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Existence of the global classical solution for a two-phase Stefan problem. (English) Zbl 0942.35168

This paper is concerned with the existence of the global classical solution for a two-phase multidimensional Stefan problem [see O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Ural’tseva, Linear and quasi-linear equations of parabolic type, Translations of Mathematical Monographs 23, AMS (1968; Zbl 0174.15403), Chapter 5, Section 9].
Precisely, let \(D=\{x\in \mathbb{R}^3: 0<R_1<|x|<R_2\}\), \(D_T=D\times (0,T)\), and \(B_i=\{x\in \mathbb{R}^3:|x|<R_i\}\) \((i=1,2)\), where \(T>0\) is an arbitrarily fixed number. The problem is to find a function \(u(x,t)\) and domains \(\Omega_T\), \(G_T\) satisfying \(\partial_t u=\Delta u\) in \(\Omega_T \cup G_T\), \[ \begin{aligned} & u= \varphi_i\text{ on }\partial B_i\times (0,T)\;(i=1,2),\\ & u^+=u^-=1,\;\sum^3_{i=1} \left({\partial u^-\over \partial x_i}-{\partial u^+ \over\partial x_i} \right) \cos(n,x_i) +\lambda \cos(n,t)= 0\text{ on }\gamma_T, \end{aligned} \] where \[ \Omega_T= \bigl\{(x,t)\in D_T:0< u(x,t) <1 \bigr \}, \;G_T=\bigl \{(x,t)\in D_T: u(x,t)>1 \bigr\}, \] \(\gamma_T= \partial \Omega_T \cap D_T= \partial G_T\cap D_T\), \(\lambda\) is a positive constant, \(n\) is the unit normal vector to \(\gamma_T\) directed tothe side of increase of \(u\), and \(u^+\), \(u^-\) are the boundary values on the surface \(\gamma_T\) taken from \(G_T\), \(\Omega_T\), respectively. The initial conditions are \[ \begin{aligned} & u(x,0)= \psi(x) \text{ in }\overline D,\;\psi(x)= \varphi_i (x,0)\text{ on }\partial B_i\;(i=1,2),\\ & 0\leq\psi(x) <1\text{ on }\partial B_1, \;\psi(x)>1 \text{ on } \partial B_2. \end{aligned} \] The function \(u\) is interpreted as the temperature of the medium, \(\gamma_T\) is the interface between the liquid and solid phases, and \(u=1\) is the temperature of melting.
In this paper the existence of the classical solution \(u\) with \(C^{2+\alpha, 1+{\alpha\over 2}}\) free boundary \(\gamma_T\) for any \(T>0\) is proved under the hypotheses \(\psi\in C^{2+ \alpha} (\overline D)\), \(\Delta\psi\leq 0\) and \({\partial\psi \over\partial |x|} >0\), in \(\overline D\), \(\varphi_1=0\) on \(\partial B_1\times (0,T)\), \(\varphi_2= \text{const}>1\) on \(\partial B_2\times (0,T)\), the compatibility conditions at \((x,0)\in \partial\Omega_T \cup\partial G_T\). By introducing a difference-differential elliptic approximation of the problem, the author obtains some uniform estimates and takes the limit in order to prove this result.

MSC:

35R35 Free boundary problems for PDEs
80A22 Stefan problems, phase changes, etc.
35A05 General existence and uniqueness theorems (PDE) (MSC2000)

Citations:

Zbl 0174.15403
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