Three results on the Energy conservation for the 3D Euler equations

TL;DR

This paper investigates energy conservation in 3D Euler equations for incompressible fluids, extending classical results to broader function spaces, analyzing gradient conditions, and discussing the Onsager singularity problem.

Contribution

It extends energy conservation results to a wider range of Besov spaces, examines gradient-based conditions, and discusses limits from Navier-Stokes to Euler solutions.

Findings

Energy conservation holds in a full scale of Besov spaces.

Gradient conditions can ensure energy conservation, similar to Navier-Stokes results.

Conditions are identified for passing from Navier-Stokes to Euler solutions while conserving energy.

Abstract

We consider the 3D Euler equations for incompressible homogeneous fluids and we study the problem of energy conservation for weak solutions in the space-periodic case. First, we prove the energy conservation for a full scale of Besov spaces, by extending some classical results to a wider range of exponents. Next, we consider the energy conservation in the case of conditions on the gradient, recovering some results which were known, up to now, only for the Navier-Stokes equations and for weak solutions of the Leray-Hopf type. Finally, we make some remarks on the Onsager singularity problem, identifying conditions which allow to pass to the limit from solutions of the Navier-Stokes equations to solution of the Euler ones, producing weak solutions which are energy conserving.