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

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.