2011, Mielke, Alexander, Müller, Ingo

We consider the passage from viscous system to rate-independent system in the limit of vanishing viscosity and for wiggly energies. Our new convergence approach is based on the (R,R*) formulation by De Giorgi, where we pass to the Γ limit in the dissipation functional. The difficulty is that the type of dissipation changes from a quadratic functional to one that is homogeneous of degree 1. The analysis uses the decomposition of the restoring force into a macroscopic part and a fluctuating part, where the latter is handled via homogenization.

2011, Glitzky, Annegret, Mielke, Alexander

We derive gradient-flow formulations for systems describing drift-diffusion processes of a finite number of species which undergo mass-action type reversible reactions. Our investigations cover heterostructures, where material parameter may depend in a nonsmooth way on the space variable. The main results concern a gradient flow formulation for electro-reaction-diffusion systems with active interfaces permitting drift-diffusion processes and reactions of species living on the interface and transfer mechanisms allowing bulk species to jump into an interface or to pass through interfaces. The gradient flows are formulated in terms of two functionals: the free energy and the dissipation potential. Both functionals consist of a bulk and an interface integral. The interface integrals determine the interface dynamics as well as the self-consistent coupling to the model in the bulk. The advantage of the gradient structure is that it automatically generates thermodynamically consistent models.

2006, Mielke, Alexander, Ortiz, Michael

This 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.

2011, Mielke, Alexander

We consider finite-dimensional, time-continuous Markov chains satisfying the detailed balance condition as gradient systems with the relative entropy E as driving functional. The Riemannian metric is defined via its inverse matrix called the Onsager matrix K. We provide methods for establishing geodesic λ-convexity of the entropy and treat several examples including some more general nonlinear reaction systems.

2010, Mielke, Alexander

The theory of GENERIC (general equations for non-equilibrium reversible irreversibel coupling) is presented in a mathematical form. It is applied first to simple mechanical systems and then generalized to standard generalized materials. It is shown that nonisothermal viscoplasticity can be cast into the form of GENERIC, if the dissipative structure is generalized from linear functionals to the more general subdifferential of convex potentials

2011, Mielke, Alexander

We consider rate-independent evolutionary systems over a physically domain Ω that are governed by simple hysteresis operators at each material point. For multiscale systems where ε denotes the ratio between the microscopic and the macroscopic length scale, we show that in the limit ε → 0 we are led to systems where the hysteresis operators at each macroscopic point is a generalized Prandtl-Ishlinskii operator.

2012, Liero, Matthias, Mielke, Alexander

We consider systems of reaction-diffusion equations as gradient systems with respect to an entropy functional and a dissipation metric given in terms of a so-called Onsager operator, which is a sum of a diffusion part of Wasserstein type and a reaction part. We provide methods for establishing geodesic lambda-convexity of the entropy functional by purely differential methods, thus circumventing arguments from mass transportation. Finally, several examples, including a drift-diffusion system, provide a survey on the applicability of the theory. We consider systems of reaction-diffusion equations as gradient systems with respect to an entropy functional and a dissipation metric given in terms of a so-called Onsager operator, which is a sum of a diffusion part of Wasserstein type and a reaction part. We provide methods for establishing geodesic lambda-convexity of the entropy functional by purely differential methods, thus circumventing arguments from mass transportation. Finally, several examples, including a drift-diffusion system, provide a survey on the applicability of the theory.

2010, Mielke, Alexander, Gajewski, Herbert

In recent years the theory of Wasserstein distances has opened up a new treatment of the diffusion equations as gradient systems, where the entropy takes the role of the driving functional and where the space is equipped with the Wasserstein metric. We show that this structure can be generalized to closed reaction-diffusion systems, where the free energy (or the entropy) is the driving functional and further conserved quantities may exists, like the total number of chemical species. The metric is constructed by using the dual dissipation potential, which is a convex function of the chemical potentials. In particular, it is possible to treat diffusion and reaction terms simultaneously. The same ideas extend to semiconductor equations involving the electron and hole densities, the electrostatic potential, and the temperature.

2010, Liero, Matthias, Mielke, Alexander

This paper is devoted to dimension reduction for linearized elastoplasticity in the rate-independent case. The reference configuration of the three-dimensional elastoplastic body has a two-dimensional middle surface and a positive but small thickness. Under suitable scalings we derive a limiting model for the case in which the thickness of the plate tends to 0. This model contains membrane and plate deformations (linear Kirchhoff--Love plate), which are coupled via plastic strains. We establish strong convergence of the solutions in the natural energy space. The analysis uses an abstract Gamma-convergence theory for rate-independent evolutionary systems that is based on the notion of energetic solutions. This concept is formulated via an energy-storage functional and a dissipation functional, such that energetic solutions are defined in terms of a stability condition and an energy balance. The Mosco convergence of the quadratic energy-storage functional follows the arguments of the elastic case. To handle the evolutionary situation the interplay with the dissipation functional is controlled by cancellation properties for Mosco-convergent quadratic energies

2011, Mielke, Alexander, Frémond, Michel

We show that many couplings between parabolic systems for processes in solids can be formulated as a gradient system with respect to the total free energy or the total entropy. This includes Allen-Cahn, Cahn-Hilliard, and reaction-diffusion systems and the heat equation. For this, we write the coupled system as an Onsager system (X,F,K) defining the evolution $dot U$= - K(U) DF(U). Here F is the driving functional, while the Onsager operator K(U) is symmetric and positive semidefinite. If the inverse G=K-1 exists, the triple (X,F,G) defines a gradient system. Onsager systems are well suited to model bulk-interface interactions by using the dual dissipation potential ?*(U, ?)= ư .