Velocity-vorticity geometric constraints for the energy conservation of 3D ideal incompressible fluids

TL;DR

This paper establishes a new criterion for energy conservation in 3D Euler equations for weak solutions with specific regularity, and applies it to Beltrami-type solutions, advancing understanding of fluid dynamics.

Contribution

It introduces a novel energy conservation criterion for weak solutions in fractional Sobolev spaces and applies it to Beltrami-type solutions using advanced functional analysis techniques.

Findings

Energy conservation criterion proven for weak solutions

Application to Beltrami-type solutions confirms energy conservation

Enhanced understanding of regularity conditions for 3D Euler equations

Abstract

In this paper we consider the 3D Euler equations and we first prove a criterion for energy conservation for weak solutions with velocity satisfying additional assumptions in fractional Sobolev spaces with respect to the space variables, balanced by proper integrability with respect to time. Next, we apply the criterion to study the energy conservation of solution of the Beltrami type, carefully applying properties of products in (fractional and possibly negative) Sobolev spaces and employing a suitable bootstrap argument.