Norm inflation with infinite loss of regularity for the generalized improved Boussinesq equation

TL;DR

This paper demonstrates that the generalized improved Boussinesq equation exhibits norm inflation with infinite loss of regularity across various function spaces, highlighting its ill-posedness in low-regularity settings.

Contribution

It establishes sharp ill-posedness results for the generalized improved Boussinesq equation in multiple function spaces, including Sobolev, Fourier-Lebesgue, modulation, and Wiener amalgam spaces.

Findings

Norm inflation with infinite loss of regularity at general initial data.

Results are sharp in the $L^2$-based Sobolev scale.

Extension of ill-posedness to multi-dimensional and other function spaces.

Abstract

In this paper, we study the ill-posedness issue for the generalized improved Boussinesq equation. In particular we prove there is norm inflation with infinite loss of regularity at general initial data in $\langle \nabla \rangle^{-s}\big(L^2 \cap L^\infty\big)(\mathbb{R})$ for any $s < 0$. This result is sharp in the $L^2$-based Sobolev scale in view of the well-posedness in $L^2(\mathbb{R}) \cap L^\infty(\mathbb{R})$. We also show that the same result applies to the multi-dimensional generalized improved Boussinesq equation. Finally, we extend our norm inflation result to Fourier-Lebesgue, modulation and Wiener amalgam spaces.