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.     

带有动力学条件的高维一相Stefan问题

本文我们证明了带有动力学条件的高维一相Ｓｔｅｆａｎ总题局部古典解的存在唯一性。     

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.     

Viscosity Solutions for the Two-Phase Stefan Problem

We introduce a notion of viscosity solutions for the two-phase Stefan problem, which incorporates possible existence of a mushy region generated by the initial data. We show that a comparison principle holds between viscosity solutions, and investigate the coincidence of the viscosity solutions and the weak solutions defined via integration by parts. In particular, in the absence of initial mushy region, viscosity solution is the unique weak solution with the same boundary data.     

一个具有动态条件的Verigin问题

本文考虑了一个具有动态条件的Verigin问题,得到了这个问题关于时间的局部古典解的存在唯一性.     

The Stefan Problem with Nonlinear Convection

In this paper, we consider the time dependent Stefan problem with convection in the fluid phase governed by the Navier-Stokes equation and with adherence of the fluid on the lateral boundaries. The existence of a weak solution is obtained via the introduction of a temperature dependent penalty term in the fluid flow equation, together with the application of various compactness arguments.     

Quasi-stationary Stefan problem as limit case of Mullins-Sekerka problem

The existence of a local classical solution to the Mullins-Sekerka problem and the convergence to the two-phase quasi-stationary Stefan problem are proved when surface tension approaches zero. This convergence gives a proof of the existence of a local classical solution of quasi-stationary Stefan problem. The methods work in all dimensions.     

A Stefan Problem with Curvature Correction in a Concentrated Capacity

We consider a Stefan problem with the curvature correction in a concentrated capacity. It is a kind of phase transition problems, in which the unknown temperature satisfies both the Stefan condition and the curvature correction on the free boundary, and also fulfills a kind of heat equations with special inner heat source. The inner beat source is related to the derivative of the temperature with respect to the direction vertical to the phase regions. The result established in this paper is the existence of a global weak solution of this problem.     

一类Stefan问题的适定性

本文讨论在烧蚀缓慢的情况下发汗控制微分方程并证明了其解的适定性。     

On the Modified Crystalline Stefan Problem with Singular Data

We study the behavior of solutions of the modified Stefan problem in the plane for polygonal interfaces. We are particularly interested in a solution near a singularity of either the loss of a facet or the breaking of a facet. We establish precise regularity results if a facet disappears. We use them to establish the existence of a weak solution with singular data, i.e., when some of the zero-crystalline-curvature facets have zero length.     

Local Existence of Bounded Solutions to the Degenerate Stefan Problem with Joule's Heating

This paper deals with the degenerate Stefan problem with Joule's heating, which describes the combined effects of heat and electrical current Rows in a metal. The local existence of a bounded weak solution for the problem in proved. Also a degenerate thermistor problem with continuous conductivity is considered.     

Determination of the Unknown Boundary Conditions in a Two-phase Stefan Problem

A two-phase Stefan problem with the heat flux boundary conditions, including an unknown function f, is considered. The existence, uniqueness, and continuous dependence upon the initial data of the solution (f, s, u_1, u_2) are proved.     

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.     

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.     

On the numerical approach of the enthalpy method for the Stefan problem

In this article an error bound is derived for a piecewise linear finite element approximation of an enthalpy formulation of the Stefan problem; we have analyzed a semidiscrete Galerkin approximation and completely discrete scheme based on the backward Euler method and a linearized scheme is given and its convergence is also proved. A second‐order error estimates are derived for the Crank‐Nicolson Galerkin method. In the second part, a new class of finite difference schemes is proposed. Our approach is to introduce a new variable and transform the given equation into an equivalent system of equations. Then, we prove that the difference scheme is second order convergent. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004     

A Degenerate Stefan Problem with Two Free Boundaries

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Asymptotic convergence of the Stefan problem to Hele-Shaw

We discuss the asymptotic behaviour of weak solutions to the Hele-Shaw and one-phase Stefan problems in exterior domains. We prove that, if the space dimension is greater than one, the asymptotic behaviour is given in both cases by the solution of the Dirichlet exterior problem for the Laplacian in the interior of the positivity set and by a singular, radial and self-similar solution of the Hele-Shaw flow near the free boundary. We also show that the free boundary approaches a sphere as , and give the precise asymptotic growth rate for the radius.

     

抛物型方程在变动区域中的Galarkin有限元法及其对Stefan问题的应用

文中叙述了具变系数抛物型方程在变动区域中的有限元半离散格式和全离散θ格式,给出了误差界和稳定性结果,并应用于求解Stefan问题。     

A boundary integral formulation of the Stefan problem

This paper considers a boundary integral formulation of the Stefan problem for two spatial dimensions. This formulation has the advantage that its numerical implementation does not require the discretization of the Stefan condition. Furthermore, the formulation is capable of solving problems with complex boundaries. Several illustrative examples are given.