Optimal control of convective Brinkman-Forchheimer equations: Dynamic programming equation and Viscosity solutions

TL;DR

This paper establishes the existence and uniqueness of viscosity solutions for the Hamilton-Jacobi-Bellman equation associated with damped convective Brinkman-Forchheimer equations in 2D and 3D, extending control theory to complex fluid models.

Contribution

It proves the existence of viscosity solutions and a comparison principle for the HJB equation related to damped CBF equations in 2D and 3D, addressing previous unresolved issues.

Findings

Existence of viscosity solutions in supercritical regimes.

Comparison principle ensuring uniqueness of solutions.

Extension of control theory to damped fluid models.

Abstract

It has been pointed out in the work [F. Gozzi et.al., \emph{Arch. Ration. Mech. Anal.} {163}(4) (2002), 295--327] that the existence and uniqueness of viscosity solutions to the first-order Hamilton-Jacobi-Bellman equation (HJBE) associated with the three-dimensional Navier-Stokes equations (NSE) have not been resolved due to the lack of global solvability and continuous dependence results. However, by adding a damping term to NSE, the so-called \emph{damped Navier-Stokes equations} fulfills the requirement of existence and uniqueness of global strong solutions. In this work, we address this issue in the context of the following two- and three-dimensional convective Brinkman-Forchheimer (CBF) equations (damped NSE) in $\mathbb{T}^d,\ d\in\{2,3\}$: \begin{align*} \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, \end{align*} where $\mu,\alpha,\beta>0$, $r\in[1,\infty)$. We first prove the existence of a viscosity solution to the infinite-dimensional HJBE in the supercritical regime. For spatial dimension $d=2$, we consider the nonlinearity exponent $r\in(3,\infty)$, while for $d=3$, due to some technical difficulty, we focus on $r\in(3,5]$. In the case $r=3$, we require the condition $2\beta\mu\geq 1$ for both $d=2$ and $d=3$. Next, we derive a comparison principle for the HJB equation covering the ranges $r\in(3,\infty)$ and $r=3$ with $2\beta\mu\geq 1$ in $d\in\{2,3\}$. It ensures the uniqueness of the viscosity solution.