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

摘要

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 H~s for s>1-α if 0≤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≤1 the corresponding modified equation (with the nonlinearity ±uuu_x) is locally well-posed in H~s for s≥1/2-α/4.