A blowup criteria along maximum points of the 3D-Navier-Stokes flow in terms of function spaces with variable growth condition

TL;DR

This paper develops a new blowup criterion for the 3D-Navier-Stokes equations based on variable growth function spaces, differing from classical criteria, and links geometric symmetry near maximum points to solution extendibility.

Contribution

It introduces a novel blowup criterion using variable growth function spaces and relates geometric symmetry near maximum points to the extension of solutions.

Findings

New blowup criterion based on variable growth function spaces

Geometric symmetry near maximum points allows solution extension

Different from classical Beale-Kato-Majda and Constantin-Fefferman criteria

Abstract

A blowup criteria along maximum point of the 3D-Navier-Stokes flow in terms of function spaces with variable growth condition is constructed. This criterion is different from the Beale-Kato-Majda type and Constantin-Fefferman type criterion. If geometric behavior of the velocity vector field near the maximum point has a kind of symmetry up to a possible blowup time, then the solution can be extended to be the strong solution beyond the possible blowup time.