Local well-posedness and blow up in the energy space for a class of L2 critical dispersion generalized Benjamin-Ono equations

TL;DR

This paper establishes local well-posedness and analyzes blow-up phenomena for a class of energy-critical dispersion generalized Benjamin-Ono equations, bridging gaps between well-studied models like gKdV and BO.

Contribution

It extends local well-posedness results to the energy space for dgBO equations and investigates blow-up behavior near solitons, adapting techniques from gKdV studies.

Findings

Proved local well-posedness in the energy space for dgBO equations.

Showed solutions with negative energy close to solitons blow up in finite or infinite time.

Extended monotonicity methods to analyze blow-up in the dgBO context.

Abstract

We consider a family of dispersion generalized Benjamin-Ono equations (dgBO) which are critical with respect to the L2 norm and interpolate between the critical modified (BO) equation and the critical generalized Korteweg-de Vries equation (gKdV). First, we prove local well-posedness in the energy space for these equations, extending results by Kenig, Ponce and Vega concerning the (gKdV) equations. Second, we address the blow up problem in the spirit of works of Martel and Merle on the critical (gKdV) equation, by studying rigidity properties of the (dgBO) flow in a neighborhood of solitons. We prove that when the model is close to critical (gKdV), solutions of negative energy close to solitons blow up in finite or infinite time in the energy space. The blow up proof requires in particular extensions to (dgBO) of monotonicity results for localized versions of L2 norms by pseudo-differential operator tools.