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Local well-posedness of the coupled KdV-KdV systems on $ \mathbb{R} $

  • *Corresponding author: Xin Yang

    *Corresponding author: Xin Yang 
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  • Inspired by the recent successful completion of the study of the well-posedness theory for the Cauchy problem of the Korteweg-de Vries (KdV) equation

    $ u_t +uu_x +u_{xxx} = 0, \quad \left. u \right |_{t = 0} = u_{0} $

    in the space $ H^{s} (\mathbb{R}) $ (or $ H^{s} (\mathbb{T}) $), we study the well-posedness of the Cauchy problem for a class of coupled KdV-KdV (cKdV) systems

    $ \left\{\begin{array}{rcl} u_t+a_{1}u_{xxx} & = & c_{11}uu_x+c_{12}vv_x+d_{11}u_{x}v+d_{12}uv_{x}, \\ v_t+a_{2}v_{xxx}& = & c_{21}uu_x+c_{22}vv_x +d_{21}u_{x}v+d_{22}uv_{x}, \\ \left. (u, v)\right |_{t = 0} & = & (u_{0}, v_{0}) \end{array}\right. $

    in the space $ \mathcal{H}^s (\mathbb{R}) : = H^s (\mathbb{R})\times H^s (\mathbb{R}) $. Typical examples include the Gear-Grimshaw system, the Hirota-Satsuma system and the Majda-Biello system, to name a few.

    In this paper we look for those values of $ s\in \mathbb{R} $ for which the cKdV systems are well-posed in $ \mathcal{H}^s ( \mathbb {R}) $. The key ingredients in the proofs are the bilinear estimates in both divergence and non-divergence forms under the Fourier restriction space norms. Sharp results are established for all four types of the bilinear estimates that are associated to the cKdV systems. In contrast to the lone critical index $ -\frac{3}{4} $ for the single KdV equation, the critical indexes for the cKdV systems are $ -\frac{13}{12} $, $ -\frac{3}{4} $, $ 0 $ and $ \frac{3}{4} $.

    As a result, the cKdV systems are classified into four classes, each of which corresponds to a unique index $ s^{*}\in\{-\frac{13}{12}, \, -\frac{3}{4}, \, 0, \, \frac{3}{4}\} $ such that any system in this class is locally analytically well-posed if $ s>s^{*} $ while the bilinear estimate fails if $ s<s^{*} $.

    Mathematics Subject Classification: Primary: 35Q53; 35E15; 35G55; 35L56; 35D30.

    Citation:

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  • Figure 1.  Range of $ s $ and $ b $ when $ s<-\frac{3}{4} $

    Table 1.  Main Results

    Case $ r=\frac{a_{2}}{a_{1}} $ Coefficients $ b_{ij} $, $ c_{ij} $ and $ d_{ij} $ $ s $
    (1) $r <0$ $(c_{ij})=0$, $d_{11}=d_{12}$ and $d_{21}=d_{22}$
    Otherwise
    $s\geq -\frac{13}{12}$
    $s>-\frac{3}{4}$
    (2) $0 <r <\frac{1}{4}$ $c_{12}=d_{21}=d_{22}=0$
    Otherwise
    $s>-\frac{3}{4}$
    $s\geq 0$
    (3) $r=\frac{1}{4}$ $c_{21}=d_{11}=d_{12}=0$
    Otherwise
    $s\geq 0$
    $s\geq\frac{3}{4}$
    (4) $\frac{1}{4} <r <1$ arbitrary $s\geq 0$
    (5) $r=1$ $b_{12}=b_{21}=0$, $d_{11}=d_{12}$ and $d_{21}=d_{22}$
    $b_{12}=b_{21}=0$, $d_{11}\neq d_{12}$ or $d_{21}\neq d_{22}$
    $s>-\frac{3}{4}$
    $s>0$
    (6) $1 <r<4$ arbitrary $s\geq 0$
    (7) $r=4$ $c_{12}=d_{21}=d_{22}=0$
    Otherwise
    $s\geq 0$
    $s\geq\frac{3}{4}$
    (8) $r>4$ $c_{21}=d_{11}=d_{12}=0$
    Otherwise
    $s>-\frac{3}{4}$
    $s\geq 0$
     | Show Table
    DownLoad: CSV

    Table 2.  LWP Results

    Case Coefficient $ a_2 $ $ s $
    (1) $ a_2\in(-\infty, 0)\cup \{1\} \cup \{4, \infty\} $ $ s>-\frac34 $
    (2) $ a_2\in(0, 1)\cup(1, 4) $ $ s\geq 0 $
    (3) $ a_2=4 $ $ s\geq\frac{3}{4} $
     | Show Table
    DownLoad: CSV

    Table 3.  GWP Results

    Case Coefficient $ a_2 $ $ s $
    (1) $ a_2=1 $ $ s>-\frac34 $
    (2) $ a_2\not\in\{1, 4\} $ $ s\geq 0 $
    (3) $ a_2=4 $ $ s\geq1 $
     | Show Table
    DownLoad: CSV

