Local profiles of self-similar solutions of the planar stationary Navier--Stokes equations
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866914304025952256 |
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| author | Li, Ming Peng, Linyu Zhang, Ping Zhang, Xin |
| author_facet | Li, Ming Peng, Linyu Zhang, Ping Zhang, Xin |
| contents | In this paper, we revisit self-similar solutions of the two-dimensional stationary incompressible Navier-Stokes equations under scaling symmetries, also known as Jeffery-Hamel solutions. We investigate the local patterns of smooth Jeffery-Hamel solutions in a conical subdomain $Ω$ with vertex at the origin, without imposing any boundary conditions on $Ω$. For radial Jeffery-Hamel solutions, we obtain all the explicit local profiles in $Ω$ with arbitrary opening angles. In the non-radial case, we show that some Jeffery-Hamel solutions can be obtained via solving a Liénard equation, and we derive new explicit local profiles expressible in terms of Weierstrass elliptic functions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_03341 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Local profiles of self-similar solutions of the planar stationary Navier--Stokes equations Li, Ming Peng, Linyu Zhang, Ping Zhang, Xin Analysis of PDEs 35Q30, 76N10 In this paper, we revisit self-similar solutions of the two-dimensional stationary incompressible Navier-Stokes equations under scaling symmetries, also known as Jeffery-Hamel solutions. We investigate the local patterns of smooth Jeffery-Hamel solutions in a conical subdomain $Ω$ with vertex at the origin, without imposing any boundary conditions on $Ω$. For radial Jeffery-Hamel solutions, we obtain all the explicit local profiles in $Ω$ with arbitrary opening angles. In the non-radial case, we show that some Jeffery-Hamel solutions can be obtained via solving a Liénard equation, and we derive new explicit local profiles expressible in terms of Weierstrass elliptic functions. |
| title | Local profiles of self-similar solutions of the planar stationary Navier--Stokes equations |
| topic | Analysis of PDEs 35Q30, 76N10 |
| url | https://arxiv.org/abs/2602.03341 |