Stationary solutions to the critical and super-critical quasi-geostrophic equation in the scaling critical Sobolev space
Mikihiro Fujii
TL;DR
The paper analyzes the stationary quasi-geostrophic equation with critical and super-critical dissipation in two dimensions, focusing on the scaling-critical Sobolev framework. It proves the unique existence of small solutions for small forcing $f$ in $\dot{H}^{-\alpha} \cap \dot{H}^{2-4\alpha}$ when $0<\alpha\leq 1/2$, and establishes continuity of the data-to-solution map, along with a non-uniform continuity phenomenon in the critical/super-critical range. For the endpoint case $\alpha=1/2$, it also obtains a Lipschitz-continuous dependence result in Besov spaces $\dot{B}_{p,1}^{2/p-1}$ to $\dot{B}_{p,1}^{2/p}$. The methods combine a Picard-type approximation with Lax–Milgram for linearized problems, energy estimates that exploit the derivative-cancellation $\langle u\cdot \nabla \theta, \theta\rangle_{L^2}=0$, Bona–Smith-type smoothing, and Besov-space bilinear bounds to handle the critical regularity and derivative loss. Collectively, the results extend well-posedness theory to the scaling-critical regime and clarify how continuity properties of the data-to-solution map degrade in the presence of derivative loss, with a sharp endpoint Besov result at the critical threshold.
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.