Lower Bounds of Potential Blow-Up Solutions of the Three-dimensional Navier-Stokes Equations in $\dot{H}^\frac{3}{2}$

TL;DR

This paper improves lower bounds for Sobolev norms of potential blow-up solutions to the 3D Navier-Stokes equations, providing new insights into the behavior of solutions near singularities.

Contribution

It presents improved lower bounds for Sobolev norms in $ abla^{3/2}$ and offers an alternative proof for bounds in $ abla^{5/2}$, advancing understanding of potential blow-up scenarios.

Findings

Enhanced lower bounds for $ abla^{3/2}$ Sobolev norms

Alternative proof for $ abla^{5/2}$ blow-up bounds

Deeper understanding of solution behavior near singularities

Abstract

We improve previous known lower bounds for Sobolev norms of potential blow-up solutions to the three-dimensional Navier-Stokes equations in $\dot{H}^\frac{3}{2}$. We also present an alternate proof for the lower bound for the $\dot{H}^\frac{5}{2}$ blow-up.