Null structures and degenerate dispersion relations in two space dimensions

TL;DR

This paper studies 2D nonlinear dispersive PDEs with special radial dispersion relations and null structures, demonstrating global scattering solutions for small data, inspired by water-wave models.

Contribution

It identifies weak null structures in cubic nonlinearities with degenerate dispersion relations, ensuring global solutions in a water-wave motivated setting.

Findings

Null structures eliminate worst nonlinear interactions.

Global scattering solutions are proven for small initial data.

Null structures naturally arise in water-wave problems.

Abstract

Motivated by water-wave problems, in this paper we consider a class of nonlinear dispersive PDEs in 2D with cubic nonlinearities, whose dispersion relations are radial and have vanishing Guassian curvature on a circle. For such a model we identify certain null structures for the cubic nonlinearity, which suffice in order to guarantee global scattering solutions for the small data problem. Our null structures in the power-type nonlinearity are weak, and only eliminate the worst nonlinear interaction. Such null structures arise naturally in some water-wave problems.