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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2504.08288 |
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| _version_ | 1866913788700131328 |
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| author | Luo, Xiaoyutao |
| author_facet | Luo, Xiaoyutao |
| contents | We prove that the incompressible Navier-Stokes equations exhibit norm inflation in $\dot B^{s}_{p,q}(\mathbb{R}^3)$ with smooth, compactly supported initial data. Such norm inflation is shown in all supercritical $\dot B^{s}_{p,q} $ near the scaling-critical line $s = -1+ \frac{3}{p}$ except at $s=0$. The growth mechanism differs depending on the sign of the regularity index $s$: forward energy cascade driven by mixing for $s>0$ and backward energy cascade caused by un-mixing for $s<0$. The construction also demonstrates arbitrarily large, finite-time growth of the vorticity, the first of such examples for the Navier-Stokes equations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_08288 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sharp norm inflation for 3D Navier-Stokes equations in supercritical spaces Luo, Xiaoyutao Analysis of PDEs We prove that the incompressible Navier-Stokes equations exhibit norm inflation in $\dot B^{s}_{p,q}(\mathbb{R}^3)$ with smooth, compactly supported initial data. Such norm inflation is shown in all supercritical $\dot B^{s}_{p,q} $ near the scaling-critical line $s = -1+ \frac{3}{p}$ except at $s=0$. The growth mechanism differs depending on the sign of the regularity index $s$: forward energy cascade driven by mixing for $s>0$ and backward energy cascade caused by un-mixing for $s<0$. The construction also demonstrates arbitrarily large, finite-time growth of the vorticity, the first of such examples for the Navier-Stokes equations. |
| title | Sharp norm inflation for 3D Navier-Stokes equations in supercritical spaces |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2504.08288 |