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Phase-field System (phase-field + system)

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Selected Abstracts

Phase-field systems for multi-dimensional Prandtl,Ishlinskii operators with non-polyhedral characteristics

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 4 2002
Jürgen Sprekels
Abstract Hysteresis operators have recently proved to be a powerful tool in modelling phase transition phenomena which are accompanied by the occurrence of hysteresis effects. In a series of papers, the present authors have proposed phase-field models in which hysteresis non-linearities occur at several places. A very important class of hysteresis operators studied in this connection is formed by the so-called Prandtl,Ishlinskii operators. For these operators, the corresponding phase-field systems are in the multi-dimensional case only known to admit unique solutions if the characteristic convex sets defining the operators are polyhedrons. In this paper, we use approximation techniques to extend the known results to multi-dimensional Prandtl,Ishlinskii operators having non-polyhedral convex characteristicsets. Copyright © 2002 John Wiley & Sons, Ltd. [source]

Stability of global and exponential attractors for a three-dimensional conserved phase-field system with memory

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 18 2009
Gianluca Mola
Abstract We consider a conserved phase-field system on a tri-dimensional bounded domain. The heat conduction is characterized by memory effects depending on the past history of the (relative) temperature ,, which is represented through a convolution integral whose relaxation kernel k is a summable and decreasing function. Therefore, the system consists of a linear integrodifferential equation for ,, which is coupled with a viscous Cahn,Hilliard type equation governing the order parameter ,. The latter equation contains a nonmonotone nonlinearity , and the viscosity effects are taken into account by a term ,,,,t,, for some ,,0. Rescaling the kernel k with a relaxation time ,>0, we formulate a Cauchy,Neumann problem depending on , and ,. Assuming a suitable decay of k, we prove the existence of a family of exponential attractors {,,,,} for our problem, whose basin of attraction can be extended to the whole phase,space in the viscous case (i.e. when ,>0). Moreover, we prove that the symmetric Hausdorff distance of ,,,, from a proper lifting of ,,,0 tends to 0 in an explicitly controlled way, for any fixed ,,0. In addition, the upper semicontinuity of the family of global attractors {,,,,,} as ,,0 is achieved for any fixed ,>0. Copyright © 2009 John Wiley & Sons, Ltd. [source]

Convergence of phase field to phase relaxation models governed by an entropy equation with memory

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 18 2006
Gianni Gilardi
Abstract The subject of the present paper consists in proving the convergence of a phase-field model, based on the entropy equation with memory, to phase relaxation. The well-posedness and the long-time behaviour of solutions for the non-linear and singular phase-field system have been recently shown by Bonetti et al. (Preprint IMATI-CNR, 2005; Discrete Contin. Dyn. Syst. Ser. B, in press). Here, we study the asymptotic behaviour of such solutions as the interfacial energy coefficient tends to zero. The limit problem is a phase relaxation problem with memory, which is new. We prove well-posedness results through convergence under rather general assumptions. However, the case of a quadratic non-linearity for the latent heat is excluded. Such a situation is dealt for the problem without memory in a generalized setting by introducing an ad hoc logarithm. Copyright © 2006 John Wiley & Sons, Ltd. [source]

Long-time convergence of solutions to a phase-field system

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 5 2001
Sergiu Aizicovici
We prove that any global bounded solution of a phase field model tends to a single equilibrium state for large times though the set of equilibria may contain a nontrivial continuum of stationary states. The problem has a partial variational structure, specifically, only the elliptic part of the first equation represents an Euler,Lagrange equation while the second does not. This requires some modifications in comparison with standard methods used to attack this kind of problems. Copyright © 2001 John Wiley & Sons, Ltd. [source]

Well-posedness and long time behavior of a parabolic-hyperbolic phase-field system with singular potentials

MATHEMATISCHE NACHRICHTEN, Issue 13-14 2007
Maurizio Grasselli
Abstract In this article, we study the long time behavior of a parabolic-hyperbolic system arising from the theory of phase transitions. This system consists of a parabolic equation governing the (relative) temperature which is nonlinearly coupled with a weakly damped semilinear hyperbolic equation ruling the evolution of the order parameter. The latter is a singular perturbation through an inertial term of the parabolic Allen,Cahn equation and it is characterized by the presence of a singular potential, e.g., of logarithmic type, instead of the classical double-well potential. We first prove the existence and uniqueness of strong solutions when the inertial coefficient , is small enough. Then, we construct a robust family of exponential attractors (as , goes to 0). (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]