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}-β\partial_{x}^{3}u +\partial_{x}(u^{2})\right)+\partial_{y}^{2}u-γu=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 $s_2\ge 0$. 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 $H^{-\frac{1}{2},0}({\mathbb{R}}^2)$.

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References in corpus (1)

• Global well-posedness for the KP-I equation on the background of a non localized solution