Scattering for defocusing generalized Benjamin-Ono equation in the energy space $H^{\frac {1}{2}}(\mathbb {R})$

Abstract:

We prove the scattering for the defocusing generalized Benjamin-Ono equation in the energy space $H^{\frac {1}{2}}(\mathbb {R})$. We first establish the monotonicity formula that describes the unidirectional propagation. More precisely, it says that the center of energy moves faster than the center of mass. This type of monotonicity was first observed by Tao in the defocusing gKdV equations.

We use the monotonicity in the setting of compactness-contradiction argument to prove the large data scattering in the energy space $H^{\frac {1}{2}}(\mathbb {R})$. On the way, we extend the critical local theory of Vento to the subcritical regime. Indeed, we obtain subcritical local theory and global well-posedness in the energy space.