Smoothing properties and existence of solutions for the generalized Benjamin-Ono equation

Abstract We study the Cauchy problem for the generalized Benjamin-Ono equation ∂tu − D |D|2μ u = DV'(u), (∗) where D = d dx , U > 0 , and V ϵ E 1( R,R ) with V(0) = V′(0) = 0. Using commutator identities and estimates, we prove that the equation (∗) exhibits smoothing properties similar to those of the generalized Korteweg-de Vries equation (GKdV), which is the special case μ = 1 of (∗). If V satisfies suitable estimates at zero and at infinity, we prove that the Cauchy problem for the equation (∗) with initial data u(0) = u0 has a solution u ϵ L∞( R , l2)∪L2loc( R , Hμloc) if u0 ϵ L2, and has a solution u ϵ L∞( R , Hμ)∪L2loc( R , H2μloc)if u0 ϵ Hμ (finite energy solution). In addition, we prove that the usual Benjamin-Ono equation ( μ = 1 2 , V′(u) = u 2 ) has a solution u ϵ L∞( R , H1)∪L2loc( R , H 3 2 loc) for u0 ϵ H1. Those results are easily extended to related equations such as the intermediate long wave equation or the Smith equation.