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

2021/10/20 by Keimer, Alexander, Pflug, Lukas · 1 citation
#34A12 #34A36 #35L03 #35L65 #35Q99 #35R09 #45K05 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2110.10503

Abstract

We study nonlocal conservation laws with a discontinuous flux function of regularity L^∞(ℝ) in the spatial variable and show existence and uniqueness of weak solutions in C([0,T];L¹_loc(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.

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