# Temporal regularity of the solution to the incompressible Euler equations in the end-point critical Triebel-Lizorkin space $F^{d+1}_{1, \infty}(\mathbb{R}^d)$

## TL;DR

This paper investigates the temporal regularity of solutions to the incompressible Euler equations in a critical Triebel-Lizorkin space, revealing discontinuity and ill-posedness in that setting.

## Contribution

It demonstrates the temporal discontinuity of solutions in the endpoint critical Triebel-Lizorkin space, establishing ill-posedness of the Euler equations in this context.

## Key findings

- Evidence of temporal discontinuity in solutions.
- Ill-posedness of the Euler equations in the critical space.
- Discussion on continuity in related function spaces.

## Abstract

An evidence of temporal dis-continuity of the solution in $F^s_{1, \infty}(\mathbb{R}^d)$ is presented, which implies the ill-posedness of the Cauchy problem for the Euler equations. Continuity and weak-type continuity of the solutions in related spaces are also discussed.