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

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.

arXiv commentcorrections in the proof of Lemma 4.2, results unchanged, 41 pages

References in corpus (6)

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Cited by in corpus (1)

• Low regularity well-posedness of KP-I equations: the three-dimensional case