A moment approach for entropy solutions to nonlinear hyperbolic PDEs *

Swann Marx (1) , Tillmann Weisser (1) , Didier Henrion (1) , Jean B Lasserre (1, 2)

1 LAAS-MAC - Équipe Méthodes et Algorithmes en Commande
2 IMT - Institut de Mathématiques de Toulouse UMR5219

We propose to solve polynomial hyperbolic partial differential equations (PDEs) with convex optimization. This approach is based on a very weak notion of solution of the nonlinear equation, namely the measure-valued (mv) solution, satisfying a linear equation in the space of Borel measures. The aim of this paper is, first, to provide the conditions that ensure the equivalence between the two formulations and, second, to introduce a method which approximates the infinite-dimensional linear problem by a hierarchy of convex, finite-dimensional, semidefinite programming problems. This result is then illustrated on the celebrated Burgers equation. We also compare our results with an existing numerical scheme, namely the Godunov scheme.

Mots clés: moments and positive polynomials; convex optimization; nonlinear partial differential equation

Domaines: Equations aux dérivées partielles [math.AP]; Optimisation et contrôle [math.OC]

Swann Marx, Tillmann Weisser, Didier Henrion, Jean B Lasserre. A moment approach for entropy solutions to nonlinear hyperbolic PDEs *. Mathematical Control and Related Fields, 2020, 10 (1), pp.113-140. ⟨10.3934/mcrf.2019032⟩. ⟨hal-01830870⟩