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}-\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 $ d = 2 $ with $ r\in[1,\infty) $ and $ d = 3 $ with $ r\in[3,\infty) $ ($ 2\beta\mu\geq 1 $ for $ d = r = 3 $), we establish the existence of a unique global strong solution for the above multi-valued problem with the help of the abstract theory of $ m $-accretive operators. Moreover, we demonstrate that the same results hold 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 $ (\boldsymbol{y}\cdot\nabla)\boldsymbol{y} $ to establish the existence and uniqueness results, while for $ r\in[3,\infty) $ ($ 2\beta\mu\geq1 $ for $ r = 3 $), we handle the Navier-Stokes nonlinearity by the nonlinear damping term $ \beta|\boldsymbol{y}|^{r-1}\boldsymbol{y} $. Finally, we discuss the applications of the above developed theory in feedback control problems like flow invariance, time optimal control and stabilization.