Longtime Dynamics of Irrotational Spherical Water Drops: Initial Notes

TL;DR

This paper introduces mathematical problems related to the oscillation of irrotational spherical water drops in zero gravity, deriving a PDE model on the sphere and discussing open problems in analysis and dynamics.

Contribution

It formulates new mathematical problems for the water drop oscillation model, including PDE inequalities, soliton existence, and lifespan estimates, linking them to Diophantine equations.

Findings

Derivation of a quasilinear dispersive PDE on the sphere

Identification of open problems in analysis of water drop oscillations

Connection of problems to Diophantine equations

Abstract

In this note, we propose several unsolved problems concerning the irrotational oscillation of a water droplet under zero gravity. We will derive the governing equation of this physical model, and convert it to a quasilinear dispersive partial differential equation defined on the sphere, which formally resembles the capillary water waves equation but describes oscillation defined on curved manifold instead. Three types of unsolved mathematical problems related to this model will be discussed in observation of hydrodynamical experiments under zero gravity: (1) Strichartz type inequalities for the linearized problem (2) existence of periodic solutons (3) normal form reduction and generic lifespan estimate. It is pointed out that all of these problems are closely related to certain Diophantine equations, especially the third one.