Abstract
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.
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