Logarithmic Continuity for the Nonlocal Degenerate Two-Phase Stefan Problem

The Classical Stefan Problem - Basic Concepts, Modelling and Analysis

Determination of one or two unknown thermal coefficients of a semi-infinite material through a two-phase stefan problem

Existence and uniqueness of classical solutions for the multidimensional expanding Hele{Shaw problem are proved.

We derive the quantitative modulus of continuity $$ \omega(r)=\left[ p+\ln \left( \frac{r_0}{r} \right) \right]^{-\alpha (n,p)}, $$ which we conjecture to be optimal, for solutions of the $p$-degenerate two-phase Stefan problem. Even in the classical case $p=2$, this represents a twofold improvement with respect to the 1984 state-of-the-art result by DiBenedetto and Friedman [J. reine angew. Math., 1984], in the sense that we discard one logarithm iteration and obtain an explicit value for the exponent $\alpha (n,p)$.

A complementary volume (co-volume) technique is used to develop a physically appealing algorithm for the solution of degenerate parabolic problems, such as the Stefan problem. It is shown that, these algorithms give rise to a discrete semigroup theory that parallels the continuous problem. In particular, the discrete Stefan problem gives rise to nonlinear semigroups in both the discreteL 1 andH ?1 spaces.

The Stefan problem is coupled with a spatially inhomogeneous and anisotropic Gibbs–Thomson condition at the phase boundary. We show the long-time existence of weak solutions for the non-degenerate Stefan problem with a spatially inhomogeneous and anisotropic Gibbs–Thomson law and a conditional existence result for the corresponding degenerate Stefan problem. To this end, approximate solutions are constructed by means of variational problems for energy functionals with spatially inhomogeneous and anisotropic interfacial energy. By passing to the limit, we establish solutions of the Stefan problem with a spatially inhomogeneous and anisotropic Gibbs–Thomson law in a weak generalisedBV-formulation.

A domain D in the (z, y) pla.ne is surrounded by curves OA, OB a.nd a line segment AB (a part of a s-traight line x = 0 > 0). For t,he following initmid-boundary value problem of the La.pla.ce hyperbolic differential inequality some results on maximum properties of Sathe? are irnyrwed in this paper, smoothness and monotonicity hypotheses in Sathera imposed on the initial curve OA and the boundary cnnre OB are relaxed, a class of strong maximum principles is established and uniqueness of solutions of the initial-boundary value. problem is prcnrecl.

Existence of a classical solution is demonstrated for an one-dimensional two-phase Stefan problem in which the spccilic heat may oscillate wildly near the melting temperature 0 in the sense that it is unbounded above and it is not bounded away from zero in a neighborhood of 0. The convective terms involved also display some type of singularity.

The method of approximate fundamental solutions (MAFS) for Stefan problems

A two-phase Stefan problem with latent heat a general power function of position is investigated. The background of the problem can be found in the soil-freezing process during the application of the artificial ground-freezing technique. After introducing the specific engineering condition, the governing equations of a two-phase Stefan problem are developed. An exact solution for the problem is established using the similarity transformation technique and the theory of the Kummer functions. It is proved that the coefficients in the solution can be appropriately determined if certain inequality is satisfied. Special cases of the solution are discussed, and several solutions reported in the literature are recovered. A similar two-phase Stefan problem involving first type of boundary condition is also introduced and solved. The coefficients in the solution can always be properly determined with no additional requirement. In the end, computational examples of the solution are presented and discussed. The exact solution provides a useful benchmark that can be used for verifying general numerical algorithms of Stefan problems, and it is also advantageous in the context of the inverse problem analysis.

The two-phase Stefan problem with anomalous diffusion

In this paper we prove existence and uniqueness of weak solutions for a general degenerate elliptic-parabolic problem with nonlinear dynamical boundary conditions. Particular instances of this problem appear in various phenomena with changes of phase like multiphase Stefan problem and in the weak formulation of the mathematical model of the so called Hele Shaw problem. Also, the problem with non-homogeneous Neumann boundary condition is included.

In this study, we propose a parametric finite element method for a degenerate multi-phase Stefan problem with triple junctions. This model describes the energy-driven motion of a surface cluster whose distributional solution was studied by Garcke and Sturzenhecker. We approximate the weak formulation of this sharp interface model by an unfitted finite element method that uses parametric elements for the representation of the moving interfaces. We establish existence and uniqueness of the discrete solution and prove unconditional stability of the proposed scheme. Moreover, a modification of the original scheme leads to a structure-preserving variant, in that it conserves the discrete analogue of a quantity that is preserved by the classical solution. Some numerical results demonstrate the applicability of our introduced schemes.

A difference method for the solution of the two-phase Stefan problem

Stefan problem is a problem involving phase transition from solid to liquid or vice versa where boundary between solid and liquid regions moves as function of time. This paper presents numerical solution of one-dimensional two-phase Stefan problem by using finite element method. The governing equations involved in Stefan problem consist of heat conduction equation for solid and liquid regions, and also transition equation in interface position (moving boundary). The equations are difficult to solve by ordinary numerical method because of the presence of moving boundary. As consequence, the equations is reformulated into the form of internal energy (enthalpy). By the enthalpy formulation, solution of the heat conduction equations is no longer concerning the phase state of material. The advantage of the enthalpy formulation is that, finite element method can be easily implemented to solve Stefan problem. Numerical simulation of interface position, temperature profile, and temperature history has good agreement with the exact solution. The approximation of interface position using finite element method was found that it is more accurate than the approximation by using Godunov method. The simulation results also reveal that the finite element method for solving Stefan problem have smaller mean absolute error than the Godunov method.