Local well-posedness for dispersion generalized Benjamin–Ono equations in Sobolev spaces

We prove that the Cauchy problem for the dispersion generalized Benjamin–Ono equation∂tu+|∂x|1+α∂xu+uux=0,u(x,0)=u0(x), is locally well-posed in the Sobolev spaces HsHs for s>1−αs>1−α if 0⩽α⩽10⩽α⩽1. The new ingredient is that we generalize the methods of Ionescu, Kenig and Tataru (2008) [13] to approach the problem in a less perturbative way, in spite of the ill-posedness results of Molinet, Saut and Tzvetkov (2001) [21]. Moreover, as a bi-product we prove that if 0<α⩽10<α⩽1 the corresponding modified equation (with the nonlinearity ±uuux±uuux) is locally well-posed in HsHs for s⩾1/2−α/4s⩾1/2−α/4.