| Abstract: |
| I present my recent work with Kihyun Kim on a scattering problem of teh defocusing generalized Benjamin-Ono equations. Due to a dispersion relation, linear Benjamin-Ono flow travels to the right direction and the higher frequency waves travel fasster. This give rise to a monotonicity formula for linear flow and is expected to hold true for the defocusing generalized Benjamin-Ono flow. We prove the monotonicity. Namely, teh center of energy moves fater than the center of mass. This type of monotonicity was first observed by Tao in the defocusing gKdV equations. Later, Dodson used it to prove the scattering of the defocusing mass-critical gKdV equations. Similarly, for gBO equations, we use the monotonicity in the setting of concentration-compactness argument to prove the large data scattering in the energy space $H^{1/2}$. |
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