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.
Toolbox of Para-differential Calculus on Compact Lie Groups
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abstract
This paper provides a toolbox of para-differential calculus on compact Lie groups. The toolbox is based on representation theory of compact Lie groups and contains exact formulas of symbolic calculus. Para-differential operators are constructed in a global, coordinate-free manner, giving lower order terms in symbolic calculus a clear form. The toolbox helps to understand non-local, nonlinear differential operators defined on certain manifolds with high symmetry.
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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.