Anomalous energy flux in critical $L^p$-based spaces

TL;DR

This paper constructs a three-dimensional vector field with positive energy flux across all scales, demonstrating optimal regularity in certain $L^p$-based spaces, which advances understanding of energy transfer in fluid dynamics.

Contribution

It introduces a vector field with positive energy flux at all scales that attains the best possible regularity in $L^p$-based spaces for $p \,\leq\, 3$, including $H^{(5/6)^-}$.

Findings

Vector field exhibits positive energy flux at every Littlewood-Paley shell.

Achieves optimal regularity in $L^p$-based spaces for $p \le 3$.

Belongs to the space $H^{(5/6)^-}$, indicating high regularity.

Abstract

We construct a three-dimensional vector field that exhibits positive energy flux at every Littlewood-Paley shell and has the best possible regularity in $L^p$-based spaces, $p \le 3$; in particular, it belongs to $H^{(\frac56)^-}$.