Gradient-Flow Characterizations of One-Dimensional Quasistatic Viscoelasticity with Bhattacharya-Like Viscosity

Abstract

We study the equation of one-dimensional quasistatic nonlinear viscoelasticity of strain-rate type with Dirichlet boundary conditions, in the particular case that the underlying dissipation geometry (provided by the viscosity) is comparable to the Bhattacharya metric on probability densities. We establish a global existence result for weak solutions, with an approach based on a spatial discretization allowing us to work directly with the Riemannian metric associated to the viscosity. Strong convergence of spatially discrete solutions is shown directly – this is possible thanks to Lipschitz estimates achieved locally on energy sublevels enabled by an explicit derivation of the stretching of tangent vectors under the flow in the discrete setting and the relationship to the Bhattacharya metric. We furthermore prove gradient-flow representations for the solutions: they are curves of maximal slope and, under a global convexity hypothesis on the energy sublevels, we prove they satisfy a metric evolutionary variational inequality.