Applications of Schochet's methods to parabolic equations

Methods used by S. Schochet in 32] enable one to find a lower bound for the life span of solutions of hyperbolic PDEs with a small parameter. We prove a similar theorem for such equations where a diffusion term has been added, with the minimal assumption on the Sobolev regularity of the initial data ( in the d-dimensional torus). When the data is smooth and under a “small divisor” assumption on the perturbation, the first term of an asymptotic expansion of the solution is computed. Those results are then applied to prove global existence theorems, for arbitrary initial data, in the 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.