Global Well-Posedness of 2D Compressible Navier–Stokes Equations with Large Data and Vacuum

c 2014 Springer Basel 1422-6928/14/030483-39 DOI 10.1007/s00021-014-0171-8

Journal of Mathematical Fluid Mechanics

Global Well-Posedness of 2D Compressible Navier–Stokes Equations with Large Data and Vacuum

Quansen Jiu, Yi Wang, and Zhouping Xin

Communicated by I. M. Gamba

Abstract.In this paper, we study the global well-posedness of the 2D compressible Navier–Stokes equations with large initial data and vacuum. It is proved that if the shear viscosityμis a positive constant and the bulk viscosityλ is the power function of the density, that is,λ(ρ) =ρ^β withβ >3, then the 2D compressible Navier–Stokes equations with the periodic boundary conditions on the torusT²admit a unique global classical solution (ρ, u) which may contain vacuums in an open set ofT². Note that the initial data can be arbitrarily large to contain vacuum states.

Keywords.Compressible Navier–Stokes equations, density-dependent viscosity, global well-posedness, vacuum.

1. Introduction

In this paper, we consider the following compressible and isentropic Navier–Stokes equations with density- dependent viscosities

∂_tρ+ div(ρu) = 0,

∂_t(ρu) + div(ρu⊗u) +∇P(ρ) =μΔu+∇((μ+λ(ρ))divu), x∈T², t >0, (1.1) where ρ(t, x)≥0,u(t, x) = (u₁, u₂)(t, x) represent the density and the velocity of the fluid, respectively.

And T² is the 2-dimensional torus [0,1]×[0,1] and t∈[0, T] for any fixed T >0. We denote the right hand side of (1.1)₂ by

L_ρu=μΔu+∇((μ+λ(ρ))divu).

Here, it is assumed that

μ= const. >0, λ(ρ) =ρ^β, β >3, (1.2) such that the operatorL_ρ is strictly elliptic.

Q. Jiu’s research is partially supported by National Natural Sciences Foundation of China (No. 11171229 and No.11231006), Hong Kong RGC Earmarked Research Grant CUHK-4048/13P, a project of Beijing Chang Cheng Xue Zhe and a grant from the Croucher Foundation.

Y. Wang’s research is partially supported by Hong Kong RGC Earmarked Research Grant CUHK4042/08P, CUHK4041/11P and National Natural Sciences Foundation of China (No. 11171326 and No. 11322106).

Z. Xin’s research is partially supported by Zheng Ge Ru Funds, Hong Kong RGC Earmarked Research Grant CUHK4042/08P, CUHK4041/11P and CUHK-4048/13P, a grant from the Croucher Foundation and a Focus Area Grant at The Chinese University of Hong Kong.

Let the pressure function be given by

P(ρ) =Aρ^γ, (1.3)

where γ >1 denotes the adiabatic exponent andA >0 is the constant. Without loss of generality,A is normalized to be 1. The initial values are given by

(ρ, u)(t= 0, x) = (ρ₀, u₀)(x). (1.4)

Here the periodic boundary conditions on the unit torusT²on (ρ, u)(t, x) are imposed to the system (1.1).

This model problem, (1.1)–(1.4), was first proposed by Vaigant–Kazhikhov in [52] where they showed the well-posedness of the classical solution to this problem provided the initial density is uniformly away from vacuum. In this paper, we study the global well-posedness of the classical solution to this problem (1.1)–(1.4) with general nonnegative initial densities.

There are extensive studies on global well-posedness of the compressible Navier–Stokes equations in the case that both the shear and the bulk viscosity are positive constants satisfying the physical restrictions. In particular, the one-dimensional theory is rather satisfactory, see [21,34,35,39] and the references therein.

In multi-dimensional case, the local well-posedness theory of classical solutions to both initial-value and initial-boundary-value problems was established by Nash [45], Itaya [27] and Tani [51] in the absence of vacuum. The short time well-posedness of either strong or classical solutions containing vacuum was studied recently by Cho-Kim [8] and Luo [41] in 3D and 2D case, respectively. In particular, Cho-Kim [8] obtained the short existence and uniqueness of the classical solution to the Cauchy problem for the isentropic CNS with general nonnegative initial density under the assumption that the initial data satisfies a natural compatibility condition [8]. One of the fundamental questions is whether these local (in time) solutions can be extended globally in time. The first pioneering work along this line is the well-known theory of Matsumura–Nishida [42], where they obtained a unique global classical solution to the CNS in H^s(R³) (s ≥3) for initial data close to its far field state which is a non-vacuum equilibrium state, and furthermore, the solution behaves diffusively toward the far field state. The proof in [42] consists of elaborate energy estimates based on the dissipative structure of the CNS and spectrum analysis for the linearized of CNS at the non-vacuum far field state. This theory has been generalized to data with discontinuities by Hoff [19] and data in Besov spaces by Danchin in [9]. It should be noted that this theory [9,19,42] requires that the solution has small oscillations from the uniform non-vacuum far field state so that the density is strictly away from the vacuum uniformly in time. A natural and important long standing open problem is whether a similar theory holds for the initial data containing vacuums.

