On well-posedness for the Benjamin-Ono equation (Q2465220)

The authors prove the existence and the uniqueness of real valued solutions for the Benjamin-Ono equation \(u_t+{\mathcal H}u_{xx}+uu_x=0\) (with the Hilbert transform \(\mathcal H\)) for initial conditions in \(H^s ({\mathbb R}), s > 1/4\). It is shown, moreover, that the flow is Hölder continuous in weaker topologies. The method of the proof is based on some ideas contained in [\textit{T. Tao}, J. Hyperbolic Differ. Equ. 1, No.~1, 27--49 (2004; Zbl 1055.35104)].