Applications of Schochet's methods to parabolic equations

Methods used by S. Schochet in [32] enable one to find a lower bound for th e life span of solutions of hyperbolic PDEs with a small parameter. We prov e a similar theorem for such equations where a diffusion in the term has be en added, with the minimal assumption on the Sobolev regularity of the init ial data (Hd/2-1 in the d-dimensional torus). When the data is smooth and u nder a "small divisor" assumption on the perturbation, the first term of an asymptotic expansion of the solution is computed. Those results are then a pplied to prove global existence theorems, for arbitrary initial data, in t he case of the primitive system of the quasigeostrophic equations, followed by the rotating fluid equations. We finally prove a more precise existence theorem for the latter, using anisotropic Sobolev and Besov spaces. (C) El sevier, Paris.