scholarly journals Uniqueness and long-time behavior for the conserved phase-field system with memory

1999 ◽  
Vol 5 (2) ◽  
pp. 375-390 ◽  
Author(s):  
Pierluigi Colli ◽  
◽  
Gianni Gilardi ◽  
Philippe Laurençot ◽  
Amy Novick-Cohen ◽  
...  
Keyword(s):  
Phase Field ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Field System ◽  
Long Time ◽  
Phase Field System ◽  
With Memory

Long time behavior of higher-order anisotropic perturbed phase field system with regular potentials

2020 ◽  
Vol 116 (3-4) ◽  
pp. 149-217
Author(s):  
Clesh Deseskel Elion Ekohela ◽  
Gabriel Bissanga ◽  
Macaire Batchi
Keyword(s):  
Phase Field ◽  
Higher Order ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Field System ◽  
Long Time ◽  
Phase Field System

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

2007 ◽  
Vol 280 (13-14) ◽  
pp. 1475-1509 ◽  
Author(s):  
Maurizio Grasselli ◽  
Alain Miranville ◽  
Vittorino Pata ◽  
Sergey Zelik
Keyword(s):  
Phase Field ◽  
Well Posedness ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Singular Potentials ◽  
Field System ◽  
Long Time ◽  
Phase Field System

scholarly journals Well-posedness and long-time behavior for the modified phase-field crystal equation

2014 ◽  
Vol 24 (14) ◽  
pp. 2743-2783 ◽  
Author(s):  
Maurizio Grasselli ◽  
Hao Wu
Keyword(s):  
Phase Field ◽  
Time Derivative ◽  
Exponential Attractor ◽  
Well Posedness ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Long Time ◽  
Global Existence And Uniqueness ◽  
And Diffusion ◽  
Phase Field Crystal

We consider a modification of the so-called phase-field crystal (PFC) equation introduced by K. R. Elder et al. This variant has recently been proposed by P. Stefanovic et al. to distinguish between elastic relaxation and diffusion time scales. It consists of adding an inertial term (i.e. a second-order time derivative) into the PFC equation. The mathematical analysis of the resulting equation is more challenging with respect to the PFC equation, even at the well-posedness level. Moreover, its solutions do not regularize in finite time as in the case of PFC equation. Here we analyze the modified PFC (MPFC) equation endowed with periodic boundary conditions. We first prove the global existence and uniqueness of a solution with initial data in a bounded energy space. This solution satisfies some uniform dissipative estimates which allow us to study the long-time behavior of the corresponding dynamical system. In particular, we establish the existence of the global attractor as well as an exponential attractor. Then we demonstrate that any trajectory originating from the bounded energy phase space converges to a single equilibrium. This is done by means of a suitable version of the Łojasiewicz–Simon inequality. An estimate on the convergence rate is also given.

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

2001 ◽  
Vol 24 (5) ◽  
pp. 277-287 ◽  
Author(s):  
Sergiu Aizicovici ◽  
Eduard Feireisl ◽  
Fran�oise Issard-Roch
Keyword(s):  
Phase Field ◽  
Convergence Of Solutions ◽  
Field System ◽  
Long Time ◽  
Phase Field System
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