = [0, π]3 . Therefore, the boundary conditions (1.2) read ui = 0 on the opposite faces {xi = 0, xj ∈ [0, π], j 6= i} ∪ {xi = π, xj ∈ [0, π], j 6= i}, j

∂u = 0 for i 6= j on ∂xi {xi = 0, xj ∈ [0, π], j 6= i} ∪ {xi = π, xj ∈ [0, π], j 6= i}. Thus, throughout this article, a suitable function-space framework is provided by the spatially periodic functions, that is, functions is defined on the torus T3 = ([−π, π]|{−π,π} )3 , and for any (t, x) ∈ Q, ρ, u, f satisfy the following geometrical conditions, ρ(t, Yi (x)) = ρ(t, x), i

i

i

i

(1.4)

u (t, Yi (x)) = −u (t, x), f (t, Yi (x)) = −f (t, x),

i

i

j 6= i,

(1.5)

i

i

j 6= i,

(1.6)

u (t, Yj (x)) = u (t, x), f (t, Yj (x)) = f (t, x),

for any i, j = 1, 2, 3, where Yi satisfies Yi [x1 , · · · , xi , · · · , x3 ] = [x1 , · · · , −xi , · · · , x3 ]. Note that u satisfies (1.5), then the Poincar´e inequality is automatically satisfied, that is, Z Z 2 |u| dx ≤ C |∇u|2 dx. (1.7) T3

T3

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 t ∈ S1 = [0, ω]|{0,ω} . Moreover, we set Q = S1 × T3 . 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 dimensional

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compressible Navier-Stokes equations under the restriction 9 . 5 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-StokesFourier 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 γ>

ρ(t + ω, x) = ρ(t, x), for the adiabatic constant satisfies γ>

u(t + ω, x) = u(t, x)

(1.8)

5 , 3

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 1.1 We 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

ρ ≥ 0, ρ ∈ L∞ S1 ; Lγ (T3 ) , ui ∈ L2 S1 ; W 1,2 (T3 ) , i = 1, 2, 3. (2) The energy E(t) is bounded a.e. t ∈ S1 and satisfies the energy inequality Z  Z    d E(t) + µ|∇u|2 + (µ + λ)| div u|2 dx ≤ C 1 + ρ|f ||u|dx dt T3 T3

in D′ (S1 ), where E(t) =

Z

T3

1

2

ρ|u|2 +

 a ργ dx. γ−1

(3) The equations of (1.1) hold in the sense of D′ (Q). (5) For any (t, x) ∈ Q, there holds Z ρ(t, x)dx = m

(1.9)

T3

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 ∂b(ρ) + div(b(ρ)u) + (b′ (ρ)ρ − b(ρ)) div u = 0 ∂t

(1.10)

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. Theorem 1.1 Let γ > 53 and suppose that f i ∈ L∞ (Q), i = 1, 2, 3, and satisfy the condition (1.6) a.e. on Q, then there exists a weak time-periodic solution (ρ, u) of the problem (1.1)–(1.3) in the sense of Definition 1.1.

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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 Z  ∂t ρ + div(ρu) = ǫ∆ρ − 2ǫρ + M ρdx , (1.11) T3

∂t (ρui ) + div(ρui u) − µ∆ui − (µ + λ)∂xi (div u) + a∂xi ργ +δ∂xi ρβ + ǫ∇ui · ∇ρ + 2ǫρui = ρf i ,

i = 1, 2, 3,

(1.12)

where ǫ, δ > 0 are small, β > 0 sufficiently large, and M (t) ∈ C∞ (R1 ),    1, t ∈ (−∞, 0], M (t) =

  

a decreasing function on (0, m),

0,

t ∈ [m, ∞).

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. Notations Throughout 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 k · kp for p ≥ 1. W k,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 : ∂t ρn + div(ρn un ) = ǫ∆ρn − 2ǫρn + M

Z

T3

 ρn dx ,

(2.1)

with the initial data ρn (0) satisfying 0 < ρ(0) ≤ ρn (0) ≤ ρ¯(0),

ρn (0) ∈ C(T3 ) ∩ W 1,2 (T3 ),

where ρ(0) and ρ¯(0) are the given constants.

