Navier--Stokes regularity criteria in sum spaces

TL;DR

This paper establishes new regularity criteria for the Navier--Stokes equations using sum spaces of scale critical Lebesgue spaces, simplifying the conditions needed for regularity.

Contribution

It introduces a novel approach using sum spaces to unify multiple scale critical regularity criteria for Navier--Stokes equations.

Findings

Proves regularity criteria based on velocity, vorticity, or strain eigenvalues in sum spaces.

Establishes a new inclusion and inequality for sum spaces of mixed Lebesgue spaces.

Demonstrates that sum space criteria cover entire families of scale critical criteria.

Abstract

In this paper, we will consider regularity criteria for the Navier--Stokes equation in mixed Lebesgue sum spaces. In particular, we will prove regularity criteria that only require control of the velocity, vorticity, or the positive part of the second eigenvalue of the strain matrix, in the sum space of two scale critical spaces. This represents a significant step forward, because each sum space regularity criterion covers a whole family of scale critical regularity criteria in a single estimate. In order to show this, we will also prove a new inclusion and inequality for sum spaces in families of mixed Lebesgue spaces with a scale invariance that is also of independent interest.