Traveling phase interfaces in viscous forward-backward diffusion equations

The viscous regularization of an ill-posed diffusion equation with bistable nonlinearity predicts a hysteretic behavior of dynamical phase transitions but a complete mathematical \mbox{understanding} of the intricate multiscale evolution is still missing. We shed light on the fine structure of \mbox...

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Bibliographic Details
Published inarXiv.org
Main Authors Geldhauser, Carina, Herrmann, Michael, Janßen, Dirk
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 20.11.2023
Subjects
Constitutive relationships
Fine structure
Phase transitions
Regularization
Traveling waves
Wave propagation
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Summary:The viscous regularization of an ill-posed diffusion equation with bistable nonlinearity predicts a hysteretic behavior of dynamical phase transitions but a complete mathematical \mbox{understanding} of the intricate multiscale evolution is still missing. We shed light on the fine structure of \mbox{propagating} phase boundaries by carefully examining traveling wave solutions in a special case. Assuming a trilinear constitutive relation we characterize all waves that possess a monotone \mbox{profile} and connect the two phases by a single interface of positive width. We further study the two sharp-interface regimes related to either vanishing viscosity or the bilinear limit.
ISSN:2331-8422