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

2024/08/29 by Kinoshita, Shinya, Sanwal, Akansha, Schippa, Robert
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2408.16348

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 ℝ² and on ℝ × \mathbbT 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 ℝ² we cover the full subcritical range, whereas on ℝ × \mathbbT 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 ℝ², the cylinder ℝ × \mathbbT, and the torus \mathbbT². Finally, we point out that on tori \mathbbT²_γ, KP-I equations are never semilinear.

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