Nonlinear Hodge maps

TL;DR

This paper studies nonlinear Hodge maps as stationary points of a specialized energy functional, analyzing their regularity, singularities, and gradient flow, with implications for quasilinear field theories.

Contribution

It introduces conditions preventing singularities in nonlinear Hodge maps and establishes smoothness results for various flow regimes.

Findings

Singular sets of certain dimensions cannot occur under specified conditions.

Smoothness of solutions is proven for sonic limits and high-dimensional flows.

Gradient flow estimates are provided for solutions.

Abstract

We consider maps between Riemannian manifolds in which the map is a stationary point of the nonlinear Hodge energy. The variational equations of this functional form a quasilinear, nondiagonal, nonuniformly elliptic system which models certain kinds of compressible flow. Conditions are found under which singular sets of prescribed dimension cannot occur. Various degrees of smoothness are proven for the sonic limit, high-dimensional flow, and flow having nonzero vorticity. The gradient flow of solutions is estimated. Implications for other quasilinear field theories are suggested.