Lower bounds for possible singular solutions for the Navier--Stokes and Euler equations revisited

TL;DR

This paper establishes optimal lower bounds on the blow-up rates of Sobolev norms for potential singular solutions of the Navier-Stokes and Euler equations, providing new insights into their singularity formation.

Contribution

It offers the first optimal lower bounds for the blow-up rates of Sobolev norms in Navier-Stokes and Euler equations, with simplified proofs for the Euler case.

Findings

Optimal lower bounds for Navier-Stokes blow-up rates in rac{1}{2}<s<rac{5}{2}

Elementary proof for lower bounds in Euler equations when s>rac{5}{2}

Enhanced understanding of potential singularity behavior in fluid dynamics equations

Abstract

In this paper we give optimal lower bounds for the blow-up rate of the $\dot{H}^{s}\left(\mathbb{T}^3\right)$-norm, $\frac{1}{2}<s<\frac{5}{2}$, of a putative singular solution of the Navier-Stokes equations, and we also present an elementary proof for a lower bound on blow-up rate of the Sobolev norms of possible singular solutions to the Euler equations when $s>\frac{5}{2}$.