STEFAN PROBLEMS WITH KINETIC CONDITIONS AT THE FREE BOUNDARY

In this paper,we prove the existence theorems of locbal or global classical solutions to Stefan problems with various kinetic conditions at the free boundary.     

Remarks on the analyticity of the free boundary for the one-dimensional Stefan problem

Sunto Si dimostra che nel problema di Stefan (con una o due fasi) nelle variabili x(-,+) e t0il contorno libero è una funzione analitica di t ^1/2 per t0,se il dato iniziale è assunto in una classe di fungioni intere. I coefficienti dello sviluppo del contorno libero in serie di poterne di t^1/2 sono determinati mediante formule ricorrenti. Un particolare esame è svolto per il caso in cui non c'è raccordo tra il dato iniziale e quello sul contorno libero, dimostrando l'esistenza di soluzioni dotate di contorno analitico, sotto opportune limitazioni sul salto tra i due dati; sono peró messi in evidenza anche casi in cui il problema non ammette soluzioni. Infine una breve analisi è svolta per i problemi in domini limitati, fornendo una dimostrazione della analiticità del contorno libero rispetto a t per t>0,alternativa a quella di [1].

---

---

Work partially supported by the Italian CNR, while the first of the Authors was visiting the Mathematical Institute «U. Dini» of the University of Florence.     

A meshless method for one-dimensional Stefan problems

The paper presents a new meshless numerical technique for solving one-dimensional problems with moving boundaries including the Stefan problems. The technique presented is based on the use of the delta-shaped functions and the method of approximate fundamental solutions (MAFS) firstly suggested for solving elliptic problems and for heat equations in domains with fixed boundaries. The numerical examples are presented and the results are compared with analytical solutions. The comparison shows that the method presented provides a very high precision in determining the position of the moving boundary even for a region that initially has zero thickness.     

Exact solutions of one-dimensional two-phase free boundary problems for parabolic equations

We construct automodel solutions for the one-dimensional two-phase Stefan, Florin, and Verigin free boundary problems for parabolic equations in the case where the initial and boundary data are not adjusted. It is shown that in the Stefan problem with “supercooling,” the liquid temperature may be less than the temperature of the phase transition, i.e., the liquid may be “supercooled” while the solid may be “superheated.” Bibliography: 8 titles. Dedicated to the memory of Olga Aleksandrovna Ladyzhenskaya __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 318, 2004, pp. 42–59.     

A comparison of numerical models for one-dimensional Stefan problems

In this paper, we present a critical comparison of the suitability of several numerical methods, level set, moving grid and phase field model, to address two well-known Stefan problems in phase transformation studies: melting of a pure phase and diffusional solid-state phase transformations in a binary system. Similarity solutions are applied to verify the numerical results. The comparison shows that the type of phase transformation considered determines the convenience of the numerical techniques. Finally, it is shown both numerically and analytically that the solid-solid phase transformation is a limiting case of the solid–liquid transformation.     

On the one-dimensional two-phase inverse Stefan problems

New formulations of the inverse nonstationary Stefan problems are considered: (a) forx [0,1] (the inverse problem IP₁; (b) forx [0, (t)] with a degenerate initial condition (the inverse problem IP). Necessary conditions for the existence and uniqueness of a solution to these problems are formulated. On the first phase {x [0, y(t)]{, the solution of the inverse problem is found in the form of a series; on the second phase {x [y(t), 1] orx [y(t), (t)]{, it is found as a sum of heat double-layer potentials. By representing the inverse problem in the form of two connected boundary-value problems for the heat conduction equation in the domains with moving boundaries, it can be reduced to the integral Volterra equations of the second kind. An exact solution of the problem IP is found for the self similar motion of the boundariesx=y(t) andx=(t).Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 8, pp. 1058–1065, August, 1993.     

A rapid method for the identification of the free boundary in two-phase Stefan problems

A simple and rapid method for the identification of the freeboundary in two-phase Stefan problems is presented. The methodconsists in solving an unconstrained convex control problemand then finding a suitable contour line. It needs additionalinformation on one part of the boundary, available through experimentalmeasurements. The basic idea is the use of a geometric controllability-typeresult for linear parabolic systems. A numerical test is discussedin detail.     

Newton’s polygon method and the local solvability of free boundary problems

For some qualitatively new problems with free (unknown) boundary, conditions of local solvability (with respect to time) are investigated. __________ Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 26, pp. 114–177, 2007.     

