Effective diffusion in thin structures via generalized gradient systems and EDP-convergence
- Frenzel, Thomas
- Liero, Matthias
2010 Mathematics Subject Classification
- 35K20 35K10 35K57 49S99
- Fokker--Planck equation, dimension reduction, sandwich structure, Γ-convergence, gradient system, EDP-convergence, Wasserstein gradient flow
The notion of Energy-Dissipation-Principle convergence (EDP-convergence) 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.