Leray-Hopf and Continuity Properties for All Weak Solutions for the 3D~Navier-Stokes Equations

TL;DR

This paper proves that all weak solutions to the 3D Navier-Stokes equations satisfy the Leray-Hopf property and are continuous in the phase space, enhancing understanding of solution regularity and properties.

Contribution

It establishes that every weak solution for the 3D Navier-Stokes system satisfies the Leray-Hopf condition and is continuous in the phase space, which was previously unconfirmed.

Findings

All weak solutions satisfy Leray-Hopf property

Weak solutions are continuous in the phase space $H$

Results improve understanding of solution regularity

Abstract

In this note we prove that each weak solution for the 3D Navier-Stokes system satisfies Leray-Hopf property. Moreover, each weak solution is rightly continuous in the standard phase space $H$ endowed with the strong convergence topology.