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Longtime Dynamics of Irrotational Spherical Water Drops: Initial Notes
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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.
Forward citations
Cited by 2 Pith papers
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Two-dimensional capillary liquid drop: Craig-Sulem formulation on $\mathbb{T}^1$ and bifurcations from multiple eigenvalues of rotating waves
For small angular momentum, there exists a unique rotation orbit of smooth rotating capillary drop solutions near the circle, built by a variational bifurcation argument from a multiple eigenvalue.
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Exponential Vorticity Hessian Growth in Capillary Liquid Drop in Two Dimensions
For the 2D free-boundary Euler equations with surface tension on a droplet, there exist arbitrarily small initial velocities such that the vorticity Hessian grows at least exponentially in time.
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