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| ∂xu+ uux = 0, u(x, 0) = u0(x), is locally well-posed in the Sobolev spaces H for s > 1 − α if 0 ≤ α ≤ 1. The new ingredient is that we develop the methods of Ionescu, Kenig and Tataru [13] to approach the problem in a less perturbative way, in spite of the ill-posedness results of Molinet, Saut and Tzvetkovin [21]. Moreover, as a bi-product we prove that if 0 < α ≤ 1 the corresponding modified equation (with the nonlinearity ±uuux) is locally well-posed in H s for s ≥ 1/2− α/4.