On the local regularity of the KP-I equation in anisotropic Sobolev space

Abstract

We prove that the KP-I initial-value problem \begin{eqnarray*} \begin{cases} \partial_tu+\partial_x^3u-\partial_x^{-1}\partial_y^2u+\partial_x(u^2/2)=0 {on}{\R}^2_{x,y}\times {\R}_t; u(x,y,0)=ϕ(x,y), \end{cases} \end{eqnarray*} is locally well-posed in the space \begin{eqnarray*} H^{1,0}(\R^2)=\{ϕ\in L^2(\R^2): \ \normϕ_{H^{1,0}(\R^2)}\approx\normϕ_{L^2}+\norm{\partial_xϕ}_{L^2}<\infty\}. \end{eqnarray*}