On the Cauchy Problem of Spherical Capillary Water Waves

TL;DR

This paper develops a new mathematical framework for analyzing spherical capillary water waves, proving local well-posedness under weaker regularity assumptions using para-differential calculus on the 2-sphere.

Contribution

It introduces a symmetrization technique for the water waves system on the sphere, enabling a novel proof of well-posedness with reduced regularity requirements.

Findings

Symmetrization of the water waves system into a dispersive para-differential equation.

Proof of local well-posedness under weaker regularity assumptions.

Application of para-differential calculus on compact Lie groups and homogeneous spaces.

Abstract

The spherical capillary water waves equation describes the motion of an almost spherical water droplet under zero gravity governed by water-air interface tension. Using para-differential calculus on compact Lie groups and homogeneous spaces developed by the author, the system is symmetrized into a quasi-linear dispersive para-differential equation of order 1.5 defined on the 2-sphere. An immediate consequence of this symmetrization is a new proof of local well-posedness of the system under much weaker regularity assumption compared to previous results.