Instantaneous continuous loss of Sobolev regularity for the 3D incompressible Euler equation
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866908482471460864 |
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| author | Jeong, In-Jee Martínez-Zoroa, Luis Ożański, Wojciech S. |
| author_facet | Jeong, In-Jee Martínez-Zoroa, Luis Ożański, Wojciech S. |
| contents | We prove instantaneous and continuous-in-time loss of supercritical Sobolev regularity for the 3D incompressible Euler equations in $\mathbb{R}^{3}$. Namely, for any $s\in (0,3/2)$ and $\varepsilon >0$, we construct a divergence-free initial vorticity $ω_0$ defined in $\mathbb{R}^{3}$ satisfying $\| ω_0 \|_{H^s}\leq \varepsilon$, as well as $T>0$, $c>0$ and a corresponding local-in-time solution $ω$ such that, for each $t\in [0,T]$, $ω(\cdot ,t ) \in {H^{\frac{s-ct}{1+ct}}}$ and $ ω(\cdot ,t ) \not \in {H^β}$ for any $β> \frac{s-ct}{1+ct} $. Moreover, $ω$ is unique among all solutions with initial condition $ω_0$ which are locally $C^2$ and belong to $C([0,T];L^p )$ for any $p>3 $. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_06333 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Instantaneous continuous loss of Sobolev regularity for the 3D incompressible Euler equation Jeong, In-Jee Martínez-Zoroa, Luis Ożański, Wojciech S. Analysis of PDEs We prove instantaneous and continuous-in-time loss of supercritical Sobolev regularity for the 3D incompressible Euler equations in $\mathbb{R}^{3}$. Namely, for any $s\in (0,3/2)$ and $\varepsilon >0$, we construct a divergence-free initial vorticity $ω_0$ defined in $\mathbb{R}^{3}$ satisfying $\| ω_0 \|_{H^s}\leq \varepsilon$, as well as $T>0$, $c>0$ and a corresponding local-in-time solution $ω$ such that, for each $t\in [0,T]$, $ω(\cdot ,t ) \in {H^{\frac{s-ct}{1+ct}}}$ and $ ω(\cdot ,t ) \not \in {H^β}$ for any $β> \frac{s-ct}{1+ct} $. Moreover, $ω$ is unique among all solutions with initial condition $ω_0$ which are locally $C^2$ and belong to $C([0,T];L^p )$ for any $p>3 $. |
| title | Instantaneous continuous loss of Sobolev regularity for the 3D incompressible Euler equation |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2508.06333 |