Title: Free boundary value problem for the radial symmetric compressible isentropic Navier-Stokes equations with density-dependent viscosity Show Chinese title

Title: 密度依赖粘性径向对称可压缩等熵纳维-斯托克斯方程的自由边界值问题

Abstract: This paper is devoted to the study of free-boundary-value problem of the compressible Naiver-Stokes system with density-dependent viscosities $\mu=const>0,\lambda=\rho^\beta$ which was first introduced by Vaigant-Kazhikhov \cite{1995 Vaigant-Kazhikhov-SMJ} in 1995. By assuming the endpoint case $\beta=1$ in the radially spherical symmetric setting, we prove the (a priori) expanding rate of the free boundary is algebraic for multi-dimensional flow, and particularly establish the global existence of strong solution of the two-dimensional system for any large initial data. This also improves the previous work of Li-Zhang \cite{2016 Li-Zhang-JDE} where they proved the similar result for $\beta>1$. The main ingredients of this article is making full use of the geometric advantange of domain as well as the critical space dimension two. Show Chinese abstract

Abstract: 本文致力于研究具有密度依赖粘性系数的可压缩纳维-斯托克斯系统的自由边界值问题$\mu=const>0,\lambda=\rho^\beta$，该问题最早由 Vaigant-Kazhikhov \cite{1995 Vaigant-Kazhikhov-SMJ}在1995年提出。通过假设径向球对称设置中的端点情况 $\beta=1$，我们证明了多维流动的自由边界扩张率为代数形式，并且特别建立了二维系统的强解对于任何大初始数据的全局存在性。这也改进了 Li-Zhang \cite{2016 Li-Zhang-JDE}的先前工作，他们在其中证明了对于 $\beta>1$的类似结果。本文的主要内容是充分利用了区域的几何优势以及临界空间维度二。