Norm Inflation for Benjamin–Bona–Mahony Equation in Fourier Amalgam and Wiener Amalgam Spaces with Negative Regularity

Divyang G. Bhimani,Saikatul Haque

doi:10.3390/math9233145

Abstract

We consider the Benjamin–Bona–Mahony (BBM) equation of the form ut+ux+uux−uxxt=0,(x,t)∈M×R where M=T or 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 strengthens several known NI results at zero initial data in Hs(T) established by Bona–Dai (2017) and the ill-posedness result established by Bona–Tzvetkov (2008) and Panthee (2011) in Hs(R). Our result is sharp with respect to the local well-posedness result of Banquet–Villamizar–Roa (2021) in modulation spaces Ms2,1(R) for s≥0.

Highlights

We study strong ill-posedness for the Benjamin–Bona–Mahony (BBM) equation of the form utu xuu xu xxt “ 0 (1)
The BBM (1) can be written as Published: 6 December 2021 iut “ φpDx quφpDx qu2, Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil
This BBM (1) model is the regularized counterpart of the Korteweg–de Vries (KdV)

Summary

Introduction

We study strong ill-posedness for the Benjamin–Bona–Mahony (BBM) equation of the form. Kishimoto [21] established norm inflation (NI) for NLS on a special ř domain (special domain: Rd1 ˆ Td2 , d “ d1 ` d2 and with non-linearity: nj“1 νj uρ j pūqσjρ j where νj P C, σj P N, ρjPNY t0u with σj ě maxpρ j , 2q) and Oh [22] established NI at general initial data for cubic NLS This approach has been used to obtain strong ill-posedness for NLW in [15,23].

Function Spaces

F Ls pMq “ f P S 1 pMq : F f xys P Lq pM

Local Well-Posedness in Wiener Algebra F L1

Estimates in w

Estimates in Ws pRq