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Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation

We consider the low regularity behavior 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*} In arXiv:1911.03253, the author showed that this equation is globally well-posed in $H^s, s\geq -\frac{1}{2}$ and ill-posedness in the sense that the solution map fails to be uniformly continuous for $-\frac{15}{14}<s<-\frac{1}{2}$. Therefore, $s=-\frac{1}{2}$ is the lowest regularity that can be handled by the contraction argument. In spite of this ill-posedness result, we obtain a priori bound below $s<-1/2$. This a priori estimate guarantees the existence of a weak solution for $-3/4<s<-1/2$. But we cannot establish full well-posedness because of the lack of energy estimate of differences of solutions. Our method is inspired by Koch-Tataru \cite{KT2007}. We use the $U^p$ and $V^p$ based spaces adapted to frequency dependent time intervals on which the nonlinear evolution can be still described by linear dynamics.