Pullback attractors for the multi-layer quasi-geostrophic equations of the ocean

This article studies the pullback asymptotic behavior of solutions for a non-autonomous multi-layer quasi-geostrophic model in a two-dimensional domain. We prove the existence of pullback attractors AV in V (the velocity has the H1-regularity) and AH in H (the velocity has the L2-regularity). Then we verify the regularity of the pullback attractors by proving that AV=AH, which implies the pullback asymptotic smoothing effect of the model in the sense that the solutions eventually become more regular than the initial data. The method used in this article is similar to the one used in Zhao and Zhou (2007) in the case of the non-autonomous incompressible non-Newtonian fluid in a two-dimensional domain. Let us mention that the non-homogeneous boundary conditions (and the non-local constraint) present in the multi-layer quasi-geostrophic model makes the estimates more complicated, see Bernier (1994). These difficulties are overcome using the new formulation presented in Tachim Medjo (2009).