Unique strong and $${ \mathbb {V} }$$ V -attractor of a three-dimensional globally modified two-phase flow model

Abstract

In this article, we study a globally modified Allen-Cahn-Navier-Stokes system in a three-dimensional domain. The model consists of the globally modified Navier-Stokes equations proposed in Caraballo et al. (Adv Nonlinear Stud 6(3):411-436, 2006) for the velocity, coupled with an Allen-Cahn model for the order (phase) parameter. We prove the existence and uniqueness of strong solutions. Using the flattening property, we also prove the existence of global V-attractors for the model. Using a limiting argument, we derive the existence of bounded entire weak solutions for the three-dimensional coupled Allen-Cahn-Navier-Stokes system with time-independent forcing.

Unique strong solutions and V-attractor of a three dimensional globally modified magnetohydrodynamic equations

“…For convergence results of solutions of the GMNSE to solutions of the three dimensional Navier-Stokes equations, see [4,19]. These results were extended in [24,25] to the case of the 3D globally modified Cahn-Hilliard-Navier-Stokes equations and the 3D globally modified Allen-Cahn-Navier-Stokes equations. The GMMHDE (2)-(3) is inspired from the globally modified Navier-Stokes equations (GMNSE) studied in [4].…”

On the convergence of a stochastic 3D globally modified two-phase flow model

“…Let us note that the coupling between the Navier-Stokes and the Allen-Cahn equations introduces in the coupled model a highly nonlinear term that makes the analysis more involved. In this article, we consider a stochastic version of the model studied in [35].…”

“…The aim of this article is to investigate the stochastic version of the GMACNSE studied in [35]. The model include an abstract and general form of random external forces depending eventually on the velocity v of the fluid and the phase function φ.…”

“…They prevent large values of v and (v, φ) U dominating the dynamics. Just like the GMNSE, the GMACNSE violates the basic laws of mechanics, but mathematically the model is well defined, [35].…”