On the propagation of regularity for solutions of the dispersion generalized Benjamin–Ono equation

Abstract

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 the Benjamin–Ono equation and the Korteweg–de Vries (KdV) equation.

Recently, it has been shown that solutions of the KdV and BO equations 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 the 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 allows us to obtain the property of propagation and explicitly the smoothing effect.