A nonlocal one-phase Stefan problem that develops mushy regions

Cristina Brändle (1) , Emmanuel Chasseigne (2, 3) , Fernando Quirós (4)

1 Departamento de Matematicas
2 FDP - Fédération de recherche Denis Poisson
3 LMPT - Laboratoire de Mathématiques et Physique Théorique
4 Departamento de Matemáticas [Madrid]

We study a nonlocal version of the one-phase Stefan problem which develops mushy regions, even if they were not present initially, a model which can be of interest at the mesoscopic scale. The equation involves a convolution with a compactly supported kernel. The created mushy regions have the size of the support of this kernel. If the kernel is suitably rescaled, such regions disappear and the solution converges to the solution of the usual local version of the one-phase Stefan problem. We prove that the model is well posed, and give several qualitative properties. In particular, the long-time behavior is identified by means of a nonlocal mesa solving an obstacle problem.

Mots clés: Stefan problem; mushy regions; nonlocal equations; degenerate parabolic equations

Domaines: Equations aux dérivées partielles [math.AP]

Cristina Brändle, Emmanuel Chasseigne, Fernando Quirós. A nonlocal one-phase Stefan problem that develops mushy regions. SIAM Journal on Mathematical Analysis, 2012, 44 (4), pp.3071-3100. ⟨10.1137/110849365⟩. ⟨hal-00626418⟩