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This is a Quantitative Study study.

October 16, 2025Open Access

Instantaneous continuous loss of Sobolev regularity for the 3D incompressible Euler equation

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This work reveals the loss of sobolev regularity in incompressible euler equations, suggesting new insights into fluid dynamics.

The study shows instantaneous loss of Sobolev regularity for 3D incompressible Euler equations in R^3.
For any s in (0, 3/2), a divergence-free initial vorticity ω_0 was constructed, satisfying ||ω_0||_{H^s} ≤ ε.
A local-in-time solution ω exists, with ω(t) belonging to H^{(s-ct)/(1+ct)} but not in H^β for any β > (s-ct)/(1+ct).
This result indicates unique solutions are locally C^2 and belong to C([0,T];L^p) for p > 3, highlighting complexity in fluid behavior.

Jeong et al. (2025) studied this question.

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