Instantaneous continuous loss of Sobolev regularity for the 3D incompressible Euler equation
Year: 2025
Type: preprint
Abstract: 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 $.
Source: arXiv (Cornell University)
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4415194994
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