Martingale solution, invariant measure and ergodicity for stochastic convective Brinkman-Forchheimer equations on general domains in ${\mathbb{R}}^d$

Abstract

The convective Brinkman-Forchheimer equations (CBFEs) \[ \frac{\partial \boldsymbol{X}}{\partial t} - μΔ\boldsymbol{X} + (\boldsymbol{X}\cdot\nabla)\boldsymbol{X} + α\boldsymbol{X} + β|\boldsymbol{X}|^{r-1}\boldsymbol{X} + \nabla p = \mathbf{F}, \qquad \nabla\cdot\boldsymbol{X}=0, \] with parameters $\hat{I}¼,\hat{I}\pm ,{\hat{I}}^2>0$ and $r\in [1,\infty )$ describe incompressible fluid motion in saturated porous media. In the stochastic setting, for $d=2,3$ and $r\in [3,\infty )$ (with $2{\hat{I}}^2\hat{I}¼\ge 1$ when $r=3$), strong pathwise solutions on general domains are already known, hence weak martingale solutions exist as well. In the same parameter regime, invariant probability measures on bounded domains have also been obtained. The present work complements and significantly extends these results. More precisely, on general domains in ${\mathbb{R}}^d$ (bounded or unbounded), for all $d\in \{2,3\}$, we prove the existence of a weak martingale solution to the stochastic CBFEs for every exponent $r\in [1,\infty )$, which includes the regimes where no strong solution theory is available. For $d=2$, $r\in [1,\infty )$, and for $d=3$, $r\in [3,\infty )$, we further show that the martingale solutions satisfy the energy equality (Itô's formula) and possess $\mathbb{H}$-valued continuous trajectories almost surely. In this regularity regime (excluding $2{\hat{I}}^2\hat{I}¼<1$ when $r=3$), we establish pathwise uniqueness and thereby, via the Yamada-Watanabe argument, obtain the existence of strong solutions and uniqueness in law, thereby recovering, in particular, the known results. Finally, for $d=2$, $r\in [1,\infty )$, and for $d=3$, $r\in [3,\infty )$ (with $2{\hat{I}}^2\hat{I}¼\ge 1$ when $r=3$), we prove the existence of an invariant probability measure for the associated Markov semigroup, while for $d=2,3$ with $r\in [3,\infty )$ (and with $2{\hat{I}}^2\hat{I}¼\ge 1$ for $r=3$), we show that at most one invariant measure can exist.