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

Abstract

We consider the Cauchy problem of the fourth-order cubic nonlinear Schrödinger equation (4NLS) \begin{align*} \begin{cases} $i\partial_tu+\{partial_x}^4u=\pm \vert u \vert^2u, \quad(t,x)\in \mathbb{R}\times \mathbb{R}\\$ $u(x,0)=u_0(x) \in H^s\left(\mathbb{R}\right)$. \end{cases} \end{align*} The main goal of this paper is to solve the low regularity well-posedness and ill-posedness problem of the fourth order NLS. We prove four results. One is the local well-posedness in $H^s\left(\mathbb{R}\right), s\geq -1/2$ via the contraction principle on the $X^{s.b}$ space, also known as Bourgain space. Another is the global well-posedness in $H^s\left(\mathbb{R}\right),s\geq -1/2$. A third is the ill-posedness in the sense that the solution map fails to be uniformly continuous for $s<-\frac{1}{2}$. Therefore, we show that $s=-1/2$ is the sharp regularity threshold for which the well-posedness problem can be dealt with the iteration argument. The method of our proof of the global well-posedness is the $I$-method with correction term, which was first developed by Colliander-Keel-Staffilani-Takaoka-Tao [10]]. To prove the ill-posedness, we follow the strategy described in Christ-Colliander-Tao [6]. In spite of this ill-posedness result, we obtain a priori bounds below $s<-1/2$. This a priori estimates guarantee the existence of weak solutions for $s<-1/2$. But we cannot establish full well-posedness because of the lack of estimates of differences of solutions. We follow the argument presented in Koch-Tataru [20].