Multiscale dynamics of hysteretic phase interfaces

• Applicant: Professor Dr. Michael Herrmann (†)
• Subject Area: Mathematics
• Term: from 2019 to 2023
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 438823303

Forward-backward diffusion equations arise in many branches of the natural and the engineering sciences but are ill-posed from a mathematical point of view as they allow for a plethora of possible solutions. The physically relevant solutions depend strongly on microscopic details, which are usually encoded by additional regularizing terms that account for small spatial scales and/or fast internal relaxation processes. The most prominent regularization of a system with forward- backward diffusion is the Cahn-Hilliard equation which has been studied intensively during the last decades. In particular, the sharp-interface limit is well understood and naturally related to mean- curvature flows and the classical Maxwell construction for admissible phase transitions.Another regularization is the so-called viscous approximation, in which the singular perturbation is given by a mixed derivative of third order. This equation has been studied by distinguished mathematicians but a complete understanding is still missing. More precisely, using formal asymptotic analysis it has been shown that the corresponding sharp-interface limit is governed by a hysteretic flow rule that predicts a more complex interface dynamics: the selection rule for admissible phase transitions now exhibits memory effects and depends not only on the current state of the system but also on its past. A rigorous justification of the hysteresis effects, however, has been open for a long time. This lack of rigorous understanding is caused by strong microscopic fluctuations, which are constantly produced near propagating phase interfaces and have a huge impact on the regularity of the PDE solutions in the small-parameter limit.In this project, we want to prove that the sharp-interface limit of the viscous regularization is in fact governed by a free boundary problem with a hysteretic Stefan condition. In a first step, we plan to study the problem under the simplifying assumptions of piecewise linear nonlinearities and well prepared single-interface initial data. Previous work on lattice systems with forward-backward diffusion as well as preliminary numerical simulations of the viscous PDE strongly indicate that these simplifications allow us to control the fluctuations up to high accuracy and to characterize their cumulative impact on large spatial and temporal scales. Essential ingredient to our asymptotic analysis are suitable time discretizations of the viscous PDE which enable us to keep track of the interface position and to handle both pinning and depinning events in a unified manner. In the second part of our project, we hope to extend our results to more general bistable nonlinearities and want to study the convergence of the underlying gradient flow structure. We further wish to investigate the viscous sharp-interface limit in two space dimensions by means of numerical simulations and formal asymptotic analysis.

• DFG Programme: Research Grants