Modified Strichartz Estimate of the Periodic Fourth Order NLS

We prove modified Strichartz estimates on the one-dimensional torus, that are adapted to a fourth-order dispersion relation, and use them to show global well-posedness of nonlinear fourth-order Schr"odinger equations. This extends the (low regularity) existence theory of the adiabatic transition of the Quantum Zakharov system to NLS. We show that the solutions behave continuously with respect to the quantum parameter in every compact time interval. Globally in time, however, we also show that such continuous dependence is generally not uniform.