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

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 [8]. 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.

The vacuum boundary problem for the spherically symmetric compressible Euler equations with positive density and unbounded entropy

“…Serre [22] and Grassin [7] proved global existence in the whole space for a special class of initial data by perturbing solutions to the vectorial Burgers equation. Recently, Rickard [20] proved globalin-time well-posedness of the Euler equations with heat transport by the pertubation of Dyson's [6] isothermal affine solutions, see below for details on affine solutions. In the other direction, Sideris [24] showed that singularities must form if the density is a strictly positive constant outside of a bounded set.…”