# Ill-posedness of Naiver-Stokes equations and critical Besov-Morrey spaces

## TL;DR

This paper investigates the ill-posedness of Navier-Stokes equations, demonstrating blow-up phenomena and norm inflation in generalized initial spaces, thereby extending previous results and highlighting the equations' sensitivity to initial conditions.

## Contribution

It establishes ill-posedness results for Navier-Stokes equations in broader initial spaces, independent of specific solution space choices, complementing prior findings.

## Key findings

- Proved blow-up in the first step of Picard's scheme.
- Demonstrated norm inflation in generalized solution spaces.
- Extended ill-posedness results beyond previously known spaces.

## Abstract

The blow up phenomenon in the first step of the classical Picard's scheme was proved in this paper. For certain initial spaces, Bourgain-Pavlovi\'c and Yoneda proved the ill-posedness of the Navier-Stokes equations by showing the norm inflation in certain solution spaces. But Chemin and Gallagher said the space $\dot{B}^{-1,\infty}_{\infty}$ seems to be optimal for some solution spaces best chosen. In this paper, we consider more general initial spaces than Bourgain-Pavlovi\'c and Yoneda did and establish ill-posedness result independent of the choice of solution space. Our result is a complement of the previous ill-posedness results on Navier-Stokes equations.