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- ItemHomogenization of Cahn-Hilliard-type equations via evolutionary Gamma-convergence(Berlin : WeierstraĂź-Institut fĂĽr Angewandte Analysis und Stochastik, 2015) Liero, Matthias; Reichelt, SinaIn this paper we discuss two approaches to evolutionary Gamma-convergence of gradient systems in Hilbert spaces. The formulation of the gradient system is based on two functionals, namely the energy functional and the dissipation potential, which allows us to employ Gamma-convergence methods. In the first approach we consider families of uniformly convex energy functionals such that the limit passage of the time-dependent problems can be based on the theory of evolutionary variational inequalities as developed by Daneri and Savare 2010. The second approach uses the equivalent formulation of the gradient system via the energy-dissipation principle and follows the ideas of Sandier and Serfaty 2004. We apply both approaches to rigorously derive homogenization limits for Cahn-Hilliard-type equations. Using the method of weak and strong two-scale convergence via periodic unfolding, we show that the energy and dissipation functionals Gamma-converge. In conclusion, we will give specific examples for the applicability of each of the two approaches.
- ItemDeriving effective models for multiscale systems via evolutionary Gamma-convergence(Berlin : WeierstraĂź-Institut fĂĽr Angewandte Analysis und Stochastik, 2015) Mielke, AlexanderWe discuss possible extensions of the recently established theory of evolutionary Gamma-convergence for gradient systems to nonlinear dynamical systems obtained by perturbation of a gradient systems. Thus, it is possible to derive effective equations for pattern forming systems with multiple scales. Our applications include homogenization of reaction-diffusion systems, the justification of amplitude equations for Turing instabilities, and the limit from pure diffusion to reaction-diffusion. This is achieved by generalizing the Gamma-limit approaches based on the energy-dissipation principle or the volutionary variational estimate.
- ItemVariational approaches and methods for dissipative material models with multiple scales(Berlin : WeierstraĂź-Institut fĂĽr Angewandte Analysis und Stochastik, 2015) Mielke, AlexanderIn a first part we consider evolutionary systems given as generalized gradient systems and discuss various variational principles that can be used to construct solutions for a given system or to derive the limit dynamics for multiscale problems. These multiscale limits are formulated in the theory of evolutionary Gamma-convergence. On the one hand we consider the a family of viscous gradient system with quadratic dissipation potentials and a wiggly energy landscape that converge to a rate-independent system. On the other hand we show how the concept of Balanced-Viscosity solution arise as in the vanishing-viscosity limit. As applications we discuss, first, the evolution of laminate microstructures in finite-strain elastoplasticity and, second, a two-phase model for shape-memory materials, where H-measures are used to construct the mutual recovery sequences needed in the existence theory.
- ItemExploring families of energy-dissipation landscapes via tilting -- Three types of EDP convergence(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Mielke, Alexander; Montefusco, Alberto; Peletier, Mark A.This paper revolves around a subtle distinction between two concepts: passing to the limit in a family of gradient systems, on one hand, and deriving effective kinetic relations on the other. The two concepts are strongly related, and in many examples they even appear to be the same. Our main contributions are to show that they are different, to show that well-known techniques developed for the former may give incorrect results for the latter, and to introduce new tools to remedy this. The approach is based on the Energy-Dissipation Principle that provides a variational formulation to gradient-flow equations that allows one to apply techniques from Γ-convergence of functional on states and functionals on trajectories.
- ItemBalanced-Viscosity solutions for multi-rate systems(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Mielke, Alexander; Rossi, Riccarda; Savaré, GiuseppeSeveral mechanical systems are modeled by the static momentum balance for the displacement u coupled with a rate-independent flow rule for some internal variable z. We consider a class of abstract systems of ODEs which have the same structure, albeit in a finite-dimensional setting, and regularize both the static equation and the rate-independent flow rule by adding viscous dissipation terms with coefficients α and , where 0<<1 and α>0 is a fixed parameter. Therefore for α different from 1 the variables u and z have different relaxation rates. We address the vanishing-viscosity analysis as tends to 0 in the viscous system. We prove that, up to a subsequence, (reparameterized) viscous solutions converge to a parameterized curve yielding a Balanced Viscosity solution to the original rate-independent system and providing an accurate description of the system behavior at jumps. We also give a reformulation of the notion of Balanced Viscosity solution in terms of a system of subdifferential inclusions, showing that the viscosity in u and the one in z are involved in the jump dynamics in different ways, according to whether α >1, α=1, or 0<α<1.