Minimal blow-up initial data in critical Fourier-Herz spaces for potential Navier-Stokes singularities

TL;DR

This paper establishes the existence of minimal blow-up initial data in critical Fourier-Herz spaces for 3D potential Navier-Stokes equations, advancing understanding of singularity formation in fluid dynamics.

Contribution

It introduces new techniques combining localization, partial regularity, and stability analysis to identify minimal blow-up data in critical Fourier-Herz spaces.

Findings

Existence of minimal blow-up initial data in critical Fourier-Herz spaces.

Development of localization and partial regularity techniques.

Insights into singularity stability and short-time behavior of solutions.

Abstract

In this paper, we mainly prove the existence of the minimal blow-up initial data in critical Fourier-Herz space $F\dot{B}^{2-{\frac3p}}_{p,q}(\RR^3)$ with $1<p\leq\infty$ and $1\leq q<\infty$ for the three dimensional incompressible potential Navier-Stokes equations by developing techniques of "localization in space" involving the partial regularity given by the De Giorgi iteration, weak-strong uniqueness, the short-time behaviour of the kinetic energy and stability of singularity of Calder\'on's solution.