Space-time localization in one-dimensional problems with free boundary for nonlinear second-order equations

We investigate the effect of space localization and stabilization for finite time for thermal and diffusion processes that take place in active media and are described by nonlinear evolution equations. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 11, pp. 1554–1558, November, 1998.     

Solution of a model problem related to singularly perturbed,free boundary,Stefan type problems

A model problem of conjunction of two phases with a small parameter κ in front of the principal term of a free boundary condition is considered. A coercive estimate for the solution with a constant independent of κ is derived in terms of a weighted H?lder space. Bibliography: 9 titles. Dedicated to Vsevolod Alekseevich Solonnikov Published in Zapiski Nauchnykh Seminarov POMI, Vol. 362, 2008, pp. 64-91.     

Investigation of solvability of the multidimensional two-phase Stefan and the nonstationary filtration Florin problems for second order parabolic equations in weighted Hölder spaces of functions

Multidimensional two-phase Stefan (k=1) and nonstationary filtration Florin (k=0) problems for second order parabolic equations in the case when the free boundary is a graph of a functionx _n =Ψ _k (x′t),x′∈ ^n?1 ,n≥2,t∈(0,T) are studied. A unique solvability theorem in weighted Hölder spaces of functions with time power weight is proved, coercive estimates for solutions are obtained. Bibliography: 30 titles.     

Explicit solutions for one-dimensional two-phase free boundary problems with either shrinkage or expansion

We consider a one-dimensional solidification of a pure substance which is initially in liquid state in a bounded interval $[0,l]$. Initially, the liquid is above the freezing temperature, and cooling is applied at $x=0$ while the other end $x=l$ is kept adiabatic. At the time $t=0$, the temperature of the liquid at $x=0$ comes down to the freezing point and solidification begins, where $x=s\left(t\right)$ is the position of the solid–liquid interface. As the liquid solidifies, it shrinks ($0<r<1$) or expands ($r<0$) and appears a region between $x=0$ and $x=rs\left(t\right)$, with $r<1$. Temperature distributions of the solid and liquid phases and the position of the two free boundaries ($x=rs\left(t\right)$ and $x=s\left(t\right)$) in the solidification process are studied. For three different cases, changing the condition on the free boundary $x=rs\left(t\right)$ (temperature boundary condition, heat flux boundary condition and convective boundary condition) an explicit solution is obtained. Moreover, the solution of each problem is given as a function of a parameter which is the unique solution of a transcendental equation and for two of the three cases a condition on the parameter must be verified by data of the problem in order to have an instantaneous phase-change process. In all the cases, the explicit solution is given by a representation of the similarity type.     

Finite-difference methods with increased accuracy and correct initialization for one-dimensional Stefan problems

Although the numerical solution of one-dimensional phase-change, or Stefan, problems is well documented, a review of the most recent literature indicates that there are still unresolved issues regarding the start-up of a computation for a region that initially has zero thickness, as well as how to determine the location of the moving boundary thereafter. This paper considers the so-called boundary immobilization method for four benchmark melting problems, in tandem with three finite-difference discretization schemes. We demonstrate a combined analytical and numerical approach that eliminates completely the ad hoc treatment of the starting solution that is often used, and is numerically second-order accurate in both time and space, a point that has been consistently overlooked for this type of moving-boundary problem.     

Regularity of weak solutions of one-dimensional two-phase Stefan problems

Sunto Si considera il problema di Stefan unidimensionale a due fasi e si dimostra l'esistenza di soluzioni classiche sotto ipotesi minimali sui dati (continuità a tratti e limitatezza). Nelle stesse ipotesi si dimostra che tali soluzioni dipendono in modo continuo dai dati, conseguendo un risultato che è più generale anche di quello noto per le soluzioni deboli.

---

---

Entrata in Redazione il 22 dicembre 1976.

---

Work partially supported by the Italian C.N.R.     

On the solvability of two-point,second-order boundary value problems

We gain solvability of a system of nonlinear, second-order ordinary differential equations subject to a range of boundary conditions. The ideas involve differential inequalities and fixed point methods. In particular, maximum principles are not employed.     

Global classical solution of quasi-stationary Stefan free boundary problem

We consider an one-phase quasi-stationary Stefan problem (Hele–Shaw problem) in multidimensional case. Under some reasonable conditions we prove that the problem has a classical solution globally in time. The method can be used in two-phase problem as well. We also discuss asymptotic behavior of solution as t→+∞. The method developed here can be extended to a general class of free boundary problems.