Local energy solutions to the Navier-Stokes equations in Wiener amalgam spaces

TL;DR

This paper introduces new local energy solutions to the Navier-Stokes equations within Wiener amalgam spaces, bridging the gap between classical Leray solutions and more general Lemarié-Rieusset solutions, and analyzing their properties.

Contribution

It establishes existence of solutions in novel Wiener amalgam space classes, extending the scope of Navier-Stokes solutions beyond traditional energy classes.

Findings

Solutions exhibit eventual regularity similar to Leray solutions.

Long-term estimates on local energy growth are established.

The new classes help identify when certain properties may fail.

Abstract

We establish existence of solutions in a scale of classes weaker than the finite energy Leray class and stronger than the infinite energy Lemari\'e-Rieusset class. The new classes are based on the $L^2$ Wiener amalgam spaces. Solutions in the classes closer to the Leray class are shown to satisfy some properties known in the Leray class but not the Lemari\'e-Rieusset class, namely eventual regularity and long time estimates on the growth of the local energy. In this sense, these solutions bridge the gap between Leray's original solutions and Lemari\'e-Rieusset's solutions and help identify scalings at which certain properties may break down.