Two-Speed Solutions to Non-convex Rate-Independent Systems

Abstract

We consider evolutionary PDE inclusions of the form

$$\begin{aligned} -\lambda {\dot{u}}_\lambda + \Delta u - \mathrm {D}W_0(u) + f \ni \partial \mathscr {R}_1({\dot{u}}) \quad \text {in}\,\,{ (0,T) \times \Omega ,} \end{aligned}$$

where $\mathscr {R}_1$ is a positively 1-homogeneous rate-independent dissipation potential and $W_0$ is a (generally) non-convex energy density. This work constructs solutions to the above system in the slow-loading limit $\lambda \downarrow 0$. Our solutions have more regularity both in space and time than those that have been obtained with other approaches. On the “slow” time scale we see strong solutions to a purely rate-independent evolution. Over the jumps, we obtain a detailed description of the behavior of the solution and we resolve the jump transients at a “fast” time scale, where the original rate-dependent evolution is still visible. Crucially, every jump transient splits into a (possibly countable) number of rate-dependent evolutions, for which the energy dissipation can be explicitly computed. This, in particular, yields a global energy equality for the whole evolution process. It also turns out that there is a canonical slow time scale that avoids intermediate-scale effects, where movement occurs in a mixed rate-dependent/rate-independent way. In this way, we obtain precise information on the impact of the approximation on the constructed solution. Our results are illustrated by examples, which elucidate the effects that can occur.