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

Abstract

We consider the Cauchy problem for the rotation-modified Kadomtsev-Petviashvili (RMKP) equation \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*} in the anisotropic Sobolev spaces $H^{s_{1},\>s_{2}}(\mathbb{R}^{2})$. When $\beta <0$ and $\gamma >0,$ we prove that the Cauchy problem is locally well-posed in $H^{s_{1},\>s_{2}}(\mathbb{R}^{2})$ with $s_{1}>-\frac{1}{2}$ and $s_{2}\geq 0$. 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 $H^{s_{1},\>0}(\mathbb{R}^{2})$ with $s_{1}<-\frac{1}{2}$ in the sense that the flow map associated to the rotation-modified Kadomtsev-Petviashvili is not $C^{3}$. When $\beta <0,\gamma >0,$ by using the $U^{p}$ and $V^{p}$ spaces, we prove that the Cauchy problem is locally well-posed in $H^{-\frac{1}{2},\>0}(\mathbb{R}^{2})$.