Norm-inflation results for the good Boussinesq equation

Bautista G.J.; Potenciano-Machado L. · Journal of Evolution Equations (2025) · DOI: 10.1007/s00028-025-01114-5

Resumen

This article studies the norm-inflation phenomena of a periodic initial-value Good Boussinesq equation in low-regularity Sobolev spaces. Particularly, this article demonstrates that the initial-value problem is ill-posed in the periodic Sobolev spaces Hp-s(0,2π)×Hp-s-2(0,2π) for all s>1/2. Our proof is based on a constructive method: we provide smooth initial data that generates solutions with arbitrarily high-norms in Hp-s(0,2π)×Hp-s-2(0,2π) and for arbitrarily short times. The result is sharp in the sense that previous work (Cerpa and Rivas in J Evol Equ 18:1501–1519, 2018; Kishimoto in J Differ Equ 254:2393–2433, 2013) has shown well-posedness for periodic Sobolev indexes of the form Hp-s(0,2π)×Hp-s-2(0,2π) with s≤1/2. © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2025.

Instituciones: Universidad Tecnológica de los Andes