Rotating Boussinesq and Quasigeostrophic Equations

The author elucidates in a concrete way dynamical challenges concerning approximate inertial manifolds (AIMS), i.e., globally invariant, exponentially attracting, finite-dimensional smooth manifolds, for nonlinear dynamical systems on Hilbert spaces. The goal of this theory is to prove the basic theorem of approximation dynamics, wherein it is shown that there is a fundamental connection between the order of the approximating manifold and the well-posedness and long-time dynamics of the rotating Boussinesq and quasigeostrophic equations. The author discusses recent progress in analytical and numerical methods covering form initial and boundary value problems, long-time dynamics and stability issues. He presents the most recent advances concerning the questions of global regularity of solutions to the 3D Navier-Stokes and Euler equations of incompressible fluids. Furthermore, he also presents recent global regularity (and finite time blow-up) results concerning the 3D quasigeostrophic and rotating Boussinesq equations describing the motion of a viscous incompressible rotating stratified fluid flow.