Large deviations for the two-time-scale stochastic convective Brinkman-Forchheimer equations

TL;DR

This paper establishes a large deviation principle for two-time-scale stochastic convective Brinkman-Forchheimer equations, extending the analysis to include small noise effects in fluid flow models within porous media.

Contribution

It introduces a Wentzell-Freidlin type large deviation framework for the SCBF equations with multiple time scales, using variational and discretization methods.

Findings

Large deviation principle proven for SCBF equations with small noise

Results applicable to 2D stochastic Navier-Stokes equations

Method combines variational approach with time discretization and stopping times

Abstract

The convective Brinkman-Forchheimer (CBF) equations characterize the motion of incompressible fluid flows in a saturated porous medium. The small noise asymptotic for the two-time-scale stochastic convective Brinkman-Forchheimer (SCBF) equations in two and three dimensional bounded domains is carried out in this work. More precisely, we establish a Wentzell-Freidlin type large deviation principle for stochastic partial differential equations with slow and fast time-scales, where the slow component is the SCBF equations in two and three dimensions perturbed by small multiplicative Gaussian noise and the fast component is a stochastic reaction-diffusion equation with damping. The results are obtained by using a variational method (based on weak convergence approach) developed by Budhiraja and Dupuis, Khasminkii's time discretization approach and stopping time arguments. In particular, the results obtained from this work are true for two dimensional stochastic Navier-Stokes equations also.