Multiscale dynamics of hysteretic phase interfaces
Final Report Abstract
Forward-backward diffusion equations with bistable nonlinearity arise in many branches of the sciences but are mathematically ill-posed as they admit a plethora of possible solutions. The physically relevant solutions depend on microscopic details, which are usually encoded by additional regularizing terms that account for small spatial scales and fast relaxation processes. A particularly interesting example is the viscous regularization as it predicts strong hysteretic effects and different types of phase boundaries. The corresponding nonlinear partial differential equations have been studied intensively during the last three decades but a complete rigorous understanding is still missing. In the main part of this project we focused on very special nonlinearities and studied two different time-discretizations of the underlying dynamical models in the limit of vanishing step size. Our first convergence result provides single-interface solutions to a free boundary problem which combines regular bulk diffusion with the Stefan condition and a hysteretic flow rule for phase boundaries. The second one establishes the existence of related solutions for the viscous regularization and also allows for both moving and standing phase interfaces. The corresponding proofs rely on a careful multi-scale analysis of mesoscopic fluctuations that propagate in space and time. We also characterized all monotone traveling wave solutions to the viscous model and investigated the stability of hysteretic phase interfaces in a nonlocal particle model. Our results are partly obtained in collaboration with Carina Geldhauser and Barbara Niethammer.
Publications
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Instability of Hysteretic Phase Interfaces in a Mean-Field Model with Inhomogeneities. SIAM Journal on Applied Mathematics, 83(4), 1422-1443.
Herrmann, Michael & Niethammer, Barbara
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Mehrskalendynamik hysteretischer Phasengrenzen in zeitdiskreten Vorwärts-Ruckwärts-Diffusionsgleichungen. doctoral thesis in mathematics, Technische Universitat Braunschweig
D. Janßen
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Traveling phase interfaces in viscous forward-backward diffusion equations
C. Geldhauser, M. Herrmann & D. Janßen
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Hysteretic dynamics of phase interfaces in bilinear forward-backward diffusion equations
M. Herrmann & D. Janßen
