scholarly journals REGULARITY OF THE SOLUTION TO A NONSTANDARD SYSTEM OF PHASE FIELD EQUATIONS

2013 ◽  
Author(s):  
Pierluigi Colli ◽  
Gianni Gilardi ◽  
Jürgen Sprekels
Keyword(s):  
Diffusion Coefficient ◽  
Asymptotic Analysis ◽  
Order Parameter ◽  
Phase Segregation ◽  
Limit Problem ◽  
System Of Differential Equations ◽  
Field Equations ◽  
Limit System ◽  
Phase Field Equations ◽  
Uniqueness Proof

A nonstandard system of differential equations describing two-species phase segregation is considered. This system naturally arises in the asymptotic analysis recently done by Colli, Gilardi, Krejčí, and Sprekels as the diffusion coefficient in the equation governing the evolution of the order parameter tends to zero. In particular, a wellposedness result is proved for the limit system. This paper deals with the above limit problem in a less general but still very significant framework and provides a very simple proof of further regularity for the solution. As a byproduct, a simple uniqueness proof is given as well.

Hugoniot-type conditions and weak solutions to the phase-field system

1999 ◽  
Vol 10 (1) ◽  
pp. 55-77 ◽  
Author(s):  
V. G. DANILOV ◽  
G. A. OMEL'YANOV ◽  
E. V. RADKEVICH
Keyword(s):  
Phase Field ◽  
Weak Solutions ◽  
Wave Train ◽  
Limit Problem ◽  
Mushy Region ◽  
Field Equations ◽  
Phase Field Equations ◽  
Phase Field System ◽  
Thomson Problem ◽  
Free Interfaces

We consider a new concept of weak solutions to the phase-field equations with a small parameter ε characterizing the length of interaction. For the standard situation of a single free interface, this concept (in contrast with the common one) leads to the well-known Stefan–Gibbs–Thomson problem as ε→0. For the case of a large number M(ε) (M(ε)→∞ as ε→0) of free interfaces, which corresponds to the ‘wave-train’ interpretation of a ‘mushy region’, this concept allows us to obtain the limit problem as ε→0.

scholarly journals Distributed optimal control of a nonstandard system of phase field equations

2011 ◽  
Vol 24 (4-6) ◽  
pp. 437-459 ◽  
Author(s):  
Pierluigi Colli ◽  
Gianni Gilardi ◽  
Paolo Podio-Guidugli ◽  
Jürgen Sprekels
Keyword(s):  
Optimal Control ◽  
Phase Field ◽  
Field Equations ◽  
Distributed Optimal Control ◽  
Phase Field Equations

A Hyperbolic Phase-Field Approach for Solidification With Supercooling

1999 ◽  
Author(s):  
Yuqi Chen ◽  
James M. McDonough ◽  
Kaveh A. Tagavi
Keyword(s):  
Phase Field ◽  
Supercooled Liquid ◽  
Field Method ◽  
Phase Field Method ◽  
Uniform Mesh ◽  
Phase Variable ◽  
Field Equations ◽  
Pure Material ◽  
Melt Temperature ◽  
Phase Field Equations

Abstract This report concerns the solidification of a “supercooled” liquid, whose temperature is initially below the equilibrium melt temperature, Tm of the solid. A new approach, the phase-field method, will be applied for this Stefan problem with supercooling, which simulates the solidification process of a pure material into a supercooled liquid in a spherical region. The advantage of the phase-field method is that it bypasses explicitly tracking the freezing front. In this approach the solid-liquid interface is treated as diffuse, and a dynamic equation for the phase variable is introduced in addition to the equation for heat flow. Thus, there are two coupled partial differential equations for temperature and phase field. In the reported study, an implicit numerical scheme using finite-difference techniques on a uniform mesh is employed to solve both Fourier phase-field equations and non-Fourier (known as damped wave or telegraph) phase-field equations. The latter gurantees a finite speed of propagation for the solidification front. Both Fourier (parabolic) and non-Fourier (hyperbolic) Stefan problems with supercooling are satisfactorily simulated and their solutions compared in the present work.

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