It has been pointed out in the work [F. Gozzi et.al., 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 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 \(\mathbb {T}^d,\ d\in \{2,3\}\) : where \(\mu ,\alpha ,\beta >0\) , \(r\in [1,\infty )\) . We study the global well-posedness of the infinite-dimensional first-order HJBE arising from an optimal control problem for CBF equations. Employing the dynamic programming approach, we prove the existence and uniqueness of viscosity solutions in both two and three-dimensions. 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 \(2\beta \mu \ge 1\) 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 \(2\beta \mu \ge 1\) in \(d\in \{2,3\}\) . It ensures the uniqueness of the viscosity solution.
