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˖ڍመ᝶஠ڙጲ

http://www.paper.edu.cn

含真空和粘性系数依赖于密度的 3-D 可压

Navier-Stokes 方程组古典解存在性

刘生全，张剑文，赵俊宁

厦门大学数学科学学院，厦门　 361005

摘要：本文研究了粘性系数依赖于密度的三维可压 Navier-Stokes 方程组的 Cauchy 问题。如

果初值满足小能量条件，作者证明了全局古典解的存在性，所得到结果包含初始密度有真空的

情况。并且本文还得到了解的大时间行为。

关键词：可压 Navier-Stokes 方程组；全局古典解；粘性系数依赖密度；真空

中图分类号： 0175;

Global classical solutions for 3D compressible

Navier-Stokes equations with vacuum and a

density-dependent viscosity coeﬃcient

LIU Sheng-Quan, ZHANG Jian-Wen, ZHAO Jun-Ning

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

Abstract: In this paper, we prove the global existence of classical solutions to the three

dimensional (3D) compressible Navier-Stokes equations with a density-dependent viscosity

coeﬃcient (λ = λ(ρ)) provided the initial data is of small energy. This in particular implies

that the solutions may have large oscillations and contain vacuum states. As a result of the

uniform estimates, the large-time behavior of the solution is also studied.

Key words: compressible Navier-Stokes equations; global classical solutions;

density-dependent viscosity; vacuum

基金项目： This work was supported by a grant from the Ph.D. Programs Foundation of Ministry of Education of China

(200803840016)

作者简介： Liu Sheng-Quan（1983-），male，Doctor，major research direction：Paritial diﬀerential equations. Zhang

Jian-Wen（1977-），male，Associate Professor，major research direction：Paritial diﬀerential equations. Correspondence

author：Zhao Jun-Ning（1945-），male，Professor，major research direction：Paritial diﬀerential equations.

- 1 -

˖ڍመ᝶஠ڙጲ

http://www.paper.edu.cn

0 Introduction

In this paper, we consider the following 3D compressible Navier-Stokes equations with a

density-dependent viscosity coeﬃcient:







+ div(ρu) = 0, x ∈ R

, t > 0,

(ρu)

+ div(ρu ⊗u) + ∇P(ρ) = µ∆u + ∇((µ + λ(ρ))divu),

(1)

where the unknowns ρ, u = (u

, u

), and P (ρ) = Aρ

with A > 0, γ > 1 are the ﬂuid

density, velocity, and pressure, respectively. The dynamic viscosity coeﬃcient µ > 0 is a

positive constant, however, the second viscosity coeﬃcient λ = λ(ρ) is a function of ρ.

We look for the global classical solutions, (ρ(x, t), u(x, t)), to the Cauchy problem of (1)

with the initial data

ρ(x, 0) = ρ

(x), u(x, 0) = u

(x), x ∈ R

, (2)

and the far ﬁeld behavior

ρ(x, t) → ˜ρ ≥ 0, u(x, t) → 0 as |x| → ∞, t > 0, (3)

where ˜ρ ≥ 0 is a ﬁxed nonnegative constant.

In the last decades, there have been a huge number of studies on the compressible Navier-

Stokes equations with constant viscosity coeﬃcients. Since the work of Kazhikov and Shelukhin

(cf. [1]), the one-dimensional compressible Navier-Stokes equations have been extensively stud-

ied by many people and signiﬁcant progress has been made on the mathematical topics. For

the multi-dimensional Navier-Stokes equations with constant viscosity coeﬃcients, when the

initial density is strictly away from vacuum, the local-in-time solvability to various initial and

boundary value problems for the compressible Navier-Stokes equations was proved by Nash [2],

Solonnikov [3], and Matsumura et al [4, 5]. When the initial vacuum is allowed, i.e., the initial

density need not be positive and may vanish in open sets, the local existence of strong/classical

solutions was shown in [6, 7]. It is worth mentioning that the result in [8] is so far the unique

theory about the global existence and uniqueness of large solutions of multi-dimensional com-

pressible Navier-Stokes equations. One of the most important breakthrough is the work of Lions

[9], who ﬁrst proved the global existence of weak solutions (the so-called “ﬁnite energy weak

solutions”) to the initial/inital-boundary value problem of (1) with generally large initial data

when the initial energy is ﬁnite and the adiabatic exponent γ is suitably large (i.e. γ > 3/2).

We also refer to [10] for the global existence of weak solutions with large symmetric data.

A lot of attention has also been paid to the compressible Navier-Stokes equations with

density-dependent viscosity coeﬃcients due to its physical importance in ocean physics and

its close connection to the shallow water equations. In the one-dimensional case, there is an

extensive literature on the density-dependent viscosity system, see, for example, [11, 12, 13,

- 2 -

˖ڍመ᝶஠ڙጲ

http://www.paper.edu.cn

14, 15] and among others. For the multi-dimensional case, Bresch et al (cf. [16]) studied the

global weak solutions for the multi-dimensional equations with the Korteweg stress tensor or

the additional quadratic friction term provided the following algebraic relation between µ(ρ)

and λ(ρ) holds,

λ(ρ) = 2(ρµ

′

(ρ) − µ(ρ)). (4)

Mellet and Vasseur [17] proved that the Korteweg or friction terms are unnecessary in the proof

of compactness for the entropy weak solution. Guo-Jiu-Xin [18] considered the global spherically

symmetric solutions to the 3D compressible Navier-Stokes equations with density-dependent

viscosity coeﬃcients satisfying (4) in a particular form.

Recently, many eﬀorts have been made to improve the global well-posedness theory of

the multi-dimensional compressible Navier-Stokes equations. The global existence of classical

solutions to the 3D compressible Navier-Stokes equations with a density-dependent viscosity

coeﬃcient λ(ρ) was studied by Zhang [19] when the initial data and solution are small in

energy-norm. It should be noted that in both [20] and [19], the density is technically required

to be strictly away from vacuum, which, as emphasized in [6, 7, 22, 23, 24, 25], is one of the

major diﬃculties in the mathematical study of compressible Navier-Stokes equations. More

recently, Huang-Li-Xin [21] considered the 3D isentropic compressible Navier-Stokes equations

with constant viscosity coeﬃcients, and established the ﬁrst existence result of classical solutions

which may have large oscillations and can contain vacuum states.

As that in [19], in this paper we focus our interesting on the compressible Navier-Stokes

equations (1) with a density-dependent viscosity coeﬃcient λ(ρ). Indeed, as aforementioned, the

system with density-dependent viscosity coeﬃcient was ﬁrstly studied by Vaigant and Kazhikov

in 1995. However, the proof in [8] is based on the particular form of the constitutive relations

and the fact that the problem is posed in two space dimensions, and thus, the method therein

cannot be adopted to deal with the 3D case. Moreover, the authors technically required in [8]

that the initial density is strictly away from vacuum. The global weak solutions, which are of

small energy but may contain vacuum states, to the 2D equations of (1) with µ = Const. and

λ = λ(ρ) were studied by Zhang and Fang [26].

For 1 < r ≤ ∞ and k ∈ Z

, we shall use the following simpliﬁed notations for the standard

homogeneous and inhomogeneous Sobolev spaces:







= L

), D

k,r



u ∈ L

loc

) : ∥∇

u∥

< ∞



, ∥u∥

k,r

= ∥∇

u∥

k,r

= L

∩ D

k,r

, H

= W

k,2

, D

= D

k,2

, D

= {u ∈ L

: ∥∇u∥

< ∞}.

The initial energy of the compressible ﬂows is deﬁned as





+ G(ρ

)



dx, (5)

- 3 -

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