Non-conservative $H^{\frac 12-}$ weak solutions of the incompressible 3D Euler equations

TL;DR

This paper constructs non-conservative weak solutions to the 3D incompressible Euler equations with regularity in $H^{eta}$ for any $eta<rac{1}{2}$, exceeding the classical $H^{1/3}$ threshold, challenging traditional turbulence theory.

Contribution

It introduces a method to explicitly construct weak solutions with regularity above $H^{1/3}$, expanding understanding of solution behaviors beyond classical turbulence limits.

Findings

Solutions are in $H^{eta}$ for all $eta<rac{1}{2}$

Solutions have regularity index strictly larger than $1/3$

Demonstrates existence of non-conservative solutions with higher regularity

Abstract

For any positive regularity parameter $\beta < \frac 12$, we construct non-conservative weak solutions of the 3D incompressible Euler equations which lie in $H^{\beta}$ uniformly in time. In particular, we construct solutions which have an $L^2$-based regularity index \emph{strictly larger} than $\frac 13$, thus deviating from the $H^{\frac{1}{3}}$-regularity corresponding to the Kolmogorov-Obhukov $\frac 53$ power spectrum in the inertial range.