Mamadou Sango

Weak Solutions of a Stochastic Model for Two-Dimensional Second Grade Fluids

We initiate the investigation of a stochastic system of evolution partial differential equations modelling the turbulent flows of a second grade fluid filling a bounded domain of . We establish the global existence of a probabilistic weak solution.

On the Stochastic 3D Navier-Stokes-α Model of Fluids Turbulence

We investigate the stochastic 3D Navier-Stokes-𝛼 model which arises in the modelling of turbulent flows of fluids. Our model contains nonlinear forcing terms which do not satisfy the Lipschitz conditions. The adequate notion of solutions is that of probabilistic weak solution. We establish the existence of a such of solution. We also discuss the uniqueness.

On the Strong Solution for the 3D Stochastic Leray-Alpha Model

We prove the existence and uniqueness of strong solution to the stochastic Leray- equations under appropriate conditions on the data. This is achieved by means of the Galerkin approximation scheme. We also study the asymptotic behaviour of the strong solution as alpha goes to zero. We show that a sequence of strong solutions converges in appropriate topologies to weak solutions of the 3D stochastic Navier-Stokes equations.

Homogenization of a stochastic nonlinear reaction–diffusion equation with a large reaction term: The almost periodic framework

Abstract Homogenization of a stochastic nonlinear reaction–diffusion equation with a large nonlinear term is considered. Under a general Besicovitch almost periodicity assumption on the coefficients of the equation we prove that the sequence of solutions of the said problem converges in probability towards the solution of a rather different type of equation, namely, the stochastic nonlinear convection–diffusion equation which we explicitly derive in terms of appropriate functionals. We study some particular cases such as the periodic framework, and many others. This is achieved under a suitable generalized concept of Σ -convergence for stochastic processes.

Splitting-up scheme for nonlinear stochastic hyperbolic equations

Abstract. In the present paper we develop the splitting-up scheme (so-called method of fractional steps) for the investigation of existence problem for a class of nonlinear hyperbolic equations containing some nonlinear terms which do not satisfy the Lipschitz condition. Through a careful blending of the numerical scheme and deep compactness results of both analytic and probabilistic nature we establish the existence of a weak probabilistic solution for the problem. Our work is the stochastic counterpart of some important results of Roger Temam obtained in the late sixties of the last century in his works on the development of the splitting-up method for deterministic evolution problems.

Homogenization of linear hyperbolic stochastic partial differential equation with rapidly oscillating coefficients: The two scale convergence method

In this paper we establish new homogenization results for stochastic linear hyperbolic equations with periodically oscillating coefficients. We first use the multiple expansion method to drive the homogenized problem. Next we use the two scale convergence method and Prokhorovs and Skorokhods probabilistic compactness results. We prove that the sequence of solutions of the original problem converges in suitable topologies to the solution of a homogenized stochastic hyperbolic problem with constant coefficients. We also prove a corrector result. Homogenization is a mathematical theory aimed at understanding the behavior of processes that take place in heterogeneous media with highly oscillating heterogeneities at the microscopic level using prop- erties of the homogeneous media obtained by homogenizing these materials. These heterogeneous ma- terials consist of finely mixed different components like soil, paper, concrete for building, fibreglass, materials used in the manufacturing of high tech equipments such as planes, rockets and so on. This signifies that almost everything around us in real life is a heterogeneous material. The physical problems described on heterogeneous materials such as heat, mechanical constraints, flow of fluids in these me- dia leads to the study of PDEs with highly oscillating coefficients depending on macroscopic scales or boundary value problems for PDEs in domain with fine grained boundaries. The main obstacle in solv- ing these problems arises either from the character of the domain or the presence of high oscillations in the coefficients of the governing equation. To this end, it is expensive to compute solutions to these type of problems. Numerical methods have proved inefficient in solving such problems due to the fact that even the most advanced parallel computers are unable to simulate schemes related to the physically interesting such problems.

Homogenization of Neumann problem for hyperbolic stochastic partial differential equations in perforated domains

In this paper, we investigate a linear hyperbolic stochastic partial differential equation (SPDE) with rapidly oscillating � -periodic coefficients in a domain with small holes (of size-� ) under Neumann conditions on the boundary of the holes and Dirichlet condition on the exterior boundary. When the number of these holes approach infinity, i.e. their sizes approach zero, the homogenized problem is a hyperbolic SPDE with constant coefficients in the domain without perforations. Moreover the convergence of the associated energy to that of the homogenized system is established.

Convergence of a sequence of solutions of the stochastic two-dimensional equations of second grade fluids

The research of the authors is supported by the University of Pretoria and the National Research Foundation South Africa. The first author is also very grateful to the support he received from The Abdus Salam International Center for Theoretical Physics.

A Tartar approach to periodic homogenization of linear hyperbolic stochastic partial differential equation

This paper deals with the homogenization of a linear hyperbolic stochastic partial differential equation (SPDE) with highly oscillating periodic coefficients. We use Tartar’s method of oscillating test functions and deep probabilistic compactness results due to Prokhorov and Skorokhod. We show that the sequence of solutions of the original problem converges in suitable topologies to the solution of a homogenized linear hyperbolic SPDE with constant coefficients. We also prove the convergence of the associated energies.

On the exponential behaviour of stochastic evolution equations for non-Newtonian fluids

We investigate the exponential long-time behaviour of the stochastic evolution equations describing the motion of a non-Newtonian fluids excited by multiplicative noise. Some results on the exponential convergence in mean square and with probability one of the weak probabilistic solution to the stationary solutions are given. We also prove an interesting result related to the stabilization of these stochastic evolution equations.