Weak solutions to Friedrichs systems with convex constraints

Bruno Després (1) , Frédéric Lagoutière (2) , Nicolas Seguin (1)

1 LJLL - Laboratoire Jacques-Louis Lions
2 LRI - Laboratoire de Recherche en Informatique

Abstract:We are interested in a problem arising for instance in elastoplasticity modeling, which consists in a system of partial differential equations and a constraint specifying that the solution should remain, for every time and every position, in a certain set. This constraint is generally incompatible with the invariant domains of the original model, thus this problem has to be precised in mathematical words. We here follow the approach proposed in [8] that furnishes a weak formulation of the constrained problem à la Kruzhkov. More precisely, the present paper deals with the study of the well-posedness of Friedrichs systems under convex constraints, in any space dimension. We prove that there exists a unique weak solution, continuous in time, square integrable in space, and with values in the constraints domain. This is done with the use of a discrete approximation scheme: we define a numerical approximate solution and prove, thanks to compactness properties, that it converges toward a solution to the constrained problem. Uniqueness is proven via energy (or entropy) estimates. Some numerical illustrations are provided.

Domaines: Analyse numérique [cs.NA]

Bruno Després, Frédéric Lagoutière, Nicolas Seguin. Weak solutions to Friedrichs systems with convex constraints. 2011. ⟨hal-00563993⟩