Stefan problem with phase relaxation. (English) Zbl 0585.35053

Visintin, Augusto

IMA J. Appl. Math. 34, 225-245 (1985).

”Taking account of a microscopical model for dynamical supercooling and superheating effects, the usual equilibrium condition prescribing a fixed temperature at the interface between two phases is replaced by relaxation dynamics for the phase variable \(\chi\), representing the concentration of one of the two phases.... The well-posedness of an initial boundary value problem is proved. The standard Stefan problem is obtained as a limit case...” (from the author’s abstract).
The initial boundary value problem with phase relaxation is, to find \(\theta\), \(\chi\) in suitable spaces of functions on a space-time cylinder, satisfying \[ \partial_ t(c\theta +L\chi)+A\theta =f,\quad \epsilon \partial_ t\chi +H^{-1}(\chi)\ni \beta (\theta),\quad \theta (0)=\theta^ 0,\quad \chi (0)=\chi^ 0. \] Here C, L, \(\epsilon\) are positive constants, A is some time-independent second order elliptic operator together with a mixed Dirichlet-Neumann condition, \(\beta\) is a real function, monotone near 0 and \(\beta (0)=0\), \(H^{-1}\) is the inverse Heaviside graph (i.e. \(H^{-1}(s)=(-\infty,0],\) \(\{\) \(0\}\), [0,\(\infty)\), \(\emptyset\) provided \(s=0\), \(s\in (0,1)\), \(s=1\) or \(| s| >1\) resp.). The unknown function \(\theta\) describes the deviation from the equilibrium temperature (melting point).
Existence is proved by a method of Rothe-type and the limit \(\epsilon\) \(\to 0\) yields the standard Stefan problem.

Reviewer: K.J.Witsch

MSC:

35K60	Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35Q99	Partial differential equations of mathematical physics and other areas of application
47H20	Semigroups of nonlinear operators
35A05	General existence and uniqueness theorems (PDE) (MSC2000)
35B65	Smoothness and regularity of solutions to PDEs

Keywords:

heat diffusion equation; relaxation dynamics; nonlinear semigroups; well- posedness; initial boundary value problem; Stefan problem; phase relaxation; time-independent second order elliptic operator; mixed Dirichlet-Neumann condition; method of Rothe-type