On the existence of self-similar solutions to the steady Navier-Stokes equations in high dimensions
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866908588887244800 |
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| author | Bang, Jeaheang Gui, Changfeng Liu, Hao Wang, Yun Xie, Chunjing |
| author_facet | Bang, Jeaheang Gui, Changfeng Liu, Hao Wang, Yun Xie, Chunjing |
| contents | We prove that the steady incompressible Navier-Stokes equations with any given $(-3)$-homogeneous, locally Lipschitz external force on $\mathbb{R}^n\setminus\{0\}$, $4\leq n\leq 16$, have at least one $(-1)$-homogeneous solution which is scale-invariant and regular away from the origin. The global uniqueness of the self-similar solution is obtained as long as the external force is small. The key observation is to exploit a nice relation between the radial component of the velocity and the total head pressure under the self-similarity assumption. It plays an essential role in establishing the energy estimates. If the external force has only the nonnegative radial component, we can prove the existence of $(-1)$-homogeneous solutions for all $n\geq 4$. The regularity of the solution follows from integral estimates of the positive part of the total head pressure, which is due to the maximum principle and a ``dimension-reduction" effect arising from the self-similarity. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_10488 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the existence of self-similar solutions to the steady Navier-Stokes equations in high dimensions Bang, Jeaheang Gui, Changfeng Liu, Hao Wang, Yun Xie, Chunjing Analysis of PDEs We prove that the steady incompressible Navier-Stokes equations with any given $(-3)$-homogeneous, locally Lipschitz external force on $\mathbb{R}^n\setminus\{0\}$, $4\leq n\leq 16$, have at least one $(-1)$-homogeneous solution which is scale-invariant and regular away from the origin. The global uniqueness of the self-similar solution is obtained as long as the external force is small. The key observation is to exploit a nice relation between the radial component of the velocity and the total head pressure under the self-similarity assumption. It plays an essential role in establishing the energy estimates. If the external force has only the nonnegative radial component, we can prove the existence of $(-1)$-homogeneous solutions for all $n\geq 4$. The regularity of the solution follows from integral estimates of the positive part of the total head pressure, which is due to the maximum principle and a ``dimension-reduction" effect arising from the self-similarity. |
| title | On the existence of self-similar solutions to the steady Navier-Stokes equations in high dimensions |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2510.10488 |