Velocity-Vorticity Geometric Constraints for the Energy Conservation of 3D Ideal Incompressible Fluids

Abstract

In this paper we consider the 3D Euler equations and we first prove a criterion for energy conservation for weak solutions, where the velocity satisfies 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.