Qualitative behavior of solutions for thermodynamically consistent Stefan problems with surface tension

The qualitative behavior of a thermodynamically consistent two-phase Stefan problem with surface tension and with or without kinetic undercooling is studied. It is shown that these problems generate local semiflows in well-defined state manifolds. If a solution does not exhibit singularities in a sense made precise below, it is proved that it exists globally in time and its orbit is relatively compact. In addition, stability and instability of equilibria is studied. In particular, it is shown that multiple spheres of the same radius are unstable, reminiscent of the onset of Ostwald ripening.Comment: 56 pages. Expanded introduction, added references. This revised version is published in Arch. Ration. Mech. Anal. (207) (2013), 611-66

Orthogonality conditions and asymptotic stability in the Stefan problem with surface tension

We prove nonlinear asymptotic stability of steady spheres in the two-phase Stefan problem with surface tension. Our method relies on the introduction of appropriate orthogonality conditions in conjunction with a high-order energy method.Comment: 25 pages, important references added, two remarks added, typos correcte

The generalized formulation for the Stefan problem with kinetic undercooling

The paper deals with the one-dimensional Stefan problem, where the phase fraction for t=0 can be a distributed function. The melting temperature satisfies a kinetic undercooling condition, which yields a Hamilton-Jacobi equation for the phase function. We prove the existence of a solution with the non-smooth initial data. For the Hamilton-Jacobi equation we find a viscosity solution, which is given as a minimum value of some functional. We study also the limit case, when a kinetic parameter tends to zero. Doing so we obtain a weak solution of the undercooled Stefan problem without the kinetic condition. (orig.)SIGLEAvailable from TIB Hannover: RN 7879(9605) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekDEGerman

Periodic solutions of the Stefan problem with hysteresis-type boundary conditions

We consider the Stefan problem with Dirichlet boundary conditions depending on a hysteresis functional where the free boundary is involved. We show existence of a positive value T and existence of a T-periodic solution of the problem, provided the Stefan number is sufficiently small and the hysteresis functional is described by the elementary rectangular hysteresis loop. If in addition the Preisach hysteresis operator is Lipschitz-continuous we prove that every periodic solution must be stationary. (orig.)Available from TIB Hannover: RO 7722(399) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman