Low regularity well-posedness for KP-I equations: the dispersion-generalized case

TL;DR

This paper establishes new well-posedness results for a family of dispersion-generalized KP-I equations in two dimensions, demonstrating global solutions for strong dispersion and limitations for small dispersion.

Contribution

It introduces novel well-posedness results for the dispersion-generalized KP-I equations, combining advanced harmonic analysis techniques and showing the limits of Picard iteration methods.

Findings

Global well-posedness in L^2 for strong dispersion

Failure of Picard iteration for small dispersion

Use of frequency-dependent time localization for small dispersion

Abstract

We prove new well-posedness results for dispersion-generalized Kadomtsev--Petviashvili I equations in $\mathbb{R}^2$, which family links the classical KP-I equation with the fifth order KP-I equation. For strong enough dispersion, we show global well-posedness in $L^2(\mathbb{R}^2)$. To this end, we combine resonance and transversality considerations with Strichartz estimates and a nonlinear Loomis--Whitney inequality. Moreover, we prove that for small dispersion, the equations cannot be solved via Picard iteration. In this case, we use an additional frequency dependent time localization.