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

The following convective Brinkman-Forchheimer (CBF) equations (or damped Navier-Stokes equations) with potential$ \begin{equation*} \frac{\partial \boldsymbol{y}}{\partial t}-\mu \Delta\boldsymbol{y}+(\boldsymbol{y}\cdot\nabla)\boldsymbol{y}+\alpha\boldsymbol{y}+\beta|\boldsymbol{y}|^{r-1}\boldsymbol{y}+\nabla p+\Psi(\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) $. For 2 3 r\in[3,\infty) ($ 2\beta\mu\geq 1 for r $), we establish the existence of unique global strong solution above multi-valued problem help abstract theory m $-accretive operators. Moreover, demonstrate that same results hold local time case r\in[1,3) 2\beta\mu<1 We explored $-accretivity nonlinear as well operators, Yosida approximations their properties, several higher order energy estimates proofs. r\in[1,3] quantize (modify) nonlinearity (\boldsymbol{y}\cdot\nabla)\boldsymbol{y} to uniqueness results, while 2\beta\mu\geq1 handle by damping term \beta|\boldsymbol{y}|^{r-1}\boldsymbol{y} Finally, discuss applications developed feedback control problems like flow invariance, optimal stabilization.