We consider the stationary problem for the quasi-geostrophic equation on the whole plane and investigate its well-posedness and ill-posedness. In the first author's previous study (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 \(1 \leqslant p \leqslant 2\) . Although the quasi-geostrophic equation has the same invariant scaling 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.
