## The Stefan problem. Translated from the Russian by Marek Niezgódka and Anna Crowley.(English)Zbl 0751.35052

De Gruyter Expositions in Mathematics. 3. Berlin etc.: Walter de Gruyter. ix, 245 p. (1992).
The classical Stefan problem consists of finding a boundary between two phases and determining the temperature distribution, when heat conduction determines the energy flow within the material.
The author points out in the introduction that several “physical” hypothesis have been introduced in the past that do not follow from any fundamental law of physics, and may be mathematically restrictive. A common assumption concerns fixed (and unique) phase change temperature. Discarding this assumption may lead to occurence of hysteresis effects. One may need an additional equation for the (variable) melting temperature. This is the Vinsintin model which presents new challenges.
Dimensionality of the problem and the number of different phases constitute sources of entirely different arguments and results in all Stefan problems. For the one-dimensional case Rubinstein proved a global existence theorem, assuming analyticity of the free boundary (around 1960). In 1965, and in early 1970-s other proofs followed (Li Shang, Cannon and Douglas, Kotlow Primicerio, Meirmanov) not requiring analyticity and relaxing other assumptions.
For multidimensional Stefan problems the only results on existence and uniqueness of generalized solutions were at that time proved by Oleinik and Kamenomotskaya. An important step was taken by Duvant in 1973, followed by results of Friedman and Kinderlehrer, using Duvant’s transform. Cafarelli, then Kinderlehrer and Nirenberg proved theorems on smoothness of the free boundary around 1978 using variational arguments. A different approach was taken by Meirmanov who introduced the von Mises variables, in which the free surface becomes a fixed surface.
A number of authors were able to apply to the Stefan problem some results of classical or functional analysis on nonlinear parabolic equations (with fixed boundaries). The author analyzes some of these results due to Duvaut, Magenes, Visintin, Niezgódka, Pawlow. The author also mentions in his introduction the “mushy phase” problem, which arises if one does not assume any axioms that are equivalent to the assumption that the melting temperature uniquely determines the specific internal energy $$U$$. Instead an interval $$U_ -\leq U\leq U_ +$$ corresponds to the unique (or average) melting temperature $$\vartheta_ *$$.
The book itself is divided into eight chapters plus an appendix authored by Götz and Meirmanov on modelling of binary alloy cristallization. Chapter I is introductory. The problem is formulated, and some known facts are reviewed from differential equations and functional analysis theory. Chapter II on classical solutions reports on construction of approximate solutions, bounds and energy estimates for a localized domain (to some interval $$[0,T])$$. Results reported here are largely those of the author. Chapter III deals with the existence of classical solutions on arbitrary time intervals. Monotonicity is assumed for the initial data. Results of Anisyutin, following the work of Caffarelli and Friedman, are reviewed. They concern primarily smoothness of the free boundary. The author shows that most of these results can be extended to a nonlinear heat equation $$a(\vartheta)\partial\vartheta/\partial t=\Delta\vartheta$$, with $$a(\cdot)$$ smooth and $$a'(\vartheta)>0$$.
In the next chapter the author poses the problem in Lagrange coordinates $$\xi$$ related to Eulerian coordinates $$x$$ through a Jacobian matrix $$M$$, with $$x=\xi$$ at $$t=0$$, and $$\vartheta(x(\xi,t),t)=u(\xi,t)$$. On the free surface $$u=0$$, the equation $$D_ tu=0$$ is valid. The problem is linearized. Estimates are derived for the linear model. Using these results the author proves the existence and uniqueness of the generalized solution to the nonlinear problem. Next (fifth) chapter returns to the one-dimensional Stefan problem. Two phase problems are reexamined under weakened hypotheses.
Next the filtration problem is studied. Results of Friedman and Jensen and some results of the author are rederived. Lagrangian formulation is used again. More specific results are offered on the one dimensional flows showing how generalized solution may become classical solutions, how connected components of the liquid become just a single component, etc. The author, Kalier, Stedry and Voivoda’s work are reviewed. Approximate models are discussed in the last (eighth) chapter.
This book, written by an outstanding researcher, is long overdue. The choice of material is restricted to author’s area of interest, but this is a positive feature, making the book readable and authoritative. The author deliberately avoided details of numerical techniques, control theoretic questions and problems with fractional order derivatives. This is a well-written monograph presenting results that are scattered through dozens of journals in one volume.

### MSC:

 35R35 Free boundary problems for PDEs 80A22 Stefan problems, phase changes, etc. 35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations 35K05 Heat equation 80A20 Heat and mass transfer, heat flow (MSC2010) 35D05 Existence of generalized solutions of PDE (MSC2000)