On a time discretization of a weakly damped forced nonlinear Schrödinger equation
Olivier Goubet Ezzeddine Zahrouni
We consider a semi-discrete in time Crank-Nicolson scheme to discretize a damped forced nonlinear Schrödinger equation. This provides us with a discrete infinite-dimensional dynamical system. We prove the existence of a finite dimensional global attractor for this dynamical system.
keywords: nonlinear Schrödinger equations. Global Attractor Crank-Nicolson scheme
Regularity of the attractor for a coupled Klein-Gordon-Schrödinger system with cubic nonlinearities in $\mathbb{R}^2$
Salah Missaoui Ezzeddine Zahrouni
we prove the existence of a global attractor to dissipative Klein-Gordon-Schrödinger (KGS) system with cubic nonlinearities in $H^1({\mathbb R}^2)\times H^1({\mathbb R}^2)\times L^2({\mathbb R}^2)$ and more particularly that this attractor is in fact a compact set of $H^2({\mathbb R}^2)\times H^2({\mathbb R}^2)\times H^1({\mathbb R}^2)$.
keywords: Klein-Gordon-Schrödinger equation global attractor asymptotic compactness.
On the Lyapunov functions for the solutions of the generalized Burgers equation
Ezzeddine Zahrouni
We derive new Lyapunov functions not arising from energy norm for global solutions of Generalized Burgers Equation with initial data in homogeneous Besov spaces.
keywords: smoothness Besov spaces Lyapunov functions. global existence generalized Burgers equation
Stability and convergence of an hybrid finite volume-finite element method for a multiphasic incompressible fluid model
Caterina Calgaro Meriem Ezzoug Ezzeddine Zahrouni

In this paper, we construct a fully discrete numerical scheme for approximating a two-dimensional multiphasic incompressible fluid model, also called the Kazhikhov-Smagulov model. We use a first-order time discretization and a splitting in time to allow us the construction of an hybrid scheme which combines a Finite Volume and a Finite Element method. Consequently, at each time step, one only needs to solve two decoupled problems, the first one for the density and the second one for the velocity and pressure. We will prove the stability of the scheme and the convergence towards the global in time weak solution of the model.

keywords: Kazhikhov-Smagulov model finite volume method finite element method stability convergence
Global existence of solutions for subcritical quasi-geostrophic equations
May Ramzi Zahrouni Ezzeddine
We prove the persistence of the regularity in the Besov norm spaces for the solutions of the subcritical Quasi-Geostrophic Equations with small size initial data in $\dot B^{-(2\alpha-1),\infty}_\infty$.
keywords: Besov spaces. Quasi-geostrophic equation

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