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Scattering of the defocusing generalized Benjamin-Ono equations.
[2018 KAIST-HKUST-NUS Joint Workshop in Mathematics : Analysis, PDE and Probability]
Date: 2018-11-17
Speaker : Soonsik Kwon (KAIST)
Abstract : I present my recent work with Kihyun Kim on a scattering problem of nonlinear dispersive equations, namely generalized Benjamin-Ono equations. Due to its dispersion relation, linear Benjamin-Ono flow travels the right and the higher frequency travel faster. This give rise to a unidirectional propagation of waves. We prove the phenomena still hold true for defocusing BO. More precisely, we show that the center of energy moves faster than the center of mass to the right. This type of monotonicity was first observed by Tao(2007) in the defocusing gKdV equations. We use the monotonicity formula as a non-rigidity theorem, in the setting of compactness-contradiction argument to prove the large data scattering in the energy space. In the talk, I will first explain the whole scheme of concentration-compactness, and the heuristics of monotonicity. Then I will discuss obstacles arising in the Benjamin-Ono setting.
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