On the Relation between Very Weak and Leray-Hopf Solutions to Navier-Stokes Equations

TL;DR

This paper establishes that very weak solutions to the Navier-Stokes equations with finite energy initial data are actually Leray-Hopf solutions, clarifying the relationship between different solution concepts.

Contribution

It proves a general result linking very weak solutions to Leray-Hopf solutions under finite energy initial conditions.

Findings

Very weak solutions coincide with Leray-Hopf solutions given finite energy initial data.

The result simplifies understanding of solution classes for Navier-Stokes equations.

Provides a unifying framework for solution concepts in fluid dynamics.

Abstract

We prove a general result that implies that very weak solutions to the Cauchy problem for the Navier-Stokes equations must be, in fact, Leray-Hopf solutions if only their initial data are (solenoidal) with finite kinetic energy.