ρn (0, Yi (x)) = ρn (0, x),

(2.2)

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The equation of un : Z d ρn un · ψdx dt 3 Z T = (ρn (un ⊗ un ) · ∇ψ − µ∇un · ∇ψ − (µ + λ) div un · div ψ) dx T3 Z
+ aργn div ψ + δρβn div ψ − ǫ∇un · ∇ρn · ψ dx 3 ZT + (−2ǫρn un · ψ + ρn fn · ψ) dx, for all ψ ∈ Xn ,

503

(2.3)

T3

with the initial condition un (0) ∈ Xn . Here, for any fixed constant n, fn ∈ C ∞ (Q),

fn satisfies (1.6)

and fni → f i strongly in L2 (Q),

kfni kL∞ (Q) ≤ kf i kL∞ (Q) ,

i = 1, 2, 3.

Moreover, Xn is the finite-dimensional space defined by n X ak [ψ j ]eik·x , where aYi [k] [ψ i ] = −ak [ψ i ], Xn = ψ = [ψ 1 , ψ 2 , ψ 3 ] : ψ j = |k|≤n

o aYj [k] [ψ i ] = ak [ψ i ], j 6= i for i = 1, 2, 3 .

Here, and in what follows, the symbols ak , k ∈ Z3 denote the Fourier coefficients. Observe that all ψ ∈ Xn satisfy (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 2.1 Suppose 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 ∈ C 1 (S1 ; C 2 (T3 )) is a classical solution of (2.1) on S1 , and there exists K depending on n such that Z ρn ≥ K > 0,

ρn (t)dx = mǫ ,

with 2ǫmǫ = |T3 |M (mǫ ).

(2.4)

T3

The energy inequality Z  Z    d 2 2 Eδ [ρn , un ] + µ|∇un | + (µ + λ)| div un | dx ≤ C 1 + ρn |fn ||un |dx dt T3 T3 holds on S1 , where Eδ [ρn , un ] =

Z

T3

1

2

ρn |un |2 +

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

 a δ ργn + ρβn dx γ−1 β−1

(2.5)

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Furthermore, there exits a constant E1 independently of n, such that Eδ [ρn , un ](0) ≤ E1 . 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. Proposition 2.2 Given γ > 53 , β > 4, δ, ǫ > 0, then there exists a time-periodic solution (ρ, u) of the problem Z  ∂t ρ + div(ρu) = ǫ∆ρ − 2ǫρ + M ρdx , a.e. on Q, (2.6) T3

∂t (ρui ) + div(ρui u) − µ∆ui − (µ + λ)∂xi (div u) + a∂xi ργ in D′ (Q),

+δ∂xi ρβ + ǫ∇ui ∇ρ + 2ǫρui = ρf i The density ρ ≥ 0 and satisfies (1.4), with Z ρ(t)dx = mǫ , ∀ t ∈ S1 , where

i = 1, 2, 3.

2ǫmǫ = |T3 |M (mǫ ).

(2.7)

(2.8)

T3

The fluid velocity u ∈ L2 (S1 ; W 1,2 (T3 )) satisfies (1.5) a.e. on Q. The energy Eδ [ρ, u] ∈ L∞ (S1 ) such that Z  Z    d 2 2 Eδ [ρ, u] + µ|∇u| + (µ + λ)| div u| dx ≤ C 1 + ρ|f ||u|dx (2.9) dt T3 T3 holds in D′ (S1 ), where Eδ [ρ, u] =

Z

1

ρ|u|2 +

2 and the constant C is independent of ǫ and δ. T3

 a δ ργ + ρβ dx γ−1 β−1

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 2.3 Given γ > 53 , β > 5, δ > 0, then there exists a time-periodic solution (ρ, u) of the problem ∂t ρ + div(ρu) = 0

in D′ (Q),

(2.10)

∂t (ρui ) + div(ρui u) − µ∆ui − (µ + λ)∂xi div u +∂xi (aργ + δρβ ) = ρf i

in D′ (Q), i = 1, 2, 3.

