Local well-posedness for the derivative nonlinear Schrödinger equation with $ L^2 $-subcritical data

Shaoming Guo; Xianfeng Ren; Baoxiang Wang

doi:10.3934/dcds.2021034

Article Contents

2021, Volume 41, Issue 9: 4207-4253. Doi: 10.3934/dcds.2021034

This issue Previous Article Partial regularity and Liouville theorems for stable solutions of anisotropic elliptic equations Next Article Turing instability and dynamic phase transition for the Brusselator model with multiple critical eigenvalues

Local well-posedness for the derivative nonlinear Schrödinger equation with $ L^2 $-subcritical data

1.
Department of Mathematics, University of Wisconsin Madison, Madison, WI, USA
2.
LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, China

^* Corresponding author: Baoxiang Wang
^* Corresponding author: Baoxiang Wang

Received: August 2020

Revised: January 2021

Early access: February 2021

Published: September 2021

The second and third named authors were supported in part by NSFC grant 11771024

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Abstract

Considering the Cauchy problem of the derivative NLS

$ \begin{align} {\rm i} u_{t} + \partial_{xx} u = {\rm i} \mu \partial_x (|u|^2u),\quad u(0,x) = u_0(x), \end{align} $

we will show its local well-posedness in modulation spaces $ M^{1/2}_{2,q}(\mathbb{R}) $ $ (4{\leqslant} q<\infty) $. It is well-known that $ H^{1/2} $ is a critical Sobolev space of the derivative NLS. Noticing that $ H^{1/2} \subset M^{1/2}_{2,q} \subset B^{1/q}_{2,q} $ $ (q{\geqslant} 2) $ are sharp inclusions, our result contains a class of functions in $ L^2\setminus H^{1/2} $.

Keywords:

Local well-posedness,
Derivative nonlinear Schrödinger equations,
Modulation spaces.

Mathematics Subject Classification: Primary: 35Q55, 42B35; Secondary: 42B37.

Citation:

Full Text(HTML)

Table 1. In $ \lambda_1,...,\lambda_3 $, there is at least one frequency near $ \lambda_0 $

$ {\rm Case} $	$ \lambda_1\in $	$ \lambda_2\in $	$ \lambda_3\in $
$ h_-h_-a $	$ [\lambda_0, 3\lambda_0/4) $	$ [\lambda_0, 3\lambda_0/4) $	$ [\lambda_0, \infty) $
$ h_-l_-l_- $	$ [\lambda_0, 3\lambda_0/4) $	$ [3\lambda_0/4, \ 0) $	$ [\lambda_0, 0) $
$ h_-l_- a_+ $	$ [\lambda_0, 3\lambda_0/4) $	$ [3\lambda_0/4,0) $	$ [0, \infty) $
$ h_-a_+ a $	$ [\lambda_0, 3\lambda_0/4) $	$ [0, \infty) $	$ [\lambda_0, \infty) $

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Table 2. $ \lambda_1,\lambda_2,\lambda_3 $ far away from $ \lambda_0 $

$ {\rm Case} $	$ \lambda_1\in $	$ \lambda_2\in $	$ \lambda_3\in $
$ l_-l_-l $	$ [3\lambda_0/4, 0) $	$ [3\lambda_0/4,0) $	$ [\lambda_0, -\lambda_0/16) $
$ l_-l_-h_+ $	$ [ 3\lambda_0/4, 0) $	$ [3\lambda_0/4, \ 0) $	$ [-\lambda_0/16, \infty) $
$ l_-a_+a_m $	$ [3\lambda_0/4, 0) $	$ [0, \infty) $	$ [\lambda_0, -\lambda_0) $
$ l_-a_+h_+ $	$ [3\lambda_0/4, 0) $	$ [0, \infty) $	$ [-\lambda_0, \infty) $

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Table 3. $ \lambda_1,\lambda_2,\lambda_3 {\geqslant} 0 $

$ {\rm Case} $	$ \lambda_1\in $	$ \lambda_2\in $	$ \lambda_3\in $
$ a_+a_+h $	$ [0, \infty) $	$ [0, \infty) $	$ [\lambda_0, \infty) $

