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

Keimer, Alexander; Pflug, Lukas

Comptes Rendus. Mathématique, Tome 361 (2023) no. G11, pp. 1723-1760

We study nonlocal conservation laws with a discontinuous flux function of regularity ${\mathsf{L}}^{\infty }\left(ℝ\right)$ in the spatial variable and show existence and uniqueness of weak solutions in $\mathsf{C}\left([0,T];{\mathsf{L}}_{\mathrm{loc}}^1\right)$, 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.

Nous étudions des lois de conservation non locales avec une fonction de flux discontinue de régularité spatiale ${\mathsf{L}}^{\infty }\left(ℝ\right)$ dans la variable spatiale et montrons l’existence et l’unicité de solutions faibles dans $\mathsf{C}\left([0,T];{\mathsf{L}}_{\mathrm{loc}}^1\right)$, ainsi que les principes de maxima connexes des principes de maximum correspondants. Nous obtenons ce caractère bien posé par une reformulation appropriée en termes d’un problème de point fixe. Nous obtenons ce caractère bien posé en reformulant de façon appropriée le problème comme un problème de point fixe. Ce problème de point fixe nécessite lui-même l’étude de l’existence, de l’unicité et de la stabilité d’une classe d’équations différentielles ordinaires discontinues. Au niveau des ODE EDO, nous comparons le type de solution défini ici avec les solutions bien connues de Carathéodory et de Filippov.

Classification : 34A12, 34A36, 35L03, 35L65, 35Q99, 35R09, 45K05

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