# C05 - Effective models for interfaces with many scales

### Project Details

**Status:**In Progress**Head(s):**Prof. Mielke**Project member(s):**Martin Heida,**Participating Institution(s):**Weierstrass Institute Berlin (WIAS)**Area:**C: Bridging the micro-macro scale range

### Project Summary

We study multiscale effects arising at interfaces or surfaces of bulk materials. Nontrivial interactions between bulk and interface phenomena will include diffusion and reaction of chemical species, elastic deformations, delamination, or dry friction. The emphasis is on the analysis of evolutionary models with interaction between competing effects, each having its own spatial or temporal scale, like the interface thickness, the length scale of structures in bulk/interface, or diffusion lengths. Using weak convergence methods such as G-convergence, homogenization, or multiscale convergence we aim to provide thermodynamically consistent root models for processes on many scales, to develop new analytical methodologies that allow us to derive and justify stationary and evolutionary scaling limits, including hybrid models connecting small-scale models and effective models in different parts of the physical domain, to apply these methodologies to particular applications motivated by the CRC.

The general philosophy of this project is to consider structured equations such as gradient systems or Hamiltonian systems with dissipation, namely GENERIC systems (General Equations for Non-Equilibrium Reversible Irreversible Coupling, cf. [Ött05]), and to pass to scaling limits within these structures. The aim is to obtain in this way not only the information about the convergence of solutions but also of other relevant quantities as for example energies, entropies, or dissipation potentials.

The general strategies will be developed on the basis of well chosen concrete examples that are motivated by the topics of the CRC. In particular, we plan to study scaling limits for bulk-interface reaction-diffusion systems coupled with elasticity, friction models, or delamination. Thereby, we will investigate both, PDE-based models with many internal scales as well as discrete or atomistic models of elastic double strings with heterogeneous interatomic potentials. From these root models the relevant scales shall be identi.ed by dimensional analysis. New methodologies will be developed in order to pass to the scaling limits in these complex evolutionary systems. The structure of the models as generalized gradient flow systems or as GENERIC systems will help to identify suitable notions of convergence for evolutionary systems. In general, the separate convergence of energy functionals and dissipation potentials will not be sufficient in order to guarantee that solutions of the corresponding systems converge to solutions of the system defined by the limit functionals.

We expect that the research program of this project will also provide valuable information to identify effective quantities that can be simulated ef.ciently and that allow for the construction of numerical approximation schemes that are thermodynamically consistent and robust with respect to the multiple scales.

**Link to the C05 Homepage at WIAS**

### Project Publications

Mielke, Alexander and Netz, Roland R. and Zendehroud, Sina (2020) *A rigorous derivation and energetics of a wave equation with fractional damping.* arXiv . pp. 1-20. (Submitted)

Heida, M. and Kornhuber, R. and Podlesny, J. (2020) *Fractal homogenization of multiscale interface problems.* Multiscale Modeling & Simulation, 18 (1). pp. 294-314. ISSN 1540-3459

Heida, M. and Neukamm, S. and Varga, M. (2019) *Stochastic homogenization of Λ-convex gradient flows.* SFB 1114 Preprint in arXiv:1905.02562 . pp. 1-28. (Unpublished)

Franchi, B. and Heida, M. and Lorenzani, S. (2019) *A Mathematical model for Alzheimer's disease: An approach via stochastic homogenization of the Smoluchowski equation.* SFB 1114 Preprint in arXiv:1904.11015 . pp. 1-43. (Unpublished)

Heida, M. and Nesenenko, S. (2019) *Stochastic homogenization of rate-dependent models of monotone type in plasticity.* Asymptotic Analysis, 112 (3-4). pp. 185-212. ISSN 0921-7134

Donati, L. and Heida, M. and Weber, M. and Keller, B. (2018) *Estimation of the infinitesimal generator by square-root approximation.* Journal of Physics: Condensed Matter, 30 (42). p. 425201. ISSN 0953-8984, ESSN: 1361-648X