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Table 4. $ \lambda_0 \ll 0 $, $ \lambda_1<\lambda_0 {\leqslant} \lambda_2\wedge \lambda_3 $

$ {\rm Case} $	$ \lambda_1\in $	$ \lambda_2\in $	$ \lambda_3\in $
$ 1 $	$ [5\lambda_0/4-C, \lambda_0) $	$ [\lambda_0, 3\lambda_0/4) $	$ [\lambda_0, 3\lambda_0/4 +C) $
$ 2 $	$ [2\lambda_0 -C, \lambda_0) $	$ [3\lambda_0/4, \ 0) $	$ [\lambda_0, C) $
$ 3 $	$ (11\lambda_0/4-C, \lambda_0) $	$ [0, -3\lambda_0/4) $	$ [\lambda_0, -3\lambda_0/4 +C) $
$ 4 $	$ (-\infty, \lambda_0) $	$ [-3\lambda_0/4, \infty) $	$ [\lambda_0, \infty) $

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Table 5. Two subcases of Case 4

$ {\rm Case} $	$ \lambda_1\in $	$ \lambda_2\in $	$ \lambda_3\in $
$ 4.1 $	$ (-\infty, \lambda_0) $	$ [-3\lambda_0/4, \infty) $	$ [\lambda_0, 0) $
$ 4.2 $	$ (-\infty, \lambda_0) $	$ [-3\lambda_0/4, \infty) $	$ [0, \infty) $

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Table 6. $ \lambda_0{\geqslant}\lambda_1{\geqslant}\lambda_3{\geqslant} \lambda_2 {\geqslant} \lambda_4{\geqslant} \lambda_5 $, only one higher frequency in $ \lambda_1,...,\lambda_5 $

$ {\rm Case} $	$ \lambda_1\in $	$ \lambda_3\in $	$ \lambda_2\in $	$ \lambda_4\in $	$ \lambda_5\in $
$ hllll $	$ h $	$ l $	$ l $	$ l $	$ l $
$ hllll_- $	$ h $	$ l $	$ l $	$ l $	$ l_- $
$ hlll_-l_- $	$ h $	$ l $	$ l $	$ l_- $	$ l_- $
$ hll_-l_-l_- $	$ h $	$ l $	$ l_- $	$ l_- $	$ l_- $
$ hl_-l_-l_-l_- $	$ h $	$ l_- $	$ l_- $	$ l_- $	$ l_- $

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Table 7. $ \lambda_0{\geqslant}\lambda_2{\geqslant}\lambda_1{\geqslant} \lambda_3 {\geqslant} \lambda_5{\geqslant} \lambda_4 $, only one higher frequency in $ \lambda_1,...,\lambda_5 $

$ {\rm Case} $	$ \lambda_2\in $	$ \lambda_1\in $	$ \lambda_3\in $	$ \lambda_5\in $	$ \lambda_4\in $
$ llllh_- $	$ l $	$ l $	$ l $	$ l $	$ h_- $
$ llll_-h_- $	$ l $	$ l $	$ l $	$ l_- $	$ h_- $
$ hlll_-l_-h_- $	$ l $	$ l $	$ l_- $	$ l_- $	$ h_- $
$ ll_-l_-l_-h_- $	$ l $	$ l_- $	$ l_- $	$ l_- $	$ h_- $
$ l_-l_-l_-l_-h_- $	$ l_- $	$ l_- $	$ l_- $	$ l_- $	$ h_- $

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DownLoad: CSV

Table 8. $ \lambda_0{\geqslant}\lambda_1{\geqslant}\lambda_2{\geqslant} \lambda_3 {\geqslant} \lambda_5{\geqslant} \lambda_4 $, only one higher frequency in $ \lambda_1,...,\lambda_5 $

