Well-posedness, regularity and ill-posedness for the nonlinear fourth-order Schr\"odinger equation

We prove the local well-posedness for the nonlinear fourth-order Schr"odinger equation (NL4S) in Sobolev spaces. We also studied the regularity of solutions in the sub-critical case. A direct consequence of this regularity is the global well-posedness above mass and energy spaces under some assumptions. Finally, we show the ill-posedness for (NL4S) in some cases of the super-critical range.