In this direction, the major breakthrough is due to Lions [38], where he obtained the existence of a renormalized weak solution with finite energy and large initial data which can contain vacuums for the isentropic CNS when the exponent γ is suitably large, see also the refinements and generalizations in [15,30]. However, little is known on the structure, regularity, and uniqueness of such a weak solution except the partial regularity estimates for 2-dimensional periodic problems in Desjardins [10] where a stronger estimate is obtained under the assumption of uniform boundedness of the density. Recently, under some additional assumptions on the viscosity coefficients, and the far fields state is a non-vacuum state, Hoff [19,20] obtained a new type of global weak solution with small total energy for the isentropic CNS, which have extra structure and regularity information (such as Lagrangian structure in the non- vacuum region) compared with the renormalized weak solutions in [15,30,38]. However, the uniqueness and regularity of those weak solutions whose existence has been proved in [15,30,38] remain completely open in general. By the weak-strong uniqueness of P. L. Lions [38], this is equivalent to the problem of global (in time) well-posedness of classical solution in the presence of vacuum. It should be pointed out that this important question is a very difficult and subtle issue since, in general, one would not expect a positive answer to this question due to the finite time blow-up results of Xin in [53], where it is shown that in the case that the initial density has compact support, any smooth solution to the Cauchy problem of the CNS without heat conduction blows up in finite time for any space dimension, see also the recent generalizations to the case for non-compact but rapidly decreasing (at far fields) initial density [47]. The mechanism for such a blow-up has also been investigated recently and various blow-up

criteria have been derived in [13,14,23,24,26,49,50]. More recently, Huang et al. [25] proved the global well-posedness of classical solutions with small energy but large oscillations which can contain vacuums to 3D isentropic compressible Navier–Stokes equations. See also the recent generalizations to 3D full compressible Navier–Stokes equations [22], the isentropic Navier–Stokes equations with potential forces [36], and 1D or spherically symmetric isentropic Navier–Stokes equations with large initial data [11,12].

The case that the viscosity coefficients depend on the density and vanish at the vacuum has re- ceived a lot attention recently, see [2–5,9,18,28–33,37,40,43,44,48,54–56] and the references therein.

Liu et al. first proposed in [40] some models of the compressible Navier–Stokes equations with density- dependent viscosities to investigate the dynamics of the vacuum. On the other hand, when deriving by Chapman–Enskog expansions from the Boltzmann equation, the viscosity of the compressible Navier–

Stokes equations depends on the temperature and thus on the density for isentropic flows. Also, the viscous Saint-Venant system for the shallow water, derived from the incompressible Navier–Stokes equa- tions with a moving free surface, is expressed exactly as in (1.1)N = 2,μ=ρ, λ= 0, and γ = 2 (see [17]). For the special case, (1.2), the global well-posedness result of Vaigant–Kazhikhov [52] is the first important surprising result for general large initial data with the only constraint that it is initially away from vacuum. However, in the presence of vacuum, there appear new mathematical challenges in deal- ing with such systems. In particular, these systems become highly degenerate. The velocity cannot even be defined in the presence of vacuum and hence it is difficult to get uniform estimates for the velocity near vacuum. Substantial achievements have been made for the one-dimensional case, such as both short time and long time existence and uniqueness for the problem of a compact of viscous fluid expands into vacuum with either stress free condition or continuity condition have been established with μ=ρ^α for suitable α, see [33,40,54,55] etc. Li et al. [37] recently proved the global existence of weak solutions to the initial-boundary value problem for such a system on a finite internal with general initial data which may contain vacuum and discovered the phenomena that all the vacuum states must vanish in finite time and any smooth solution blows up near the time of vacuum vanishing which are in sharp contrast to the case of constant viscosity coefficients, which have been extended to the Cauchy problem on R¹ for arbitrary initial data with a uniform non-vacuum state at far fields by Jiu and Xin [33]. In the case that a basic nonlinear wave pattern is the rarefaction wave, whose nonlinear asymptotic stability has been proved in [31,32] for the one-dimensional isentropic CNS system with density-dependent viscosity in the framework of weak solutions even the rarefaction wave is connecting to the vacuum [39]. Note also that in the case that the initial data is strictly away from vacuum, Mellet and Vasseur [44] has obtained the existence and uniqueness of global strong solution to the one-dimensional Cauchy problem. However, the progress is very limited for multi-dimensional problems. Even the short time well-posedness of classical solutions has not been established for such a system in the presence of vacuum. The global existence of general weak solutions to the compressible Navier–Stokes equations with density-dependent viscosities or the viscous Saint-Venant system for the shallow water model in the multi-dimensional case remains open, and one can refer to [4,5,18,43] for recent developments along this line. Note also that Zhang and Fang [56] proved the existence of global weak solution with small energy to 2D Vaigant–Kazhikhov model [52]

in the framework of [20] and presented the vanishing vacuum behavior. However, the uniqueness of this weak solution is open.

