Sharp well-posedness and ill-posedness of the stationary quasi-geostrophic equation

Abstract

We consider the stationary problem for the quasi-geostrophic equation on the whole plane and investigate its well-posedness and ill-posedness. In[Fujii, Ann. PDE 10, 10 (2024)], it was shown that the two-dimensional stationary Navier--Stokes equations are ill-posed in the critical Besov spaces ${\dot{B}}_{p,1}^{\frac{2}{p}-1}({\mathbb{R}}^2)$ with $1\le p\le 2$. Although the quasi-geostrophic equation has the same invariant scale structure as the Navier--Stokes equations, we reveal that the quasi-geostrophic equation is well-posed in the scaling critical Besov spaces ${\dot{B}}_{p,q}^{\frac{2}{p}-1}({\mathbb{R}}^2)$ with $(p,q)\in [1,4)\times [1,\infty ]$ or $(p,q)=(4,2)$ due to the better properties of the nonlinear structure of the quasi-geostrophic equation compared to that of the Navier--Stokes equations. Moreover, we also prove the optimality for the above range of $(p,q)$ ensuring the well-posedness in the sense that the stationary quasi-geostrophic equation is ill-posed for all the other cases.