On some regularity properties for the dispersive generalized Benjamin-Ono-Zakharov-Kuznetsov Equation

TL;DR

This paper investigates the propagation of regularity in solutions to the dispersive generalized Benjamin-Ono-Zakharov-Kuznetsov equation, showing that initial smoothness on certain regions extends infinitely fast throughout the solution.

Contribution

It establishes the propagation of regularity for solutions of the dispersive generalized Benjamin-Ono-Zakharov-Kuznetsov equation based on new formulas relating derivatives in specific regions.

Findings

Regularity propagates with infinite speed from initial data.

Propagation of smoothness is valid regardless of the regularity scale.

Derived formulas relate derivatives in different regions of the plane.

Abstract

This work aims to study some smoothness properties concerning the initial value problem associated to the dispersive generalized Benjamin-Ono-Zakharov-Kuznetsov equation. More precisely, we prove that the solutions to this model satisfy the so-called propagation of regularity. Roughly speaking, this principle states that if the initial data enjoys some extra smoothness prescribed on a family of half-spaces, then the regularity is propagated with infinite speed. In this sense, we prove that regardless of the scale measuring the extra regularity in such hyperplane collection, then all this regularity is also propagated by solutions of this model. Our analysis is mainly based on the deduction of propagation formulas relating homogeneous and non-homogeneous derivatives in certain regions of the plane.