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

A central question that originates in the celebrated work in the 1980's of DiPerna and Majda asks what is the optimal decay (Formula presented.) such that uniform rates (Formula presented.) of the vorticity maximal functions guarantee strong convergence without concentrations of approximate solutions to energy-conserving weak solutions of the 2D Euler equations with vortex sheet initial data. A famous result of Majda (1993) shows (Formula presented.), (Formula presented.), as the optimal decay for distinguished sign vortex sheets. In the general setting of mixed sign vortex sheets, DiPerna and Majda (1987) established (Formula presented.) with (Formula presented.) as a sufficient condition for the lack of concentrations, while the expected gap (Formula presented.) remains as an open question. In this paper, we resolve the DiPerna–Majda 2D gap problem: In striking contrast to the well-known case of distinguished sign vortex sheets, we identify (Formula presented.) 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 2D Euler equations in such a way that wild behavior creates within the relevant geometry of 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 where strong connections between energy conservation and sparseness are established.