Illposedness of incompressible fluids in supercritical Sobolev spaces
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866929358922317824 |
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| author | Luo, Xiaoyutao |
| author_facet | Luo, Xiaoyutao |
| contents | We prove that the 3D Euler and Navier-Stokes equations are strongly illposed in supercritical Sobolev spaces. In the inviscid case, for any $0 < s < \frac{5}{2} $, we construct a $C^\infty_c$ initial velocity field with arbitrarily small $H^{s}$ norm for which the unique local-in-time smooth solution of the 3D Euler equation develops large $\dot{H}^{s}$ norm inflation almost instantaneously. In the viscous case, the same $\dot{H}^{s}$ norm inflation occurs in the 3D Navier-Stokes equation for $0< s < \frac{1}{2} $, where $s = \frac{1}{2}$ is scaling critical for this equation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_07813 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Illposedness of incompressible fluids in supercritical Sobolev spaces Luo, Xiaoyutao Analysis of PDEs We prove that the 3D Euler and Navier-Stokes equations are strongly illposed in supercritical Sobolev spaces. In the inviscid case, for any $0 < s < \frac{5}{2} $, we construct a $C^\infty_c$ initial velocity field with arbitrarily small $H^{s}$ norm for which the unique local-in-time smooth solution of the 3D Euler equation develops large $\dot{H}^{s}$ norm inflation almost instantaneously. In the viscous case, the same $\dot{H}^{s}$ norm inflation occurs in the 3D Navier-Stokes equation for $0< s < \frac{1}{2} $, where $s = \frac{1}{2}$ is scaling critical for this equation. |
| title | Illposedness of incompressible fluids in supercritical Sobolev spaces |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2404.07813 |