A sparse resolution of the DiPerna-Majda gap problem for $2$D Euler equations

TL;DR

This paper determines the precise decay rate of vorticity maximal functions that guarantees the absence of concentration phenomena in solutions to the 2D Euler equations with vortex sheet initial data, resolving a long-standing open problem.

Contribution

It identifies the optimal decay rate for mixed sign vortex sheets, specifically $f(r) = [ ext{log}(1/r)]^{-1}$, that prevents concentrations, advancing understanding of vortex sheet regularity.

Findings

Established the optimal decay rate $f(r) = [ ext{log}(1/r)]^{-1}$ for mixed sign vortex sheets.

Developed a novel explicit construction of solutions exhibiting controlled wild behavior.

Connected sparseness of geometric structures to energy conservation in Euler solutions.

Abstract

A central question which originates in the celebrated work in the 1980's of DiPerna and Majda asks what is the optimal decay $f > 0$ such that uniform rates $|\omega|(Q) \leq f(|Q|)$ of the vorticity maximal functions guarantee strong convergence without concentrations of approximate solutions to energy-conserving weak solutions of the $2$D Euler equations with vortex sheet initial data. A famous result of Majda (1993) shows $f(r) = [\log (1/r)]^{-1/2}$, $r<1/2$, as the optimal decay for \emph{distinguished} sign vortex sheets. In the general setting of \emph{mixed} sign vortex sheets, DiPerna and Majda (1987) established $f(r) = [\log (1/r)]^{-\alpha}$ with $\alpha > 1$ as a sufficient condition for the lack of concentrations, while the expected gap $\alpha \in (1/2, 1]$ remains as an open question. In this paper we resolve the DiPerna-Majda $2$D gap problem: In striking contrast to the well-known case of distinguished sign vortex sheets, we identify $f(r) = [\log (1/r)]^{-1}$ as the optimal regularity for mixed sign vortex sheets that rules out concentrations. For the proof, we propose a novel method to construct explicitly solutions with mixed sign to the $2$D Euler equations in such a way that wild behaviour creates within the relevant geometry of \emph{sparse} cubes (i.e., these cubes are not necessarily pairwise disjoint, but their possible overlappings can be controlled in a sharp fashion). Such a strategy is inspired by the recent work of the first author and Milman \cite{DM} where strong connections between energy conservation and sparseness are established.