Kolmogorov equations for 2D stochastic convective Brinkman-Forchheimer equations: Analysis and Applications

TL;DR

This paper analyzes the Kolmogorov equations related to 2D stochastic convective Brinkman-Forchheimer equations, establishing existence, identities, and applications in control and obstacle problems under certain conditions.

Contribution

It provides the first rigorous analysis of the Kolmogorov equation for 2D SCBF equations, including existence, derivative estimates, and applications to control and obstacle problems.

Findings

Proved existence of solutions to the Kolmogorov equation in the invariant measure space.

Established the 'carré du champs' identity for the Kolmogorov operator.

Demonstrated solutions for control and obstacle problems related to 2D SCBF equations.

Abstract

In this work, we consider the following 2D stochastic convective Brinkman-Forchheimer (SCBF) equations in a bounded smooth domain $\mathcal{O}$: \begin{align*} \mathrm{d}\boldsymbol{u}+\left[-\mu \Delta\boldsymbol{u}+(\boldsymbol{u}\cdot\nabla)\boldsymbol{u}+\alpha\boldsymbol{u}+\beta|\boldsymbol{u}|^{r-1}\boldsymbol{u}+\nabla p\right]\mathrm{d}t=\sqrt{\mathrm{Q}}\mathrm{W}, \ \nabla\cdot\boldsymbol{u}=0, \end{align*} where $\mu,\alpha,\beta>0$, $r\in\{1,2,3\}$, $\mathrm{Q}$ is a non-negative operator of trace class, $\mathrm{W}$ is a cylindrical Wiener process in a Hilbert space $\mathbb{H}$. Under the following assumption on the viscosity co-efficient $\mu$ and the Darcy co-efficient $\alpha$: for some positive constant $\gamma_1$, \begin{equation*} \mu(\mu+\alpha)^2>\gamma_1\max\{4\mathrm{Tr}(\mathrm{Q}),\mathrm{Tr}(\mathrm{A}^{2\delta}\mathrm{Q})\}, \end{equation*} where $\mathrm{A}$ is the Stokes operator and $\delta\in(0,\frac{1}{2})$, our primary goal is to solve the corresponding Kolmogorov equation in the space $\mathbb{L}^2(\mathbb{H};\eta),$ where $\eta$ is the unique invariant measure associated with 2D SCBF equations. Then, we establish the well-known ``carr\'e du champs'' identity. Some sharp estimates on the derivatives of the solution constitute the key component of the proofs. We take into consideration two control problems from the application point of view. The first is an infinite horizon control problem for which we establish the existence of a solution for the Hamilton-Jacobi-Bellman equation associated with it. Finally, by exploiting $m$-accretive theory, we demonstrate the existence of a unique solution for an obstacle problem associated with the Kolmogorov operator corresponding to the stopping-time problem for 2D SCBF equations.