We examine the Kolmogorov equation corresponding to the following stochastic two- and three-dimensional incompressible convective Brinkman-Forchheimer equations, also known as the damped Navier-Stokes equations, driven by L\`evy noise on the torus:
\begin{align*}
\d\boldsymbol{u}+[-\mu\Delta\boldsymbol{u}+(\boldsymbol{u}\cdot\nabla)\boldsymbol{u}+\alpha\boldsymbol{u}+\beta|\boldsymbol{u}|^{r-1}\boldsymbol{u}+\nabla p]\d t
=\sqrt{\Q}\d\W+\int_{Z}\sigma(t,z)\widetilde{\pi}(\d t,\d z),
\end{align*}
where $\mu,\alpha,\beta>0$ are physical constants; $\mathrm{Q}$ is a non-negative, trace-class operator; $\mathrm{W}$ is a cylindrical Wiener process on a Hilbert space; $\sigma$ represents the jump-noise coefficient; $(Z,\mathscr{B}(Z))$ is a measurable space; $\pi$ is a time-homogeneous Poisson random measure; and $\widetilde{\pi}$ denotes its compensator. We establish the essential $m$-dissipativity of the associated Kolmogorov operator without requiring exponential moments.
In particular, for $r>3$ in two dimensions and for $3< r \leq 5$, as well as for $r=3=$ under the condition $2\beta\mu \geq 1$, in three dimensions, the absorption term supplies enough regularisation to eliminate the need for exponential moment estimates.
We also establish the ``Carr\`e du Champ identity,`` derive perturbation results for the corresponding Kolmogorov operator, and apply the developed framework to an infinite-horizon stochastic optimal control problem, demonstrating the solvability of the associated infinite-dimensional Hamilton-Jacobi-Bellman (integro-differential) equation.