Partial regularity for the Navier-Stokes-Fourier system

TL;DR

This paper proves partial regularity results for weak solutions of a heat-conductive incompressible Navier-Stokes-Fourier system with temperature-dependent viscosity, analyzing the nature of potential singularities in space and time.

Contribution

It establishes partial regularity for proper weak solutions of the coupled Navier-Stokes-Fourier system with temperature-dependent viscosity and heat transfer.

Findings

Partial regularity for velocity in the Navier-Stokes-Fourier system.

Characterization of the singular set in space and time.

Extension of regularity results to heat-conductive fluids.

Abstract

This paper addresses a nonstationary flow of heat-conductive incompressible Newtonian fluid with temperature-dependent viscosity coupled with linear heat transfer with advection and a viscous heat source term, under Navier/Dirichlet boundary conditions. The partial regularity for the velocity of the fluid is proved to each proper weak solution, that is, for such weak solutions which satisfy some local energy estimates in a similar way to the suitable weak solutions of the Navier-Stokes system. Finally, we study the nature of the set of points in space and time upon which proper weak solutions could be singular.