Para-differential Calculus on Compact Lie Groups and Spherical Capillary Water Waves

Abstract

This paper provides a para-differential calculus toolbox on compact Lie groups and homogeneous spaces. It helps to understand non-local, nonlinear partial differential operators with low regularity on manifolds with high symmetry. In particular, the paper provides a para-linearization formula for the Dirichlet-Neumann operator of a distorted 2-sphere, a key ingredient in understanding long-time behaviour of spherical capillary water waves. As an initial application, the paper provides a new proof of local well-posedness for spherical capillary water waves equation under weaker regularity assumptions compared to previous results.