Energy conservation for 3D Euler and Navier-Stokes equations in a bounded domain. Applications to Beltrami flows

TL;DR

This paper investigates energy conservation in 3D Euler and Navier-Stokes equations within bounded domains, establishing criteria for energy conservation and analyzing Beltrami flows, highlighting differences between energy conservation and regularity conditions.

Contribution

It extends energy conservation criteria to bounded domains with slip boundary conditions and applies these findings to Beltrami flows, revealing distinct conditions for energy conservation and regularity.

Findings

Energy conservation criteria are similar for Euler and Navier-Stokes with slip boundaries.

Beltrami flows exhibit unique energy conservation conditions.

Energy conservation and regularity conditions differ significantly for Beltrami solutions.

Abstract

In this paper we consider the incompressible 3D Euler and Navier-Stokes equations in a smooth bounded domain. First, we study the 3D Euler equations endowed with slip boundary conditions and we prove the same criteria for energy conservation involving the gradient, already known for the Navier-Stokes equations. Subsequently, we utilise this finding, which is based on a proper approximation of the velocity (and doesn't require estimates or additional assumptions on the pressure), to explore energy conservation for Beltrami flows. Finally, we explore Beltrami solutions to the Navier-Stokes equations and demonstrate that conditions leading to energy conservation are significantly distinct from those implying regularity. This remains true even when making use of the bootstrap regularity improvement, stemming from the solution being a Beltrami vector field.