A Verification Theorem for an Optimal Control Problem Governed by the Convective Brinkman--Forchheimer Equations

This article establishes a verification theorem for an optimal control problem governed by the two- and three-dimensional convective Brinkman--Forchheimer equations on the $d$-dimensional torus, $d\in\{2,3\}$:
$$\frac{\partial\mathfrak{u}}{\partial t}
-\mu\Delta\mathfrak{u}
+(\mathfrak{u}\cdot\nabla)\mathfrak{u}
+\alpha\mathfrak{u}
+\beta|\mathfrak{u}|^{r-1}\mathfrak{u}
+\nabla\mathfrak{p}
=\boldsymbol{f},
\qquad
\nabla\cdot\mathfrak{u}=0,$$
where $\mu,\alpha,\beta>0$ and $r\in[1,\infty)$. We derive the Pontryagin maximum principle and develop a verification framework for the associated control problem, a topic that has received comparatively little attention for fluid models of Navier--Stokes type. A major challenge in establishing the verification theorem and the corresponding feedback characterization for the CBF system is that the analysis requires a substantially different regularity framework from that used for the two-dimensional Navier--Stokes equations. In particular, the present approach relies on strong solution theory, a delicate treatment of the nonlinear absorption term, novel estimates in negative-order Sobolev spaces, and continuous dependence estimates in stronger topologies, especially in the three-dimensional setting. A distinctive feature of the present work is that the verification framework is developed not only in two dimensions, but also in the three-dimensional supercritical regime, corresponding to $r\in(3,5]$, and in the critical case $r=3$ under the condition $2\beta\mu\geq1$. Consequently, the feedback characterization and verification arguments can be rigorously justified in both two and three dimensions.