Nonlocal degenerate parabolic hyperbolic equations on bounded domains

Abstract: We study well-posedness of degenerate mixed-type parabolic-hyperbolic equations $$\partial_t u+\mathrm{div}\big(f(u)\big)=\mathcal{L}[b(u)]$$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 and existence of bounded entropy solutions under general assumptions. 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. The existence proof is based on fixed point iteration for zero-order operators $\mathcal{L}$, and then extended to more general operators through approximations, weak-$\star$ compactness of approximate solutions $u_n$, and \textit{strong} compactness of $b(u_n)$. Strong compactness follows from energy estimates and arguments we introduce to transfer weak regularity from $\partial_t u_n$ to $\partial_t b(u_n)$.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, and as opposed to previous nonlocal theories, play an essential role in our arguments. Several results of independent interest are established, including a characterization of the $\mathcal{L}$'s for which the corresponding energy/Sobolev-space compactly embeds into $L^2$.