Existence and uniqueness results for viscous, heat-conducting 3-D fluid with vacuum

TL;DR

This paper establishes local existence, uniqueness, and blow-up criteria for strong solutions to 3-D viscous, heat-conducting compressible Navier-Stokes equations with vacuum, considering variable coefficients and unbounded domains.

Contribution

It proves the local existence and uniqueness of strong solutions with vacuum and variable coefficients, and provides blow-up criteria for these solutions.

Findings

Proved local existence and uniqueness of strong solutions with vacuum.

Established blow-up criteria for solutions.

Demonstrated blow-up of solutions with specific initial conditions.

Abstract

We consider the 3-D full Navier-Stokes equations whose the viscosity coefficients and the thermal conductivity coefficient depend on the density and the temperature. We prove the local existence and uniqueness of the strong solution in a domain $\Omega\subset\mathbb{R}^3$. The initial density may vanish in an open set and $\Omega$ could be a bounded or unbounded domain. We also prove a blow-up criterion for the solution. Finally, we show the blow-up of the smooth solution to the compressible Navier-Stokes equations in $\mathbb{R}^n$ ($n\geq1$) when the initial density has compactly support and the initial total momentum is nonzero.