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

This work was supported in part by the Labex CEMPI (ANR-11-LABX-0007-01)

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  • We consider the cubic nonlinear fourth-order Schrödinger equation

    $ i \partial_t u - \Delta^2 u + \mu \Delta u = \pm |u|^2 u, \quad \mu \geq 0 $

    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.

    Mathematics Subject Classification: Primary: 35Q55; Secondary: 35A01.


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