The Cauchy problem for the periodic Kadomtsev--Petviashvili--II equation below L²

2024/07/17 by Sebastian Herr, Robert Schippa, Herr, Sebastian +3
Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2407.12222

openalex publication_date 2024/07/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30

Abstract

We extend Bourgain's L²-wellposedness result for the KP-II equation on \mathbbT² to initial data with negative Sobolev regularity. The key ingredient is a new linear L⁴-Strichartz estimate which is effective on frequency-dependent time scales. The L⁴-Strichartz estimates follow from combining an ℓ²-decoupling inequality recently proved by Guth--Maldague--Oh with semiclassical Strichartz estimates. Moreover, we rely on a variant of Bourgain's bilinear Strichartz estimate on frequency-dependent times, which is proved via the Córdoba--Fefferman square function estimate.

Related

• Bilinear Strichartz estimates for the KdV equation on the torus
• Well-posedness for the Kadomtsev-Petviashvili II equation and generalisations
• Refinements of Strichartz estimates on tori and applications
• Strichartz estimates for quasi-periodic functions and applications
• Strichartz estimates for Schrödinger equations in weighted L² spaces and their applications