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

Abstract

We prove that the Cauchy problem for the dispersion generalized Benjamin-Ono equation \[\partial_t u+|\partial_x|^{1+α}\partial_x u+uu_x=0,\ u(x,0)=u_0(x),\] is locally well-posed in the Sobolev spaces $H^s$ for $s>1-\alpha$ if $0\le \alpha \le 1$. The new ingredient is that we develop the methods of Ionescu, Kenig and Tataru \cite{IKT} to approach the problem in a less perturbative way, in spite of the ill-posedness results of Molinet, Saut and Tzvetkovin \cite{MST}. Moreover, as a bi-product we prove that if $0<\alpha \le 1$ the corresponding modified equation (with the nonlinearity $\pm uu{u}_x$) is locally well-posed in $H^s$ for $s\ge 1/2-\alpha /4$.

33 pages

References in corpus (3)

• Decay estimates for a class of wave equations
• Nonlinear wave interactions for the Benjamin-Ono equation
• Local Well-posedness and a priori bounds for the modified Benjamin-Ono equation without using a gauge transformation

Cited by in corpus (1)

• On the local regularity of the KP-I equation in anisotropic Sobolev space