摘要
主要介绍了近年来等熵可压缩Navier-Stokes方程组真空问题的主要研究进展,涉及粘性系数为常数以及粘性系数依赖于密度函数两种情形.由于当真空出现时,可压缩Navier-Stokes方程组有较强的退化性,其解的适定性相当复杂,有许多不同于非真空情形的新现象产生.文献显示,一维等熵可压缩Navier-Stokes方程组的真空问题研究成果比较丰富,高维情形进展相对缓慢,许多重要问题都没有解决.本文的最后给出了有关粘性系数依赖于密度的等熵可压缩Navier-Stokes方程组的一些公开问题.
In this paper,we survey the recent major progress on studies of the isentropic compressible Navier-Stokes equations with vacuum.Both the constant viscosity coefficients and the density-dependent viscosity coefficients are considered.In the presence of the vacuum,the compressible Navier-Stokes equations have strong degeneracy which lead to considerable complexities in the analysis of the well-posedness of solutions,and many new phenomena occur which are different from the non-vacuum case.Though substantial progress on the Cauchy problem and free boundary problem has been achieved for the one-dimensional isentropic compressible Navier-Stokes systems,the multi-dimensional case is much more difficult and there remain many important challenging problems to be solved.These difficulties are discussed in this paper.Finally,we list several open problems concerning the vacuum dynamics for the compressible Navier-Stokes equations.
作者
郭真华
李自来
辛周平
Guo Zhenhua;Li Zilai;Xin Zhouping(College of Mathematics, Northwest University, Xi′an 710127, China;School of Mathematics and Information Science,Henan Polytechnic University, Jiaozuo 454000, China;The Institute of Mathematical Sciences,The Chinese University of Hong Kong, Hong Kong, China)
出处
《纯粹数学与应用数学》
2018年第4期382-399,共18页
Pure and Applied Mathematics
基金
国家自然科学基金(11331005
11671319
11601128)
RGC研究基金(CUHK14305315
CUHK14300917
CUHK14302917)
NSFC-RGC联合基金N-CUHK443-14