(2.11)

Moreover, ρ, u satisfy (1.4)–(1.5), and ρ ∈ L∞ (S1 ; Lβ (T3 )) ∩ Lβ+1 (Q),

u ∈ L2 (S1 ; W 1,2 (T3 )).

The equation (2.10) holds in the sense of renormalized solutions and the energy Eδ [ρ, u] satisfies Z Z   d 2 2 Eδ [ρ, u] + µ|∇u| + (µ + λ)| div u| dx ≤ C(1 + ρ|f ||u|dx) (2.12) dt T3 T3 in D′ (S1 ), where Eδ [ρ, u] =

Z

1

ρ|u|2 +

T3 2 and the constant C is independent of δ.

 a δ ργ + ρβ dx ∈ L∞ (S1 ) γ−1 β−1

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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 L1 convergence (see [9] Lemma 1.1). Lemma 3.1 If ψ : R → (−∞, +∞] is a proper, lower semi-continuous, and strictly convex function, D ⊂ Rm is a domain with bounded measure, and sup kvk kp < +∞, k

weakly in L1 (D),

vk → v

ψ(vk ) → ψ(v) weakly in L1 (D), Z Z ψ(v)dx = ψ(v)dx, D

D

with p > 1, then it holds that vk → v

strongly in L1 (D).

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 3.2 Let ρ ≥ 0, u satisfy ρ ∈ L∞ (S1 ; Lγ (T3 )),

sup kρk1 ≤ m,

u ∈ L2 (S1 ; W 1,2 (T3 )).

t∈S1

Then, there holds sup E(t) ≤ C(1 + t∈S1

Z

P (ρ(t))dxdt),

(3.1)

Q

where C is a constant depending on µ, λ, kf kL∞ (Q) , P denotes a convex function such that P (ρ) ≥ and E(t) =

Z

1

a 5 ργ for γ > , γ−1 3

 ρ(t)|u(t)|2 + P (ρ(t)) dx ∈ L1 (S1 )

T3 2 satisfying the energy inequality Z Z   d 2 2 E(t) + (µ|∇u| + (µ + λ)| div u| dx ≤ C 1 + ρ|f ||u|dx . (3.2) dt T3 T3 According, the weak periodic solutions constructed in Proposition 2.3 will be denoted by (ρδ , uδ ). We now first derive the estimates of ρδ , uδ independent of δ > 0, where the technique is inspired by [3]. Consider the operators Ai [v] = ∆−1 [∂xi v], i = 1, 2, 3,

where ∆−1 stands for the inverse of the Laplacian on the space of spatially periodic functions with zero mean. We have Z 1 ∂xi Ai [v] = v − 3 vdx, |T | T3

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with the standard elliptic regularity results: kAi [v]kW 1,s (T3 ) ≤ C(s, T3 )kvkLs (T3 ) ,

1 < s < ∞,

kAi [v]kLq (T3 ) ≤ C(s, q, T3 )kvkLs (T3 ) ,

for q finite, provided

kAi [v]kL∞ (T3 ) ≤ C(s, T3 )kvkLs (T3 ) ,

1 1 1 ≥ − , q s 3

(3.3)

if s > 3.

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 3.3 Let (ρδ , uδ ) be the sequence of weak time-periodic solutions of problem (2.10), (2.11) obtained in Proposition 2.3, then Z aργ+ϑ + δρβ+ϑ dxdt, δ δ Q

sup Eδ [ρδ , uδ ], t∈S1

kuδ kL2 (S1 ;W 1,2 (T3 ))

1 are bounded independently of δ, where ϑ = min{ 2γ−3 3γ , 4 }.

