In [2] M. Jaoua et al. studied the linear approximation of Robin problem on Ω an open bounded domain of R d , and they given some important results. In this paper, we study a nonlinear approximation of an elliptic problem with a nonlinear Robin boundary condition in a domain of R 2 . We prove the existence and uniqueness of solution by an iterative construction method with admissible condition on ∂Ω.

In \cite{YZ}, the author proved the global existence of the two-dimensional anisotropic quasi-geostrophic equations with condition on the parameters $\alpha,$ $\beta$ in the Sobolev spaces $H^s( \R^2)$; $s\geq 2$. In this paper, we show that this equations has a global solution in the spaces $H^s(\R^2)$, where $\max\{2-2\alpha,2-2\beta\}< s<2$, with additional condition over $\alpha$ and $\beta$. The proof is based on the Gevrey-class regularity of the solution in neighborhood of zero.