Existence of strong solutions with critical regularity to a polytropic model for radiating flows

Abstract

This paper is the continuation of our recent work Danchin and Ducomet (J Evol Equ 14:155-195, 2013) devoted to barotropic radiating flows. We here aim at investigating the more physically relevant situation of polytropic flows. More precisely, we consider a model arising in radiation hydrodynamics which is based on the full Navier-Stokes-Fourier system describing the macroscopic fluid motion, and a P1-approximation (see below) of the transport equation modeling the propagation of radiative intensity. In the strongly underrelativistic situation, we establish the global-in-time existence and uniqueness of solutions with critical regularity for the associated Cauchy problem with initial data close to a stable radiative equilibrium. We also justify the nonrelativistic limit in that context. For smoother (possibly) large data bounded away from the vacuum and more general physical coefficients that may depend on both the density and the temperature, the local existence of strong solutions is shown.

Large time behavior of solutions to a diffusion approximation radiation hydrodynamics model

“…Where the global-in-time existence of strong small perturbation solution was established in the critical Besov spaces Ḃn/2 2,1 (R n ). Moreover, the global existence of the solution in critical Besov space for radiation hydrodynamics model (1.2) also has been achieved in [6] recently. It is noticed that the large time behavior of the solutions have not been given in both of these two papers.…”

“…The proof can be done by using the standard iteration arguments and fixed point theorem. One also refer to [6] (see Section 2.1). We omit the details for simplicity of presentation.…”

Partially dissipative systems in the critical regularity setting, and strong relaxation limit

“…systems related to the description of plasma or radiative phenomena, see e.g. [16]). As said before, having a 'compensating function' at hand allows to construct an energy functional that encodes the dissipative properties of the system.…”

“…The class that is considered contains the isentropic Euler equations with relaxation. We expect the whole strategy modified accordingly to be adaptable to hyperbolic-parabolic systems, to operators of any order and to more complex situations where the partially dissipative terms have mixed orders (see recent examples in [16] and [8]). It would also be of interest to study to what extent it may be adapted to situations where pseudo-differential operators depending on the space variable come into play.…”

Unique solvability of a stationary radiative‐conductive heat transfer problem in a system consisting of an absolutely black body and several semitransparent bodies

“…The same field of studies comprises the solvability problems in radiation gas‐ and hydrodynamics 65–81 . We also mention works dealing with homogenization of radiative‐conductive heat exchange problems 82–92 and with optimal control in complex heat exchange problems 93–102 .…”