Sharp well-posedness and ill-posedness of the stationary quasi-geostrophic equation
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , |
|---|---|
| Format: | Preprint |
| Publié: |
2025
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866916652527910912 |
|---|---|
| author | Fujii, Mikihiro Iwabuchi, Tsukasa |
| author_facet | Fujii, Mikihiro Iwabuchi, Tsukasa |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_11092 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sharp well-posedness and ill-posedness of the stationary quasi-geostrophic equation Fujii, Mikihiro Iwabuchi, Tsukasa Analysis of PDEs 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. |
| title | Sharp well-posedness and ill-posedness of the stationary quasi-geostrophic equation |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2503.11092 |