Stationary solutions to the critical and super-critical quasi-geostrophic equation in the scaling critical Sobolev space

Abstract

We consider the stationary problem for the quasi-geostrophic equation with the critical and super-critical dissipation and prove the unique existence of small solutions for given small external force in the scaling critical Sobolev spaces framework. Moreover, we also show that the data-to-solution map is continuous. Since the critical and super-critical case involves the derivative loss, which affects the class of the continuity of the data-to-solution map, we reveal that the map is no longer uniform continuous, in contrast to the sub-critical case, where the Lipschitz continuity holds.