Global solutions to the compressible Euler equations with heat transport by convection around Dyson's isothermal affine solutions

TL;DR

This paper proves the existence of global solutions to the compressible Euler equations with heat convection by perturbing Dyson's isothermal affine solutions, overcoming challenges posed by vacuum behavior at infinity.

Contribution

It introduces a novel stability analysis incorporating Gaussian functions and finite propagation results to handle unbounded terms, advancing the mathematical understanding of heat transport in fluid dynamics.

Findings

Established global existence of solutions under perturbations

Developed a finite propagation result for Gaussian-involved stability

Utilized heat transport formulation with new time weight techniques

Abstract

Global solutions to the compressible Euler equations with heat transport by convection in the whole space are shown to exist through perturbations of Dyson's isothermal affine solutions. This setting presents new difficulties because of the vacuum at infinity behavior of the density. In particular, the perturbation of isothermal motion introduces a Gaussian function into our stability analysis and a novel finite propagation result is proven to handle potentially unbounded terms arising from the presence of the Gaussian. Crucial stabilization-in-time effects of the background motion are mitigated through the use of this finite propagation result however and a careful use of the heat transport formulation in conjunction with new time weight manipulations are used to establish global existence. The heat transport by convection offers unique physical insights into the model and mathematically, we use a controlled spatial perturbation in the analysis of this feature of our system which leads us to exploit source term estimates as part of our techniques.