Improved well-posedness for quasilinear and sharp local well-posedness for semilinear KP-I equations

TL;DR

This paper establishes new well-posedness results for generalized KP-I equations with increased dispersion, identifying sharp thresholds for quasilinear and semilinear regimes using advanced Fourier analysis and geometric inequalities.

Contribution

It provides the first sharp well-posedness results for dispersion-generalized KP-I equations in anisotropic Sobolev spaces, including full subcritical range on  and strict subcriticality on  .

Findings

Sharp dispersion rate identified for generalized KP-I equations.

Improved well-posedness results using short-time Fourier restriction.

Nonlinear Loomis-Whitney inequalities adapted to different geometries.

Abstract

We show new well-posedness results in anisotropic Sobolev spaces for dispersion-generalized KP-I equations with increased dispersion compared to the KP-I equation. We obtain the sharp dispersion rate, below which generalized KP-I equations on $\mathbb{R}^2$ and on $\mathbb{R} \times \mathbb{T}$ exhibit quasilinear behavior. In the quasilinear regime, we show improved well-posedness results relying on short-time Fourier restriction. In the semilinear regime, we show sharp well-posedness with analytic data-to-solution mapping. On $\mathbb{R}^2$ we cover the full subcritical range, whereas on $\mathbb{R} \times \mathbb{T}$ the sharp well-posedness is strictly subcritical. Nonlinear Loomis-Whitney inequalities are one ingredient. These are presently proved for Borel measures with growth condition reflecting the different geometries of the plane $\mathbb{R}^2$, the cylinder $\mathbb{R} \times \mathbb{T}$, and the torus $\mathbb{T}^2$. Finally, we point out that on tori $\mathbb{T}^2_\gamma$, KP-I equations are never semilinear.