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

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 ${\mathbb{T}}_{\hat{I}}^3$, KP-I equations are never semilinear.

72 pages