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| Format: | Preprint |
| Veröffentlicht: |
2025
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| Online-Zugang: | https://arxiv.org/abs/2512.13456 |
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| _version_ | 1866915676120154112 |
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| author | Egamberganov, Khakim Yao, Yao |
| author_facet | Egamberganov, Khakim Yao, Yao |
| contents | We consider the axisymmetric Euler equations in $\mathbb{R}^3$ without swirl, and establish several upper and lower bounds for the growth of solutions. On the one hand, we obtain an upper bound $t^2$ for the radial moment $\int_{\mathbb{R}^3} rω^θdx$, which is the conjectured optimal rate by Childress (Phys. D 237(14-17):1921-1925, 2008). On the other hand, for all initial data satisfying certain symmetry and sign conditions, we prove that the radial moment grows at least like $t/\log t$ as time goes to infinity, and $\|ω(\cdot,t)\|_{L^p(\mathbb{R}^3)}$ exhibits at least $t^{1/4}$ growth in the limsup sense for all $1\leq p\leq \infty$. To the best of our knowledge, this is the first result to establish power-law $L^p$-norm growth for smooth, compactly supported initial vorticity in $\mathbb{R}^3$. For these initial data, we also show that nearly all vorticity must eventually escape to $r\to\infty$ in the time-integral sense. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_13456 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Growth estimates for axisymmetric Euler equations without swirl Egamberganov, Khakim Yao, Yao Analysis of PDEs 35Q31 We consider the axisymmetric Euler equations in $\mathbb{R}^3$ without swirl, and establish several upper and lower bounds for the growth of solutions. On the one hand, we obtain an upper bound $t^2$ for the radial moment $\int_{\mathbb{R}^3} rω^θdx$, which is the conjectured optimal rate by Childress (Phys. D 237(14-17):1921-1925, 2008). On the other hand, for all initial data satisfying certain symmetry and sign conditions, we prove that the radial moment grows at least like $t/\log t$ as time goes to infinity, and $\|ω(\cdot,t)\|_{L^p(\mathbb{R}^3)}$ exhibits at least $t^{1/4}$ growth in the limsup sense for all $1\leq p\leq \infty$. To the best of our knowledge, this is the first result to establish power-law $L^p$-norm growth for smooth, compactly supported initial vorticity in $\mathbb{R}^3$. For these initial data, we also show that nearly all vorticity must eventually escape to $r\to\infty$ in the time-integral sense. |
| title | Growth estimates for axisymmetric Euler equations without swirl |
| topic | Analysis of PDEs 35Q31 |
| url | https://arxiv.org/abs/2512.13456 |