Hagen Papenburg | Searcharxiv

arXiv2025

The Benjamin-Ono equation with quasi-periodic data

We construct local solutions to the Benjamin-Ono equation for quasi-periodic initial data. The solution is unique among limits of smooth solutions and depends continuously on the data. Our result applies to a richer class of quasi-periodic functions than previous theorems. Central to the argument is an a-priori estimate, the proof of which utilizes Strichartz estimates for quasi-periodic functions obtained recently via decoupling, and a quasi-periodic extension of Tao's gauge transform. As a byproduct of our method, we also establish new local wellposedness results in certain anisotropic Sobolev spaces.

math.AP↗

arXiv2024

Local Wellposedness of dispersive equations with quasi-periodic initial data

We prove unconditional local well-posedness in a space of quasi-periodic functions for dispersive equations of the form $$\partial_tu + Lu + \partial_x(u^{p+1})=0,$$ where $L$ is a multiplier operator with purely imaginary symbol which grows at most exponentially. The class of equations to which our method applies includes the generalized Korteweg-de Vries equation, the generalized Benjamin-Ono equation, and the derivative nonlinear Schrödinger equation. We also discuss well-posedness of some dispersive models which do not have a problematic derivative in the nonlinearity, namely, the nonlinear Schrödinger equation and the generalized Benjamin-Bona-Mahony equation, with quasi-periodic initial data. In this way, we recover and improve upon results from arXiv:1212.2674v3 [math.AP], arXiv:2110.11263v1 [math.AP] and arXiv:2201.02920v1 [math.AP] by shorter arguments.

math.AP↗