In this paper, we investigate the global existence of the classical solution to 2-dimensional Vaigant–

Kazhikhov model [52], that is, CNS system (1.1)–(1.4) with periodic boundary condition and general nonnegative initial density. It should be noted that for the 2-dimensional problem, the basic reformulation of Vaigant–Kazhikhov [52] and the formulation in terms of the material derivative used in [19,25] are equivalent (see [2.1]). The new ingredient of this paper is that we are able to derive the uniform upper bound of the density under the assumptions that the initial density is nonnegative. In our proof, we will approximate the initial data in an appropriate way such that the approximate initial density is away from the vacuum and the approximate initial velocity is constructed by solving an elliptic problem with periodic boundary condition uniquely (see [3.2]) to keep up the compatibility conditions (1.6). Based on [52], there exists an unique and smooth approximate solution to (1.1) with the approximate initial data. To get the uniform upper bound of the density, we first obtain the uniformL^k(k≥1) estimates of the density and

first order derivative estimates of the velocity, following the approaches of Vaigant–Kazhikhov [52]. Then, in the second order derivative estimates of the velocity, the compatibility will be crucially applied. As pointed out in [6–8] and [25], the compatibility conditions are necessary and sufficient conditions when we consider the classical solutions and if the initial data are permitted to include vacuum. We observe that the imposed conditions on the initial data in Vaigant–Kazhikhov [52] in the estimates of the second order derivative of the velocity are consistent with the compatibility conditions while we deal with the vacuum problem, see Step 5 in Sect.3. Finally, in this paper, we derive the higher order estimates to the solution to guarantee the existence of the global classical solution.

Now we give some comments about the conditionβ >3 in (1.2) which is crucially needed in Vaigant–

Kazhikhov [52] and in the present paper. In fact, when obtaining the L^k(k ≥ 1) integrability of the density (Lemma 3.4), we only require that β >1. The condition β > 3 is essentially used in Step 4 in Sect. 3 to get the first-order derivative estimates of the velocity. More precisely, when estimating the first-order derivative estimates of the velocity, we obtain an ODE inequality with a little bit supercritical power which can not be dealt with by usual Gronwall inequality (see [3.60]). However, by directly solving this ODE inequality, we obtain the expected estimates under the restriction β > 3. And it would be interesting to relax this restriction in future works.

Then the main results of the present paper can be stated in the following.

Theorem 1.1. If the initial values(ρ₀, u₀)(x) satisfy that

0≤(ρ₀(x), P(ρ₀)(x))∈W^2,q(T²)×W^2,q(T²), u₀(x)∈H²(T²),

T²

ρ₀(x)dx >0 (1.5) for someq >2and the compatibility condition

Lρ0u₀− ∇P(ρ₀) =√

ρ₀g(x) (1.6)

with some g∈L²(T²), then there exists a unique global classical solution (ρ, u)(t, x) to the compressible Navier–Stokes equations (1.1)–(1.4)with

0≤ρ(t, x)≤C, ∀(t, x)∈[0, T]×T², (ρ, P(ρ))(t, x)∈C([0, T];W^2,q(T²)), u∈C([0, T];H²(T²))∩L²(0, T;H³(T²)), √

tu∈L^∞(0, T;H³(T²)),

tu∈L^∞(0, T;W^3,q(T²)), u_t∈L²(0, T;H¹(T²)), (1.7)

√tu_t ∈L²(0, T;H²(T²))∩L^∞(0, T;H¹(T²)), tu_t∈L^∞(0, T;H²(T²)),

√t√

ρu_tt ∈L²(0, T;L²(T²)), t√

ρu_tt∈L^∞(0, T;L²(T²)), t∇u_tt∈L²(0, T;L²(T²)).

Remark 1.1. From the regularity of the solution (ρ, u)(t, x), it can be shown that (ρ, u) is a classical solution of the system (1.1) in [0, T]×T² (see the details in Section 5).

Remark 1.2. If the initial data contains vacuum, then it is natural to impose the compatibility (1.6) as the case of constant viscosity coefficients in [8].

Remark 1.3. In Theorem 1.1, it is not clear whether or notu_tt∈L²(0, T;L²(T²)) even though one has the regularityt∇u_tt∈L²(0, T;L²(T²)).

Remark 1.4. It is as yet open to get the similar theory to the Dirichlet problem to the 2D compressible Navier–Stokes equations (1.1).

Remark 1.5. Note that Gamba–Morawetz [16] constructed a steady solution for potential (irrotational) flows under a broad set of boundary conditions by taking a limit of the corresponding viscous steady solution. It would be very interesting to study the existence and regularity of steady state solutions to the current system with periodic boundary conditions and steady periodic forcing. Other boundary conditions involving non-slip boundary conditions should bring more difficulties. Some new ideas are needed to handle these cases. This will be left for future studies.

