# On the propagation of regularity for solutions of the Dispersion Generalized Benjamin-Ono Equation

## TL;DR

This paper investigates how regularity propagates in solutions of the dispersive generalized Benjamin-Ono equation, revealing a smoothing effect and addressing challenges posed by its nonlocal term.

## Contribution

It introduces a novel approach combining commutator expansions with weighted energy estimates to analyze regularity propagation in the dispersive generalized Benjamin-Ono equation.

## Key findings

- Regularity propagates with infinite speed for solutions.
- A new method effectively handles the nonlocal term.
- Explicit smoothing effects are established.

## Abstract

In this paper we study some properties of propagation of regularity of solutions of the dispersive generalized Benjamin-Ono (BO) equation. This model defines a family of dispersive equations, that can be seen as a dispersive interpolation between Benjamin-Ono equation and Korteweg-de Vries (KdV) equation. Recently, it has been showed that solutions of the KdV equation and Benjamin-Ono equation, satisfy the following property: if the initial data has some prescribed regularity on the right hand side of the real line, then this regularity is propagated with infinite speed by the flow solution. In this case the nonlocal term present in the dispersive generalized Benjamin-Ono equation is more challenging that the one in BO equation. To deal with this a new approach is needed. The new ingredient is to combine commutator expansions into the weighted energy estimate. This allow us to obtain the property of propagation and explicitly the smoothing effect.