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Author: Mielke, Alexander × End date: 2016 × Start date: 2010 × Subject: Gamma convergence ×

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    Passing to the limit in a Wasserstein gradient flow : from diffusion to reaction
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) Arnrich, Steffen; Mielke, Alexander; Peletier, Mark A.; Savar´e, Giuseppe; Veneroni, Marco
    We study a singular-limit problem arising in the modelling of chemical reactions. At finite e>0, the system is described by a Fokker-Planck convection-diffusion equation with a double-well convection potential. This potential is scaled by 1/e, and in the limit eto0, the solution concentrates onto the two wells, resulting into a limiting system that is a pair of ordinary differential equations for the density at the two wells. This convergence has been proved in Peletier, Savaré, and Veneroni, em SIAM Journal on Mathematical Analysis, 42(4):1805--1825, 2010, using the linear structure of the equation. In this paper we re-prove the result by using solely the Wasserstein gradient-flow structure of the system. In particular, we make no use of the linearity, nor of the fact that it is a second-order system. The first key step in this approach is a reformulation of the equation as the minimization of an action functional that captures the propety of being a emphcurve of maximal slope in an integrated form. The second important step is a rescaling of space. Using only the Wasserstein gradient-flow structure, we prove that the sequence of rescaled solutions is pre-compact in an appropriate topology. ...
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    Linearized elasticity as Mosco-limit of finite elasticity in the presence of cracks
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Gussmann, Pascal; Mielke, Alexander
    The small-deformation limit of finite elasticity is considered in presence of a given crack. The rescaled finite energies with the constraint of global injectivity are shown to Gamma-converge to the linearized elastic energy with a local constraint of non-interpenetration along the crack.
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    Deriving amplitude equations via evolutionary [Gamma]-convergence
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Mielke, Alexander
    We discuss the justification of the GinzburgLandau equation with real coefficients as an amplitude equation for the weakly unstable one-dimensional SwiftHohenberg equation. In contrast to classical justification approaches we employ the method of evolutionary [Gamma]-convergence by reformulating both equation as gradient systems. Using a suitable linear transformation we show [Gamma]-convergence of the associated energies in suitable function spaces. The limit passage of the time-dependent problem relies on the recent theory of evolutionary variational inequalities for families of uniformly convex functionals as developed by Daneri and Savare 2010. In the case of a cubic energy it suffices that the initial conditions converge strongly in L2, while for the case of a quadratic nonlinearity we need to impose weak convergence in H1. However, we do not need wellpreparedness of the initial conditions.
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    On evolutionary [Gamma]-convergence for gradient systems : in memory of Eduard, Waldemar, and Elli Mielke
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Mielke, Alexander
    In these notes we discuss general approaches for rigorously deriving limits of generalized gradient flows. Our point of view is that a generalized gradient system is defined in terms of two functionals, namely the energy functional E and the dissipation potential R or the associated dissipation distance. We assume that the functionals depend on a small parameter and that the associated gradient systems have solutions u. We investigate the question under which conditions the limits u of (subsequences of) the solutions u are solutions of the gradient system generated by the [Gamma]-limits E0 and R0. Here the choice of the right topology will be crucial awell as additional structural conditions. We cover classical gradient systems, where R is quadratic, and rate-independent systems as well as the passage from classical gradient to rate-independent systems. Various examples, such as periodic homogenization, are used to illustrate the abstract concepts and results.