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

Kihoon Seong

doi:10.3934/cpaa.2020247

Article Contents

2020, Volume 19, Issue 12: 5437-5473. Doi: 10.3934/cpaa.2020247

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

Kihoon Seong

Department of Mathematical Sciences, Korea Advanced Institute of Science and Technology, 291 Daehak-ro, Yuseong-gu, Daejeon 34141, Republic of Korea

Received: December 31, 2019

Revised: June 30, 2020

Early access: October 2020

Published: December 2020

The author is supported by NRF-2018R1D1A1A09083345

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Abstract

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 [29], the author showed that this equation is globally well-posed in $ H^s(\mathbb{R}), s\\\geq -\frac{1}{2} $ and mildly ill-posed in the sense that the solution map fails to be locally 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 mild ill-posedness result, we obtain an a priori bound below $ s<-1/2 $. This an a priori estimate guarantees the existence of a weak solution for $ -3/4<s<-1/2 $. Our method is inspired by Koch-Tataru [17]. We use the $ U^p $ and $ V^p $ based spaces adapted to frequency dependent time intervals on which the nonlinear evolution can still be described by linear dynamics.

Keywords:

fourth order nonlinear Schrödinger equation,
low regularity solutions,
frequency dependent time scale,
$ U^p $ and $ V^p $ spaces,
normal form reduction.

Mathematics Subject Classification: 35Q55.

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References

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