# American Institute of Mathematical Sciences

June  2012, 32(6): 1997-2025. doi: 10.3934/dcds.2012.32.1997

## On a phase field model for solid-liquid phase transitions

 1 Université de Lyon, CNRS UMR 5208 & Université Lyon 1, Institut Camille Jordan, 43 bd du 11 novembre 1918, F-69622 Villeurbanne cedex, France, France 2 Université Blaise Pascal & CNRS UMR 6620, Laboratoire de Mathématiques, Campus des Cézeaux, B.P. 80026, F-63177 Aubière cedex, France 3 CEA-Grenoble (DEN/DTP/SMTH), 17, rue des martyrs, F-38054 Grenoble cedex 9, France

Received  January 2011 Revised  July 2011 Published  February 2012

A new phase field model is introduced, which can be viewed as a nontrivial generalisation of what is known as the Caginalp model. It involves in particular nonlinear diffusion terms. By formal asymptotic analysis, it is shown that in the sharp interface limit it still yields a Stefan-like model with: 1) a generalized Gibbs-Thomson relation telling how much the interface temperature differs from the equilibrium temperature when the interface is moving or/and is curved with surface tension; 2) a jump condition for the heat flux, which turns out to depend on the latent heat and on the velocity of the interface with a new, nonlinear term compared to standard models. From the PDE analysis point of view, the initial-boundary value problem is proved to be locally well-posed in time (for smooth data).
Citation: Sylvie Benzoni-Gavage, Laurent Chupin, Didier Jamet, Julien Vovelle. On a phase field model for solid-liquid phase transitions. Discrete & Continuous Dynamical Systems - A, 2012, 32 (6) : 1997-2025. doi: 10.3934/dcds.2012.32.1997
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