On the propagation of regularities in solutions of the Benjamin-Ono equation

TL;DR

This paper investigates how regularity properties of solutions to the Benjamin-Ono equation propagate over time, showing that certain regularities in initial data spread infinitely fast to the left as solutions evolve.

Contribution

It proves that specific regularities in initial data for the Benjamin-Ono equation propagate instantly to the entire left half-line over time.

Findings

Regularity in initial data propagates to the left infinitely fast.

Solutions gain regularity in the entire left half-line for positive times.

Regularity properties are preserved and spread over time.

Abstract

We shall deduce some special regularity properties of solutions to the IVP associated to the Benjamin-Ono equation. Mainly, for datum $u_0\in H^{3/2}(\mathbb R)$ whose restriction belongs to $H^m((b,\infty))$ for some $m\in\mathbb Z^+,\,m\geq 2,$ and $b\in \mathbb R$ we shall prove that the restriction of the corresponding solution $u(\cdot,t)$ belongs to $H^m((\beta,\infty))$ for any $\beta\in \mathbb R$ and any $t>0$. Therefore, this type of regularity of the datum travels with infinite speed to its left as time evolves.