On strong solutions of the multi-layer quasi-geostrophic equations of the ocean

In this article, we study the multi-layer quasi-geostrophic equations of the ocean. The existence of strong solutions is proved. We also prove the existence of a maximal attractor in L2(Ω)L2(Ω) and we derive estimates of its Hausdorff and fractal dimensions in terms of the data. Our estimates rely on a new formulation that we introduce for the multi-layer quasi-geostrophic equation of the ocean, which replaces the nonhomogeneous boundary conditions (and the nonlocal constraint) on the stream-function by a simple homogeneous Dirichlet boundary condition. This work improves the results given in [C. Bernier, Existence of attractor for the quasi-geostrophic approximation of the Navier–Stokes equations and estimate of its dimension, Adv. Math. Sci. Appl. 4 (2) (1994) 465–489].