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

Abstract

<p style='text-indent:20px;'>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</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ u_t +uu_x +u_{xxx} = 0, \quad \left. u \right |_{t = 0} = u_{0} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'> in the space <inline-formula><tex-math id="M1">\begin{document}$ H^{s} (\mathbb{R}) $\end{document}</tex-math></inline-formula> (or <inline-formula><tex-math id="M2">\begin{document}$ H^{s} (\mathbb{T}) $\end{document}</tex-math></inline-formula>), we study the well-posedness of the Cauchy problem for a class of coupled KdV-KdV (cKdV) systems</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ \left\{\begin{array}{rcl} u_t+a_{1}u_{xxx} &amp; = &amp; c_{11}uu_x+c_{12}vv_x+d_{11}u_{x}v+d_{12}uv_{x}, \\ v_t+a_{2}v_{xxx}&amp; = &amp; c_{21}uu_x+c_{22}vv_x +d_{21}u_{x}v+d_{22}uv_{x}, \\ \left. (u, v)\right |_{t = 0} &amp; = &amp; (u_{0}, v_{0}) \end{array}\right. $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>in the space <inline-formula><tex-math id="M3">\begin{document}$ \mathcal{H}^s (\mathbb{R}) : = H^s (\mathbb{R})\times H^s (\mathbb{R}) $\end{document}</tex-math></inline-formula>. Typical examples include the Gear-Grimshaw system, the Hirota-Satsuma system and the Majda-Biello system, to name a few.</p><p style='text-indent:20px;'> In this paper we look for those values of <inline-formula><tex-math id="M4">\begin{document}$ s\in \mathbb{R} $\end{document}</tex-math></inline-formula> for which the cKdV systems are well-posed in <inline-formula><tex-math id="M5">\begin{document}$ \mathcal{H}^s ( \mathbb {R}) $\end{document}</tex-math></inline-formula>. The key ingredients in the proofs are the bilinear estimates in both divergence and non-divergence forms under the Fourier restriction space norms. Sharp results are established for all four types of the bilinear estimates that are associated to the cKdV systems. In contrast to the lone critical index <inline-formula><tex-math id="M6">\begin{document}$ -\frac{3}{4} $\end{document}</tex-math></inline-formula> for the single KdV equation, the critical indexes for the cKdV systems are <inline-formula><tex-math id="M7">\begin{document}$ -\frac{13}{12} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M8">\begin{document}$ -\frac{3}{4} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M9">\begin{document}$ 0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M10">\begin{document}$ \frac{3}{4} $\end{document}</tex-math></inline-formula>.</p><p style='text-indent:20px;'> As a result, the cKdV systems are classified into four classes, each of which corresponds to a unique index <inline-formula><tex-math id="M11">\begin{document}$ s^{*}\in\{-\frac{13}{12}, \, -\frac{3}{4}, \, 0, \, \frac{3}{4}\} $\end{document}</tex-math></inline-formula> such that any system in this class is locally analytically well-posed if <inline-formula><tex-math id="M12">\begin{document}$ s&gt;s^{*} $\end{document}</tex-math></inline-formula> while the bilinear estimate fails if <inline-formula><tex-math id="M13">\begin{document}$ s&lt;s^{*} $\end{document}</tex-math></inline-formula>.</p>