  We consider multi-dimensional extensions of Maxwell's seminal rheo-logical
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 propagates 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.
