In this paper, we considered the isentropic Navier-Stokes equations for compressible fluids with density-dependent viscosities in $\mathbb{R}^3$. These systems come from the Boltzmann equations through the Chapman-Enskog expansion to the second order, cf.\cite{tlt}, and are degenerate when vacuum appears. We firstly establish the existence of the unique local regular solution (see Definition \ref{d1} or \cite{sz3}) when the initial data are arbitrarily large with vacuum at least appearing in the far field. Moreover it is interesting to show that we could't obtain any global regular solution that the $L^\infty$ norm of $u$ decays to zero as time $t$ goes to infinity.
Global unique solvability of the initial-boundary value problem for the equations of one-dimensional polytropic flows of viscous compressible multifluids
Global solvability of the initial boundary value problem for a model system of one-dimensional equations of polytropic flows of viscous compressible fluid mixtures
Stability of Superposition of Viscous Contact Wave and Rarefaction Waves for Compressible Navier-Stokes System with Degenerate Heat-Conductivity and Large-Data
We consider the one-dimensional compressible Navier-Stokes system with constant viscosity and the nonlinear heat conductivity being proportional to a positive power of the temperature which may be degenerate. This problem is imposed on the stress-free boundary condition, which reveals the motion of a viscous heat-conducting perfect polytropic gas with adiabatic ends putting into a vacuum. We prove that the solution of one dimensional compressible Navier-Stokes system with the stress-free boundary condition shares the same large-time behavior as the case of constant heat conductivity.
Expanding solutions to the compressible Navier-Stokes equations with degenerate viscosities in spherical symmetry: global existence and inviscid limit
The Institute of Mathematical Sciences, The Chinese University of Hong Kong（香港中文大学数学科学研究所）；School of Mathematical Sciences, South China Normal University（华南师范大学数学科学学院）
We are concerned about the barotropic compressible Navier-Stokes equations with density-dependent viscosities which may degenerate in vacuum. We show that any classical solution to barotropic compressible Navier-Stokes equations in bounded domains will blow up, when the initial density admits an isolated mass group and the viscousity coefficients satisfy some conditions. A new condition on viscosities is first put forward in this paper.
Two-velocity hydrodynamics in fluid mechanics: Part II Existence of global $κ$-entropy solutions to compressible Navier-Stokes systems with degenerate viscosities
本文讨论了退化粘性可压缩正压流体Navier-Stokes方程的所谓$κ$ -熵解的整体存在性问题。本文考虑三维空间区域中具有周期边界条件的问题。我们的解决方案满足弱制定的质量和动量守恒方程，也是一个推广的BD-熵的身份称为：$κ$ -熵。这个新的熵涉及两个速度$\bf{u}$和$\bf{u} + 2\nablaφ(\varrho)$之间的混合参数$κ\in (0,1)$（后者由前两位作者在[C. R. Acad. Sci. Paris 2004]），其中$\bf{u}$是速度场，$φ$是由$φ'(s)=μ'(s)/s$定义的密度$\varrho$的函数。作为存在性证明的副产品，我们表明，双速流体力学（在S.C. Shugrin 1994）是一种可能的具有退化粘性的正压可压缩流模型。它用于构造基于保守增广近似格式的近似解。
Well-posedness of three-dimensional isentropic compressible Navier-Stokes equations with degenerate viscosities and far field vacuum
本文考虑三维等熵可压缩Navier-斯托克斯方程的Cauchy问题。当粘性系数是密度幂的常数倍数时（$ρ^δ$ with $0<δ<1$），在对该系统的非线性结构进行分析的基础上，通过引入新的变量和初始相容条件，确定了在非齐次索伯列夫空间中存在具有远场真空和有限能量的局部正则解的初始数据类，该方法解决了Bresh-Desjardins-Metivier [3]、Jiu-Wang-Xin [11]等部分提到的退化粘性流的一个公开问题。我们证明了当时间$t$趋于无穷大时，不可能得到任何整体正则解的$u$的$L^\infty$范数衰减到零。
It has been known, since the pioneering works by Serre, Hoff, Vaĭgant-Kazhikhov, Lions and Feireisl, among others, the regularizing properties of the effective viscous flux and its characterization as the function whose gradient is the gradient part in the Hodge decomposition of the Newtonian force of the fluid, when the shear viscosity of the fluid is constant. In this article, we explore further the connection between the Hodge decomposition of the Newtonian force and the regularizing properties of its gradient part, by addressing the problem of the global existence of weak solutions for com…
We consider the flow of two viscous and incompressible fluids within a bounded domain modeled by means of a two-phase Navier-Stokes system. The two fluids are assumed to be immiscible, meaning that they are separated by an interface. With respect to the motion of the interface, we consider pure transport by the fluid flow. Along the boundary of the domain, a complete slip boundary condition for the fluid velocities and a constant ninety degree contact angle condition for the interface are assumed. The main result of the present work establishes in 2D a weak-strong uniqueness result in terms of…
