Global Weak Solutions to the Compressible Navier-Stokes Equations in the Exterior Domain with Spherically Symmetric Data

In this paper, we prove the global existence of weak solutions to the full compressible Navier-Stokes equations in the domain exterior to a ball in R ⁿ (n=2,3) and with spherically symmetric data.     

一维可压Navier—Stokes方程初边值问题局部解的存在性

我们证明了半空间中一维可压Navier—Stokes方程初边值问题局部解的存在性,证明主要是利用了能量方法．     

The Initial Boundary Value Problem for Symmetric Long Wave Equations with Nonhomogeneous Boundary Value

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具超音速边界可压Navier-Stokes方程解的指数衰减

考虑了二维空间上具超音速物理边界的可压Navier-Stokes方程的初边值问题.给定常数平衡态(ρ~*,0),得到了所考虑问题解的整体存在性.在平衡态附近的小扰动下,利用加权能量估计方法得到解的指数衰减性.     

A Vacuum Problem for the One-Dimensional Compressible Navier-Stokes Equations with Density-Dependent Viscosity

This paper is concerned with the free boundary problem for the one-dimensional compressible Navier-Stokes equations with density-dependent viscosity. A local (in time) existence result is established when the initial density is of compact support and connects to the vacuum continuously.     

三维区域中的Navier-Stokes方程的初边值问题

本文在初边值适当小的假设下，建立了任意三维区域中Navier－Stokes方程初边值问题整体强解的存在性定理．     

可压缩Navier-Stokes方程组的指数稳定性

We establish the exponential stability of global solutions and Co-semigroup for the compressible Navier-Stokes equations of a viscous polytropic ideal gas in both bounded domain in R1 and bounded annular domains in Rn(n=2,3).     

一个与硅氧化有关的粘性可压缩流体的自由边界问题

本文研究一个描述硅的氧化过程的自由边界问题．它的数学模型是一个可压缩的Ｎａｖｉｏｒ－Ｓｔｏｋｅｓ方程与一个抛物方程以及一个双曲方程的耦合，其中在自由边界上存在表面张力并且密度方程是非齐次的．本文将证明只要已知数据满足相容性条件，则上述问题有唯一局部强解．     

可压缩Navier-Stokes方程组球对称弱解的大时间行为

研究了可压缩Navier-Stokes方程组 球对称弱解的大时间行为. 假设压强 $p(\varrho)=\varrho^\gamma$, 绝热指数$\gamma>1$, 外力是球对称的. 证明了假如外力满足一定的正则性及某种结构性条件, 则当时间 趋于无穷大时, 密度将趋于其对应的静止问题的唯一解.     

Weak Dissipative Structure for Compressible Navier-Stokes Equations

This paper concerns the Cauchy problem for compressible Navier-Stokes equations.The weak dissipative structure is explored and a new proof for the classical solutions are shown to exist globally in time if the initial data is sufficiently small.     

Traveling Wave Solutions to the Free Boundary Incompressible Navier-Stokes Equations

In this paper we study a finite-depth layer of viscous incompressible fluid in dimension $n\ge 2$, modeled by the Navier-Stokes equations. The fluid is assumed to be bounded below by a flat rigid surface and above by a free, moving interface. A uniform gravitational field acts perpendicularly to the flat surface, and we consider the cases with and without surface tension acting on the free interface. In addition to these gravity-capillary effects, we allow for a second force field in the bulk and an external stress tensor on the free interface, both of which are posited to be in traveling wave form, i.e., time-independent when viewed in a coordinate system moving at a constant velocity parallel to the rigid lower boundary. We prove that, with surface tension in dimension $n\ge 2$ and without surface tension in dimension $n=2$, for every nontrivial traveling velocity there exists a nonempty open set of force and stress data that give rise to traveling wave solutions. While the existence of inviscid traveling waves is well-known, to the best of our knowledge this is the first construction of viscous traveling wave solutions. Our proof involves a number of novel analytic ingredients, including: the study of an overdetermined Stokes problem and its underdetermined adjoint problem, a delicate asymptotic development of the symbol for a normal-stress to normal-Dirichlet map defined via the Stokes operator, a new scale of specialized anisotropic Sobolev spaces, and the study of a pseudodifferential operator that synthesizes the various operators acting on the free surface functions. © 2022 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.     

A Steady Three-Dimensional Noncompact Free Boundary-Value Problem for the Navier-Stokes Equations

A steady three-dimensional flow of a viscous incompressible fluid with a noncompact free boundary above a fixed unbounded bottom is studied. It is assumed that the motion of the fluid is generated by sources and sinks situated in a bounded part of the bottom and having zero total flux. The existence of a unique solution to this problem with small data is proved and the asymptotic of the solution is constructed. Bibliography: 33 titles.This paper is dedicated to Prof. V. A. Solonnikov’s 70th birthday__________Published in Zapiski Nauchnykh Seminarov POMI, Vol. 306, 2003, pp. 134–164.     

可压缩Navier-Stokes方程的压力梯度局部投影间断有限元法

将压力梯度投影与间断有限元法相结合,对可压缩线性化N-S方程提出了一种稳定化间断有限元格式．证明了此格式在速度和压力有限元空间无需满足B-B型条件的情况下,解的存在性和唯一性,以及相应的误差估计．     

非线性波动方程外区域初边值问题整体存在定理

在三维空间Ｒ~３中讨论非线性波动方程外区域初边值问题．当外区域　和初值ф、Ф及非线性项Ｆ满足一定条件时,利用线性化问题的衰减估计和Ｎａｓｈ－Ｍｏｓｅｒ技巧,得到了整体解存在定理．     

On the Navier-Stokes Equations for Exothermically Reacting Compressible Fluids

Abstract We analyze mathematical models governing planar flow of chemical reaction from unburnt gasesto burnt gases in certain physical regimes in which diffusive effects such as viscosity and heat conduction aresignificant. These models can be then formulated as the Navier-Stokes equations for exothermically reactingcompressible fluids. We first establish the existence and dynamic behavior, including stability, regularity, andlarge-time behavior, of global discontinuous solutions of large oscillation to the Navier-Stokes equations withconstant adiabatic exponent γ and specific heat C_v. Our approach for the existence and regularity is to combinethe difference approximation techniques with the energy methods, total variation estimates, and weak conver-gence argumeots to deal with large jump discontinuities; and for large-time behavior is an a posteriori argumentdirectly from the weak form of the equations. The approach and ideas we develop here can be applied to solvinga more complicated model where γ