If the initial values are much more regular, based on Theorem1.1, we can prove Theorem 1.2. If the initial values(ρ₀, u₀)(x) satisfy that

0≤(ρ₀(x), P(ρ₀)(x))∈H³(T²)×H³(T²), u₀(x)∈H³(T²),

T²

ρ₀(x)dx >0 (1.8) and the compatibility condition (1.6), then there exists a unique global classical solution(ρ, u)(t, x)to the compressible Navier–Stokes equations (1.1)–(1.4)satisfying all the properties listed in(1.7)in Theorem1.1 with any2< q <∞. Furthermore, it holds that

u∈L²(0, T;H⁴(T²)), (ρ, P(ρ))∈C([0, T];H³(T²)),

ρu∈C([0, T];H³(T²)), √ρ∇³u∈C([0, T];L²(T²)). (1.9) Remark 1.6. In fact, the conditions on the initial velocity u₀ can be weakened to u₀ ∈ H²(T²) and

√ρ₀∇³u₀∈L²(T²) to get (1.9).

Remark 1.7. In Theorem 1.2, it is not clear whether or not u∈C([0, T];H³(T²)) even though one has ρu∈C([0, T];H³(T²)).

Remark 1.8. It is noted that in Theorem 1.2, the compatibility condition (1.6) is exactly same as in Theorem 1.1.

Notations.Throughout this paper, positive generic constants are denoted bycandC, which are indepen- dent of δ, m andt ∈[0, T], without confusion, andC(·) stands for some generic constant(s) depending only on the quantity listed in the parenthesis. For function spaces, L^p(T²),1≤p≤ ∞, denote the usual Lebesgue spaces onT²and · _pdenotes itsL^p norm.W^k,p(T²) denotes thek^thorder Sobolev space and H^k(T²) :=W^k,2(T²).

2. Preliminaries

As in [52], we introduce the following variables. First denote the effective viscous flux by F = (2μ+λ(ρ))divu−P(ρ),

and the vorticity by

ω=∂_x1u₂−∂_x2u₁. Also, we define that

H = 1

ρ(μω_x1+F_x2), L= 1

ρ(−μω_x2+F_x1).

Then the momentum equation (1.1)₂ can be rewritten as

u_1t+u· ∇u1= ¹_ρ(−μωx2+F_x1) =L,

u_2t+u· ∇u₂= ¹_ρ(μω_x1+F_x2) =H. (2.1) Then the effective viscous flux F and the vorticityω solve the following system:

ω_t+u· ∇ω+ωdivu=H_x1−L_x2,

F+P(ρ) 2μ+λ(ρ)

t+u· ∇

F+P(ρ) 2μ+λ(ρ)

+ (u_1x1)²+ 2u_1x2u_2x1+ (u_2x2)²=H_x2+L_x1. (2.2) Due to the continuity equation (1.1)₁, it holds that

⎧⎪

⎨

⎪⎩

ω_t+u· ∇ω+ωdivu=H_x1−L_x2, F_t+u· ∇F−ρ(2μ+λ(ρ))[F

1 2μ+λ(ρ)

+

P(ρ) 2μ+λ(ρ)

]divu

+(2μ+λ(ρ))[(u_1x1)²+ 2u_1x2u_2x1+ (u_2x2)²] = (2μ+λ(ρ))(H_x2+L_x1).

(2.3)

Furthermore, the system for (H, L) can be derived as

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎩

ρH_t+ρu· ∇H−ρHdivu+u_x2· ∇F+μu_x1· ∇ω+μ(ωdivu)_x1

−

ρ(2μ+λ(ρ))[F 1

2μ+λ(ρ)

+

P(ρ) 2μ+λ(ρ)

]divu

x2

+

(2μ+λ(ρ))[(u_1x1)²+ 2u_1x2u_2x1+ (u_2x2)²]

x2

= [(2μ+λ(ρ))(H_x2+L_x1)]_x2+μ(H_x1−L_x2)_x1,

ρL_t+ρu· ∇L−ρLdivu+u_x1· ∇F−μu_x2· ∇ω−μ(ωdivu)_x2

−

ρ(2μ+λ(ρ))[F 1

2μ+λ(ρ)

+

P(ρ) 2μ+λ(ρ)

]divu

x1

+

(2μ+λ(ρ))[(u_1x1)²+ 2u_1x2u_2x1+ (u_2x2)²]

x1

= [(2μ+λ(ρ))(H_x2+L_x1)]_x1−μ(H_x1−L_x2)_x2.

(2.4)

In the following, we will utilize the above systems in different steps. Note that these systems are equivalent to each other for the smooth solution to the original system (1.1).

Several elementary Lemmas are needed later. The first one is the Gagliardo-Nirenberg inequality which can be found in [46].

Lemma 2.1. ∀h∈W₀^1,m(T²)orh∈W^1,m(T²)with

T²h dx= 0, it holds that

h_q≤C∇h^α_mh^1−α_r , (2.5)

where α = (¹_r − ¹_q)(¹_r −_m¹ +¹₂)⁻¹, and if m < 2, then q is between r and _2−m^2m , that is, q ∈ [r,_2−m^2m ] if r < _2−m^2m , q ∈ [_2−m^2m , r] if r ≥ _2−m^2m , if m = 2, then q ∈ [r,+∞), if m > 2, then q ∈ [r,+∞].

Consequently,∀h∈W^1,m(T²), one has

h_q≤C(h₁+∇h^α_mh^1−α_r ), (2.6)

where C is a constant which may depend onq.

