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

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.