Weak solutions to Friedrichs systems with convex constraints

Abstract

We are interested in a problem arising, for instance, in elastoplasticity modelling, 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 specified in mathematical terms. Here we follow the approach proposed in Després (2007 Arch. Ration. Mech. Anal. 186 275–308) that furnishes a weak formulation of the constrained problem à la Kruzhkov. More precisely, this 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 towards a solution to the constrained problem. Uniqueness is proven via energy (or entropy) estimates. Some numerical illustrations are provided.