Global Strong Solution for 3D Viscous Incompressible Heat Conducting Navier-Stokes Flows with Non-negative Density

TL;DR

This paper proves the existence and uniqueness of global strong solutions for 3D heat-conducting Navier-Stokes flows with non-negative density, including vacuum states, under certain conditions.

Contribution

It establishes the global strong solution existence for nonhomogeneous heat-conducting Navier-Stokes flows with vacuum, extending previous results to more general initial conditions.

Findings

Global strong solutions exist with vacuum initial density under Serrin's condition.

Uniqueness of solutions is proven under smallness assumptions.

Energy estimates and regularity properties are key to the proofs.

Abstract

We are concerned with an initial boundary value problem for the nonhomogeneous heat conducting Navier-Stokes flows with non-negative density. First of all, we show that for the initial density allowing vacuum, the strong solution exists globally if the velocity satisfies the Serrin's condition. Then, under some smallness condition, we prove that there is a unique global strong solution to the 3D viscous nonhomogeneous heat conducting Navier-Stokes flows. Our method relies upon the delicate energy estimates and regularity properties of Stokes system and elliptic equation.