The structure of weak solutions to the Navier-Stokes equations

TL;DR

This paper characterizes the structure of weak solutions to the Navier-Stokes equations, showing they can be represented as transformations of solutions satisfying a local pressure expansion, thus providing a new framework for understanding these solutions.

Contribution

It introduces a representation theorem for weak solutions of Navier-Stokes equations as transgalilean transformations of solutions with local pressure expansion.

Findings

Weak solutions can be obtained via transgalilean transformations.

A sufficient condition for local pressure expansion is established.

Provides a new structural understanding of weak solutions.

Abstract

The existence of superfluous solutions to the Navier-Stokes equations in the whole space implies that not all solutions with uniformly locally bounded energy satisfy a useful local pressure expansion. We prove that every weak solution in a parabolic uniformly local $L^2$ class can be obtained as a transgalilean transformation of a solution satisfying the local pressure expansion in a distributional sense. This gives a powerful representation theorem for a large class of solutions. We use this structure to obtain a sufficient condition for the local pressure expansion.