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

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 $m\in {\mathbb{Z}}^{+},m\ge 2,$ 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.