For gradient systems depending on a microstructure, it is desirable to derive a macroscopic gradient structure describing the effective behavior of the microscopic scale on the macroscopic evolution. We introduce a notion of evolutionary Gamma-convergence that relates the microscopic energy and the microscopic dissipation potential with their macroscopic limits via Gamma-convergence. This new notion generalizes the concept of EDP-convergence, which was introduced in [26], and is now called relaxed EDP-convergence. Both notions are based on De Giorgi’s energy-dissipation principle (EDP), however the special structure of the dissipation functional in terms of the primal and dual dissipation potential is, in general, not preserved under Gamma-convergence. By using suitable tiltings we study the kinetic relation directly and, thus, are able to derive a unique macroscopic dissipation potential. The wiggly-energy model of Abeyaratne-Chu-James (1996) serves as a prototypical example where this nontrivial limit passage can be fully analyzed.

The notion of Energy-Dissipation-Principle convergence (EDPconvergence) is used to derive effective evolution equations for gradient systems describing diffusion in a structure consisting of several thin layers in the limit of vanishing layer thickness. The thicknesses of the sublayers tend to zero with different rates and the diffusion coefficients scale suitably. The Fokker-Planck equation can be formulated as gradient-flow equation with respect to the logarithmic relative entropy of the system and a quadratic Wasserstein-type gradient structure. The EDP-convergence of the gradient system is shown by proving suitable asymptotic lower limits of the entropy and the total dissipation functional. The crucial point is that the limiting evolution is again described by a gradient system, however, now the dissipation potential is not longer quadratic but is given in terms of the hyperbolic cosine. The latter describes jump processes across the thin layers and is related to the Marcelin-de Donder kinetics.

We present analytical and numerical investigations of two anti-symmetrically coupled 1D Swift-Hohenberg equations (SHEs) with cubic nonlinearities. The SHE provides a generic formulation for pattern formation at a characteristic length scale. A linear stability analysis of the homogeneous state reveals a wave instability in addition to the usual Turing instability of uncoupled SHEs. We performed weakly nonlinear analysis in the vicinity of the codimension-two point of the Turingwave instability, resulting in a set of coupled amplitude equations for the Turing pattern as well as left and right traveling waves. In particular, these complex Ginzburg-Landau-type equations predict two major things: there exists a parameter regime where multiple different patterns are stable with respect to each other; and that the amplitudes of different patterns interact by local mutual suppression. In consequence, different patterns can coexist in distinct spatial regions, separated by localized interfaces. We identified specific mechanisms for controlling the position of these interfaces, which distinguish what kinds of patterns the interface connects and thus allow for global pattern selection. Extensive simulations of the original SHEs confirm our results. Originally set up to describe convection instabilities, the Swift-Hohenberg equation (SHE)generically captures the formation of stationary patterns in nonequilibrium systems. A scalar parameter in this equation controls the bifurcation of a spatially homogeneous steady state to a so-called Turing-pattern, a stable stationary pattern with a finite length scale. In this paper, we consider two antisymmetrically coupled SHEs in one dimension. For comparable intrinsic length scales, the coupling gives rise to wave solutions emerging from an additional linear instability. A linear stability analysis of the homogeneous state allowed us to identify the codimension-two point where both patterns become unstable. Moreover, we have performed a weakly nonlinear analysis in the vicinity of this point to derive so-called amplitude equations which describe the evolution of patterns on large temporal and spatial scales. The amplitude dynamics of left and right traveling waves as well as the Turing pattern is captured in three mutually coupled complex Ginzburg-Landau equations. Numerical simulations of these equations and the coupled SHEs coincide, but the former provide substantially more insight into the coexistence of patterns. Each pattern locally suppresses other patterns and tends to saturate to a constant amplitude solution. In a certain range of parameters, both wave and Turing patterns are stable. In this bistable regime, random initial conditions typically result in the coexistence of different patterns in different spatial domains which are separated by distinct interfaces: wave-Turing interfaces as well as sources and sinks which separate left and right traveling waves, respectively. Now, the properties of these interfaces are pivotal for local control of globally competing patters: ...

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