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Author: Mielke, Alexander × End date: 2022 × Start date: 2020 × Type: Text × Subject: energy-dissipation principle ×

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    Balanced-Viscosity solutions to infinite-dimensional multi-rate systems
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Mielke, Alexander; Rossi, Riccarda
    We consider generalized gradient systems with rate-independent and rate-dependent dissipation potentials. We provide a general framework for performing a vanishing-viscosity limit leading to the notion of parametrized and true Balanced-Viscosity solutions that include a precise description of the jump behavior developing in this limit. Distinguishing an elastic variable $u$ having a viscous damping with relaxation time $eps^alpha$ and an internal variable $z$ with relaxation time $eps$ we obtain different limits for the three cases $alpha in (0,1)$, $alpha=1$ and $alpha>1$. An application to a delamination problem shows that the theory is general enough to treat nontrivial models in continuum mechanics.
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    EDP-convergence for nonlinear fast-slow reaction systems with detailed balance
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Mielke, Alexander; Peletier, Mark A.; Stephan, Artur
    We consider nonlinear reaction systems satisfying mass-action kinetics with slow and fast reactions. It is known that the fast-reaction-rate limit can be described by an ODE with Lagrange multipliers and a set of nonlinear constraints that ask the fast reactions to be in equilibrium. Our aim is to study the limiting gradient structure which is available if the reaction system satisfies the detailed-balance condition. The gradient structure on the set of concentration vectors is given in terms of the relative Boltzmann entropy and a cosh-type dissipation potential. We show that a limiting or effective gradient structure can be rigorously derived via EDP convergence, i.e. convergence in the sense of the Energy-Dissipation Principle for gradient flows. In general, the effective entropy will no longer be of Boltzmann type and the reactions will no longer satisfy mass-action kinetics.