The following Lemma is the Poicare inequality.

Lemma 2.2. ∀h∈W₀^1,m(T²)orh∈W^1,m(T²)with

T²h dx= 0, if 1≤m <2,then

h_2−m^2m ≤C(2−m)⁻¹²∇hm, (2.7)

where the positive constant C is independent ofm.

The following Lemma follows from Lemma2.2, of which proof can be found in [52].

Lemma 2.3. ∀h∈W^1,m+η2m (T²)withm≥2 and0< η≤1, we have h_2m≤C(h₁+m¹²h^s_2(1−ε)∇h^1−s2m

m+η), (2.8)

where ε∈[0,¹₂], s=^{(1−ε)(1−η)}_{m−η(1−ε)} and the positive constantC is independent of m.

3. Approximate Solutions

In this section, we construct a sequence of approximate solutions by making use of the theory of Vaigant–

Kazhikhov [52] and derive some uniform a-priori estimates which are necessary to prove Theorem1.1. To this end, we need a careful approximation of the initial data.

Step 1. Approximation of initial data: To apply the theory of Vaigant–Kazhikhov [52], we approximate of the initial data in (1.8) as follows. First. the initial density and pressure can be approximated as

ρ^δ₀=ρ₀+δ, P₀^δ =P(ρ₀) +δ, (3.1)

for any small positive constant δ >0. To approximate the initial velocity, we defineu^δ₀to be the unique solution to the following elliptic problem

L_ρδ

0u^δ₀=∇P₀^δ+√

ρ₀g (3.2)

with the periodic boundary conditions onT²and

T²u^δ₀dx=

T²u₀dx:= ¯u₀.It should be noted thatu^δ₀ is uniquely determined due to the compatibility condition (1.6).

It follows from (3.2) that

Lρ0u^δ₀=−∇

(λ(ρ^δ₀)−λ(ρ₀))divu^δ₀

+∇P₀^δ+√

ρ₀g. (3.3)

By the elliptic regularity, it holds that u^δ₀−u¯_0H²_(T²₎

≤C

λ(ρ^δ₀)−λ(ρ₀)_∞∇(divu^δ₀)₂+∇(λ(ρ^δ₀)−λ(ρ₀))_∞divu^δ₀₂+∇P₀^δ₂+√ ρ₀g₂

≤C

δu^δ_0H²_(T²₎+P_0H¹_(T²₎+√

ρ_0L^∞_(T²₎g₂

≤C

δu^δ_0H²_(T²₎+ 1

. (3.4)

where the generic positive constant Cis independent ofδ >0.

Therefore, ifδ1, then (3.4) yields that

u^δ_0H²_(T²₎≤C (3.5)

where the positive constantC is independent of 0< δ1.

Due to the compatibility condition (1.6) and (3.2), it holds that Lρ0(u^δ₀−u₀) =−∇

(λ(ρ^δ₀)−λ(ρ₀))divu^δ₀

:= Θ^δ. (3.6)

Therefore, by the elliptic regularity, (3.1) and (3.5), one can get that u^δ₀−u_0H²_(T²₎

≤CΘ^δ₂≤C

λ(ρ^δ₀)−λ(ρ₀)_L^∞_(T²₎∇²u^δ₀₂+∇(λ(ρ^δ₀)−λ(ρ₀))_L^∞_(T²₎divu^δ₀₂

≤C

λ(ρ^δ₀)−λ(ρ₀)_L^∞_(T²₎+∇(λ(ρ^δ₀)−λ(ρ₀))_L^∞_(T²₎

≤Cδ →0, asδ→0. (3.7)

For the initial data (ρ^δ₀, P₀^δ, u^δ₀) constructed above for each fixed δ > 0, it is proved in [52] that the compressible Navier–Stokes equations (1.1) withβ >3 has a unique global strong solution (ρ^δ, u^δ) such that c_δ ≤ρ^δ ≤C_δ for some positive constantsc_δ, C_δ depending onδ. In the following, we will derive the uniform bound to (ρ^δ, u^δ) with respect toδ and then pass the limit δ→0 to get the classical solution which may contain vacuum states in an open set of T². It should be noted that in comparison with estimates presented in [52], we will obtain uniform estimates with respect to the lower bound of the density such that vacuum is permitted in these estimates. To this end, the compatibility condition (1.6) will be crucial.

For simplicity of notations, we will omit the superscript^δ of (ρ^δ, u^δ) in the following in the case of no confusions.

Step 2. Elementary energy estimates:

Lemma 3.1. There exists a positive constantC depending on(ρ₀, u₀), such that sup

t∈[0,T]

√ρu²₂+ρ^γ_γ +

T 0

∇u²₂+ω²₂+(2μ+λ(ρ))¹²divu²₂

dt≤C. (3.8) Proof. Multiplying the equation (2.1)_i by ρu_i,(i = 1,2), summing the resulting equations and then integrating overT² and using the continuity equation (1.1)₁, it holds that

d dt

ρ|u|²dx+

(μω²+ (2μ+λ(ρ))(divu)²)dx+

u· ∇P dx= 0.

