12 research outputs found
Homogenization of nonlinear stochastic partial differential equations in a general ergodic environment
In this paper, we show that the concept of sigma-convergence associated to
stochastic processes can tackle the homogenization of stochastic partial
differential equations. In this regard, the homogenization problem for a
stochastic nonlinear partial differential equation is studied. Using some deep
compactness results such as the Prokhorov and Skorokhod theorems, we prove that
the sequence of solutions of this problem converges in probability towards the
solution of an equation of the same type. To proceed with, we use a suitable
version of sigma-convergence method, the sigma-convergence for stochastic
processes, which takes into account both the deterministic and random
behaviours of the solutions of the problem. We apply the homogenization result
to some concrete physical situations such as the periodicity, the almost
periodicity, the weak almost periodicity, and others.Comment: To appear in: Stochastic Analysis and Application
Some qualitative properties of the solutions of the Magnetohydrodynamic equations for nonlinear bipolar fluids
In this article we study the long-time behaviour of a system of nonlinear
Partial Differential Equations (PDEs) modelling the motion of incompressible,
isothermal and conducting modified bipolar fluids in presence of magnetic
field. We mainly prove the existence of a global attractor denoted by \A for
the nonlinear semigroup associated to the aforementioned systems of nonlinear
PDEs. We also show that this nonlinear semigroup is uniformly differentiable on
\A. This fact enables us to go further and prove that the attractor \A is
of finite-dimensional and we give an explicit bounds for its Hausdorff and
fractal dimensions.Comment: The final publication is available at Springer via
http://dx.doi.org/10.1007/s10440-014-9964-
Homogenization of a stochastic nonlinear reaction-diffusion equation with a large reaction term: the almost periodic framework
Homogenization of a stochastic nonlinear reaction-diffusion equation with a
large non- linear 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 non-
linear convection-diffusion equation which we explicitly derive in terms of
appropriated functionals. We study some particular cases such as the periodic
framework, and many others. This is achieved under a suitable generalized
concept of sigma-convergence for stochastic processes.Comment: 34 page
Stochastic Reaction-diffusion Equations Driven by Jump Processes
We establish the existence of weak martingale solutions to a class of second
order parabolic stochastic partial differential equations. The equations are
driven by multiplicative jump type noise, with a non-Lipschitz multiplicative
functional. The drift in the equations contains a dissipative nonlinearity of
polynomial growth.Comment: See journal reference for teh final published versio
Existence and large time behaviour for a stochastic model of a modified magnetohydrodynamic equations
In this paper we initiate the mathematical analysis of a system of nonlinear
Stochastic Partial Differential equations describing the motion of turbulent
Non-Newtonian media in the presence of fluctuating magnetic field. The system
is basically obtained by a coupling of the dynamical equations of a
Non-Newtonian fluids having -structure and the Maxwell equations. We mainly
show the existence of weak martingale solutions and their exponential decay
when time goes to infinity.Comment: This paper needs some revisio