Data assimilation for a 3D Brinkman-Forchheimer-extended Darcy model of porous media with convection

We consider three-dimensional Boussinesq convection system of a fluid in a closed sample of a porous medium. Specifically, our aim is to introduce and analyze a modified three-dimensional Bnard convection problem describing the behavior of a fluid with a high flow rate between two plates heated from the bottom and cooled from the top. The first objective of this project is to show the well-posedness of the associated Cauchy problem in the adequate energy space. The second objective is to design a continuous data assimilation (downscaling) algorithm for this system using only partial coarse mesh measurements. In particular, we shall use only measurements of two components of the velocity. The idea is to prove that under an appropriate choice of damping and nudging parameters, and small enough measurements mesh size, the solution of the algorithm converges asymptotically in time at an exponential rate to the unique exact unknown reference solution of the original modified three- dimensional Extended Bnard convection system.