Entropy-bounded solutions to the one-dimensional heat conductive compressible Navier-Stokes equations with far field vacuum

TL;DR

This paper proves the global existence and entropy boundedness of solutions to the one-dimensional heat conductive compressible Navier-Stokes equations with vacuum at far fields, using weighted energy estimates and De Giorgi iteration.

Contribution

It introduces a novel approach combining singularly weighted energy estimates and De Giorgi iteration to handle entropy bounds with far field vacuum in compressible flows.

Findings

Global well-posedness of strong solutions established.

Uniform boundedness of entropy proven under specific decay conditions.

Entropy remains bounded for solutions with vacuum at far fields.

Abstract

In the presence of vacuum, the physical entropy for polytropic gases behaves singularly and it is thus a challenge to study its dynamics. It is shown in this paper that the boundedness of the entropy can be propagated up to any finite time provided that the initial vacuum presents only at far fields with sufficiently slow decay of the initial density. More precisely, for the Cauchy problem of the one dimensional heat conductive compressible Navier-Stokes equations, the global well-posedness of strong solutions and uniform boundedness of the corresponding entropy are established, 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 of proving the uniform boundedness of the entropy are some singularly weighted energy estimates carefully designed for the heat conductive compressible Navier-Stokes equations and an elaborate De Giorgi type iteration technique for some classes of degenerate parabolic equations. The De Giorgi type iterations are carried out to different equations in establishing the lower and upper bounds of the entropy.