About the shock formation in multi-dimensional scalar conservation laws

Denis Serre (1)

1 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées

We establish a space-time integral estimate of the difference of solutions of non-degenerate scalar conservation laws. It is valid over the maximal domain $(0,T_{\rm max})\times\R^d$ in which the solutions are shock-free, and it fails beyond $T_{\rm max}$. When comparing a solution with its shifts, we deduce a Besov-like estimate at the development of its first singularity.

Mots clés: Divergence-free tensors; Compensated Integrability; Scalar conservation laws; Burgers equation; shock formation

Mots clés: Tenseurs à Divergence nulle; Intégrabilit\é par Compensation; lois de conservation scalaires; équation de Burgers; formation des chocs

Domaines: Mathématiques [math]; Equations aux dérivées partielles [math.AP]

Denis Serre. About the shock formation in multi-dimensional scalar conservation laws. 2023. ⟨ensl-03967004⟩