Multiplying the continuity equation (1.1)₁by _γ−1^γ ρ^γ−1 and then integrating overT²yield that d

dt ρ^γ

γ−1dx+

Pdivu dx= 0.

Therefore, combining the above two estimates and then integrating over [0, t] with respect tot, we obtain 1

2ρ|u|²+ 1 γ−1ρ^γ

dx+

T 0

(μω²+ (2μ+λ(ρ))(divu)²)dx dt

= 1

2ρ^δ₀|u^δ₀|²+ 1 γ−1(ρ^δ₀)^γ

dx≤C

ρ^δ_0W^2,q_(T²₎u^δ₀²_H2(T²)+ρ^δ₀^γ_W2,q(T²)

≤C. (3.9)

Denote

φ(t) =

(μω²+ (2μ+λ(ρ))(divu)²)dx, t∈[0, T]. (3.10) Then

∇u²₂(t)≤C

ω²₂(t) +divu²₂(t)

≤Cφ(t)∈L¹(0, T).

Thus the proof of Lemma3.1is completed.

Step 3. Density estimates:Applying the operatordivto the momentum equation (1.1)₂, we have

[div(ρu)]_t+ div[div(ρu⊗u)] = ΔF. (3.11)

Consider the following two elliptic problems:

Δξ= div(ρu),

ξ dx= 0, (3.12)

Δη = div[div(ρu⊗u)],

η dx= 0, (3.13)

both with the periodic boundary condition on the torusT². By the elliptic estimates and H¨older inequality, it holds that Lemma 3.2. (1) ∇ξ_2m≤Cmρ^2mk

k−1u_2mk, for anyk >1, m≥1;

(2) ∇ξ_2−r≤C√ρu₂ρ¹²2−r

r , for any0< r <1;

(3) η_2m≤Cmρ^2mk

k−1u²_4mk,for any k >1, m≥1;

whereC are positive constants independent ofm, kandr.

Proof. (1) By the elliptic estimates to Eq. (3.12) and then using the H¨older inequality, we have for any k >1, m≥1,

∇ξ2m≤Cmρu2m≤Cmρ^2mk_k−1u2mk.

Similarly, the statements (2) and (3) can be proved.

Based on Lemmas2.1,2.3and 3.2, it holds that

Lemma 3.3. (1) ξ2m≤Cm¹²∇ξ_m+1^2m ≤Cm¹²ρm¹², for anym≥2;

(2) u2m≤C

m¹²∇u2+ 1

,for any m≥2;

(3) ∇ξ_2m≤C

m³²k¹²ρ^2mk

k−1φ(t)¹²+mρ^2mk

k−1

, for anyk >1, m≥1;

(4) η2m≤C

m²kρ^2mk_k−1φ(t) +mρ^2mk_k−1

,for any k >1, m≥1;

whereC are positive constants independent ofm, k.

Proof. (1) By Lemmas2.2and3.2(2), it holds that ξ_2m≤Cm¹²∇ξ ^2m

m+1 ≤Cm¹²√ρu₂ρm¹²

≤Cm¹²ρm¹²,

where in the last inequality one has used the elementary energy estimates (3.9).

(2). From the conservative form of the compressible Navier–Stokes equations (1.1) and the periodic boundary conditions, we have

d dt

ρ(t, x)dx= d dt

ρu(t, x)dx= 0, that is,

ρ(t, x)dx=

ρ₀(x)dx,

ρu(t, x)dx=

ρ₀u₀(x)dx, ∀t∈[0, T].

By Lemma2.2, it follows that

u_2m≤ u−u¯ _2m+¯u_2m≤Cm¹²∇u ^2m

m+1+|¯u|, (3.14)

where m >2 and ¯u= ¯u(t) =

u(t, x)dx.

On the other hand, we have

ρ(u−u)¯ dx

≤ ργu−u¯ _γ−1^γ ≤C∇u2, (3.15)

where in the last inequality we have used the elementary energy estimates (3.9) and the Poincare inequal- ity.

Note that

ρ(u−u)¯ dx =

ρ₀u₀dx−u¯

ρ₀(x)dx ≥ |¯u|

ρ₀dx−

ρ₀u₀dx

. (3.16)

Combining (3.15) with (3.16) implies that

|¯u| ≤ |

ρ₀u₀dx|

ρ₀dx +C∇u ₂

ρ₀dx . (3.17)

Substituting (3.17) into (3.14) completes the proof of Lemma3.3(2).

The assertions (3) and (4) in Lemma 3.3are direct consequences of Lemma 3.3(2) and Lemma 3.2 (1), (3), respectively. Thus the proof of Lemma3.3is completed.

