On a perturbation of the Benjamin–Ono equation

We prove using the Fourier restriction norm method that the initial value problem associated to a perturbation of the Benjamin–Ono equation ut+uux+βHuxx+η(Hux−uxx)=0ut+uux+βHuxx+η(Hux−uxx)=0, where x∈Rx∈R, t≥0t≥0, η>0η>0 and HH denotes the usual Hilbert transform, is locally well-posed in the Sobolev space Hs(R)Hs(R) for any s>−1/2s>−1/2 and globally well-posed in Hs(R)Hs(R) when s≥0s≥0. For s<−1s<−1 we also prove some ill-posedness issues.