2D and 3D convective Brinkman-Forchheimer equations perturbed by a subdifferential and applications to control problems

Abstract

The following convective Brinkman-Forchheimer (CBF) equations (or damped Navier-Stokes equations) with potential \begin{equation*} \frac{\partial \boldsymbol{y}}{\partial t}-μΔ\boldsymbol{y}+(\boldsymbol{y}\cdot\nabla)\boldsymbol{y}+α\boldsymbol{y}+β|\boldsymbol{y}|^{r-1}\boldsymbol{y}+\nabla p+Ψ(\boldsymbol{y})\ni\boldsymbol{g},\ \nabla\cdot\boldsymbol{y}=0, \end{equation*} in a $d$-dimensional torus is considered in this work, where $d\in \{2,3\}$, $\mu ,\alpha ,\beta >0$ and $r\in [1,\infty )$. For $d=2$ with $r\in [1,\infty )$ and $d=3$ with $r\in [3,\infty )$ ($2\beta \mu \ge 1$ for $d=r=3$), we establish the existence of \textsf{\emph{a unique global strong solution}} for the above multi-valued problem with the help of the \textsf{\emph{abstract theory of $m$-accretive operators}}. %for nonlinear differential equations of accretive type in Banach spaces. Moreover, we demonstrate that the same results hold \textsf{\emph{local in time}} for the case $d=3$ with $r\in [1,3)$ and $d=r=3$ with $2\beta \mu <1$. We explored the $m$-accretivity of the nonlinear as well as multi-valued operators, Yosida approximations and their properties, and several higher order energy estimates in the proofs. For $r\in [1,3]$, we {quantize (modify)} the Navier-Stokes nonlinearity $(\mathbf{y}\cdot \nabla )\mathbf{y}$ to establish the existence and uniqueness results, while for $r\in [3,\infty )$ ($2\beta \mu \ge 1$ for $r=3$), we handle the Navier-Stokes nonlinearity by the nonlinear damping term $\beta |\mathbf{y}{|}^{r-1}\mathbf{y}$. Finally, we discuss the applications of the above developed theory in feedback control problems like flow invariance, time optimal control and stabilization.