Continuity of the data-to-solution map for the FORQ equation in Besov Spaces

TL;DR

This paper investigates the FORQ equation in Besov spaces, demonstrating that the data-to-solution map lacks uniform continuity under certain regularity conditions, using advanced harmonic analysis techniques.

Contribution

It establishes non-uniform dependence of solutions on initial data for the FORQ equation in specific Besov spaces, extending understanding of well-posedness issues.

Findings

Data-to-solution map is not uniformly continuous in specified Besov spaces.

Proof utilizes approximate solutions and Littlewood-Paley decomposition.

Results highlight limitations of solution stability in certain regularity regimes.

Abstract

For Besov spaces $B^s_{p,r}(\rr)$ with $s>\max\{ 2 + \frac1p , \frac52\} $, $p \in (1,\infty]$ and $r \in [1 , \infty)$, it is proved that the data-to-solution map for the FORQ equation is not uniformly continuous from $B^s_{p,r}(\rr)$ to $C([0,T]; B^s_{p,r}(\rr))$. The proof of non-uniform dependence is based on approximate solutions and the Littlewood-Paley decomposition.