Abstract. In the nonconvex case, solutions of rate-independent systems may develop jumps as a function of time. To model such jumps, we adopt the philosophy that rate-independence should be considered as limit of systems with smaller and smaller viscosity. For the finite-dimensional case we study the vanishing-viscosity limit of doubly nonlinear equations given in terms of a differentiable energy functional and a dissipation potential that is a viscous regularization of a given rate-independent dissipation potential. The resulting definition of "BV solutions" involves, in a nontrivial way, both the rate-independent and the viscous dissipation potential, which play crucial roles in the description of the associated jump trajectories. We shall prove general convergence results for the time-continuous and for the time-discretized viscous approximations and establish various properties of the limiting BV solutions. In particular, we shall provide a careful description of the jumps and compare the new notion of solutions with the related concepts of energetic and local solutions to rate-independent systems.Mathematics Subject Classification. 49Q20, 58E99.

Abstract. This paper addresses the Cauchy problem for the gradient flow equation in a Hilbert Our analysis is also motivated by some models describing phase transitions phenomena, leading to systems of evolutionary PDEs which have a common underlying gradient flow structure: in particular, we will focus on quasistationary models, which exhibit highly non convex Lyapunov functionals.Mathematics Subject Classification. 35A15, 35K50, 35K85, 58D25, 80A22.

We analyze a rate-independent model for damage evolution in elastic bodies. The central quantities are a stored energy functional and a dissipation functional, which is assumed to be positively homogeneous of degree one. Since the energy is not simultaneously (strictly) convex in the damage variable and the displacements, solutions may have jumps as a function of time. The latter circumstance makes it necessary to recur to suitable notions of weak solution. However, the by-now classical concept of global energetic solution fails to describe accurately the behavior of the system at jumps.Hence, we consider rate-independent damage models as limits of systems driven by viscous, ratedependent dissipation. We use a technique for taking the vanishing viscosity limit, which is based on arc-length reparameterization. In this way, in the limit we obtain a novel formulation for the rateindependent damage model, which highlights the interplay of viscous and rate-independent effects in the jump regime, and provides a better description of the energetic behavior of the system at jumps.

In this paper we analyze a PDE system modelling (non-isothermal) phase transitions and damage phenomena in thermoviscoelastic materials. The model is thermodynamically consistent: in particular, no small perturbation assumption is adopted, which results in the presence of quadratic terms on the right-hand side of the temperature equation, only estimated in L 1 . The whole system has a highly nonlinear character.We address the existence of a weak notion of solution, referred to as "entropic", where the temperature equation is formulated with the aid of an entropy inequality, and of a total energy inequality. This solvability concept reflects the basic principles of thermomechanics, as well as the thermodynamical consistency of the model. It allows us to obtain global-in-time existence theorems without imposing any restriction on the size of the initial data.We prove our results by passing to the limit in a time-discretization scheme, carefully tailored to the nonlinear features of the PDE system (with its "entropic" formulation), and of the a priori estimates performed on it. Our time-discrete analysis could be useful towards the numerical study of this model.for all t ∈ (0, T ] and almost all s ∈ (0, t), with ξ a selection in the (convex analysis) subdifferential ∂ β( χ ) = ∂I [0,+∞) ( χ ) of I [0,+∞) . In [47, Prop. 2.14] (see also [24]), we prove that, under additional regularity properties any weak solution in fact fulfills (1.3) pointwise.Let us also mention that other approaches to the weak solvability of coupled PDE systems with an L 1 -righthand side are available in the literature: in particular, we refer here to [54] and [49]. In [54], the notion of renormalized solution has been used in order to prove a global-in-time existence result for a nonlinear system in thermoviscoelasticity. In [49] the focus is on rate-independent processes coupled with viscosity and inertia in the displacement equation, and with the temperature equation. There the internal variable equation is not of gradient-flow type as (1.3), but instead features a 1-positively homogeneous dissipation potential. For the resulting PDE system, a weak solution concept partially mutuated from the theory of rate-independent processes by A. Mielke (cf., e.g., [39]) is analyzed. An existence result is proved combining techniques for rate-independent evolution, with Boccardo-Gallouët type estimates of the temperature gradient in the heat equation with L 1 -right-hand side. Our existence results. The main results of this paper, Theorems 1 and 2, state the existence of entropic solutions for system (1.1-1.3), supplemented with the boundary conditions (1.4) (cf. Remark 2.12), in the irreversible (µ = 1) and reversible (µ = 0) cases.More precisely, in the case of unidirectional evolution for χ we can prove the existence of a global-intime entropic solution (i.e. satisfying the entropy (1.6) and the total energy (1.7) inequalities, the (pointwise) momentum balance (1.2), the one-sided variational inequality (1.9) and the energy (1.10) inequalities for...

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.Comment: 45 page

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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 BVestimates that are of independent interest.whose support function is Ψ. Following [MRS12a], we say that a curve u ∈ BV([0, T ]; V ) is a BV solution to the RIS (V, E, Ψ, Φ), if it fulfills the following local stability conditionwhere J u is the jump set of u, and the Energy-Dissipation Balance Var f (u; [0, t]) + E t (u(t)) = E 0 (u(0)) + t 0

We address a model for adhesive unilateral frictionless Signorini-type contact between bodies of heatconductive viscoelastic material, in the linear Kelvin-Voigt rheology, undergoing thermal expansion. The flow-rule for debonding the adhesion is considered rate-independent and unidirectional, and a thermodynamically consistent model is derived and analysed as far as the existence of a weak solution is concerned.