Global Lipschitz continuity of free boundaries in the one-phase Stefan problem

1985 ◽  
Vol 142 (1) ◽  
pp. 197-213 ◽  
Author(s):  
Chen Ya-zhe
Keyword(s):  
Stefan Problem ◽  
Lipschitz Continuity ◽  
Free Boundaries ◽  
The One

Remarks on the quasi-steady one phase Stefan problem

1986 ◽  
Vol 102 (3-4) ◽  
pp. 263-275 ◽  
Author(s):  
Bento Louro ◽  
José-Francisco Rodrigues
Keyword(s):  
Free Boundary ◽  
Variational Inequalities ◽  
Specific Heat ◽  
Rate Of Convergence ◽  
Stefan Problem ◽  
Free Boundaries ◽  
Variational Solutions ◽  
The One ◽  
Regularity Results ◽  
Boundary Estimates

SynopsisThis paper presents some regularity results on the solution and on the free boundary for the one phase Stefan problem with zero specific heat in the framework of the variational inequalities formulation. In particular we show the Hölder continuity of the free boundary. Estimates on the rate of convergence when the specific heat vanishes are given for the variational solutions and for the free boundaries.

scholarly journals The one-phase fractional Stefan problem

2020 ◽  
pp. 1-49
Author(s):  
Félix del Teso ◽  
Jørgen Endal ◽  
Juan Luis Vázquez
Keyword(s):  
Free Boundary ◽  
Stefan Problem ◽  
Dimensional Space ◽  
Continuous Functions ◽  
Finite Difference Schemes ◽  
Free Boundaries ◽  
Finite Speed Of Propagation ◽  
Bounded Data ◽  
Speed Of Propagation ◽  
The One

We study the existence and properties of solutions and free boundaries of the one-phase Stefan problem with fractional diffusion posed in [Formula: see text]. In terms of the enthalpy [Formula: see text], the evolution equation reads [Formula: see text], while the temperature is defined as [Formula: see text] for some constant [Formula: see text] called the latent heat, and [Formula: see text] stands for the fractional Laplacian with exponent [Formula: see text]. We prove the existence of a continuous and bounded selfsimilar solution of the form [Formula: see text] which exhibits a free boundary at the change-of-phase level [Formula: see text]. This level is located at the line (called the free boundary) [Formula: see text] for some [Formula: see text]. The construction is done in 1D, and its extension to [Formula: see text]-dimensional space is shown. We also provide well-posedness and basic properties of very weak solutions for general bounded data [Formula: see text] in several dimensions. The temperatures [Formula: see text] of these solutions are continuous functions that have finite speed of propagation, with possible free boundaries. We obtain estimates on the growth in time of the support of [Formula: see text] for solutions with compactly supported initial temperatures. Besides, we show the property of conservation of positivity for [Formula: see text] so that the support never recedes. On the contrary, the enthalpy [Formula: see text] has infinite speed of propagation and we obtain precise estimates on the tail. The limits [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] are also explored, and we find interesting connections with well-studied diffusion problems. Finally, we propose convergent monotone finite-difference schemes and include numerical experiments aimed at illustrating some of the obtained theoretical results, as well as other interesting phenomena.

scholarly journals On the Two-phase Fractional Stefan Problem

2020 ◽  
Vol 20 (2) ◽  
pp. 437-458 ◽  
Author(s):  
Félix del Teso ◽  
Jørgen Endal ◽  
Juan Luis Vázquez
Keyword(s):  
Free Boundary ◽  
Stefan Problem ◽  
Free Boundary Problems ◽  
Classical Problem ◽  
Nonlocal Diffusion ◽  
Two Phase ◽  
Boundary Problems ◽  
Numerical Studies ◽  
The One ◽  
Evolution Type

AbstractThe classical Stefan problem is one of the most studied free boundary problems of evolution type. Recently, there has been interest in treating the corresponding free boundary problem with nonlocal diffusion. We start the paper by reviewing the main properties of the classical problem that are of interest to us. Then we introduce the fractional Stefan problem and develop the basic theory. After that we center our attention on selfsimilar solutions, their properties and consequences. We first discuss the results of the one-phase fractional Stefan problem, which have recently been studied by the authors. Finally, we address the theory of the two-phase fractional Stefan problem, which contains the main original contributions of this paper. Rigorous numerical studies support our results and claims.

scholarly journals An integral relationship for a fractional one-phase Stefan problem

2018 ◽  
Vol 21 (4) ◽  
pp. 901-918 ◽  
Author(s):  
Sabrina Roscani ◽  
Domingo Tarzia
Keyword(s):  
Boundary Condition ◽  
Stefan Problem ◽  
One Dimensional ◽  
Similarity Type ◽  
Integral Relationship ◽  
Temperature Boundary ◽  
Liouville Derivative ◽  
The One ◽  
Stefan Condition ◽  
Temperature Boundary Condition

Abstract A one-dimensional fractional one-phase Stefan problem with a temperature boundary condition at the fixed face is considered by using the Riemann–Liouville derivative. This formulation is more convenient than the one given in Roscani and Santillan (Fract. Calc. Appl. Anal., 16, No 4 (2013), 802–815) and Tarzia and Ceretani (Fract. Calc. Appl. Anal., 20, No 2 (2017), 399–421), because it allows us to work with Green’s identities (which does not apply when Caputo derivatives are considered). As a main result, an integral relationship between the temperature and the free boundary is obtained which is equivalent to the fractional Stefan condition. Moreover, an exact solution of similarity type expressed in terms of Wright functions is also given.

Oscillatory instabilities and dynamics of multi-spike patterns for the one-dimensional Gray-Scott model

2009 ◽  
Vol 20 (2) ◽  
pp. 187-214 ◽  
Author(s):  
WAN CHEN ◽  
MICHAEL J. WARD
Keyword(s):  
Hopf Bifurcation ◽  
Stefan Problem ◽  
Differential Algebraic Equations ◽  
Finite Domain ◽  
Quasi Equilibrium ◽  
Algebraic Equations ◽  
Source Terms ◽  
One Dimensional ◽  
The One ◽  
Oscillatory Instabilities

The dynamics and oscillatory instabilities of multi-spike solutions to the one-dimensional Gray-Scott reaction–diffusion system on a finite domain are studied in a particular parameter regime. In this parameter regime, a formal singular perturbation method is used to derive a novel ODE–PDE Stefan problem, which determines the dynamics of a collection of spikes for a multi-spike pattern. This Stefan problem has moving Dirac source terms concentrated at the spike locations. For a certain subrange of the parameters, this Stefan problem is quasi-steady and an explicit set of differential-algebraic equations characterizing the spike dynamics is derived. By analysing a nonlocal eigenvalue problem, it is found that this multi-spike quasi-equilibrium solution can undergo a Hopf bifurcation leading to oscillations in the spike amplitudes on an O(1) time scale. In another subrange of the parameters, the spike motion is not quasi-steady and the full Stefan problem is solved numerically by using an appropriate discretization of the Dirac source terms. These numerical computations, together with a linearization of the Stefan problem, show that the spike layers can undergo a drift instability arising from a Hopf bifurcation. This instability leads to a time-dependent oscillatory behaviour in the spike locations.

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