Sharp well-posedness of the Cauchy problem for the rotation-modified Kadomtsev-Petviashvili equation in anisotropic Sobolev spaces

Abstract

<p style='text-indent:20px;'>We consider the Cauchy problem for the rotation-modified Kadomtsev-Petviashvili (RMKP) equation</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{align*} \partial_{x}\left(u_{t}-\beta\partial_{x}^{3}u +\partial_{x}(u^{2})\right)+\partial_{y}^{2}u-\gamma u = 0 \end{align*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>in the anisotropic Sobolev spaces <inline-formula><tex-math id="M1">\begin{document}$ H^{s_{1},s_{2}}(\mathbb{R}^{2}) $\end{document}</tex-math></inline-formula>. When <inline-formula><tex-math id="M2">\begin{document}$ \beta &lt;0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ \gamma &gt;0, $\end{document}</tex-math></inline-formula> we prove that the Cauchy problem is locally well-posed in <inline-formula><tex-math id="M4">\begin{document}$ H^{s_{1}, s_{2}}(\mathbb{R}^{2}) $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M5">\begin{document}$ s_{1}&gt;-\frac{1}{2} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M6">\begin{document}$ s_{2}\geq 0 $\end{document}</tex-math></inline-formula>. Our result considerably improves the Theorem 1.4 of R. M. Chen, Y. Liu, P. Z. Zhang(Transactions of the American Mathematical Society, 364(2012), 3395–3425.). The key idea is that we divide the frequency space into regular region and singular region. We further prove that the Cauchy problem for RMKP equation is ill-posed in <inline-formula><tex-math id="M7">\begin{document}$ H^{s_{1},0}(\mathbb{R}^{2}) $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M8">\begin{document}$ s_{1}&lt;-\frac{1}{2} $\end{document}</tex-math></inline-formula> in the sense that the flow map associated to the rotation-modified Kadomtsev-Petviashvili is not <inline-formula><tex-math id="M9">\begin{document}$ C^{3} $\end{document}</tex-math></inline-formula>. When <inline-formula><tex-math id="M10">\begin{document}$ \beta &lt;0,\gamma &gt;0, $\end{document}</tex-math></inline-formula> by using the <inline-formula><tex-math id="M11">\begin{document}$ U^{p} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M12">\begin{document}$ V^{p} $\end{document}</tex-math></inline-formula> spaces, we prove that the Cauchy problem is locally well-posed in <inline-formula><tex-math id="M13">\begin{document}$ H^{-\frac{1}{2},0}(\mathbb{R}^{2}) $\end{document}</tex-math></inline-formula>.</p>