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

[摘要] We consider the Cauchy problem for the fourth order cubic nonlinear Schrodinger 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 L is locally well-posed in H-s (R),s >= - 1/2 using the Fourier restriction norm method. Second, we show that ( Ei) is globally well-posed in H-s (R),s >= - 1/2. 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 H-s (R),s >= - 1/2. Therefore, these results show that s = -1/2 is the sharp regularity threshold for which the well-posedness problem can be dealt with an iteration argument. (C) 2021 Elsevier Inc. All rights reserved.

[发布日期] 2021-12-01

[关键词] Low regularity solutions;Local well-posedness;Global well-posedness;Mild ill-posedness;t-method;Correction term