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Nonsmooth analysis of doubly nonlinear evolution equations
In this paper we analyze a broad class of abstract doubly nonlinear evolution equations in Banach spaces, driven by nonsmooth and nonconvex energies. We provide some general sufficient conditions, on the dissipation potential and the energy functional, for existence of solutions to the related Cauchy problem. We prove our main existence result by passing to the limit in a time-discretization scheme with variational techniques. Finally, we discuss an application to a material model in finite-strain elasticity.
On the vanishing viscosity limit in parabolic systems with rate independent dissipation terms
We consider quasilinear parabolic systems with a nonsmooth rate-independent dissipation term in the limit of very slow loading rates, or equivalently with fixed loading and vanishing viscosity $varepsilon>0$. Because for nonconvex energies the solutions will develop jumps, we consider the vanishing-viscosity limit for the graphs of the solutions in the extended state space in arclength parametrization, where the norm associated with the viscosity is used to keep the subdifferential structure of the problem. A crucial point in the analysis are new a priori estimates that are rate independent and that allows us to show that the total length of the graph remains bounded in the vanishing-viscosity limit. To derive these estimates we combine parabolic regularity estimates with ideas from rate-independent systems
Balanced viscosity (BV) solutions to infinite-dimensional rate-independent systems
Balanced Viscosity solutions to rate-independent systems arise as limits of regularized rate-independent flows by adding a superlinear vanishing-viscosity dissipation. We address the main issue of proving the existence of such limits for infinite-dimensional systems and of characterizing them by a couple of variational properties that combine a local stability condition and a balanced energy-dissipation identity. A careful description of the jump behavior of the solutions, of their differentiability properties, and of their equivalent representation by time rescaling is also presented. Our techniques rely on a suitable chain-rule inequality for functions of bounded variation in Banach spaces, on refined lower semicontinuity-compactness arguments, and on new BV-estimates that are of independent interest.
An existence result and evolutionary [Gamma]-convergence for perturbed gradient systems
We consider the initial-value problem for the perturbed gradient flows, where a differential inclusion is formulated in terms of a subdifferential of an energy functional, a subdifferential of a dissipation potential and a more general perturbation, which is assumed to be continuous and to satisfy a suitable growth condition. Under additional assumptions on the dissipation potential and the energy functional, existence of strong solutions is shown by proving convergence of a semi-implicit discretization scheme with a variational approximation technique.
Variational convergence of gradient flows and rate-independent evolutions in metric spaces
We study the asymptotic behaviour of families of gradient flows in a general metric setting, when the metric-dissipation potentials degenerate in the limit to a dissipation with linear growth. We present a general variational definition of BV solutions to metric evolutions, showing the different characterization of the solution in the absolutely continuous regime, on the singular Cantor part, and along the jump transitions. By using tools of metric analysis, BV functions and blow-up by time rescaling, we show that this variational notion is stable with respect to a wide class of perturbations involving energies, distances, and dissipation potentials. As a particular application, we show that BV solutions to rate-independent problems arise naturally as a limit of p-gradient flows, p>1, when the exponents p converge to 1.