Random data theory for the cubic fourth-order nonlinear Schrödinger equation

Abstract

We consider the cubic nonlinear fourth-order Schrödinger equation \begin{document}$ i \partial_t u - \Delta^2 u + \mu \Delta u = \pm |u|^2 u, \quad \mu \geq 0 $\end{document} on $ \mathbb R^N, N\geq 5 $ with random initial data. We prove almost sure local well-posedness below the scaling critical regularity. We also prove probabilistic small data global well-posedness and scattering. Finally, we prove the global well-posedness and scattering with a large probability for initial data randomized on dilated cubes.