Improved a priori bounds for thermal fluid equations

TL;DR

This paper establishes improved a priori bounds for 3D hydrodynamic fluid models with temperature-dependent viscosity and conductivity, providing new insights into the control of enstrophy in thermal fluid flows.

Contribution

It introduces stronger a priori bounds for both incompressible and compressible models with temperature-dependent properties, advancing theoretical understanding of thermal fluid equations.

Findings

Proved a strong a priori bound on enstrophy weighted by temperature.

Bounded solutions for arbitrary initial data under specific temperature and conductivity conditions.

Extended bounds to both incompressible and compressible fluid models.

Abstract

We consider two hydrodynamic model problems (one incompressible and one compressible) with three dimensional fluid flow on the torus and temperature-dependent viscosity and conductivity. The ambient heat for the fluid is transported by the flow and fed by the local energy dissipation, modeling the transfer of kinetic energy into thermal energy through fluid friction. Both the viscosity and conductivity grow with the local temperature. We prove a strong a priori bound on the enstrophy of the velocity weighed against the temperature for initial data of arbitrary size, requiring only that the conductivity be proportionately larger than the viscosity (and, in the incompressible case, a bound on the temperature as a Muckenhoupt weight).