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

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

paper · doi:10.48550/arxiv.2503.11092

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 B_p,1^(2)/(p)-1(ℝ²) with 1 ≤ p ≤ 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 B_p,q^(2)/(p)-1(ℝ²) with (p,q) ∈ [1,4) × [1,∞] 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.

Related

• Global well-posedness for the 2 D quasi-geostrophic equation in a critical Besov space
• On the global well-posedness of the critical quasi-geostrophic equation
• Ill-posedness for the stationary Navier-Stokes equations in critical Besov spaces
• On the global solutions of the super-critical 2D quasi-geostrophic equation in Besov spaces
• Global well-posedness for a Modified 2D dissipative quasi-geostrophic equation with initial data in the critical Sobolev space H¹