Preprint C1453/2019
ON THE PROPAGATION OF REGULARITY IN SOME NON-LOCAL DISPERSIVE MODELS
Argenis José Méndez García
Keywords: Propagation of regularity | Dispersive generalized Benjamin-Ono equation | Fractional Korteweg-de Vries equation

In this thesis we study some properties of propagation of regularity of solutions associated
to some nonlocal dispersive models. The first analysis, corresponds to the dispersive generalized
Benjamin-Ono (DGBO) equation, that can be seen as a dispersive interpolation between Benjamin-
Ono (BO) equation and the Korteweg-de Vries (KdV) equation. The second one, is the fractional
Korteweg-de Vries (fKdV) equation, that unlike the DGBO, this can be interpreted as a dispersive
perturbation of the Burger’s equation. Recently, it has been shown that solutions of the KdV
equation and BO 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 the case that attain us, the nonlocal term present in the dispersive
part in both equations, the DGBO and the fKdV equations, are more challenging than for KdV and
BO equation. In the case of the DGBO, we firstly prove a local well-posedness result associated
to the initial value problem, by using a compactness argument. As a second part of the thesis
we study the propagation of regularity for solutions of the DGBO equation. To deal with this a
new argument is employed. In fact, the approach used to deal with the later equations cannot
be applied directly and some new ideas have to be introduced. The new ingredient combines
commutator expansions into the weighted energy estimate that allow us to obtain the property
of propagation. Finally, the technique used in the propagation of regularity in solutions of the
DGBO allow us to verify that the solutions of the fKdV also satisfy the propagation of regularity
phenomena. Here the argument is more involved since the dispersion is very weak.