Martingale solutions of two and three dimensional stochastic convective Brinkman-Forchheimer equations forced by Lévy noise (2109.05510v1)

Abstract: The convective Brinkman-Forchheimer equations given by $$ \frac{\partial \boldsymbol{u}}{\partial t}-\mu \Delta\boldsymbol{u}+(\boldsymbol{u}\cdot\nabla)\boldsymbol{u}+\alpha\boldsymbol{u}+\beta|\boldsymbol{u}|^{r-1}\boldsymbol{u}+\nabla p=\boldsymbol{f}, \ \nabla\cdot\boldsymbol{u}=0, $$ where $\mu,\alpha,\beta>0$ and $r\in[1,\infty)$ describe the motion of incompressible fluid flows in a saturated porous medium. In bounded domains (for $d=2,3$ and $r\in[1,\infty)$), the existence of a weak martingale solution for stochastic convective Brinkman-Forchheimer equations forced by L\'evy noise consisting of a $\mathrm{Q}$-Wiener process and a compensated time homogeneous Poisson random measure is established in this work. Using the classical Faedo-Galerkin approximation, a compactness method and a version of the Skorokhod embedding theorem for nonmetric spaces, we prove the existence of a weak martingale solution. For $d=2$, $r\in[1,\infty)$ and $d=3$, $r\in[3,\infty)$, we prove that the martingale solution has stronger regularity properties such as it satisfies the energy equality (It^o's formula) and hence the trajectories are equal almost everywhere to an $\mathbb{H}$-valued c`adl`ag function defined on $[0, T ]$, ${\mathbb{P}}$-a.s. Furthermore, for $d=2$, $r\in[1,\infty)$ and $d=3$, $r\in[3,\infty)$ ($2\beta\mu\geq 1$ for $r=3$), we show the pathwise uniqueness of solutions and use the classical Yamada-Watanabe argument to derive the existence of a strong solution and uniqueness in law.