📝 SummarXiv

Abstract

We prove that singularities propagate globally for viscosity solutions of Hamilton-Jacobi equations related to magnetic mechanical systems on closed Riemannian manifolds. Our main result shows that for any weak KAM solution $u$, the singular set $\text{Sing}\,(u)$ remains invariant under the generalized gradient flow dynamics. The proof combines three key elements: (1) reduction from magnetic to Riemannian systems, (2) analysis of reparameterized flows, and (3) regularization techniques. Compared to previous analytic approaches, our geometric method provides clearer insights into the underlying Riemannian structure. We also establish necessary conditions for singularity existence, particularly when the Euler characteristic is nonzero and the magnetic form is non-exact. This approach does not extend directly to Finsler metrics due to structural differences.