Stationary equation of the relativistic heat diffusion in transparent media having $L^1$--data

TL;DR

This paper proves the existence of solutions to a stationary relativistic heat diffusion equation in transparent media with $L^1$ data, extending previous results to less regular data and exploring solution regularity.

Contribution

It introduces new methods to establish existence of solutions with $L^1$ data where traditional theories do not apply, and analyzes their regularity.

Findings

Existence of solutions with $L^1$ data in bounded domains.

Extension of regularity results to $L^p$ data for $1<p<N$.

Development of new analytical tools beyond Anzellotti theory.

Abstract

Our objective is to prove existence of a solution to the Dirichlet problem for an equation arising in the theory of radiation hydrodynamics to deal with the radiating energy in transparent media. We study its stationary equation with $L^1$--datum in a bounded domain. This problem was addressed in [11] for regular data (data belonging to $L^N(\Omega)$) and a bounded solution was obtained. In our framework, the proof of existence is far from trivial since the solution sought cannot be bounded. Consequently, the Anzellotti theory of pairings does not apply and we have to use new developments to introduce the meaning of solution. We also study the regularity of solutions when data belong to $L^p(\Omega)$, with $1<p<N$. Our result is coherent with the regularity found in [11].