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### Open Access Journals

CPAA

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.

CPAA

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 $H^2({\mathbb R}^2)\times H^2({\mathbb R}^2)\times H^1({\mathbb R}^2)$.

CPAA

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

CPAA

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

CPAA

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 $\dot B^{-(2\alpha-1),\infty}_\infty$.

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