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.