Gradient-flow characterizations of one-dimensional quasistatic viscoelasticity with Bhattacharya-like viscosity

TL;DR

This paper analyzes one-dimensional quasistatic viscoelasticity using gradient flow methods, establishing existence, convergence, and variational characterizations of solutions with a focus on Bhattacharya-like viscosity metrics.

Contribution

It introduces a novel approach combining spatial discretization and Riemannian geometry to study viscoelasticity equations with gradient flow techniques.

Findings

Proved global existence of weak solutions.

Established strong convergence of discrete solutions.

Demonstrated gradient flow and variational inequality representations.

Abstract

We study the equation of one-dimensional quasistatic nonlinear viscoelasticity 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.