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**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.

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) |