Lower bound on the blow-up rate of the 3D Navier-Stokes equations in H^{5/2}

TL;DR

This paper establishes new lower bounds on the blow-up rate of solutions to the 3D Navier-Stokes equations in certain Sobolev spaces, addressing an open question and providing optimal bounds in some cases.

Contribution

It introduces a novel approach to lower bound the blow-up rate in Sobolev spaces, specifically in H^{5/2}, and answers an open question from prior research.

Findings

Derived lower bounds for blow-up rates in H^{5/2} in both whole space and periodic cases.

Provided optimal lower bounds for blow-up rates in rac{3}{2} and H^{1} spaces.

Addressed an open problem posed by James et al. (2012).

Abstract

Under assumption that $T^{\ast}$ is the maximal time of existence of smooth solution of the 3D Navier-Stokes equations in the Sobolev space $H^{s}$, we establish lower bounds for the blow-up rate of the type$\ \left( T^{\ast }-t\right) ^{-\varphi\left( n\right) }$, where $n$ is a natural number independent of $s$ and $\varphi$ is a linear function. Using this new type in the 3D Navier-Stokes equations in the $H^{5/2}$, both on the whole space and in the periodic case, we give an answer to a question left open by James et al (2012, J. Math. Phys.). We also prove optimal lower bounds for the blow-up rate in $\dot{H}^{3/2}$ and in $H^{1}$.