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Author: Mielke, Alexander × Subject: generalized gradient flows ×

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Now showing 1 - 5 of 5
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    Variational convergence of gradient flows and rate-independent evolutions in metric spaces
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2012) Mielke, Alexander; Rossi, Riccarda; Savaré, Giuseppe
    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.
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    BV solutions and viscosity approximations of rate-independent systems
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2009) Mielke, Alexander; Rossi, Riccarda; Savaré, Giuseppe
    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 which 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 a crucial role in the description of the associated jump trajectories. We shall prove a general convergence result 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.
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    An existence result and evolutionary [Gamma]-convergence for perturbed gradient systems
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Bacho, Aras; Emmrich, Etienne; Mielke, Alexander
    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.
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    Balanced viscosity (BV) solutions to infinite-dimensional rate-independent systems
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Mielke, Alexander; Rossi, Riccarda; Savaré, Giuseppe
    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.
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    Nonsmooth analysis of doubly nonlinear evolution equations
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) Mielke, Alexander; Rossi, Riccarda; Savaré, Giuseppe
    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.