Moderate deviation principle for the 2D stochastic convective Brinkman–Forchheimer equations

Abstract

This work addresses some asymptotic behavior of solutions to the stochastic convective Brinkman-Forchheimer (SCBF) equations perturbed by multiplicative Gaussian noise in bounded domains. Using a weak convergence approach of Budhiraja and Dupuis, we establish the Laplace principle for the strong solution to the SCBF equations in a suitable Polish space. Then, the Wentzell-Freidlin large deviation principle is derived using the well known results of Varadhan and Bryc. The large deviations for short time are also considered in this work. Furthermore, we study the exponential estimates on certain exit times associated with the solution trajectory of the SCBF equations. Using contraction principle, we study these exponential estimates of exit times from the frame of reference of Freidlin-Wentzell type large deviations principle. This work also improves several LDP results available in the literature for the tamed Navier-Stokes equations as well as Navier-Stokes equations with damping in bounded domains.

Convergence of random attractors towards deterministic singleton attractor for 2D and 3D convective Brinkman-Forchheimer equations

“…It can be easily seen that the map C(•) : V ∩ L r+1 → V + L r+1 r . For any r ∈ [1, ∞) and u 1 , u 2 ∈ V ∩ L r+1 , we have (see subsection 2.4, [39])…”

“…We estimate the final term from (42) using Hölder's and Young's inequalities as (similarly as in [39])…”

“…For deterministic CBF equations, the result on the existence of unique weak as well as strong solutions is established in [1,18,24,37,38], etc. For stochastic CBF equations, the existence of strong solutions (in the probabilistic sense) is obtained in [39,42] (Gaussian and pure jump noise), martingale solutions is established in [36,44] (Gaussian and Lévy noise), and mild solution is proved in [43] (Lévy noise). The existence of a unique pathwise strong solution for 3D stochastic NSE is a wellknown open problem, similarly it is open for 3D CBF equations for 1 ≤ r < 3.…”

“…The existence of a unique pathwise strong solution for 3D stochastic NSE is a wellknown open problem, similarly it is open for 3D CBF equations for 1 ≤ r < 3. Therefore, for n = 3, we restrict ourselves to r ≥ 3 (r > 3, for any µ, β > 0, and r = 3, for 2βµ ≥ 1) (see [39] for more details).…”

Large deviation principle for stochastic convective Brinkman–Forchheimer equations perturbed by pure jump noise

“…The Wentzell-Freidlin type LDP for SCBF equations perturbed by Gaussian noise is established in [50]. The author in [51] obtained a moderate deviations principle (MDP) for 2D SCBF equations with the absorption exponent r ∈ [1,3] driven by Gaussian noise.…”