Well‐posedness of a stochastic phase‐field problem with multiplicative noises

2020 ◽  
Vol 43 (15) ◽  
pp. 8538-8567
Author(s):  
Perla El Kettani
Keyword(s):  
Phase Field ◽  
Well Posedness ◽  
Field Problem ◽  
Multiplicative Noises

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.

scholarly journals Towards a 3-Dimensional Phase-Field Model of Non-Isothermal Alloy Solidification

2014 ◽  
Vol 783-786 ◽  
pp. 2166-2171 ◽  
Author(s):  
Andrew M. Mullis ◽  
Peter C. Bollada ◽  
Peter K. Jimack
Keyword(s):  
Phase Field ◽  
Adaptive Mesh Refinement ◽  
Phase Field Model ◽  
Lewis Number ◽  
Adaptive Mesh ◽  
Implicit Time ◽  
Dimensional Problem ◽  
Alloy Solidification ◽  
Field Problem ◽  
3 Dimensional

We review the application of advanced numerical techniques such as adaptive mesh refinement, implicit time-stepping, multigrid solvers and massively parallel implementations as a route to obtaining solutions to the 3-dimensional phase field problem for coupled heat and solute transport during non-isothermal alloy solidification. Using such techniques it is shown that such models are tractable for modest values of the Lewis number (ratio of thermal to solutal diffusivities). Solutions to the 3-dimensional problem are compared with existing solutions to the equivalent 2-dimensional problem.

scholarly journals An Efficient Algorithm for Reconstruction Images Corrupted by Some Multiplicative Noises

2019 ◽  
Vol 5 (2) ◽  
pp. 263-278
Author(s):  
L. Ziad ◽  
O. Oubbih ◽  
F. Sniba
Keyword(s):  
Multiplicative Noise ◽  
Slow Growth ◽  
Mathematical Framework ◽  
Well Posedness ◽  
Multiplicative Noises ◽  
Proposed Model ◽  
Staircase Effect ◽  
Map Estimator ◽  
Denoised Image ◽  
Numerical Approaches

AbstractIn this paper, we propose a novel hybrid model for restoration of images corrupted by multiplicative noise. Using a MAP estimator, we can derive a functional whose minimizer corresponds to the denoised image we want to recover. The energies studied here are inspired by image restoration with non linear variable exponent [1, 2], and it is a combination of fast growth with respect to low gradient and slow growth when the gradient is large. We study a mathematical framework to prove the well posedness of the minimizer problem and we introduce the associated evolution problem, for which we derive numerical approaches. At last, compared experimental results distinctly demonstrate the superiority of the proposed model, in term of removing some muliplicative noise while preserving the edges and reducing the staircase effect.

An Adaptive Finite Element Multi-Mesh Approach for Interacting Deformable Objects in Flow

2016 ◽  
Vol 16 (3) ◽  
pp. 475-484 ◽  
Author(s):  
Siqi Ling ◽  
Wieland Marth ◽  
Simon Praetorius ◽  
Axel Voigt
Keyword(s):  
Phase Field ◽  
Adaptive Mesh Refinement ◽  
Adaptive Mesh ◽  
Numerical Approach ◽  
General Concept ◽  
Field Variable ◽  
Deformable Objects ◽  
Adaptive Finite Element ◽  
Field Problem ◽  
Multi Phase

AbstractWe consider a hydrodynamic multi-phase field problem to model the interaction of deformable objects. The numerical approach considers one phase field variable for each object and allows for an independent adaptive mesh refinement for each variable. Using the special structure of various terms allows interpolating the solution on one mesh onto another without loss of information. Together with a general multi-mesh concept for the other terms speedup by a factor of two can be demonstrated which improves with the number of interacting objects. The general concept is demonstrated on an example describing the interaction of red blood cells in an idealized vessel.

Global and exponential attractors for a Caginalp type phase-field problem

2013 ◽  
Vol 11 (9) ◽  
Author(s):  
Brice Bangola
Keyword(s):  
Phase Field ◽  
Phase Field Model ◽  
Neumann Boundary ◽  
Spatial Behavior ◽  
Infinite Cylinder ◽  
Time Behavior ◽  
Exponential Attractors ◽  
Field Problem ◽  
Long Time ◽  
Well Posed

AbstractWe deal with a generalization of the Caginalp phase-field model associated with Neumann boundary conditions. We prove that the problem is well posed, before studying the long time behavior of solutions. We establish the existence of the global attractor, but also of exponential attractors. Finally, we study the spatial behavior of solutions in a semi-infinite cylinder, assuming that such solutions exist.

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