Proof

Integrating the energy inequality (2.12) over S1 and by (1.7), we obtain Z Z   √ √ kuδ k2W 1,2 (T3 ) dt ≤ C 1 + k ρδ kL2 (T3 ) k ρδ uδ kL2 (T3 ) dt 1 S1 ZS   √ ≤C 1+ k ρδ kL3 (T3 ) kuδ kL6 (T3 ) dt , S1

which implies

Z

S1

kuδ k2W 1,2 (T3 ) dt ≤ C(1 + sup kρδ k 23 ).

(3.4)

S1

By the fact that ρδ is a renormalized solution of (2.10), we see that for some ϑ > 0, ∂t ρϑδ + div(ρϑδ uδ ) + (ϑ − 1)ρϑδ div uδ = 0 in D′ (Q).

(3.5)

In view of (3.3) and the regularity results achieved in Proposition 2.3, we are allowed to take φi = Ai [ρϑδ ], i = 1, 2, 3, as a test function for (2.11). Thus, we have Z aργ+ϑ + δρβ+ϑ dxdt δ δ Q Z Z Z Z 1 γ β ϑ (aρδ + δρδ )dx ρδ dxdt + (λ + 2µ) ρϑ div udxdt = 3 |T | S1 T3 T3 Q Z (3.6) +(ϑ − 1) ρδ uiδ Ai [ρϑδ (div uδ )] + uiδ Qi,j [ρϑδ , ρδ ujδ ] − ρδ f i Ai [ρϑδ ]dxdt, Q

where the bilinear operator Qi,j [v, w] = vRi,j [w] − wRi,j [v] and Ri,j = ∂xi ∆−1 ∂xj . 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 the

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fact that sup kρk1 is bounded independently of δ, the H¨older inequality, the Sobolev embedding t∈S1

theorems together with the estimates for Ai presented in (3.3). Therefore, one has Z Z ρδ ui Ai [ρϑ (div uδ )]dxdt ≤ kρδ kγ kuδ k6 kAi [ρϑδ div uδ ]k 6γ dt δ δ 5γ−6 Q S1 Z ≤C kρδ kγ kuδ k6 kρϑδ div uδ k 6γ dt 7γ−6 S1 Z ≤C kρδ kγ kuδ k6 kρϑδ k 3γ k∇uδ k2 dt 2γ−3 S1 Z ≤ C sup kρδ kγ kuδ k2W 1,2 dt, t∈S1

(3.7)

S1

where the constant C is independent of δ provided ϑ ≤

2γ − 3 . 3γ

Z Z uiδ Qi,j [ρϑδ , ρδ uj ]dxdt ≤ kρϑδ k 3γ kuδ k26 kρδ kγ dt δ 2γ−3 1 Q S Z kuδ k2W 1,2 dt, ≤ C sup kρδ kγ t∈S1

(3.8)

S1

where the constant C is independent of δ provided ϑ ≤

2γ − 3 . 3γ

Z Z δf i Ai [ρϑδ ]dxdt ≤ kf k∞ kρδ k1 kAi [ρϑδ ]k∞ dt Q S1 Z ≤C kρδ k1 kρϑδ ks dt S1

≤ C,

(3.9)

where s > 3, so the constant C is independent of δ provided ϑ ≤ 1s < 13 . Hence, it follows from (3.6)–(3.9) that Z Z   β+ϑ 2 aργ+ϑ + δρ dxdt ≤ C 1 + sup kρ k ku k dt . 1,2 δ γ δ W δ δ t∈S1

Q

(3.10)

S1

In view of (3.4) and the interpolation inequality, we have Z   sup kρδ kγ kuδ k2W 1,2 dt ≤ C 1 + sup kρδ kγ sup kρδ k 23 t∈S1

t∈S1

S1



≤ C 1 + sup kρδ kγ t∈S1

t∈S1

4γ−3  3γ−3

.

