Propagation of uniform boundedness of entropy and inhomogeneous regularities for viscous and heat conductive gases with far field vacuum in three dimensions

TL;DR

This paper proves that for viscous, heat-conductive gases with vacuum at far fields, the entropy remains uniformly bounded and certain regularities are preserved, using weighted energy estimates and De Giorgi iteration techniques.

Contribution

It establishes the propagation of entropy boundedness and regularities for compressible Navier-Stokes equations with vacuum at far fields, under specific decay conditions.

Findings

Entropy remains uniformly bounded over time.

Velocity and temperature regularities are preserved.

Results apply to initial densities vanishing at a rate no faster than 1/|x|^2.

Abstract

Due to the highly degeneracy and singularities of the entropy equation, the physical entropy for viscous and heat conductive polytropic gases behave 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 $\mathbb R^3$, 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 $L^2$ 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 no more than $O(\frac{1}{|x|^2})$. The main tools are some singularly weighted energy estimates and an elaborate De Giorgi type iteration technique. The De Giorgi type iterations are carried out to different equations in establishing the lower and upper bounds of the entropy.