Discontinuous nonlocal conservation laws and related discontinuous ODEs -- Existence, Uniqueness, Stability and Regularity

Abstract

We study nonlocal conservation laws with a discontinuous flux function of regularity ${\mathsf{L}}^{\infty }(\mathbb{R})$ in the spatial variable and show existence and uniqueness of weak solutions in $\mathsf{C}([0,T];{\mathsf{L}}_{\text{loc}}^1(\mathbb{R}))$, as well as related maximum principles. We achieve this well-posedness by a proper reformulation in terms of a fixed-point problem. This fixed-point problem itself necessitates the study of existence, uniqueness and stability of a class of discontinuous ordinary differential equations. On the ODE level, we compare the solution type defined here with the well-known Carathéodory and Filippov solutions.

50 pages, 3 figures