Combining this with (3.10), it yields Z  4γ−3  aργ+ϑ + δρβ+ϑ dxdt ≤ C 1 + sup kρδ kγ3γ−3 . δ δ t∈S1

Q

Moreover, interpolating between the space L1 and Lγ+ϑ , we deduce kρδ kγ ≤ kρδ k1−θ kρδ kθγ+ϑ , 1

θ=

(γ − 1)(γ + ϑ) . γ(γ + ϑ − 1)

Finally, by virtue of Lemma 3.2 and the above inequality, formula (3.12) reads Z 4γ−3   Z  3γ(γ−1) β+ϑ γ aργ+ϑ + δρ dxdt ≤ C 1 + kρ k dt δ γ δ δ Q

S1

(3.11)

(3.12)

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≤ C 1+

Z

(γ−1)(γ+ϑ)

S1



≤ C 1+

Z

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S1

γ+ϑ−1 kρδ kγ+ϑ dt

kρδ kγ+ϑ γ+ϑ dt

4γ−3   3γ(γ−1)

4γ−3  3γ(γ+ϑ−1) 

.

(3.13)

1 As the exponent ϑ = min{ 2γ−3 3γ , 4 }, we easily get

4γ − 3 < 1. 3γ(γ + ϑ − 1) R Consequently, (3.13) implies that Q aργ+ϑ + δρβ+ϑ dxdt is bound independently of δ. And the δ δ boundedness of sup Eδ [ρδ , uδ ], kuδ kL2 (S1 ;W 1,2 (T3 )) follows from Lemma 3.2 and (3.4). Hence, t∈S1

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 δρβδ → 0

in L1 (Q).

Similarly as in [3] Lemma 4.3, the other information obtained from the uniform energy estimates of Lemma 3.3 is ρδ → ρ in C(S1 ; Lγweak (T3 )), uiδ

→u

ρδ uiδ

i

2

1,2

2γ γ+1

3

weakly in L (S ; W

→ ρu

i

1

(3.14)

1

(T )), i = 1, 2, 3,

in C(S ; Lweak (T )),

ρδ uiδ ujδ → ρui uj

3

i = 1, 2, 3,

in D′ (Q) i, j = 1, 2, 3.

(3.15) (3.16) (3.17)

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

weakly in L

γ+ϑ γ

(Q),

but not more. So, the limit δ → 0+ is quite clear, and the only last proof will consist in showing that the weak limit ργ of ργδ is in fact equal to ργ . Letting δ → 0 in (2.10), (2.11), we obtain the weak time-periodic solutions ρ, u satisfying ∂t ρ + div(ρu) = 0 in D′ (Q), ∂t (ρui ) + div(ρui u) − µ∆ui − (λ + µ)∂xi (div u) + a∂xi (ργ ) = ρf i in D′ (Q),

(3.18) (3.19)

where i = 1, 2, 3. In the rest of this section, we will prove ργ = ργ . To 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   z if z ≤ 1; T (z) =  2 if z ≥ 3,

then build a sequence of functions Tk defined as follows z Tk (z) = kT ( ) for k = 1, 2, · · · . k Recall that ρδ , uδ is a renormalized solution to (2.10) implies

∂t (Tk (ρδ )) + div (Tk (ρδ )uδ ) + (Tk′ (ρδ )ρδ − Tk (ρδ )) div uδ = 0 in D′ (Q).

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Passing to the limit for δ → 0+, we deduce that
∂t Tk (ρ) + div (Tk (ρ)u) + Tk′ (ρ)ρ − Tk (ρ) div u = 0 in D′ (Q),

where

Tk (ρδ ) → Tk (ρ) in C(S1 ; Lpweak (T3 )) for all 1 ≤ p < ∞, and


(Tk′ (ρδ )ρδ − Tk (ρδ )) div uδ → Tk′ (ρ)ρ − Tk (ρ) div u weakly in L2 (Q).

