Non-critical fractional conservation laws in domains with boundary

We study bounded solutions for a multidimensional conservation law coupled with a power $s\in (0,1)$ of the Dirichlet laplacian acting in a domain. If $s \leq 1/2$ then the study centers on the concept of entropy solutions for which existence and uniqueness are proved to hold. If $s >1/2$ then the focus is rather on the $C^\infty$-regularity of weak solutions. This kind of results is known in $\mathbb{R}^N$ but perhaps not so much in domains. The extension given here relies on an abstract spectral approach, which would also allow many other types of nonlocal operators.