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

TL;DR

This paper establishes local well-posedness for the KP-I equation in an anisotropic Sobolev space, advancing understanding of its regularity properties and solution behavior in a specific functional setting.

Contribution

It proves local well-posedness of the KP-I initial-value problem in the anisotropic Sobolev space H^{1,0}, a novel result for this equation's regularity analysis.

Findings

Proves local well-posedness in H^{1,0} space.

Establishes regularity properties of solutions.

Advances understanding of KP-I equation in anisotropic Sobolev spaces.

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)=\phi(x,y), \end{cases} \end{eqnarray*} is locally well-posed in the space \begin{eqnarray*} H^{1,0}(\R^2)=\{\phi\in L^2(\R^2): \ \norm{\phi}_{H^{1,0}(\R^2)}\approx\norm{\phi}_{L^2}+\norm{\partial_x\phi}_{L^2}<\infty\}. \end{eqnarray*}