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 |