Well-posedness and asymptotic behavior of the stochastic convective Brinkman-Forchheimer equations perturbed by pure jump noise

TL;DR

This paper establishes the well-posedness and analyzes the long-term behavior of stochastic convective Brinkman-Forchheimer equations with pure jump noise, including stability and invariant measures, in bounded or periodic domains.

Contribution

It proves existence, uniqueness, and regularity of solutions for the first time for these equations with jump noise, and investigates their stability and ergodic properties.

Findings

Existence and uniqueness of strong solutions established.

Exponential stability of stationary solutions proved.

Existence of a unique ergodic invariant measure demonstrated.

Abstract

This paper is concerned about the stochastic convective Brinkman-Forchheimer (SCBF) equations subjected to multiplicative pure jump noise in bounded or periodic domains. Our first goal is to establish the existence of a pathwise unique strong solution satisfying the energy equality (It\^o's formula) to the SCBF equations. We resolve the issue of the global solvability of SCBF equations, by using a monotonicity property of the linear and nonlinear operators and a stochastic generalization of the Minty-Browder technique. The major difficulty is that an It\^o's formula in infinite dimensions is not available for such systems. This difficulty is overcame by approximating the solution using approximate functions composing of the elements of eigenspaces of the Stokes operator in such a way that the approximations are bounded and converge in both Sobolev and Lebesgue spaces simultaneously. Due to technical difficulties, we discuss about the global in time regularity results of such strong solutions in periodic domains only. Once the system is well-posed, we look for the asymptotic behavior of strong solutions. The exponential stability results (in mean square and pathwise sense) for the stationary solutions is established in this work for large effective viscosity. Moreover, a stabilization result of the SCBF equations by using a multiplicative pure jump noise is also obtained. Finally, we prove the existence of a unique ergodic and strongly mixing invariant measure for the SCBF equations subject to multiplicative pure jump noise, by using the exponential stability of strong solutions.