Substituting (3.12) and (3.13) into (3.11) yields that Δ

ξ_t+η−F+

F(t, x)dx

= 0. (3.18)

Thus, it holds that

ξ_t+η−F+

F(t, x)dx= 0. (3.19)

It follows from the definition of the effective viscous fluxF that ξ_t−(2μ+λ(ρ))divu+P(ρ) +η+

F(t, x)dx= 0. (3.20)

Then the continuity equation (1.1)₁ yields that ξ_t+2μ+λ(ρ)

ρ (ρ_t+u· ∇ρ) +P(ρ) +η+

F(t, x)dx= 0. (3.21)

Define

θ(ρ) = ρ 1

2μ+λ(s)

s ds= 2μlnρ+ 1

β(ρ^β−1). (3.22)

Then we obtain the following transport equation

(ξ+θ(ρ))_t+u· ∇(ξ+θ(ρ)) +P(ρ) +η−u· ∇ξ+

F(t, x)dx= 0. (3.23) Lemma 3.4. For any k≥1 andβ >1, it holds that

sup

t∈[0,T]ρ(t,·)_k ≤Ck^β−12 . (3.24)

Proof. Multiplying Eq. (3.23) byρ[(ξ+θ(ρ))₊]^2m−1 withm≥4 being integer, here and in what follows, the notation (· · ·)₊ denotes the positive part of (· · ·), one can get that

1 2m

d dt

ρ[(ξ+θ(ρ))₊]^2mdx+

ρP(ρ)[(ξ+θ(ρ))₊]^2m−1dx=−

ρη[(ξ+θ(ρ))₊]^2m−1dx +

ρu· ∇ξ[(ξ+θ(ρ))₊]^2m−1dx−

F(t, x)dx

ρ[(ξ+θ(ρ))₊]^2m−1dx. (3.25) Denote

f(t) =

ρ[(ξ+θ(ρ))₊]^2mdx _2m¹

, t∈[0, T]. (3.26)

Now we estimate the terms on the right hand side of (3.25). First, −

ρη[(ξ+θ(ρ))₊]^2m−1dx ≤

ρ^2m1 |η|

ρ(ξ+θ(ρ))^2m₊ ^2m−1_2m dx

≤ ρ_2mβ+1^2m1 η_2m+β¹ρ(ξ+θ(ρ))^2m_{+ 1}^2m−12m

≤Cρ_2mβ+1^2m1

(m+ 1

2β)²kρ2(m+ 12β)k k−1

φ(t) + (m+ 1

2β)ρ2(m+ 12β)k k−1

f(t)^2m−1

≤Cρ¹⁺_2mβ+1^2m1 f(t)^2m−1

m²φ(t) +m ,

(3.27)

where φ(t) is defined as in (3.10) and in the last inequality we have takenk= _β−1^β . Next, for _2mβ+1¹ +_p¹+¹_q = 1 with p, q≥1, one has

ρu· ∇ξ[(ξ+θ(ρ))₊]^2m−1dx ≤

ρ^2m1 |u||∇ξ|

ρ(ξ+θ(ρ))^2m₊ ^2m−1_2m dx

≤ ρ_2mβ+1^2m1 u_2mp∇ξ_2mqρ(ξ+θ(ρ))^2m_{+ 1}^2m−12m

≤Cρ_2mβ+1^2m1

(mp)¹²∇u₂+ 1 (mq)³²k¹²ρ2mqk

k−1φ(t)¹² +mρ2mqk k−1

f(t)^2m−1

≤Cρ¹⁺_2mβ+1^2m1 f(t)^2m−1

m¹²φ(t)¹² + 1 m³²φ(t)¹² +m

≤Cρ¹⁺_2mβ+1^2m1 f(t)^2m−1

m²φ(t) +m ,

(3.28)

where in the third inequality one has chosenp=q= ^2mβ+1_mβ and k= _β−1^β .

Then it follows that −

F(t, x)dx

ρ[(ξ+θ(ρ))₊]^2m−1dx

≤

|(2μ+λ(ρ))divu−P(ρ)|dx

ρ^2m1

ρ(ξ+θ(ρ))^2m₊ ^2m−1_2m dx

≤

(2μ+λ(ρ))(divu)²dx ¹₂

(2μ+λ(ρ))dx ¹₂

+

P(ρ)dx

ρ₁^2m1 ρ(ξ+θ(ρ))^2m_{+ 1}^2m−12m

≤C

φ(t)¹² +φ(t)¹²(

ρ^βdx)¹² + 1

f(t)^2m−1

≤C

φ(t)¹² +φ(t)¹²ρ_2mβ+1^β2 + 1

f(t)^2m−1. (3.29)

Substituting (3.27), (3.28) and (3.29) into (3.25) yields that 1

2m d

dt(f^2m(t)) +

ρP(ρ)[(ξ+θ(ρ))₊]^2m−1dx

≤Cρ¹⁺_2mβ+1^2m1 f(t)^2m−1

m²φ(t) +m +C

φ(t)¹² +φ(t)¹²ρ_2mβ+1^β2 + 1

f(t)^2m−1. (3.30) Then it holds that

d

dtf(t)≤C

1 +φ(t)¹²+φ(t)¹²ρ_2mβ+1^β2 +

m²φ(t) +m

ρ¹⁺_2mβ+1^2m1

. (3.31)

Integrating the above inequality over [0, t] gives that f(t)≤f(0) +C

⎡

⎣1 + t 0

φ(τ)¹²ρ_2mβ+1^β2 (τ)dτ + t 0

m²φ(τ) +m

ρ¹⁺_2mβ+1^2m1 (τ)dτ

⎤

⎦. (3.32)

Now we calculate the quantity

f(0) =

ρ^δ₀[(ξ₀^δ+θ(ρ^δ₀))₊]^2mdx _2m¹

.

