Evolution of 3-D crystals from supersaturated vapor with modified Stefan condition: Galerkin method approach

We examine the evolution of crystals in three dimensions. We assume that the Wulff shape is a prism with a hexagonal base. We include the Gibbs-Thomson law on the crystal surface and the so-called Stefan condition. We show local in time existence of solutions assuming that the initial crystal has admissible shape.     

Two-Phase Stefan Problem as the Limit Case of Two-Phase Stefan Problem with Kinetic Condition

Both one-dimensional two-phase Stefan problem with the thermodynamic equilibrium condition u(R(t),t)=0 and with the kinetic rule u_ε(R_ε(t),t)=εR_ε′(t) at the moving boundary are considered. We prove, when ε approaches zero, R_ε(t) converges to R(t) in C^1+δ/2[0,T] for any finite T>0, 0<δ<1.     

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.     

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 stationary Stefan problem with convection governed by a non‐linear Darcy's law

We consider the bidimensional stationary Stefan problem with convection. The problem is governed by a coupled system involving a non‐linear Darcy's law and the energy balance equation with second member in L¹. We prove existence of at least one weak solution of the problem, using the penalty method and the Schauder fixed point principle. Copyright © 1999 John Wiley & Sons, Ltd.     

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.     

一个具有内部物质对流的两项连铸Stefan问题

考虑了一个具有内部物质对流和非线性边界热交换的多维连铸Stefan问题,并得到了这个问题整体弱解的存在性、唯一性和对初边界条件的连续依赖性。本项工作改进和推广了J.F.Fodri-gues＆F.Yi的结果,放宽了他们对内部流和边界条件的一些不太符合实际的限制。     

An explicit solution for an instantaneous two-phase Stefan problem with nonlinear thermal coefficients

We consider a nonlinear heat conduction problem for a semi-infinitematerial x > 0, with phase-change temperature T₁, an initialtemperature T₂ (> T₁) and a heat flux of the type q (t) =q₀/t imposed on the fixed face x = 0. We assume that the volumetricheat capacity and the thermal conductivity are particular nonlinearfunctions of the temperature in both solid and liquid phases. We determine necessary and/or sufficient conditions on the parametersof the problem in order to obtain the existence of an explicitsolution for an instantaneous nonlinear twophase Stefan problem(solidification process).     

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 a multiple time-scales perturbation analysis of a Stefan problem with a time-dependent Dirichlet boundary condition

In this paper, a classical Stefan problem with a prescribed and small time-dependent temperature at the boundary is studied. By using a multiple time-scales perturbation method, it is shown analytically how the moving boundary profile is influenced by the prescribed temperature at the boundary and the initial conditions. Only a few exact solutions are available for this type of problems and it turns out that the constructed approximations agree very well with these exact solutions. In particular, approximations of solutions for this type of problems, with periodic and decaying temperatures at the boundary, are constructed. Furthermore, these approximations are valid on a long time scale, and seems to be not available in the literature.     

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.     

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.     

On the refined integral method for the one-phase Stefan problem with time-dependent boundary conditions

Refined integral heat balance is developed for Stefan problem with time-dependent temperature applied to exchange surface. The method is applied to phase change in the half-plane and ordinary differential equation is obtained for the solid/liquid interface. The results are compared to those obtained by heat balance integral, perturbation and numerical methods.     

A Stefan problem modelling crystal dissolution and precipitation

A simple 1D model for crystal dissolution and precipitationis presented. The model equations resemble a one-phase Stefanproblem and involve non-linear and multivalued exchange ratesat the free boundary. The original equations are formulatedon a variable domain. By transforming the model to a fixed domainand applying a regularization, we prove the existence and uniquenessof a solution. The paper is concluded by numerical simulations.     

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.     

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.     

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.     

On the domain dependence of solutions to the two-phase Stefan problem

We prove that solutions to the two-phase Stefan problem defined on a sequence of spatial domains _n ^N converge to a solution of the same problem on a domain where is the limit of _n in the sense of Mosco. The corresponding free boundaries converge in the sense of Lebesgue measure on ^N.     

On Free Boundary Problem for the Non-Newtonian Shear Thickening Fluids

The aim of this paper is to explore the free boundary problem for the NonNewtonian shear thickening fluids. These fluids not only have vacuum, but also have strong nonlinear properties. In this paper, a class of approximate solutions is first constructed, and some uniform estimates are obtained for these approximate solutions. Finally, the existence of free boundary problem solutions is proved by these uniform estimates.