Type: Article
Publication Date: 2020-11-23
Citations: 2
DOI: https://doi.org/10.3934/dcds.2020382
<p style='text-indent:20px;'>We consider the initial value problem associated to a coupled system of modified Korteweg-de Vries type equations <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \begin{cases} \partial_tv + \partial_x^3v + \partial_x(vw^2) = 0,&v(x,0) = \phi(x),\\ \partial_tw + \alpha\partial_x^3w + \partial_x(v^2w) = 0,& w(x,0) = \psi(x), \end{cases} \end{equation*} $\end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>and prove the local well-posedness results for a given data in low regularity Sobolev spaces <inline-formula><tex-math id="M1">\begin{document}$ H^{s}( \rm{I}\! \rm{R})\times H^{k}( \rm{I}\! \rm{R}) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ s,k> -\frac12 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ |s-k|\leq 1/2 $\end{document}</tex-math></inline-formula>, for <inline-formula><tex-math id="M4">\begin{document}$ \alpha\neq 0,1 $\end{document}</tex-math></inline-formula>. Also, we prove that: (I) the solution mapping that takes initial data to the solution fails to be <inline-formula><tex-math id="M5">\begin{document}$ C^3 $\end{document}</tex-math></inline-formula> at the origin, when <inline-formula><tex-math id="M6">\begin{document}$ s<-1/2 $\end{document}</tex-math></inline-formula> or <inline-formula><tex-math id="M7">\begin{document}$ k<-1/2 $\end{document}</tex-math></inline-formula> or <inline-formula><tex-math id="M8">\begin{document}$ |s-k|>2 $\end{document}</tex-math></inline-formula>; (II) the trilinear estimates used in the proof of the local well-posedness theorem fail to hold when (a) <inline-formula><tex-math id="M9">\begin{document}$ s-2k>1 $\end{document}</tex-math></inline-formula> or <inline-formula><tex-math id="M10">\begin{document}$ k<-1/2 $\end{document}</tex-math></inline-formula> (b) <inline-formula><tex-math id="M11">\begin{document}$ k-2s>1 $\end{document}</tex-math></inline-formula> or <inline-formula><tex-math id="M12">\begin{document}$ s<-1/2 $\end{document}</tex-math></inline-formula>; (c) <inline-formula><tex-math id="M13">\begin{document}$ s = k = -1/2 $\end{document}</tex-math></inline-formula>;