Well-posedness and ill-posedness for the fourth order cubic nonlinear Schrödinger equation in negative Sobolev spaces

We consider the Cauchy problem for the fourth order cubic nonlinear Schrödinger equation (4NLS). The main goal of this paper is to prove low regularity well-posedness and mild ill-posedness for (4NLS). We prove three results. First, we show that (4NLS) is locally well-posed in Hs(R),s≥−12 using the Fourier restriction norm method. Second, we show that (4NLS) is globally well-posed in Hs(R),s≥−12. To prove this, we use the I-method with the correction term strategy presented in Colliander-Keel-Staffilani-Takaoka-Tao [7]. Finally, we prove that (4NLS) is mildly ill-posed in the sense that the flow map fails to be locally uniformly continuous in Hs(R),s<−12. Therefore, these results show that s=−12 is the sharp regularity threshold for which the well-posedness problem can be dealt with an iteration argument.