Energy Conservation for 3D Euler and Navier-Stokes Equations in a Bounded Domain: Applications to Beltrami Flows

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 utilize this finding, which is based on a proper approximation of the velocity (and does not require any estimate or additional assumption 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.