On Right Continuity in $(L^2)^3$ at All the Points of Energy-regularized Solutions for the 3D Navier-Stokes Equations

TL;DR

This paper introduces energy-regularized solutions for the 3D Navier-Stokes equations, proving their right continuity in the phase space and their adherence to the Leray-Hopf property, thus advancing understanding of solution regularity.

Contribution

It defines ER-solutions for the 3D Navier-Stokes equations and proves their right continuity and Leray-Hopf property, providing new insights into solution regularity.

Findings

ER-solutions can be obtained via Galerkin methods.

ER-solutions satisfy the Leray-Hopf property.

ER-solutions are right continuous in the phase space.

Abstract

In this note I provide the notion of energy-regularized solutions (ER-solutions) of the 3D Navier-Stokes equations. These solutions can be obtained via the standard Galerkin arguments. I prove that each ER-solution for the 3D Navier-Stokes system satisfies Leray-Hopf property. Moreover, each ER-solution is rightly continuous in the standard phase space $H$ endowed with the strong convergence topology.