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

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.