In this paper 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 Benjamin-Ono equation and Korteweg-de Vries (KdV) equation. Recently, it has been showed that solutions of the KdV equation and BenjaminOno 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 this case the nonlocal term present in the dispersive generalized BenjaminOno equation is more challenging that the one in 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 allow us to obtain the property of propagation and explicitly the smoothing effect.

中文翻译：

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关于色散广义 Benjamin-Ono 方程解的规律性传播

在本文中，我们研究了色散广义 Benjamin-Ono (BO) 方程解的正则性传播的一些性质。该模型定义了一系列色散方程，可以将其视为 Benjamin-Ono 方程和 Korteweg-de Vries (KdV) 方程之间的色散插值。最近，已经证明 KdV 方程和 BenjaminOno 方程的解满足以下性质：如果初始数据在实线的右手边有一些规定的规律，那么这个规律通过流动以无限速度传播解决方案。在这种情况下，离散广义 BenjaminOno 方程中存在的非局部项比 BO 方程中的非局部项更具挑战性。为了解决这个问题，需要一种新的方法。新成分是将换向器扩展组合到加权能量估计中。这使我们能够获得传播的属性和明确的平滑效果。