Lower bounds on the blowup rate of vorticity in the Euler equations
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866918401175191552 |
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| author | Ingimarson, Benjamin Kukavica, Igor |
| author_facet | Ingimarson, Benjamin Kukavica, Igor |
| contents | Under the assumption that a solution to the 3D incompressible Euler equations blows up at a time $T_\ast$ and that $T_\ast $ is the first such time, we establish lower bounds on the rate of blow-up of the maximum norm of the vorticity. In particular, when the domain is $\mathbb{R}^3$ or $\mathbb{T}^3$, we provide lower bounds on $\int_{0}^{t}\Vert ω\Vert_{L^\infty}\,ds$ and $\sup_{s\in[0,t]}\|ω\|_{L^\infty}$ for $t$ sufficiently close to~$T_\ast$. Notably, this gives a quantitative description of the BKM blow-up criterion. Moreover, we provide pointwise-in-time lower bounds on~$\|D^k ω\|_{L^\infty}$. Finally, we state some consequences on the blow-up rate of the derivative of the deformation tensor. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_17431 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Lower bounds on the blowup rate of vorticity in the Euler equations Ingimarson, Benjamin Kukavica, Igor Analysis of PDEs Under the assumption that a solution to the 3D incompressible Euler equations blows up at a time $T_\ast$ and that $T_\ast $ is the first such time, we establish lower bounds on the rate of blow-up of the maximum norm of the vorticity. In particular, when the domain is $\mathbb{R}^3$ or $\mathbb{T}^3$, we provide lower bounds on $\int_{0}^{t}\Vert ω\Vert_{L^\infty}\,ds$ and $\sup_{s\in[0,t]}\|ω\|_{L^\infty}$ for $t$ sufficiently close to~$T_\ast$. Notably, this gives a quantitative description of the BKM blow-up criterion. Moreover, we provide pointwise-in-time lower bounds on~$\|D^k ω\|_{L^\infty}$. Finally, we state some consequences on the blow-up rate of the derivative of the deformation tensor. |
| title | Lower bounds on the blowup rate of vorticity in the Euler equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2603.17431 |