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On the local well-posedness of the Benjamin-Ono and modified Benjamin-Ono equations. (English) Zbl 1044.35072

Summary: We prove that the Benjamin-Ono equation \[ \partial_t u+ H\partial^2_x u+ u\partial_x u= 0 \] is locally well-posed in \(H^s(\mathbb{R})\) for \(s> 9/8\) and that for arbitrary initial data, the modified (cubic nonlinearity) Benjamin-Ono equation \[ \partial_t u+ H\partial^2_x u+ u^2\partial_x u= 0 \] is locally well-posed in \(H^s(\mathbb{R})\) for \(s\geq 1\).

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids
76B55 Internal waves for incompressible inviscid fluids
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