Long-time behavior of the higher-order anisotropic Caginalp phase-field systems based on the Cattaneo law

2021 ◽  
pp. 1-30
Author(s):  
Armel Judice Ntsokongo ◽  
Christian Tathy
Keyword(s):  
Phase Field ◽  
Higher Order ◽  
Well Posedness ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Fourier Law ◽  
Finite Dimensional ◽  
Long Time ◽  
Field Systems

The aim of this paper is to study higher-order Caginalp phase-field systems based on the Maxwell–Cattaneo law, instead of the classical Fourier law. More precisely, one obtains well-posedness results, as well as the existence of finite-dimensional attractors.

The coupled Cahn–Hilliard/Allen–Cahn system with dynamic boundary conditions

2021 ◽  
pp. 1-27
Author(s):  
Ahmad Makki ◽  
Alain Miranville ◽  
Madalina Petcu
Keyword(s):  
Fractal Dimension ◽  
Boundary Conditions ◽  
Well Posedness ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Dynamic Boundary Conditions ◽  
Finite Dimensional ◽  
Dynamic Boundary ◽  
Long Time ◽  
Finite Fractal Dimension

In this article, we are interested in the study of the well-posedness as well as of the long time behavior, in terms of finite-dimensional attractors, of a coupled Allen–Cahn/Cahn–Hilliard system associated with dynamic boundary conditions. In particular, we prove the existence of the global attractor with finite fractal dimension.

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 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 Blow-Up Phenomena and Global Existence to a Weakly Dissipative Shallow Water Equation

2014 ◽  
Vol 12 ◽  
pp. 10-20
Author(s):  
Jiang Bo Zhou ◽  
Jun De Chen ◽  
Wen Bing Zhang
Keyword(s):  
Global Existence ◽  
Shallow Water ◽  
Blow Up ◽  
Shallow Water Equation ◽  
Well Posedness ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Holm Equation ◽  
Special Cases ◽  
Long Time

We first establish the local well-posedness for a weakly dissipative shallow water equation which includes both the weakly dissipative Camassa-Holm equation and the weakly dissipative Degasperis-Procesi equation as its special cases. Then two blow-up results are derived for certain initial profiles. Finally, We study the long time behavior of the solutions.

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