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].
이 논문에서는 낮은 정칙성 하에서 4차 슈뢰딩거 방정식을 다룬다. 본 연구에서는 총 4가지 결과를 보인다. 우선 정칙성이 -1/2보다 크거나 같을때 local well posedness를 보이고 그 다음에는 global well posedness를 보인다. 또한 정칙성이 -1/2보다 작을 때는 ill posedness 결과를 보임으로써 우리의 결과가 최적임을 보인다.
정칙성이 -1/2보다 작을 때 ill posedness 결과가 있음에도 불구하고 우리는 -1/2보다 낮은 정칙성 하에서 a priori estimates 를 보임으로써 모든 시간 약해의 존재성을 보인다. 하지만 우리의 방법론이 해의 유일성과 초기 조건의 연속 의존성에 대한 얘기는 하지 못하기 때문에 완전한 well posedness 결과를 얻어내진 못한다.
