The Limit of the Stefan Problem with Surface Tension and Kinetic Undercooling on the Free Boundary

In this paper we consider the Stefan problem with surface tension and kinetic undercooling effects, that is with the temperature u satisfying the condition u = -σK - εV_n on the interface Γ_t, σ, ε = const. ≥ 0 where K and V_n are the mean curvature and the normal velocity of Γ_t, respectively. In any of the following situations: (1) σ > 0 fixed, ε > 0, (2) σ = ε → 0; (3) σ → 0, ε = 0, we shall prove the convergence of the corresponding local (in time) classical solution of the Stefan problem.     

The order of convergence in the stefan problem with vanishing specific heat

The paper is concerned with the two-phase Stefan problem with a small parameter ϵ, which coresponds to the specific heat of the material. It is assumed that the initial condition does not coincide with the solution for t = 0 of the limit problem related to ε = 0. To remove this discrepancy, an auxiliary boundary layer type function is introduced. It is proved that the solution to the two-phase Stefan problem with parameter ϵ differs from the sum of the solution to the limit Hele–Shaw problem and a boundary layer type function by quantities of order O(ϵ). The estimates are obtained in H?lder norms. Bibliography: 13 titles.     

Justification of a quasistationary approximation for the Stefan problem

Unique solvability of the one-phase Stefan problem with a small multiplier ε at the time derivative in the equation is proved on a certain time interval independent of ε for ε ∈ (0, ε₀). The solution to the Stefan problem is compared with the solution to the Hele-Show problem, which describes the process of melting materials with zero specific heat ε and can be regarded as a quasistationary approximation for the Stefan problem. It is shown that the difference of the solutions has order . This provides a justification of the quasistationary approximation. Bibliography: 23 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 348, 2007, pp. 209–253.     

Stefan problem with a kinetic and the classical conditions at the free boundary

The Stefan problem is considered with the kinetic condition u⁺=u^–=k(y, )-v at the phase interface, where k(y, ) is the half-sum of the principal curvatures of the free boundary and v is the speed of its shifting in the direction of a normal. The solvability of a modified Stefan problem in spaces of smooth functions and the convergence of its solutions as 0 to a solution of the classical Stefan problem are proved.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 2, pp. 155–166, February, 1992.     

On a free interface problem for linear ordinary differential equations and the one-phase Stefan problem

A numerical method for the solution of the one-phase Stefan problem is discussed. By discretizing the time variable the Stefan problem is reduced to a sequence of free boundary value problems for ordinary differential equations which are solved by conversion to initial value problems. The numerical solution is shown to converge to the solution of the Stefan problem with decreasing time increments. Sample calculations indicate that the method is stable provided the proper algorithm is chosen for integrating the initial value problems.     

Homotopy perturbation method for a limit case Stefan problem governed by fractional diffusion equation

The work presents a mathematical model describing the time fractional anomalous-diffusion process of a generalized Stefan problem which is a limit case of a shoreline problem. In this model, the governing equations include a fractional time derivative of order 0 < α ? 1 and variable latent heat. The approximate solution of the problem is obtained by homotopy perturbation method. The results thus obtained are compared graphically with the exact solutions. A brief sensitivity study is also performed.     

Two-phase Stefan problem with vanishing specific heat

The unique solvability of the two-phase Stefan problem with a small parameter ε ∈ [0; ε ₀] at the time derivative in the heat equations is proved. The solution is obtained on a certain time interval [0; t ₀] independent of ε. The solution of the Stefan problem is compared with the solution to the Hele–Shaw problem corresponding to the case ε = 0. The solutions of both problems are not assumed to coincide at the initial moment of time. Bibliography: 18 titles. Dedicated to Vsevolod Alekseevich Solonnikov on the occasion of his jubilee Published in Zapiski Nauchnykh Seminarov POMI, Vol. 362, 2008, pp. 337–363.     

A bidimensional inverse Stefan problem: Identification of boundary value

We propose to regularize the bidimensional inverse Stefan problem that is to determine the boundary temperature u(x,0,t) in the liquid phase in a medium of water and melting ice. This ill-posed problem is regularized by means of a convolution equation and an error estimate in L²(R²) is obtained. Numerical results are given.     

Multidimensional two-phase quasistationary stefan problem

Existence and uniqueness of the classic solution to a two-dimensional quasistationary Stefan problem are considered. The family of model problems dependent on the parameter ε>0 which defines a heat conductivity of a matter in the direction of thex-axis is analysed. When ε→0 it is approximated by the approximate model problem having less dimensions. Analogous results are also valid for a three-dimensional problem.     

Stability in the Stefan Problem with Surface Tension (I)

We develop a high-order energy method to prove asymptotic stability of flat steady surfaces for the Stefan problem with surface tension – also known as the Stefan problem with Gibbs–Thomson correction.     

