Time asymptotics, time regularity and separation rates for Navier-Stokes flows in supercritical solution classes
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866918110341103616 |
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| author | Bradshaw, Zachary Hudson, Joshua |
| author_facet | Bradshaw, Zachary Hudson, Joshua |
| contents | This paper extends the weak solution theory for the 3D Navier-Stokes equations of Barker, Seregin and Sverak from a critical setting to a supercritical setting making sure to include a useful a priori energy bound as well as a statement about stability under weak-star convergence. Two applications of the a priori bound are then explored. The first provides a spatially local, short-time asymptotic expansion in the time variable starting at $t=0$ which, as a corollary, provides an upper bound on how fast hypothetical non-unique solutions to the Navier-Stokes equations can separate locally. The second establishes higher-order time regularity at a singular time and at spatial points positioned away from the singularity. This quantifies the degree to which the non-local nature of the pressure allows a far flung singularity to disrupt the time regularity at a regular point. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_00714 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Time asymptotics, time regularity and separation rates for Navier-Stokes flows in supercritical solution classes Bradshaw, Zachary Hudson, Joshua Analysis of PDEs This paper extends the weak solution theory for the 3D Navier-Stokes equations of Barker, Seregin and Sverak from a critical setting to a supercritical setting making sure to include a useful a priori energy bound as well as a statement about stability under weak-star convergence. Two applications of the a priori bound are then explored. The first provides a spatially local, short-time asymptotic expansion in the time variable starting at $t=0$ which, as a corollary, provides an upper bound on how fast hypothetical non-unique solutions to the Navier-Stokes equations can separate locally. The second establishes higher-order time regularity at a singular time and at spatial points positioned away from the singularity. This quantifies the degree to which the non-local nature of the pressure allows a far flung singularity to disrupt the time regularity at a regular point. |
| title | Time asymptotics, time regularity and separation rates for Navier-Stokes flows in supercritical solution classes |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2508.00714 |