On Classical Solutions for Viscous Polytropic Fluids with Degenerate Viscosities and Vacuum

被引：32

作者：

Li, Yachun ^{[1
,2
]}

Pan, Ronghua ^{[3
]}

Zhu, Shengguo ^{[4
,5
]}

机构：

[1] Shanghai Jiao Tong Univ, Sch Math Sci, MOE LSC, Shanghai 200240, Peoples R China

[2] Shanghai Jiao Tong Univ, SHL MAC, Shanghai 200240, Peoples R China

[3] Georgia Tech, Sch Math, Atlanta, GA 30332 USA

[4] Shanghai Jiao Tong Univ, Sch Math Sci, Shanghai 200240, Peoples R China

[5] Univ Oxford, Math Inst, Oxford OX2 6GG, England

来源：

ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS | 2019年 / 234卷 / 03期

基金：

中国国家自然科学基金; 美国国家科学基金会;

关键词：

NAVIER-STOKES EQUATIONS; SHALLOW-WATER EQUATIONS; GLOBAL WEAK SOLUTIONS; SMOOTH SOLUTIONS; WELL-POSEDNESS; CAUCHY-PROBLEM; BLOW-UP; EULER EQUATIONS; LOCAL EXISTENCE; KORTEWEG;

D O I：

10.1007/s00205-019-01412-6

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

In this paper, we consider the three-dimensional isentropic Navier-Stokes equations for compressible fluids allowing initial vacuum when viscosities depend on density in a superlinear power law. We introduce the notion of regular solutions and prove the local-in-time well-posedness of solutions with arbitrarily large initial data and a vacuum in this class, which is a long-standing open problem due to the very high degeneracy caused by a vacuum. Moreover, for certain classes of initial data with a local vacuum, we show that the regular solution that we obtained will break down in finite time, no matter how small and smooth the initial data are.

引用

页码：1281 / 1334

页数：54

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