Publication: The classical one-phase Stefan problem: a catalog of interface behaviors
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2001
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Springer
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Abstract
It is well known that the ice-water phase transition can be modelled by the following set of equations:
∂T ∂t =ΔTfor x∈R N ∖Ω(t),t>0, (1)
T=0for x∈∂Ω(t),t>0, (2)
vn =−∂T ∂n for x∈∂Ω(t),t>0, (3)
where T(x,t) stands for the temperature at the point x and time t and Ω(t)⊂R N (N=2,3) denotes the ice region (with T=0), while the heat equation (1) governs the water region (with T>0 in the interior). Of course, the physical constants are normalized (e.g., condition (2) means that the melting temperature T m is assumed to be 0). Moreover n stands for the unit outer normal vector at any given point in ∂Ω(t) and the local velocity of the moving interface ∂Ω(t) (the free boundary) is represented by vn. When the system (1)-(2)-(3) is complemented with suitable initial and boundary conditions, it is called the one-phase Stefan problem. We also have to mention a couple of related situations. The first occurs when the heat equation (1) governs the evolution of both phases (clearly with different diffusion coefficients) and it is termed the two-phase Stefan problem. The second is the so-called supercooled Stefan problem and concerns the case when the ice grows in a medium with cooled (T<0) water. The authors outline that a major drawback to getting global results to system (1)-(2)-(3) consists in showing that an initially smooth interface stays smooth for all subsequent times. For instance, it is known that it is true (in some sense) locally in time. Hence, an interface, regular at time t=0, can develop singularities in a finite time t=t ∗ >0. Indeed, here it is proved that this occurs and a complete classification of the singularities is provided. Without entering into details, we only mention that stable and unstable singularities are carefully studied in two and three space dimensions. As for the tools of the procedure, a first step consists of the application of the so-called Baiocchi transformation to T(x,t). It leads to the following semilinear parabolic equation—instead of (1)—
∂u ∂t −Δu+H(u)=0for x∈R N ,0<t<t ∗ , (4)
where H denotes the Heaviside function. Some particular solutions of (4) play an essential role in the analysis. Although the authors actually neglect some physical phenomena (surface tension anisotropy, kinetic and molecular effects), addressing only the simple situation (1)-(2)-(3), their work should contribute to some real progress in the analysis of the general question of the occurrence of singularities in free boundary problems.