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 partial derivative(t)u + vertical bar partial derivative(x)vertical bar(1+alpha)partial derivative(x)u + uu(x) = 0, u(x,0) = u(0)(x), is locally well-posed in the Sobolev spaces H-s for s > 1 - alpha if 0 <= alpha <= 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 < alpha <= 1 the corresponding modified equation (with the nonlinearity +/- uuu(x)) is locally well-posed in H-s for s >= 1/2 - alpha/4. (C) 2011 Elsevier Inc. All rights reserved.