arXiv subjects

T. Tachim Medjo

Publications and source records attributed to T. Tachim Medjo.

4 recordsLinked to original sources

arXiv2026

On a stochastic Cahn-Hilliard-Brinkman model

In this paper, we consider a stochastic version of the Cahn-Hilliard-Brinkman model in a smooth two- or three-dimensional domain with dynamical boundary conditions. The system describes creeping two-phase flows and is basically a coupling of the Brinkman equation for the velocity field that governs the flow through the porous media coupled with convective Cahn-Hilliard equations for the phase field, both with two independent sources of randomness given by general multiplicative-type Wiener noises in the Cahn-Hilliard equations. The existence of a weak solution, both in the probabilistic and PDEs sense, is proved. Our construction of a solution is based on the classical Faedo-Galerkin approximation, the Yosida approximation and uses a compactness method. Our paper is the first attempt to generalize the paper \cite{Colli+Knopf+Schimperna+Signor_2024} to a stochastic setting.

math.PR↗

arXiv2024

Optimal Boundary Control for the Stochastic Allen-Cahn Navier-Stokes system in two dimensions

In this work, we study an optimal boundary control for the stochastic Allen Cahn Navier Stokes system. The governing system of nonlinear partial differential equations consists of the stochastic Navier Stokes equations with non homogeneous Navier slip boundary condition coupled with a phase-field equation, which is the convective Allen Cahn equation type. We investigate the well posedness of the nonlinear system. More precisely, the existence and uniqueness of global strong solution in dimension two is established and we prove the existence of an optimal solution to the control problem.

math.AP↗

arXiv2014

On a regularized family of models for homogeneous incompressible two-phase flows

We consider a general family of regularized models for incompressible two-phase flows based on the Allen-Cahn formulation in n-dimensional compact Riemannian manifolds for n=2,3. The system we consider consists of a regularized family of Navier-Stokes equations (including the Navier-Stokes-α-like model, the Leray-α model, the Modified Leray-α model, the Simplified Bardina model, the Navier-Stokes-Voight model and the Navier-Stokes model) for the fluid velocity suitably coupled with a convective Allen-Cahn equation for the (phase) order parameter. We give a unified analysis of the entire three-parameter family of two-phase models using only abstract mapping properties of the principal dissipation and smoothing operators, and then use assumptions about the specific form of the parametrizations, leading to specific models, only when necessary to obtain the sharpest results. We establish existence, stability and regularity results, and some results for singular perturbations, which as special cases include the inviscid limit of viscous models and the α->0 limit in α-models. Then, we also show the existence of a global attractor and exponential attractor for our general model, and then establish precise conditions under which each trajectory converges to a single equilibrium by means of a LS inequality. We also derive new results on the existence of global and exponential attractors for the regularized family of Navier-Stokes equations and magnetohydrodynamics models which improve and complement the results of Holst et. al. [J. Nonlinear Science 20, 2010, 523-567]. Finally, our analysis is applied to certain regularized Ericksen-Leslie (RSEL) models for the hydrodynamics of liquid crystals in n-dimensional compact Riemannian manifolds.

math.AP↗

arXiv2013

Long time stability of a classical efficient scheme for an incompressible two-phase flow model

In this article we consider the implicit Euler scheme for a homogeneous two-phase flow model in a two-dimensional domain and with the aid of the discrete Gronwall lemma and of the discrete uniform Gronwall lemma we prove that the global attractors generated by the numerical scheme converge to the global attractor of the continuous system as the time-step approaches zero.

math.NA↗