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 3.4 Let ρδ , uδ be the solutions obtained in Proposition 2.3, then it holds that Z Z γ lim (aρδ − (λ + 2µ) div uδ )Tk (ρδ )dxdt = (aργ − (λ + 2µ) div u)Tk (ρ)dxdt. δ→0+

Q

Q

Lemma 3.5 There exists a constant C independent of k such that lim sup kTk (ρδ ) − Tk (ρ)kLγ+1 (Q) ≤ C

for any k ≥ 1.

δ→0+

The proof of the following lemma can be proved in a similar way as [4]; the details are omitted. Lemma 3.6 The limit functions ρ, u obtained in this section is a renormalized solution to (3.18), that is, ∂t b(ρ) + div(b(ρ)u) + (b′ (ρ)ρ − b(ρ)) div u = 0 holds in D′ (Q) for any b ∈ C 1 (R+ ) with b′ (z) = 0, for z is large. To complete the proof of Theorem 1.1, we now introduce the functions   for 0 ≤ z < k,  z ln z, Z z Lk (z) = Tk (s)   z ln k + z ds, for z ≥ k. s2 k

It is not difficult to see that Lk can be written as

˜ k (z), Lk (z) = βk z + L ˜ ′ (z) = 0 for z ≥ 3k. Then, ρδ , uδ is a renormalized solution to (2.10) with respect to where L k the function Lk (z). This reads as in D′ (Q),

∂t Lk (ρδ ) + div(Lk (ρδ )u) + Tk (ρδ ) div uδ = 0

(3.20)

by L′k (z)z − Lk (z) = Tk (z). Similarly, by (3.18) and Lemma 3.6, we obtain ∂t Lk (ρ) + div(Lk (ρ)u) + Tk (ρ) div u = 0

in D′ (Q).

(3.21)

Additionally, by virtue of (3.20), we have Lk (ρδ ) → Lk (ρ) in C(S1 ; Lγweak (T3 )),

(3.22)

where 3 Lk (ρ) ∈ BC(S1 ; Lα weak (T )),

1≤α<γ

(3.23)

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and 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, 3 ρδ log ρδ → ρ log ρ in C(S1 ; Lα weak (T ))

(3.24)

for 1 ≤ α < γ by approximating z log z ≈ Lk (z). In particular, Lk (ρδ ), Lk (ρ) ∈ C(S1 ; Lγweak (T3 )), so taking φ ∈ D(T3 ), φ = 1 for x ∈ T3 as the test function for the difference of (3.20) and (3.21), then, for any τ1 < τ2 , integrating with respect to t, we have Z Z (Lk (ρδ ) − Lk (ρ))(τ2 )dx − (Lk (ρδ ) − Lk (ρ))(τ1 )dx 3 T3 ZTτ2 Z = (Tk (ρ) div u − Tk (ρδ ) div uδ )dxdt. T3

τ1

Passing to the limit for δ → 0 and by (3.22), we have Z Z (Lk (ρ) − Lk (ρ))(τ2 )dx − (Lk (ρ) − Lk (ρ))(τ1 )dx T3 T3 Z τ2 Z = lim (Tk (ρ) div u − Tk (ρδ ) div uδ )dxdt. δ→0+

τ1

(3.25)

T3

In view of Lemma 3.4, we can estimate the right-hand side of the above inequality as Z τ2 Z (Tk (ρ) div u − Tk (ρδ ) div uδ )dxdt lim δ→0+ τ1 T3 Z τ2 Z Z τ2 Z Tk (ρδ ) div uδ dxdt Tk (ρ) div udxdt − lim = δ→0+ τ T3 τ T3 1 Z 1τ2 Z Z τ2 Z 1 Tk (ρ) div udxdt + = (aργδ − (λ + 2µ) div uδ )Tk (ρδ )dxdt lim λ + 2µ δ→0+ τ1 T3 τ1 T3 Z τ2 Z 1 − aργ Tk (ρδ )dxdt lim λ + 2µ δ→0+ τ1 T3 δ Z τ2 Z Z τ2 Z 1 Tk (ρ) div udxdt + = (aργ − (λ + 2µ) div u)Tk (ρ)dxdt λ + 2µ τ1 T3 τ1 T3 Z τ2 Z 1 − aργ Tk (ρδ )dxdt, lim λ + 2µ δ→0+ τ1 T3 δ which, together with (3.25), implies Z Z (Lk (ρ) − Lk (ρ))(τ2 )dx − (Lk (ρ) − Lk (ρ))(τ1 )dx T3 T3 Z τ2 Z 1 aργ Tk (ρδ ) − aργ Tk (ρ)dxdt + lim λ + 2µ δ→0+ τ1 T3 δ Z τ2 Z = (Tk (ρ) − Tk (ρ)) div udxdt. τ1

