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

TL;DR

This work analyzes the dispersion_generalized KP_I equation in two dimensions, $\partial_t u - D_x^{\alpha}\partial_x u - \partial_x^{-1}\partial_y^2 u = u\partial_x u$, for $2<\alpha<4$, focusing on low_regular_ity well_posedness and the interplay of resonance with dispersion. It develops a cohesive framework combining resonance/transversality analysis, linear Strichartz estimates, and nonlinear Loomis_Whitney inequalities with short_time Fourier restriction spaces to manage derivative losses in the nonlinearity. The authors establish $C^2$ ill_posedness for small dispersion ($\alpha<7/3$), quasilinear local well_posedness in $H^{s,0}$ for $s>5-2\alpha$ (2<\alpha\le 5/2), and semilinear local well_posedness for $\alpha>5/2$ in $H^{s,0}$ with $s>5/4-\alpha/2$, alongside global results in $L^2$ for strong dispersion and persistence. The results bridge known KP_I theory with higher_order KP_I regimes, clarify when Picard iteration fails, and provide robust tools (transversality, LoomisWhitney, short_time spaces) for analyzing dispersive PDEs with resonant nonlinearities.

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.

Theorems & Definitions (50)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Corollary 1.4
• Lemma 2.1: cf. IKT
• Lemma 2.2: cf. IKT
• Theorem 2.3: Linear Strichartz estimate, cf. Hadac
• Lemma 2.4: Strichartz estimates for low frequencies
• proof
• Corollary 2.5