Well-posedness and regularity for the fractional Navier-Stokes equations

TL;DR

This paper investigates the well-posedness and regularity of the fractional Navier-Stokes equations, extending previous results and clarifying their relation to known ill-posedness outcomes.

Contribution

It generalizes Koch-Tataru's well-posedness results to fractional Navier-Stokes equations and discusses regularity and analyticity without contradicting Bourgain-Pavlovic's ill-posedness findings.

Findings

Established well-posedness for fractional Navier-Stokes equations

Clarified the relation to ill-posedness in certain function spaces

Analyzed regularity and analyticity in space variables

Abstract

We consider the wellposedness of the fractional Navier-Stokes as a generalization of the wellposedness result in Koch-Tataru's paper. An interesting remark is that our result does not contradict to the well-known ill-posedness result for Navier-Stokes with initial data in $\dot B^{-1, \infty}_\infty$ by Bourgain-Pavlovic. In the end, we also discuss the regularity and analyticity in the space variable.