RESONANCES AND REGULARITY FOR BOUSSINESQ EQUATIONS

We consider classical Boussinesq equations for a rotating stably strat ified fluid with large N-0 = N/F, where N-0 describes the stratificati on, Omega(0) = N Omega is the rate of rotation, and N is a large scali ng parameter. Solutions of full nonlinear Boussinesq equations have a decomposition of the form U(t, x(1), x(2), x(3)) = (U) over tilde(t, x (1), x(2), x(3)) + W-g(t, x(1), x(2), x(3)) + r, where (U) over tilde is a solution of the quasigeostrophic equation and r is a remainder, w hich is uniformly bounded above by a majorant of the order of 1/N. The vector field W-g(t, x(1), x(2), x(3)) describes the rapidly oscillati ng gravity wave component. The ''amplitude'' of this component describ es the propagation of slow waves, and it satisfies a linear equation w ith coefficients determined by the quasigeostrophic component found fr om the nonlinear quasigeostrophic equations. The control of the error r based on estimates related to small denominators, for generic values of parameters, is used to prove the existence, on a long time interva l T, of regular solutions to classical Boussinesq equations with gene ral initial data (T --> +infinity as N --> infinity).