Nonlocal degenerate parabolic hyperbolic equations on bounded domains

TL;DR

This paper establishes the well-posedness of degenerate mixed-type parabolic-hyperbolic equations with nonlocal diffusion on bounded domains, extending existing theories and introducing an entropy solution framework that ensures uniqueness.

Contribution

It introduces a novel entropy solution formulation for nonlocal degenerate equations on bounded domains, proving uniqueness without relying on prior energy estimates.

Findings

Proved uniqueness of bounded entropy solutions.

Developed an entropy formulation that implies energy estimates.

Extended nonlocal theories from the whole space to bounded domains.

Abstract

We study well-posedness of degenerate mixed-type parabolic-hyperbolic equations $$ \partial_tu+\text{div}\big(f(u)\big)=\mathcal{L}[b(u)] $$ on bounded domains with general Dirichlet boundary/exterior conditions. The nonlocal diffusion operator $\mathcal{L}$ can be any symmetric L{\'e}vy operator (e.g. fractional Laplacians) and $b$ is nondecreasing and allowed to have degenerate regions ($b'=0$). We propose an entropy solution formulation for the problem and show uniqueness of bounded entropy solutions under general assumptions. Existence of solutions is shown in a separate paper. The uniqueness proof is based on the Kru\v{z}kov doubling of variables technique and incorporates several a priori results derived from our entropy formulation: an $L^\infty$-bound, an energy estimate, strong initial trace, weak boundary traces, and a \textit{nonlocal} boundary condition. Our work can be seen as both extending nonlocal theories from the whole space to domains and local theories on domains to the nonlocal case. Unlike local theories our formulation does not assume energy estimates. They are now a consequence of the formulation, but as opposed to previous nonlocal theories, they play an essential role in our arguments.