Classification of finite-time blow-up of strong solutions to the incompressible free boundary Euler equations with surface tension
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arXiv
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| Format: | Preprint |
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2025
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| author | Hao, Chengchun Luo, Tao Yang, Siqi |
| author_facet | Hao, Chengchun Luo, Tao Yang, Siqi |
| contents | We establish the first complete classification of finite-time blow-up scenarios for strong solutions to the three-dimensional incompressible Euler equations with surface tension in a bounded domain possessing a closed, moving free boundary. Uniquely, we make \textit{no} assumptions on symmetry, periodicity, graph representation, or domain topology (simple connectivity). At the maximal existence time $T<\infty$, up to which the velocity field and the free boundary can be continued in $H^3\times H^4$, blow-up must occur in at least one of five mutually exclusive ways: (i) self-intersection of the free boundary for the first time; (ii) loss of mean curvature regularity in $H^{\frac{3}{2}}$, or the free boundary regularity in $H^{2+\varepsilon}$ (for any sufficiently small constant $\varepsilon>0$); (iii) loss of $H^{\frac{5}{2}}$ regularity for the normal boundary velocity; (iv) the $L^1_tL^\infty$-blow-up of the tangential velocity gradient on the boundary; or (v) the $L^1_tL^\infty$-blow-up of the full velocity gradient in the interior. Furthermore, for simply connected domains, blow-up scenario (v) simplifies to a vorticity-based Beale-Kato-Majda criterion, and in particular, irrotational flows admit blow-up only at the free boundary. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_10032 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Classification of finite-time blow-up of strong solutions to the incompressible free boundary Euler equations with surface tension Hao, Chengchun Luo, Tao Yang, Siqi Analysis of PDEs 35Q35, 35R35, 35B44, 76B03, 76B45 We establish the first complete classification of finite-time blow-up scenarios for strong solutions to the three-dimensional incompressible Euler equations with surface tension in a bounded domain possessing a closed, moving free boundary. Uniquely, we make \textit{no} assumptions on symmetry, periodicity, graph representation, or domain topology (simple connectivity). At the maximal existence time $T<\infty$, up to which the velocity field and the free boundary can be continued in $H^3\times H^4$, blow-up must occur in at least one of five mutually exclusive ways: (i) self-intersection of the free boundary for the first time; (ii) loss of mean curvature regularity in $H^{\frac{3}{2}}$, or the free boundary regularity in $H^{2+\varepsilon}$ (for any sufficiently small constant $\varepsilon>0$); (iii) loss of $H^{\frac{5}{2}}$ regularity for the normal boundary velocity; (iv) the $L^1_tL^\infty$-blow-up of the tangential velocity gradient on the boundary; or (v) the $L^1_tL^\infty$-blow-up of the full velocity gradient in the interior. Furthermore, for simply connected domains, blow-up scenario (v) simplifies to a vorticity-based Beale-Kato-Majda criterion, and in particular, irrotational flows admit blow-up only at the free boundary. |
| title | Classification of finite-time blow-up of strong solutions to the incompressible free boundary Euler equations with surface tension |
| topic | Analysis of PDEs 35Q35, 35R35, 35B44, 76B03, 76B45 |
| url | https://arxiv.org/abs/2507.10032 |