Sharp well-posedness and ill-posedness of the Navier-Stokes initial value problem in Besov-type spaces

TL;DR

This paper investigates the well-posedness and ill-posedness of the Navier-Stokes initial value problem within specific Besov-type spaces, establishing critical thresholds for solution behavior using advanced bilinear estimates.

Contribution

It identifies precise conditions under which the Navier-Stokes problem is well-posed or ill-posed in logarithmically refined Besov spaces, employing sharp bilinear estimates and refined analytical techniques.

Findings

Well-posedness in Besov spaces with second index above critical value

Ill-posedness when second index is below critical value

Use of sharp bilinear estimates from Hardy-Littlewood inequalities

Abstract

We prove that the Navier-Stokes initial value problem is well-posed in the logrithmically refined Besov spaces when the second index is not less than certain critical value, and ill-posed in such spaces when the second index is less than this critical value. The well-posedness result is proved by using some sharp bilinear estimates obtained from some Hardy-Littlewood type inequalities. The ill-posedness assertion is proved by refining the arguments of Wang [18] and Yoneda [20].