Norm inflation for BBM equation in Fourier amalgam and Wiener amalgam spaces with negative regularity

TL;DR

This paper proves that the BBM equation exhibits norm inflation with infinite regularity loss in Fourier and Wiener amalgam spaces with negative regularity, extending known ill-posedness results to broader function spaces.

Contribution

It establishes sharp norm inflation results for the BBM equation in Fourier and Wiener amalgam spaces with negative regularity, strengthening previous ill-posedness findings.

Findings

Norm inflation occurs with infinite regularity loss in specified spaces.

Results extend and strengthen previous ill-posedness results.

The findings are sharp relative to existing local well-posedness in modulation spaces.

Abstract

We consider Benjamin-Bona-Mahony (BBM) equation of the form $$ u_t+u_x+uu_x-u_{xxt}=0, \quad (x, t)\in \mathcal{M}\times \mathbb R $$ where $\mathcal{M}= \mathbb T$ or $\mathbb R.$ We establish norm inflation (NI) with infinite loss of regularity at general initial data in Fourier amalgam and Wiener amalgam spaces with negative regularity. This strengthen several known NI results at zero initial data in $H^s(\mathbb T)$ established by Bona-Dai (2017) and ill-posedness result established by Bona-Tzvetkov (2008) and Panthee (2011) in $H^s(\mathbb R).$ Our result is sharp with respect to local well-posedness result of Banquet-Villamizar-Roa (2021) in modulation spaces $M^{2,1}_s(\mathbb R)$ for $s\geq 0$.