Heida, M. and Patterson, R. I. A and Renger, M. (2018) *Topologies and measures on the space of functions of bounded variation taking values in a Banach or metric space.* J. Evol. Equ. . pp. 1-42. ISSN Online: 1424-3202 Print: 1424-3199

Heida, M. (2018) *Convergences of the squareroot approximation scheme to the Fokker–Planck operator.* Mathematical Models and Methods in Applied Sciences, 28 (13). pp. 2599-2635. ISSN 0218-2025, ESSN: 1793-6314

Mielke, A. and Rossi, R. and Savaré, G. (2018) *Global existence results for viscoplasticity at finite strain.* Archive for Rational Mechanics and Analysis, 227 (1). pp. 423-475. ISSN Print: 0003-9527; Online: 1432-0673

Heida, M. and Neukamm, S. and Varga, M. (2017) *Stochastic unfolding and homogenization.* SFB 1114 Preprint at WIAS 12/2017 . pp. 1-45. (Unpublished)

Liero, M. and Mielke, A. and Savaré, G. (2017) *Optimal Entropy-Transport problems and a new Hellinger-Kantorovich distance between positive measures.* Invent. math. . pp. 1-149. ISSN 1432-1297 (online)

Mielke, A. and Patterson, R. I. A and Peletier, M. A. and Renger, M. (2017) *Non-equilibrium thermodynamical principles for chemical reactions with mass-action kinetics.* SIAM Journal on Applied Mathematics, 77 (4). pp. 1562-1585. ISSN 1095-712X (online)

Heida, M. (2017) *Stochastic homogenization of rate-independent systems and applications.* Continuum Mech. Thermodyn., 29 (3). pp. 853-894. ISSN 1432-0959 (online) 0935-1175 (print)

Gussmann, P. and Mielke, A. (2017) *Linearized elasticity as Mosco-limit of finite elasticity in the presence of cracks.* Adv. Calc. Var. . (Submitted)

Flegel, F. and Heida, M. and Slowik, M. (2017) *Homogenization theory for the random conductance model with degenerate ergodic weights and unbounded-range jumps.* SFB 1114 Preprint in arXiv:1702.02860 . (Unpublished)

Heida, M. and Schweizer, B. (2017) *Stochastic homogenization of plasticity equations.* ESAIM: Control, Optimisation and Calculus of Variations . pp. 1-30. (Submitted)

Liero, M. and Mielke, A. and Peletier, M. A. and Renger, M. (2017) *On microscopic origins of generalized gradient structures.* Discrete and Continuous Dynamical Systems - Series S, 10 (1).

Heida, M. and Mielke, A. (2017) *Averaging of time-periodic dissipation potentials in rate-independent processes.* Discrete and Continuous Dynamical Systems - Series S, 10 (6). pp. 1303-1327.

Mielke, A. (2017) *Three examples concerning the interaction of dry friction and oscillations.* In: Trends on Application of Mathematics to Mechanics. Springer INdAM series. (In Press)

Mielke, A. and Mittnenzweig, M. (2017) *Convergence to Equilibrium in Energy-Reaction–Diffusion Systems Using Vector-Valued Functional Inequalities.* Journal of Nonlinear Science . pp. 1-42. ISSN 1432-1467 (online)

Bonetti, E. and Rocca, E. and Rossi, R. and Thomas, M. (2016) *A rate-independent gradient system in damage coupled with plasticity via structured strains.* ESAIM: Proceedings and Surveys, 54 . pp. 54-69.

Liero, M. and Mielke, A. and Savaré, G. (2016) *Optimal Transport in Competition with Reaction: The Hellinger--Kantorovich Distance and Geodesic Curves.* SIAM J. Math. Anal., 48 (2). pp. 2869-2911. ISSN 1095-7154 (online)

Mielke, A. and Peletier, M. A. and Renger, M. (2016) *A generalization of Onsager's reciprocity relations to gradient flows with nonlinear mobility.* Journal of Non-Equilibrium Thermodynamics, 41 (2).

Mielke, A. and Rossi, R. and Savaré, G. (2016) *Balanced-Viscosity solutions for multi-rate systems.* Journal of Physics: Conference Series, 727 . pp. 1-27.