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.
To my parents
Keywords
dispersive generalized Benjamin–Ono equation,
well-posedness, propagation of regularity, refined
Strichartz