In this paper, the Cauchy problem for the three-dimensional (3-D) full compressible Navier-Stokes equations (CNS) with zero thermal conductivity is considered. First, when shear and bulk viscosity coefficients both depend on the absolute temperature $θ$ in a power law ($θ^ν$ with $ν>0$) of Chapman-Enskog, based on some elaborate analysis of this system's intrinsic singular structures, we identify one class of initial data admitting a local-in-time regular solution with far field vacuum in terms of the mass density $ρ$, velocity $u$ and entropy $S$. Furthermore, it is shown that within its life…
On a Local Existence Theorem for the Evolution Equation of Viscous Gaseous Stars in a Physical Vacuum
School of Mathematical Sciences, Shanghai Jiao Tong University（上海交通大学数学科学学院）；CMA–Shanghai, and MOE–LSC, Shanghai Jiao Tong University（上海数学中心与教育部高等学校动力学国家重点实验室，上海交通大学）
We study the Cauchy problem for the isentropic hypo-viscous compressible Navier-Stokes equations (CNS) under general pressure laws in all dimensions $d\geq 2$. For all hypo-viscosities $(-Δ)^α$ with $α\in (0,1)$, we prove that there exist infinitely many weak solutions with the same initial data. This provides the first non-uniqueness result of weak solutions to viscous compressible fluid. Our proof features new constructions of building blocks for both the density and momentum, which respect the compressible structure. It also applies to the compressible Euler equations and the hypo-viscous i…
On the global-in-time inviscid limit of the 3D isentropic compressible Navier-Stokes equations with degenerate viscosities and vacuum
本文研究了三维等熵可压缩Navier-斯托克斯方程的无粘边界问题。首先，当粘性系数被给定为密度幂的常数倍数时，（$(ρ^ε)^δ$ with $δ>1$），对于相应Cauchy问题的正则解，通过引入一个“拟对称双曲”-“退化椭圆”耦合结构来控制真空附近速度的行为，我们建立了$H^3$和$H^2$中的$(ρ^ε)^{\frac{δ-1}{2}}$的局部声速关于任意大时间的粘性系数的一致能量估计假设初始密度。其次，充分利用该结构的拟对称性质和弱光滑效应，证明了退化粘性流正则解在$H^2$中对任意长时间的强收敛性.本文的结果适用于一类退化的依赖于密度的粘性系数，不依赖于粘性的B-D关系，并且似乎是第一个关于三维空间中可压缩流的具有大速度和真空状态的光滑解的整体时间无粘极限的结果，而不需要任何对称性假设.
Well-posedness of a diffuse interface model for Hele-Shaw flows
本文研究了Hele-Shaw盒中两种粘性流体在表面张力驱动下的扩散界面模型。整个系统由Cahn-Hilliard方程和Darcy定律组成。我们解决的物理相关的情况下，两种流体具有不同的粘度（不匹配的粘度的情况下）和自由能密度是对数亥姆霍兹势。在二维情形下，我们证明了弱解在正则性准则下的唯一性，以及整体强解的存在唯一性。在三维空间中，我们证明了强解的存在性和唯一性，对于大数据，强解在时间上是局部的，对于适当的小数据，强解在时间上是全局的。这些结果推广了Giorgini、Grasselli和Wu（Ann. Inst. H. J. P.）庞加莱肛门。Non Linéaire 35（2018），318-360）。此外，我们利用对数势的多项式逼近证明了二维弱解的唯一性。
Global dynamics of viscous gaseous stars in a physical vacuum
School of Mathematical Sciences, Shanghai Jiao Tong University（上海交通大学数学科学学院）；CMA-Shanghai, and MOE-LSC, Shanghai Jiao Tong University（上海中心数学实验室和教育部线性代数与计算科学重点实验室，上海交通大学）
In this paper, we provide rigorous justification of the hydrostatic approximation and the derivation of primitive equations as the small aspect ratio limit of the incompressible three-dimensional Navier-Stokes equations in the anisotropic horizontal viscosity regime. Setting $\varepsilon >0$ to be the small aspect ratio of the vertical to the horizontal scales of the domain, we investigate the case when the horizontal and vertical viscosities in the incompressible three-dimensional Navier-Stokes equations are of orders $O(1)$ and $O(\varepsilon^α)$, respectively, with $α>2$, for which the limi…
Exponential stability of the linearized viscous Saint-Venant equations using a quadratic Lyapunov function
Ecole Nationale des Ponts et Chaussées, Institut Polytechnique de Paris（国立路桥学校，巴黎理工学院）；Korean Institute for Advanced Study（韩国高等研究院）；University of California, Berkeley（加州大学伯克利分校）
Nonlinear Stability for the Superposition of Viscous Contact Wave and Rarefaction Waves to Non-isentropic Compressible Navier-Stokes System with General Initial Perturbations
We are concerned with the inviscid limit of the Navier-Stokes equations to the Euler equations for barotropic compressible fluids in $\mathbb{R}^3$. When the viscosity coefficients obey a lower power-law of the density (i.e., $ρ^δ$ with $0<δ<1$), we identify a quasi-symmetric hyperbolic--singular elliptic coupled structure of the Navier-Stokes equations to control the behavior of the velocity of the fluids near the vacuum. Then this structure is employed to prove that there exists a unique regular solution to the corresponding Cauchy problem with arbitrarily large initial data and far-field va…
Due to the highly degeneracy and singularities of the entropy equation, the physical entropy for viscous and heat conductive polytropic gases behave singularly in the presence of vacuum and it is thus a challenge to study its dynamics. It is shown in this paper that the uniform boundedness of the entropy and the inhomogeneous Sobolev regularities of the velocity and temperature can be propagated for viscous and heat conductive gases in $\mathbb R^3$, provided that the initial vacuum occurs only at far fields with suitably slow decay of the initial density. Precisely, it is proved that for any …
Global Existence and Asymptotic Behavior of Large Strong Solutions to the 3D Full Compressible Navier-Stokes Equations with Density-dependent Viscosities