(3.26)

T3

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 3.7 There exists a constant d > 0, such that Z τ2 Z lim aργδ Tk (ρδ ) − aργ Tk (ρ)dxdt + r(k)(τ2 − τ1 ) δ→0+ τ1 T3 Z τ2  Z  ≥d Ψ Lk (ρ) − Lk (ρ)dx dt, τ1

T3

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where r(k) → 0

for k → ∞

and Ψ is the convex function from Lemma 3.8. 1 Lemma 3.8 Fix η ∈ ( γ+1 , 1) and consider the function Ψ (depending on η) determined by the relation 1

Ψ(z η + z η ) = z γ+1

for all z ≥ 0.

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 Z Z
sup ρ log ρ − Lk (ρ) dx ≤ sup ρ log ρdx t∈S1 T3 t∈S1 {ρ≥k} Z p2 p1 ≤ sup (meas{ρ ≥ k}) ργ dx t∈S1

≤C

T3

 m p1

→ 0,

k

as k → ∞,

for certain p1 , p2 > 0 independent of k. In addition, by the definition of Lk (ρδ ), we also have Z Z
sup Lk (ρδ ) − ρδ log ρδ dx ≤ sup t∈S1

t∈S1

T3

≤ sup t∈S1

≤k

Lk (ρδ )dx

{ρ≥k}

Z

ργδ γ−1−ε dx {ρ≥k} ρδ

−γ+1+ε

≤ Ck

(3.27)

sup

t∈S1 −γ+1+ε

Z

ργ dx

{ρ≥k}

,

(3.28)

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 Z
sup Lk (ρ) − ρ log ρ dx ≤ Ck −γ+1+ε → 0 as k → ∞. (3.29) t∈S1

T3

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 Z Z (ρ log ρ − ρ log ρ)(τ2 , x)dx − (ρ log ρ − ρ log ρ)(τ1 , x)dx T3 T3 Z τ2  Z  d + Ψ (ρ log ρ − ρ log ρ)(t, x)dx dt λ + 2µ τ1 T3 Z τ2 Z ≤ lim sup (Tk (ρ) − Tk (ρ)) div udxdt , (3.30) k→∞

τ1

T3

where the term on the right-hand side can be estimated as follows, Z τ2 Z (Tk (ρ) − Tk (ρ)) div udxdt τ1 T3  Z τ2 Z  12  Z τ2 Z  12 ≤ |Tk (ρ) − Tk (ρ)|2 dxdt | div u|2 dxdt τ1

T3

τ1

T3

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≤

Z

τ2

×

|Tk (ρ) − Tk (ρ)|dxdt

T3

τ1

Z

Z τ2

τ1

Z

Z  γ−1 2γ

τ2

τ1

| div u|2 dxdt T3

 12

Z

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|Tk (ρ) − Tk (ρ)|γ+1 dxdt

T3

1  2γ

(3.31)

.

