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

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.