Viscoelastic flows of Maxwell fluids with conservation laws

We consider multi-dimensional extensions of Maxwell's seminal rheological equation for 1D viscoelastic flows. We aim at a causal model for compressible flows, defined by semi-group solutions given initial conditions, and such that perturbations propagate at finite speed. We propose a symmetric hyperbolic system of conservation laws that contains the Upper-Convected Maxwell (UCM) equation as causal model. The system is an extension of polyconvex elastodynamics, with an additional material metric variable that relaxes to model viscous effects. Interestingly, the framework could also cover other rheological equations, depending on the chosen relaxation limit for the material metric variable. We propose to apply the new system to incompressible free-surface gravity flows in the shallow-water regime, when causality is important. The system reduces to a viscoelastic extension of Saint-Venant 2D shallow-water system that is symmetric-hyperbolic and that encompasses our previous viscoelastic extensions of Saint-Venant proposed with F. Bouchut.