Well-Posedness and Exponential Decay for the Navier-Stokes Equations of Viscous Compressible Heat-Conductive Fluids with Vacuum

TL;DR

This paper proves the existence, uniqueness, and exponential decay of solutions to the Navier-Stokes equations for viscous, heat-conductive compressible fluids with vacuum, under less restrictive initial conditions.

Contribution

It extends the well-posedness theory for compressible Navier-Stokes equations to less regular data and establishes exponential decay of solutions.

Findings

Existence of local-in-time solutions with less regular data.

Global classical solutions with large oscillations and vacuum.

Exponential decay of solutions over time.

Abstract

This paper is concerned with the Cauchy problem of Navier-Stokes equations for compressible viscous heat-conductive fluids with far-field vacuum at infinity in $\R^3$. For less regular data and weaker compatibility condition than those proposed by Cho-Kim \cite{CK2006}, we first prove the existence of local-in-time solutions belonging to a larger class of functions in which the uniqueness can be shown to hold. The local solution is in fact a classical one away from the initial time, provided the initial density is regular. We also establish the global well-posedness of classical solutions with large oscillations and vacuum in the case when the initial total energy is suitably small. The exponential decay estimates of the global solutions are obtained.