$ {\rm Case} $	$ \lambda_1\in $	$ \lambda_2\in $	$ \lambda_3\in $	$ \lambda_5\in $	$ \lambda_4\in $
$ hllll $	$ h $	$ l $	$ l $	$ l $	$ l $
$ hllll_- $	$ h $	$ l $	$ l $	$ l $	$ l_- $
$ hlll_-l_- $	$ h $	$ l $	$ l $	$ l_- $	$ l_- $
$ hll_-l_-l_- $	$ h $	$ l $	$ l_- $	$ l_- $	$ l_- $
$ hl_-l_-l_-l_- $	$ h $	$ l_- $	$ l_- $	$ l_- $	$ l_- $

| Show Table

DownLoad: CSV

Table 9. $ \lambda_0{\geqslant}\lambda_1{\geqslant}\lambda_2{\geqslant} \lambda_3 {\geqslant} \lambda_5{\geqslant} \lambda_4 $, only one higher frequency in $ \lambda_1,...,\lambda_5 $

$ {\rm Case} $	$ \lambda_1\in $	$ \lambda_2\in $	$ \lambda_3\in $	$ \lambda_5\in $	$ \lambda_4\in $
$ llllh_- $	$ l $	$ l $	$ l $	$ l $	$ h_- $
$ llll_-h_- $	$ l $	$ l $	$ l $	$ l_- $	$ h_- $
$ hlll_-l_-h_- $	$ l $	$ l $	$ l_- $	$ l_- $	$ h_- $
$ ll_-l_-l_-h_- $	$ l $	$ l_- $	$ l_- $	$ l_- $	$ h_- $
$ l_-l_-l_-l_-h_- $	$ l_- $	$ l_- $	$ l_- $	$ l_- $	$ h_- $

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DownLoad: CSV

Table 10. $ \lambda_0{\geqslant}\lambda_1{\geqslant}\lambda_3{\geqslant} \lambda_5 {\geqslant} \lambda_2{\geqslant} \lambda_4 $, only one higher frequency in $ \lambda_1,...,\lambda_5 $

$ {\rm Case} $	$ \lambda_1\in $	$ \lambda_3\in $	$ \lambda_5\in $	$ \lambda_2\in $	$ \lambda_4\in $
$ hllll $	$ h $	$ l $	$ l $	$ l $	$ l $
$ hllll_- $	$ h $	$ l $	$ l $	$ l $	$ l_- $
$ hlll_-l_- $	$ h $	$ l $	$ l $	$ l_- $	$ l_- $
$ hll_-l_-l_- $	$ h $	$ l $	$ l_- $	$ l_- $	$ l_- $
$ hl_-l_-l_-l_- $	$ h $	$ l_- $	$ l_- $	$ l_- $	$ l_- $

| Show Table

DownLoad: CSV

Table 11. $ \lambda_0{\geqslant}\lambda_1{\geqslant}\lambda_3{\geqslant} \lambda_5 {\geqslant} \lambda_2{\geqslant} \lambda_4 $, only one higher frequency in $ \lambda_1,...,\lambda_5 $

$ {\rm Case} $	$ \lambda_1\in $	$ \lambda_3\in $	$ \lambda_5\in $	$ \lambda_2\in $	$ \lambda_4\in $
$ llllh_- $	$ l $	$ l $	$ l $	$ l $	$ h_- $
$ llll_-h_- $	$ l $	$ l $	$ l $	$ l_- $	$ h_- $
$ hlll_-l_-h_- $	$ l $	$ l $	$ l_- $	$ l_- $	$ h_- $
$ ll_-l_-l_-h_- $	$ l $	$ l_- $	$ l_- $	$ l_- $	$ h_- $
$ l_-l_-l_-l_-h_- $	$ l_- $	$ l_- $	$ l_- $	$ l_- $	$ h_- $

| Show Table

DownLoad: CSV

Table 12. $ \lambda_0{\geqslant}\lambda_1{\geqslant}\lambda_3{\geqslant} \lambda_2 {\geqslant} \lambda_5 {\geqslant} \lambda_4 $, only one higher frequency in $ \lambda_1,...,\lambda_5 $

