## The ill-posed Hele-Shaw model and the Stefan problem for supercooled water.(English)Zbl 0621.35102

Let D,G be open sets with regular boundaries, such that $$\bar G\subset D\subset {\mathbb{R}}^ n$$. Set $$\Omega =D\setminus \bar G$$ and $$\Omega_ T=\Omega \times (0,T]$$, $$0<T<\infty$$. The following system is a classical multidimensional scheme for the Hele-Shaw cell $$(n=2$$ is the physical case). Find $$T>0$$ (the extinction time) and a pair of functions u(x,t), $$\Phi$$ (x,t) defined on $${\bar \Omega}_ T$$, such that (i) for $$0<t\leq T$$, u is harmonic in the set $$\Omega$$ (t)$$\subset \Omega$$ where $$\Phi (x,t)<0$$, (ii) $$u=0$$ and $$\nabla_ xu\cdot \nabla_ x\Phi =\Phi_ t$$ on $$\partial \Omega (t)\setminus \partial G$$, (iii) $$\partial u/\partial \nu =Q>0$$ on $$\partial G\times (0,T)$$, (iv) $$u\leq 0$$ in $$\Omega_ T$$, $$u(x,0)<0$$ in $$\Omega$$, (v) $$u(x,T)=0$$ in $$\Omega$$.
In (iii), $$\nu$$ is the outward normal to $$\partial G$$. The function u is related to pressure within the liquid in the Hele-Shaw flow, and $$Q>0$$ means that the flow is produced by suction at the boundary $$\partial G$$. In such a situation the problem is known to be ill posed. The weak formulation is based on the introduction of the function $$v(x,t)=\int^{T}_{t}u(x,\tau)d\tau$$ and of the additional unknown $$\xi$$ (x)$$\in H(u(x,T))$$ (H denotes the Heaviside function). The weak equation is written in the sense of distributions -$$\Delta$$ $$v\in H(v)- \xi (x)$$, $$t\in [0,T]$$, and T is determined by integration: $$Q| \partial G| T=\int_{\Omega}(1-\xi (x))dx.$$
The function $$\xi$$ is called ”the terminal phase” and can be associated to the dendritic structure of the terminal configuration. It is proved that if (T,$$\xi$$,v) is a weak solution, then $$V(x)=v(x,0)$$ has to satisfy a variational equality, thus implying that the set $$\Omega$$ (0) has to belong to a particular class. Conversely, for $$\Omega$$ given in this class and $$\xi$$ prescribed in a convenient set is it proved that for any $$T>0$$ there exists a unique weak solution (T,$$\xi$$,v) with $$v,v_ t\Delta v$$ satisfying a priori bounds and $$\nabla_ xv$$ being Hölder continuous. A penalization method is used. Analyticity of the free boundary with respect to space variables and Hölder continuity with respect to time is then proved under special assumptions on $$\xi$$ and on G. The results allow the authors to interpret the phenomenon of fingering in the sense that any fingered shape (depending on the choice of $$\xi)$$ can develop from a nearly spherical initial domain $$\Omega$$.
A parallel analysis is carried out for the so-called Stefan problem for supercooled water, differing from the previous one in the fact that the equation $$u_ t-\Delta u=0$$ replaces $$-\Delta u=0$$ in $$\Omega$$ (t), requiring the specification of $$u(x,0)=u_ 0(x)$$. Of course the function $$u_ 0$$ enters into the definition of the class of admissible initial configurations.
This paper is very interesting and contributes to a better understanding of the problems considered.

### MSC:

 35R35 Free boundary problems for PDEs 35K05 Heat equation 80A20 Heat and mass transfer, heat flow (MSC2010)
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