By Lemma3.2(1) witht= 0, we can easily get

ξ₀^δ_L^∞ ≤C.

Furthermore, by the definition ofθ(ρ^δ₀) = 2μlnρ^δ₀+_β¹((ρ^δ₀)^β−1), we have ξ₀^δ+θ(ρ^δ₀)→ −∞, as ρ^δ₀→0 +. Thus there exists a positive constant σ, such that if 0≤ρ^δ₀≤σ, then

(ξ₀^δ+θ(ρ^δ₀))₊≡0.

Now one has

f(0)=

⎡

⎢⎣

⎛

⎜⎝

[0≤ρ0≤σ]

+

[σ≤ρ^δ₀≤M]

⎞

⎟⎠ρ^δ₀(ξ₀^δ+θ(ρ^δ₀))^2m₊ dx

⎤

⎥⎦

2m1

=

⎡

⎢⎣

[σ≤ρ^δ0≤M]

ρ^δ₀(ξ^δ₀+θ(ρ^δ₀))^2m₊ dx

⎤

⎥⎦

2m1

≤C(σ, M),

(3.33)

where the positive constantC(σ, M) is independent ofδandm.

It follows from (3.32) and (3.33) that f(t)≤C

⎡

⎣1 + t 0

φ(τ)¹²ρ_2mβ+1^β2 (τ)dτ+ t 0

m²φ(τ) +m

ρ¹⁺_2mβ+1^2m1 (τ)dτ

⎤

⎦. (3.34)

Set Ω₁(t) ={x∈T²|ρ(t, x)>2}and Ω₂(t) ={x∈Ω₁(t)|ξ(t, x) +θ(ρ)(t, x)>0}. Then one has

ρ^β_2mβ+1(t) = ρ^2mβ+1dx _2mβ+1^β

=

⎛

⎜⎝

Ω1(t)

ρ^2mβ+1dx+

T²\Ω1(t)

ρ^2mβ+1dx

⎞

⎟⎠

2mβ+1β

≤

⎛

⎜⎝

Ω1(t)

ρ^2mβ+1dx

⎞

⎟⎠

2mβ+1β

+C≤C

⎛

⎜⎝

Ω1(t)

ρ|θ(ρ)|^2mdx

⎞

⎟⎠

2mβ+1β

+C

=C

⎛

⎜⎝

Ω2(t)

ρ|θ(ρ) +ξ−ξ|^2mdx+

Ω1(t)\Ω2(t)

ρ|θ(ρ)|^2mdx

⎞

⎟⎠

2mβ+1β

+C

≤C

⎛

⎜⎝

Ω2(t)

ρ(θ(ρ) +ξ)^2mdx+

Ω2(t)

ρ|ξ|^2mdx+

Ω1(t)\Ω2(t)

ρ|ξ|^2mdx

⎞

⎟⎠

2mβ+1β

+C

≤C

⎛

⎝f(t)^2m+

T²

ρ|ξ|^2mdx

⎞

⎠

2mβ+1β

+C≤C

⎡

⎢⎣f(t) +

⎛

⎝

T²

ρ|ξ|^2mdx

⎞

⎠

2mβ+1β

+ 1

⎤

⎥⎦.

(3.35) Note that

⎛

⎝

T²

ρ|ξ|^2mdx

⎞

⎠

2mβ+1β

≤ ρ_2mβ+1^2mβ+1β |ξ|^2m2mβ+12mβ+1^β 2mβ

=ρ_2mβ+1^2mβ+1β ξ_2m+^2mβ+12mβ1 β

≤ ρ_2mβ+1^2mβ+1β

C

m+ 1 2β

¹₂

ρ_m+¹² 1 2β

_2mβ+1^2mβ

≤Cm¹²ρ_2mβ+1^{β(m+1)2mβ+1}, (3.36) Then one can get

ρ^β_2mβ+1(t)≤C

1 +f(t) +m¹²ρ_2mβ+1^{β(m+1)2mβ+1}(t)

≤ 1

2ρ^β_2mβ+1(t) +C

1 +f(t) +m

mβ+ 12 m(2β−1)

.

(3.37) Thus it holds that

ρ^β_2mβ+1(t)≤C

f(t) +m^2β−1β

≤C

⎡

⎣m^2β−1β + t 0

φ(τ)¹²ρ_2mβ+1^β2 (τ)dτ + t 0

m²φ(τ) +m

ρ¹⁺_2mβ+1^2m1 (τ)dτ

⎤

⎦

≤C

⎡

⎣m^2β−1β + t 0

ρ^β_2mβ+1(τ)dτ + t 0

m²φ(τ) +m

ρ¹⁺_2mβ+1^2m1 (τ)dτ

⎤

⎦. (3.38)