Energy Decay of Solutions to the Boussinesq, Primitive, and Planetary Geostrophic Equations

In this paper we study the long time behavior of the energy of solutions to the Boussinesq, planetary geostrophic, and primitive equations. The equations are considered in the whole space R3. The asymptotic behavior will depend on the type of data and how many damping constants are nonzero in the equations. In several cases we are able to establish an algebraic rate of decay of the same order as the solutions of the underlying linear equations. In the case with less damping our results establish that either the energy of the solutions decays with no rate to an equilibrium or it will be oscillating. Q 1999 Academic Press Key Words: partial differential equations; Navier]Stokes equations; estimates. 1