# Remarks on the well-posedness of the Euler equations in the Triebel-Lizorkin spaces

## TL;DR

This paper establishes the continuous dependence of solutions to the Euler equations within critical Triebel-Lizorkin spaces, filling a gap in the mathematical understanding of these equations' well-posedness.

## Contribution

It extends the well-posedness results for Euler equations to Triebel-Lizorkin spaces using the Bona-Smith method, which was previously applied to Besov spaces.

## Key findings

- Proves continuous dependence of solutions in Triebel-Lizorkin spaces
- Uses Bona-Smith method for the proof
- Fills a gap in the mathematical theory of Euler equations

## Abstract

We prove the continuous dependence of the solution maps for the Euler equations in the (critical) Triebel-Lizorkin spaces, which was not shown in the previous works(\cite{Ch02, Ch03, ChMiZh10}). The proof relies on the classical Bona-Smith method as \cite{GuLiYi18}, where similar result was obtained in critical Besov spaces $B^1_{\infty,1}$.