On the ill-posedness for the Navier--Stokes equations in the weakest Besov spaces
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2024
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866917562285031424 |
|---|---|
| author | Yu, Yanghai Li, Jinlu |
| author_facet | Yu, Yanghai Li, Jinlu |
| contents | It is proved in \cite{IO21} that the Cauchy problem for the full compressible Navier--Stokes equations of the ideal gas is ill-posed in $\dot{B}_{p, q}^{2 / p}(\mathbb{R}^2) \times \dot{B}_{p, q}^{2 / p-1}(\mathbb{R}^2) \times \dot{B}_{p, q}^{2 / p-2}(\mathbb{R}^2) $ with $1\leq p\leq \infty$ and $1\leq q<\infty$. In this paper, we aim to solve the end-point case left in \cite{IO21} and prove that the Cauchy problem is ill-posed in $\dot{B}_{p, \infty}^{d / p}(\mathbb{R}^d) \times \dot{B}_{p, \infty}^{d / p-1}(\mathbb{R}^d) \times \dot{B}_{p, \infty}^{d / p-2}(\mathbb{R}^d)$ with $1\leq p\leq\infty$ by constructing a sequence of initial data which shows that the solution map is discontinuous at zero. As a by-product, we demonstrate that the incompressible Navier--Stokes equations is also ill-posed in $\dot{B}_{p,\infty}^{d/p-1}(\mathbb{R}^d)$, which is an interesting open problem in itself. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_04387 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the ill-posedness for the Navier--Stokes equations in the weakest Besov spaces Yu, Yanghai Li, Jinlu Analysis of PDEs It is proved in \cite{IO21} that the Cauchy problem for the full compressible Navier--Stokes equations of the ideal gas is ill-posed in $\dot{B}_{p, q}^{2 / p}(\mathbb{R}^2) \times \dot{B}_{p, q}^{2 / p-1}(\mathbb{R}^2) \times \dot{B}_{p, q}^{2 / p-2}(\mathbb{R}^2) $ with $1\leq p\leq \infty$ and $1\leq q<\infty$. In this paper, we aim to solve the end-point case left in \cite{IO21} and prove that the Cauchy problem is ill-posed in $\dot{B}_{p, \infty}^{d / p}(\mathbb{R}^d) \times \dot{B}_{p, \infty}^{d / p-1}(\mathbb{R}^d) \times \dot{B}_{p, \infty}^{d / p-2}(\mathbb{R}^d)$ with $1\leq p\leq\infty$ by constructing a sequence of initial data which shows that the solution map is discontinuous at zero. As a by-product, we demonstrate that the incompressible Navier--Stokes equations is also ill-posed in $\dot{B}_{p,\infty}^{d/p-1}(\mathbb{R}^d)$, which is an interesting open problem in itself. |
| title | On the ill-posedness for the Navier--Stokes equations in the weakest Besov spaces |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2401.04387 |