Next, observe that Tk (ρδ ) − ρδ → Tk (ρ) − ρ in C(S1 ; Lγweak (T3 )), and hence it follows that when 1 ≤ p ≤ γ, kTk (ρ) − ρkLp (Q) ≤ lim inf kTk (ρδ ) − ρδ kLp(Q) . δ→0+

On the other hand, we have for 1 ≤ p < γ, kTk (ρδ ) − ρδ kLp (Q) ≤ 2

p

Z

(3.32)

|ρδ |p dxdt

{ρδ ≥k}

≤ 2p

Z

ργδ

{ρδ ≥k}

≤ 2p k −(γ−p)

ργ−p Zδ

dxdt

{ρδ ≥k}

ργδ dxdt

≤ C2p k −(γ−p) .

(3.33)

Noting that Tk is concave, and therefore, Tk (ρ) ≥ Tk (ρ), also, by (3.32), (3.33) and Tk (ρ) ≤ ρ, we have Z τ2 Z Z τ2 Z |Tk (ρ) − Tk (ρ)|dxdt ≤ |ρ − Tk (ρ)|dxdt τ1

T3

τ1

≤ 2Ck

T3 −(γ−1)

→ 0 as k → ∞.

(3.34)

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 ρ ∈ BC(S1 ; Lα (T3 )) for 1 ≤ α < γ, in particular, ρ log ρ ∈ BC(S1 ; Lα (T3 )) for 1 ≤ α < γ.

(3.35)

Therefore, it follows from (3.23), (3.29), (3.35), and the convexity of ρ log ρ, we know that the function Z
D(t) = ρ log ρ − ρ log ρ dx T3

is continuous, bounded, and nonegative on S1 . Furthermore, (3.30) implies that for any τ1 < τ2 , Z τ2 d D(τ2 ) − D(τ1 ) + Ψ(D(t))dt ≤ 0. λ + 2µ τ1

Consequently, D ≡ 0, then, by Lemma 3.1, we have the strong convergence of the sequence ρδ , that is, ρδ → ρ in L1 (Q), which means that ργ = ργ This completes the proof of Theorem 1.1.

a.e. on Q.

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References [1] Diperna R J, Lions P L. Ordinary differential equations, transport theory and Sobolev spaces. Invent Math, 1989, 98: 511–547 [2] Feireisl E, Mucha Piotr B, Novotn´ y A, Pokorn´ y M. Time-periodic solutions to the full Navier-Stokes-Fourier system. Arch Ration Mech Anal, 2012, 204: 745–786 [3] Feireisl E, Ne˘ casov´ a S M, Petzeltov´ a H, Stra˘skraba I. On the motion of a viscous compressible fluid driven by a time periodic external force. Arch Rational Mech Anal, 1999, 149: 69–96 [4] Feireisl E, Novotn´ y A, Petzeltov´ a H. On the existence of globally defined weak solutions to the Navier-Stokes equations. J Math Fluid Mech, 2001, 3: 358–392 [5] Feireisl E, Petzltov´ a H. Asymptotic compactness of globally trajectories generated by the Navier-Stokes equations of a compressible fluid. J Differential Equations, 2001, 173: 390–409 [6] Feireisl E. Dynamics of viscous compressible fluids. Oxford Lecture Series in Mathematics and its Applications. Vol 26. Oxford: Oxford University Press, 2004 [7] Lions P L. Mathematical topics in fluid dynamics. Vol 2. Compressible models. Oxford: Oxford University Press. 1998 [8] Lions J L. Quelques m´ ethodes de r´ esolution des probl` emes aux limites non lin´ eaires. Dunod-GauthierVillars, 1969 [9] Kobayashi T, Suzuki T. Weak solutions to the Navier-Stokes-Poisson equations. Adv Math Sci Appl, 2008, 18(1): 141–168 [10] Kobayashi T. Time periodic solutions of the Navier-Stokes equations under general outflow condition. Tokyo J Math, 2009, 32(2): 409–424 [11] Novotn´ y A, Stra˘skraba I. Introduction to the Mathematical Theory of Compressible Flow. London: Oxford University Press, 2004 [12] Temam R. Navier-Stokes equations. North-Holland: Amsterdom, 1070 [13] Yan W P. Motion of compressible magnetic fluids in T3 . Electron J Differential Equations, 2013, 232: 1–29