Local and global well-posedness of entropy-bounded solutions to the compressible Navier-Stokes equations in multi-dimensions

被引:0
作者
Jinkai Li
Zhouping Xin
机构
[1] South China Normal University,South China Research Center for Applied Mathematics and Interdisciplinary Studies, School of Mathematical Sciences
[2] The Chinese University of Hong Kong,The Institute of Mathematical Sciences
来源
Science China Mathematics | 2023年 / 66卷
关键词
heat conductive compressible Navier-Stokes equations; strong solutions; far field vacuum; uniform boundedness of entropy; inhomogeneous Sobolev spaces; 35Q30; 76N99;
D O I
暂无
中图分类号
学科分类号
摘要
Due to the high degeneracy and singularities of the entropy equation, the physical entropy for viscous and heat conductive polytropic gases behaves 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 ℝ3, 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 strong solution to the Cauchy problem of the heat conductive compressible Navier-Stokes equations, the corresponding entropy keeps uniformly bounded and the L2 regularities of the velocity and temperature can be propagated, up to the existing time of the solution, as long as the initial density vanishes only at far fields with a rate not faster than O(1|x|2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O\left( {{1 \over {{{\left| x \right|}^2}}}} \right)$$\end{document}. The main tools are some singularly weighted energy estimates and an elaborate De Giorgi type iteration technique. We apply the De Giorgi type iterations to different equations in establishing the lower and upper bounds of the entropy.
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页码:2219 / 2242
页数:23
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