$ {\rm Case} $	$ \lambda_1\in $	$ \lambda_3\in $	$ \lambda_2\in $	$ \lambda_5\in $	$ \lambda_4\in $
$ hllll $	$ h $	$ l $	$ l $	$ l $	$ l $
$ hllll_- $	$ h $	$ l $	$ l $	$ l $	$ l_- $
$ hlll_-l_- $	$ h $	$ l $	$ l $	$ l_- $	$ l_- $
$ hll_-l_-l_- $	$ h $	$ l $	$ l_- $	$ l_- $	$ l_- $
$ hl_-l_-l_-l_- $	$ h $	$ l_- $	$ l_- $	$ l_- $	$ l_- $

| Show Table

DownLoad: CSV

Table 13. $ \lambda_0{\geqslant}\lambda_1{\geqslant}\lambda_3{\geqslant} \lambda_2 {\geqslant} \lambda_5 {\geqslant} \lambda_4 $, only one higher frequency in $ \lambda_1,...,\lambda_5 $

$ {\rm Case} $	$ \lambda_1\in $	$ \lambda_3\in $	$ \lambda_2\in $	$ \lambda_5\in $	$ \lambda_4\in $
$ llllh_- $	$ l $	$ l $	$ l $	$ l $	$ h_- $
$ llll_-h_- $	$ l $	$ l $	$ l $	$ l_- $	$ h_- $
$ hlll_-l_-h_- $	$ l $	$ l $	$ l_- $	$ l_- $	$ h_- $
$ ll_-l_-l_-h_- $	$ l $	$ l_- $	$ l_- $	$ l_- $	$ h_- $
$ l_-l_-l_-l_-h_- $	$ l_- $	$ l_- $	$ l_- $	$ l_- $	$ h_- $

| Show Table

DownLoad: CSV

Table 14. $ \lambda_1{\geqslant}\lambda_0{\geqslant}\lambda_2{\geqslant} \lambda_3 {\geqslant} \lambda_4 {\geqslant} \lambda_5 $, only one higher frequency in $ \lambda_1,...,\lambda_5 $

$ {\rm Case} $	$ \lambda_1\in $	$ \lambda_2\in $	$ \lambda_3\in $	$ \lambda_4\in $	$ \lambda_5\in $
$ 2hllll $	$ [\lambda_0, 20\lambda_0] $	$ l $	$ l $	$ l $	$ l $
$ 2hllll_- $	$ [\lambda_0, 20\lambda_0] $	$ l $	$ l $	$ l $	$ l_- $
$ 2hlll_-l_- $	$ [\lambda_0, 20\lambda_0] $	$ l $	$ l $	$ l_- $	$ l_- $
$ 2hll_-l_-l_- $	$ [\lambda_0, 20\lambda_0] $	$ l $	$ l_- $	$ l_- $	$ l_- $
$ 2hl_-l_-l_-l_- $	$ [\lambda_0, 20\lambda_0] $	$ l_- $	$ l_- $	$ l_- $	$ l_- $

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Table 15. $ \lambda_0{\geqslant}\lambda_1{\geqslant}\lambda_2{\geqslant} \lambda_3 {\geqslant} \lambda_5 {\geqslant} \lambda_4 $, only one higher frequency in $ \lambda_1,...,\lambda_5 $

$ {\rm Case} $	$ \lambda_1\in $	$ \lambda_2\in $	$ \lambda_3\in $	$ \lambda_5\in $	$ \lambda_4\in $
$ 3llllh_- $	$ l $	$ l $	$ l $	$ l $	$ [-20\lambda_0,-\lambda_0] $
$ 3llll_-h_- $	$ l $	$ l $	$ l $	$ l_- $	$ [-20\lambda_0,-\lambda_0] $
$ 3lll_-l_-h_- $	$ l $	$ l $	$ l_- $	$ l_- $	$ [-20\lambda_0,-\lambda_0] $
$ 3ll_-l_-l_-h_- $	$ l $	$ l_- $	$ l_- $	$ l_- $	$ [-20\lambda_0,-\lambda_0] $
$ 2l_-l_-l_-l_-h_- $	$ l_- $	$ l_- $	$ l_- $	$ l_- $	$ [-20\lambda_0,-\lambda_0] $

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