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

2025/03/14 by Fujii, Mikihiro
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2503.11095

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.

Related

• Dissipative quasi-geostrophic equations in critical Sobolev spaces: smoothing effect and global well-posedness
• Asymptotic study of a global solution of super-critical Quasi-Geostrophic equation
• Asymptotic behavior of critical dissipative quasi-geostrophic equation\n in Fourier space
• On the critical dissipative quasi-geostrophic equation
• Existence and Uniqueness of the Solution to the Anisotropic Quasi-Geostrophic Equations in the Sobolev Space