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Low regularity a priori estimates for the fourth order cubic nonlinear Schr\"odinger equation
Analysis of PDEs · Submitted 11 Nov 2019
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim This a priori estimate guarantees the existence of a weak solution for $-3/4<s<-1/2$.
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We consider the low regularity behavior of the fourth order cubic nonlinear Schr\"odinger 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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We consider the low regularity behavior of the fourth order cubic nonlinear Schr\"odinger 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). | background 0.88 | 0.38 | 0.03 | asserted 0.63 |
| s1 | \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 $-\frac{15}{14}<s<-\frac{1}{2}$. | background 0.98 | 0.12 | 0.10 | asserted 1.00 |
| s2 | Therefore, $s=-\frac{1}{2}$ is the lowest regularity that can be handled by the contraction argument. | interpretation 0.68 | 0.14 | 0.07 | asserted 1.00 |
| s3 | In spite of this ill-posedness result, we obtain a priori bound below $s<-1/2$. | result 0.53 | 0.93 | 0.40 | asserted 1.00 |
| s4 | This a priori estimate guarantees the existence of a weak solution for $-3/4<s<-1/2$. | result 0.82 | 0.92 | 0.54 | asserted 1.00 |
| s5 | But we cannot establish full well-posedness because of the lack of energy estimate of differences of solutions. | limitation 1.00 | 0.89 | 0.10 | asserted 1.00 |
| s6 | Our method is inspired by Koch-Tataru \cite{KT2007}. | method 0.90 | 0.17 | 0.02 | asserted 1.00 |
| s7 | 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. | method 1.00 | 0.82 | 0.03 | asserted 1.00 |
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Original abstract
We consider the low regularity behavior of the fourth order cubic nonlinear Schr\"odinger 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.
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