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

TL;DR

This work proves the existence of nonnegative stationary solutions to the relativistic transparent media equation $-div\left(u^m\frac{Du}{|Du|}\right)=f$ with $f\ge0$ in $L^1(\Omega)$ and Dirichlet boundary data, for $m>0$. It introduces a generalized pairing framework for unbounded divergence-measure fields to give meaning to the flux $z=u^m w$ and establishes a solution $u\in DTBV(\Omega)$ with a flux $z\in \mathcal{DM}^1(\Omega)$ and a weak boundary trace, along with a boundary condition in the trace sense. The paper further proves key a priori estimates and derives a sharp flux–gradient relation, together with a Gauss–Green formula in this generalized setting. Regularity results show that, for $f\in L^p(\Omega)$ with $1<p<N$, the diffusion flux satisfies $u^m\in L^{\frac{Np}{N-p}}(\Omega)$, and in the critical regime $p\ge\frac{N(m+1)}{Nm+1}$, one has $u^{m+1}\in BV(\Omega)$, aligning with known limits and extending the BV framework to $L^1$ data. Overall, the work extends the BV/DTBV theory to unbounded fluxes, enabling existence and regularity analyses for a class of relativistic diffusion operators relevant in radiation hydrodynamics.

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 $L^N(Ω)$) 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 $L^p(Ω)$, with $1<p<N$. Our result is coherent with the regularity found in [11].

Theorems & Definitions (59)

• Lemma 2.1
• Lemma 2.2
• Theorem 2.3
• Lemma 2.4
• Lemma 2.5
• Lemma 2.6
• Lemma 3.1
• Lemma 3.2
• Lemma 3.3
• Lemma 3.4