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

  • * Corresponding author: Baoxiang Wang

    * Corresponding author: Baoxiang Wang

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

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  • 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} $.

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

    Citation:

    \begin{equation} \\ \end{equation}
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  • 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) $
     | Show Table
    DownLoad: CSV

    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) $
     | Show Table
    DownLoad: CSV

    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) $
     | Show Table
    DownLoad: CSV

    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) $
     | Show Table
    DownLoad: CSV

    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) $
     | Show Table
    DownLoad: CSV

    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_- $
     | Show Table
    DownLoad: CSV

    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_- $
     | Show Table
    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_- $
     | Show Table
    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_- $
     | Show Table
    DownLoad: CSV

    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] $
     | Show Table
    DownLoad: CSV
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