On analysis of the exponential map of volume-preserving diffeomorphism group on closed orientable surfaces through the vorticity

2022/04/20 by Siran Li, Li, Siran · 1 citation
Mathematics · #Geometric Analysis and Curvature Flows #Advanced Differential Equations and Dynamical Systems #Navier-Stokes equation solutions

paper · pdf · doi:10.48550/arxiv.2204.09497

Abstract

We study the exponential map of group of volume-preserving diffeomorphisms on closed orientable surfaces via the vorticity formulation of the incompressible Euler equation. We present an alternative, fluid dynamical proof of the theorem of Ebin--Misiołek--Preston: the exponential is a nonlinear Fredholm mapping of index zero. We extend Shnirelman's rigidity result for the exponential map from 2-dimensional flat torus to arbitrary orientable closed surfaces. That is, we prove that the exponential map is Fredholm quasiregular.

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• Singular Sets of Riemannian Exponential Maps in Hydrodynamics

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