    Table 4.  LWP Results

    Case Coefficients $ a_1 $ and $ c_{12} $ $ s $
    (1) $ a_1\in(-\infty, 0)\cup(0, \frac14) $ $ s>-\frac34 $
    (2) $ a_1\in(\frac14, 1)\cup(1, \infty) $ $ s\geq 0 $
    (3) $ a_1=1 $ $ s>0 $
    (4) $ a_1=\frac14 $ $ s\geq\frac{3}{4} $
     | Show Table
    DownLoad: CSV

    Table 5.  GWP Results

    Case Coefficients $ a_1 $ and $ c_{12} $ $ s $
    (1) $ a_1\not\in\{\frac14, 1\} $, $ c_{12}>0 $ $ s\geq 0 $
    (2) $ a_1=\frac14 $, $ c_{12}>0 $ $ s\geq 1 $
     | Show Table
    DownLoad: CSV

    Table 6.  LWP Results

    Case $ \rho_1 $, $ \rho_2 $ and $ \sigma_{i} (1\leq i\leq 4) $ $ s $
    (1) $ \sigma_3=0 $, $ \rho_1=1 $ $ s>-\frac34 $
    (2) $ \rho_2\sigma_3^2>1 $ $ s>-\frac34 $
    (3) $ \rho_2\sigma_3^2<1 $, (1.13) fails $ s\geq 0 $
    (4) $ \rho_2\sigma_3^2<1 $, (1.13) holds $ s\geq\frac{3}{4} $
     | Show Table
    DownLoad: CSV

    Table 7.  GWP Results

    Case $ \rho_1 $, $ \rho_2 $ and $ \sigma_{i} (1\leq i\leq 4) $ $ s $
    (1) $ \rho_2\sigma_3^2\neq 1 $, (1.13) fails $ s\geq 0 $
    (2) $ \rho_2\sigma_3^2\neq 1 $, (1.13) holds $ s\geq 1 $
     | Show Table
    DownLoad: CSV

    Table 8.  Bilinear Estimates

    Type $ r<0 $ $ 0<r<\frac{1}{4} $ $ r=\frac{1}{4} $ $ r>\frac{1}{4} $, $ r\neq 1 $ $ r=1 $
    (D1): (3.4) $ s>-\frac{3}{4} $ $ s>-\frac{3}{4} $ $ s\geq \frac{3}{4} $ $ s\geq 0 $ $ s>-\frac{3}{4} $
    (D2): (3.5) $ s>-\frac{3}{4} $ $ s\geq \frac{3}{4} $ $ s\geq 0 $ $ s>-\frac{3}{4} $
    (ND1): (3.6) $ s>-\frac{3}{4} $ $ s>-\frac{3}{4} $ $ s\geq \frac{3}{4} $ $ s\geq 0 $ $ s>0 $
    (ND2): (3.7) $ s>-\frac{3}{4} $ $ s>-\frac{3}{4} $ $ s\geq \frac{3}{4} $ $ s\geq 0 $ $ s>0 $
     | Show Table
    DownLoad: CSV

    Table 9.  Sharpness of Bilinear Estimates

    Type $ r<0 $ $ 0<r<\frac{1}{4} $ $ r=\frac{1}{4} $ $ r>\frac{1}{4} $, $ r\neq 1 $ $ r=1 $
    (D1): (3.4) $ s<-\frac{3}{4} $ $ s<-\frac{3}{4} $ $ s< \frac{3}{4} $ $ s< 0 $ $ s<-\frac{3}{4} $
    (D2): (3.5) $ s<-\frac{3}{4} $ $ s< \frac{3}{4} $ $ s< 0 $ $ s<-\frac{3}{4} $
    (ND1): (3.6) $ s<-\frac{3}{4} $ $ s<-\frac{3}{4} $ $ s< \frac{3}{4} $ $ s< 0 $ $ s<0 $
    (ND2): (3.7) $ s<-\frac{3}{4} $ $ s<-\frac{3}{4} $ $ s<\frac{3}{4} $ $ s< 0 $ $ s<0 $
     | Show Table
    DownLoad: CSV

    Table 10.  Troubles and Critical Indexes ($ r = \frac{{\alpha}_2}{{\alpha}_1} $)

    $ r<0 $ $ 0<r<\frac{1}{4} $ $ r=\frac{1}{4} $ $ r>\frac{1}{4} $, $ r\neq 1 $ $ r=1 $
    (D1): (3.4) (T2)
    $-\frac{3}{4}$
    (T2)
    $-\frac{3}{4}$
    (T1)+(T2)
    $\frac{3}{4}$
    (T1)
    $ 0$
    (T2)
    $-\frac{3}{4}$
    (D2): (3.5) None
    $ -\frac{13}{12}$
    (T2)
    $-\frac{3}{4}$
    (T1)+(T2)
    $ \frac{3}{4}$
    (T1)
    $ 0$
    (T2)
    $-\frac{3}{4}$
    (ND1): (3.6) (T3)
    $-\frac{3}{4}$
    (T2) or (T3)
    $-\frac{3}{4}$
    (T1)+(T2)
    $ \frac{3}{4}$
    (T1)
    $ 0$
    (T2)+(T3)
    $0$
    (ND2): (3.7) (T3)
    $-\frac{3}{4}$
    (T2) or (T3)
    $-\frac{3}{4}$
    (T1)+(T2)
    $ \frac{3}{4}$
    (T1)
    $ 0$
    (T2)+(T3)
    $0$
     | Show Table
    DownLoad: CSV
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