Stefan Problem with Change Density Upon Change of Phase (I)

In this paper, we establish the existence of one-dimensional classical solution of one-phase problem and its continuous dependence. In addition, we prove that if ε → 0, the free boundary X(t) withdraws and solution converges to the solution of classical Stefan problem. The two-phase problem wiU be discussed in the coming paper.     

Approximation of a Two-phase Continuous Casting Stefan Problem

The continuous casting Stefan problem is a mathematical model describing the solidification with convection of a material being cast continuously with a prescribed velocity. We propose a practical piecewise linear finite element scheme motivated by the characteristic finite element method and derive an error estimate for the scheme which is of the same convergence order as that proved for Stefan problem without convection.     

A Quasisteady Stefan Problem with Curvature Correction and Kinetic Undercooling

A quasisteady Stefan problem with curvature correction and kinetic undercooling is considered. It is a problem with phase transition, in which not only the Stefan condition, but also the curvature correction and kinetic undercooling effect hold on the free boundary, and in phase regions elliptic equations are satisfied by the unknown temperature at each time. The existence and uniqueness of a local classical solution of this problem are obtained.     

CLASSICAL SOLUTION OF QUASI-STATIONARY STEFAN PROBLEM

This paper considers the quasi-stationary Stefan problem:△u(x,t)=0 in space-time domain,u=0 and Vv (?)u/(?)u=0 on the free boundary.Under the natural conditions the existence of classical solution locally in time is proved bymaking use of the property of Frechet derivative operator and fixed point theorem. For thesake of simplicity only the one-phase problem is dealt with. In fact two-phase problem can bedealt with in a similar way with more complicated calculation.     

Explicit solution for a Stefan problem with variable latent heat and constant heat flux boundary conditions

In Voller, Swenson and Paola [V.R. Voller, J.B. Swenson, C. Paola, An analytical solution for a Stefan problem with variable latent heat, Int. J. Heat Mass Transfer 47 (2004) 5387-5390], and Lorenzo-Trueba and Voller [J. Lorenzo-Trueba, V.R. Voller, Analytical and numerical solution of a generalized Stefan problem exhibiting two moving boundaries with application to ocean delta formation, J. Math. Anal. Appl. 366 (2010) 538-549], a model associated with the formation of sedimentary ocean deltas is studied through a one-phase Stefan-like problem with variable latent heat. Motivated by these works, we consider a two-phase Stefan problem with variable latent of fusion and initial temperature, and constant heat flux boundary conditions. We obtain the sufficient condition on the data in order to have an explicit solution of a similarity type of the corresponding free boundary problem for a semi-infinite material. Moreover, the explicit solution given in the first quoted paper can be recovered for a particular case by taking a null heat flux condition at the infinity.     

Pulsating traveling waves in the singular limit of a reaction–diffusion system in solid combustion

We consider a coupled system of parabolic/ODE equations describing solid combustion. For a given rescaling of the reaction term (the high activation energy limit), we show that the limit solution solves a free boundary problem which is to our knowledge new.In the time-increasing case, the limit coincides with the Stefan problem with spatially inhomogeneous coefficients. In general it is a parabolic equation with a memory term.In the first part of our paper we give a characterization of the limit problem in one space dimension. In the second part of the paper, we construct a family of pulsating traveling waves for the limit one phase Stefan problem with periodic coefficients. This corresponds to the assumption of periodic initial concentration of reactant.     

L p -theory for the Stefan problem

Local solvability of the one-phase Stefan problem is established in anisotropic Sobolev spaces. There is no loss of regularity. Hanzawa transformation of the Stefan problem to a problem in a domain with a fixed boundary is modified. Bibliography: 27 titles. Dedicated to the memory of A. P. Oskolkov Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 243, 1997, pp. 299–323. Translated by E. V. Frolova.     

Quasistationary Approximation for the Stefan Problem

To justify the quasistationary approximation for the Stefan problem, the difference between the solution to the Hele-Show problem and the solution to the Stefan problem with small parameter ε at the time-derivative in the equation is considered. Bibliography: 7 titles. __________ Translated from Problemy Matematicheskogo Analiza, No. 31, 2005, pp. 167–178.     

A regularization of a reaction–diffusion system approximation to the two-phase Stefan problem

Reaction–diffusion system approximations to the classical two-phase Stefan problem are considered in the present study. A reaction–diffusion system approximation to the Stefan problem has been proposed by Hilhorst et al. from an ecological point of view, and they have given convergence results for the system. In the present study, a new reaction–diffusion system approximation to the Stefan problem is proposed based on regularization of the enthalpy–temperature constitutive relation. For a deeper understanding of the approximation mechanism by means of reaction–diffusion systems, the rates of convergence for both the solutions and the free boundaries are investigated.     

On a model of phase relaxation for the hyperbolic Stefan problem

In this paper we study a model of phase relaxation for the Stefan problem with the Cattaneo–Maxwell heat flux law. We prove an existence and uniqueness result for the resulting problem and we show that its solution converges to the solution of the Stefan problem as the two relaxation parameters go to zero, provided a relation between these parameters holds.