Local well-posedness of spatially quasiperiodic gravity water waves in two dimensions

TL;DR

This paper proves local well-posedness for 2D gravity water waves with spatially quasiperiodic initial conditions using holomorphic coordinates and specialized harmonic analysis tools.

Contribution

It is the first to establish local well-posedness for quasiperiodic water waves, introducing a conformal mapping approach and adapted Littlewood-Paley analysis.

Findings

Established local well-posedness for quasiperiodic initial data

Developed a framework for Dirichlet-Neumann operator in quasiperiodic setting

Derived energy estimates for linearized equations with quasiperiodic data

Abstract

We provide the first proof of local well-posedness for the two-dimensional gravity water wave equations with spatially quasi-periodic initial conditions. We represent the solution using holomorphic coordinates, which are equivalent to a conformal mapping formulation of the equations of motion. This allows us to compute the Dirichlet-Neumann operator via the Hilbert transform, which has a simple form in the spatially quasiperiodic setting. We use a Littlewood-Paley decomposition adapted to the quasiperiodic setting and establish multiplicative and commutator estimates in this framework. The key step of the proof is the derivation of quasilinear energy estimates for the linearized water wave equations with quasiperiodic initial data.