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

TL;DR

This paper investigates the nonlinear fourth-order Schrödinger equation, establishing local and global well-posedness results in Sobolev spaces, analyzing solution regularity, and identifying ill-posedness in super-critical cases.

Contribution

It provides new well-posedness and regularity results for NL4S and characterizes ill-posedness in super-critical regimes.

Findings

Local well-posedness in Sobolev spaces

Global well-posedness above mass and energy spaces

Ill-posedness in super-critical cases

Abstract

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.