Stability of Equilibria for the Stefan Problem With Surface Tension

Abstract

Abstract. We characterize the equilibrium states for the two-phase Stefan problem with surface tension and with or without kinetic undercooling, and we analyze their stability in dependence of physical and geometric quantities.

On convergence of solutions to equilibria for quasilinear parabolic problems

“…[26,Theorem 2.1]) that the spectrum of L consists of countably many real nonnegative eigenvalues of finite algebraic multiplicity, and that 0 is a semi-simple eigenvalue of L with multiplicity n +1, see also [15,Proposition 5.4 and Lemma 6.1]. Moreover, the kernel of L is given by N(L) = span{Y 0 , Y 1 , .…”

Center manifolds and dynamics near equilibria of quasilinear parabolic systems with fully nonlinear boundary conditions

“…Moreover, there are important applications where M cu consists of equilibria only, see e.g. [10,Prop.6.4], [15]. Thus it is quite possible that one can check the stability of u * with respect to the semiflow on M cu generated by (1.1) without knowing a priori that u * is stable with respect to the full semiflow of (1.1) on M. In Theorem 6.1 below we show that u * is stable on M under the following conditions: s(−A 0 ) ≤ 0, u * is stable on M cu = M c , P cu = P c has finite rank, and the additional regularity assumption (RR) holds.…”

“…The results of the present paper do not directly cover such problems, but we think that our methods can be generalized in order to deal with moving boundaries and transmission problems in future work. We note that the recent work [15] already contains the linear spectral analysis which is necessary for applications of center manifold theory to the Stefan problem with surface tension.…”

Preface

“…He obtained deep results on the Cahn-Hilliard equation [68,70,73,94] and on the Stefan problem [58,79,85,104,109,113,115]. In [86], boundary conditions of relaxation type were introduced for the first time, building a general framework for a large class of nonstandard boundary value problems, including free boundary value problems and diffusion on the boundary.…”

“…This result turned out to have wide applications. Over the last 15 years, Jan has devoted his mathematical interests to the study of moving boundary problems in fluid flows and phase transitions [58,65,79,85,95,96,[98][99][100]103,104,106,108,111,113,115,118,120,122,123,129]. While processes with moving surfaces are omnipresent in nature, it turns out that their mathematical analysis poses great challenges.…”