Energy conservation for weak solutions of incompressible fluid equations: the H\"older case and connections with Onsager's conjecture

TL;DR

This paper provides elementary proofs of energy conservation for weak solutions of Euler and Navier-Stokes equations with H"older continuity, relaxing assumptions and extending results to boundary conditions, connecting to Onsager's conjecture.

Contribution

It offers simplified proofs of energy conservation under H"older conditions, including boundary cases, without extra pressure assumptions, advancing understanding of fluid equations.

Findings

Energy conservation holds for H"older continuous weak solutions of Euler and Navier-Stokes.

New proofs relax assumptions on time regularity and integrability.

First known result for viscous case with H"older regularity and boundary conditions without pressure assumptions.

Abstract

In this paper we give elementary proofs of energy conservation for weak solutions to the Euler and Navier-Stokes equations in the class of H\"older continuous functions, relaxing some of the assumptions on the time variable (both integrability and regularity at initial time) and presenting them in a unified way. Then, in the final section we prove (for the Navier-Stokes equations) a result of energy conservation in presence of a solid boundary and with Dirichlet boundary conditions. This result seems the first one -- in the viscous case -- with H\"older type assumptions, but without additional assumptions on the pressure.