The 3D Navier-Stokes Equations with Damping in Bounded Domains

The Navier-Stokes equations with damping describe the flow with the resistance to motion. From a mathematical viewpoint, the equations can be viewed as a modification of the classical Navier-Stokes equations with the regularizing term. This monograph concerns the initial-boundary value problems for the Navier-Stokes equations and the Boussinesq system with damping in three-dimensional bounded domains, respectively. First, the existence of strong solutions of both problems in a certain range of parameters is proved. Second, under the same conditions on parameters, the uniqueness of Leray-Hopf weak solutions of the Navier-Stokes equations with damping is proved and the result is extended to the Boussinesq system with damping.Finally, the conditions on parameters to guarantee the existence and globally asymptotic stability of the time-periodic solutions of both problems are found, respectively.