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- ItemPassing 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, MarcoWe 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. ...
- ItemLinearized elasticity as Mosco-limit of finite elasticity in the presence of cracks(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Gussmann, Pascal; Mielke, AlexanderThe 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.
- ItemFrom large deviations to Wasserstein gradient flows in multiple dimensions(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Erbar, Matthias; Maas, Jan; Renger, D.R. MichielWe study the large deviation rate functional for the empirical measure of independent Brownian particles with drift. In one dimension, it has been shown by Adams, Dirr, Peletier and Zimmer [ADPZ11] that this functional is asymptotically equivalent (in the sense of -convergence) to the JordanKinderlehrerOtto functional arising in the Wasserstein gradient flow structure of the FokkerPlanck equation. In higher dimensions, part of this statement (the lower bound) has been recently proved by Duong, Laschos and Renger, but the upper bound remained open, since the proof in [DLR13] relies on regularity properties of optimal transport maps that are restricted to one dimension. In this note we present a new proof of the upper bound, thereby generalising the result of [ADPZ11] to arbitrary dimensions.
- ItemMulticomponent incompressible fluids -- An asymptotic study(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Bothe, Dieter; Dreyer, Wolfgang; Druet, Pierre-ÉtienneThis paper investigates the asymptotic behavior of the Helmholtz free energy of mixtures at small compressibility. We start from a general representation for the local free energy that is valid in stable subregions of the phase diagram. On the basis of this representation we classify the admissible data to construct a thermodynamically consistent constitutive model. We then analyze the incompressible limit, where the molar volume becomes independent of pressure. Here we are confronted with two problems: (i) Our study shows that the physical system at hand cannot remain incompressible for arbitrary large deviations from a reference pressure unless its volume is linear in the composition. (ii) As a consequence of the 2nd law of thermodynamics, the incompressible limit implies that the molar volume becomes independent of temperature as well. Most applications, however, reveal the non-appropriateness of this property. According to our mathematical treatment, the free energy as a function of temperature and partial masses tends to a limit in the sense of epi-- or Gamma--convergence. In the context of the first problem, we study the mixing of two fluids to compare the linearity with experimental observations. The second problem will be treated by considering the asymptotic behavior of both a general inequality relating thermal expansion and compressibility and a PDE-system relying on the equations of balance for partial masses, momentum and the internal energy.
- ItemFast reaction limits via $Gamma$-convergence of the flux rate functional(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Peletier, Mark A.; Renger, D. R. MichielWe study the convergence of a sequence of evolution equations for measures supported on the nodes of a graph. The evolution equations themselves can be interpreted as the forward Kolmogorov equations of Markov jump processes, or equivalently as the equations for the concentrations in a network of linear reactions. The jump rates or reaction rates are divided in two classes; `slow' rates are constant, and `fast' rates are scaled as 1/∈, and we prove the convergence in the fast-reaction limit ∈ → 0. We establish a Γ-convergence result for the rate functional in terms of both the concentration at each node and the flux over each edge (the level-2.5 rate function). The limiting system is again described by a functional, and characterizes both fast and slow fluxes in the system. This method of proof has three advantages. First, no condition of detailed balance is required. Secondly, the formulation in terms of concentration and flux leads to a short and simple proof of the Γ-convergence; the price to pay is a more involved compactness proof. Finally, the method of proof deals with approximate solutions, for which the functional is not zero but small, without any changes.
- ItemA class of minimum principles for characterizing the trajectories and the relaxation of dissipative systems(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Mielke, Alexander; Ortiz, MichaelThis work is concerned with the reformulation of evolutionary problems in a weak form enabling consideration of solutions that may exhibit evolving microstructures. This reformulation is accomplished by expressing the evolutionary problem in variational form, i.e., by identifying a functional whose minimizers represent entire trajectories of the system. The particular class of functionals under consideration is derived by first defining a sequence of time-discretized minimum problems and subsequently formally passing to the limit of continuous time. The resulting functionals may be regarded as elliptic regularizations of the original evolutionary problem. We find that the $Gamma$-limits of interest are highly degenerate and provide limited information regarding the limiting trajectories of the system. Instead we seek to characterize the minimizing trajectories directly. The special class of problems characterized by a rate-independent dissipation functional is amenable to a particularly illuminating analysis. For these systems it is possible to derive a priori bounds that are independent of the regularizing parameter, whence it is possible to extract convergent subsequences and find the limiting trajectories. Under general assumptions on the functionals, we show that all such limits satisfy the energetic formulation (S) & (E) for rate-independent systems. Moreover, we show that the accumulation points of the regularized solutions solve the associated limiting energetic formulation.
- Item[Gamma]-limits and relaxations for rate-independent evolutionary problems(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Mielke, Alexander; Toubíček, Tomáš; Stefanelli, UlisseThis work uses the energetic formulation of rate-independent systems that is based on the stored-energy functionals ε and the dissipation distance D. For sequences (ε k)k ∈ ℕ and (D k)k ∈ ℕ we address the question under which conditions the limits q∞ of solutions qk: [0,T] → Q satisfy a suitable limit problem with limit functionals ε∞ and D∞, which are the corresponding Γ-limits. We derive a sufficient condition, called emphconditional upper semi-continuity of the stable sets, which is essential to guarantee that q∞ solves the limit problem. In particular, this condition holds if certain emphjoint recovery sequences exist. Moreover, we show that time-incremental minimization problems can be used to approximate the solutions. A first example involves the numerical approximation of functionals using finite-element spaces. A second example shows that the stop and the play operator convergece if the yield sets converge in the sense of Mosco. The third example deals with a problem developing microstructure in the limit k → ∞, which in the limit can be described by an effective macroscopic model.
- ItemDeriving amplitude equations via evolutionary [Gamma]-convergence(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Mielke, AlexanderWe 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.
- ItemOn evolutionary [Gamma]-convergence for gradient systems : in memory of Eduard, Waldemar, and Elli Mielke(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Mielke, AlexanderIn 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.