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

Xin Yang; Bing-Yu Zhang

doi:10.3934/eect.2022002

Article Contents

2022, Volume 11, Issue 5: 1829-1871. Doi: 10.3934/eect.2022002

This issue Previous Article Exponential stabilization of the problem of transmission of wave equation with linear dynamical feedback control Next Article

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

Xin Yang^1, , and
Bing-Yu Zhang^2,

1.
Department of Mathematics, University of California, Riverside, Riverside, CA 92521, USA
2.
Department of Mathematical Sciences, University of Cincinnati, Cincinnati, OH 45221, USA

^*Corresponding author: Xin Yang
^*Corresponding author: Xin Yang

Received: August 2021

Revised: November 2021

Early access: January 2022

Published: October 2022

Abstract / Introduction Full Text(HTML) Figure(1) / Table(10) Related Papers Cited by

Abstract

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^{*} $.

Keywords:

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

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

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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	$s>-\frac{3}{4}$ $s\geq 0$

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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} $

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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 $

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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} $

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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 $

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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} $

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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 $

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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 $

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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 $

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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) $0$

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