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

Mikihiro Fujii, Tsukasa Iwabuchi

TL;DR

This work classifies the well-posedness versus ill-posedness of the stationary quasi-geostrophic equation on $\mathbb{R}^2$ within scaling-critical Besov spaces. By reformulating the nonlinear term as a double-divergence operator $\mathcal{B}[\cdot,\cdot]$, the authors unveil a favorable structure that enables sharp a priori estimates and a contraction mapping proof of well-posedness for $(p,q)\in([1,4)\times[1,\infty])\cup\{(4,2)\}$. They also construct forcing sequences that drive ill-posedness outside this range, demonstrating norm inflation and discontinuity of the data-to-solution map in various Besov topologies. The results clearly separate the well-posed regime from the ill-posed regime and establish the sharp boundary, contributing a precise counterpart to known NS ill-posedness results in two dimensions. The methods combine paraproduct calculus, bilinear Fourier multipliers, and a tailored fixed-point argument to obtain a complete classification.

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 $1 \leq p \leq 2$. 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.

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

TL;DR

This work classifies the well-posedness versus ill-posedness of the stationary quasi-geostrophic equation on within scaling-critical Besov spaces. By reformulating the nonlinear term as a double-divergence operator , the authors unveil a favorable structure that enables sharp a priori estimates and a contraction mapping proof of well-posedness for . They also construct forcing sequences that drive ill-posedness outside this range, demonstrating norm inflation and discontinuity of the data-to-solution map in various Besov topologies. The results clearly separate the well-posed regime from the ill-posed regime and establish the sharp boundary, contributing a precise counterpart to known NS ill-posedness results in two dimensions. The methods combine paraproduct calculus, bilinear Fourier multipliers, and a tailored fixed-point argument to obtain a complete classification.

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 with . 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 with or 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 ensuring the well-posedness in the sense that the stationary quasi-geostrophic equation is ill-posed for all the other cases.

Paper Structure

This paper contains 4 sections, 4 theorems, 101 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem \ref{['thm:WP']}
  4. Proof of Theorem \ref{['thm:IP']}

Key Result

Theorem 1.2

Let $1 \leqslant p < 4$ and $1 \leqslant q \leqslant \infty$, or $p=4$ and $q =2$. Then, there exist positive constants $\delta_0=\delta_0(p,q)$ and $\varepsilon_0=\varepsilon_0(p,q)$ such that if the external force $f \in D_{p,q}(\mathbb{R}^2)$, where then there exists a unique solution $\theta \in S_{p,q}(\mathbb{R}^2)$, where Moreover, the solution map $\Phi:D_{p,q}(\mathbb{R}^2) \ni f \mapst

Theorems & Definitions (9)

  • Definition 1.1
  • Theorem 1.2: Well-posedness of \ref{['eq:QG']}
  • Theorem 1.3: Ill-posedness of \ref{['eq:QG']}
  • Remark 1.4
  • Lemma 3.1
  • Lemma 3.2: Gra-Miy-Tom-13
  • proof
  • proof : Proof of Theorem \ref{['thm:WP']}
  • proof : Proof of Theorem \ref{['thm:IP']}