Variational construction of singular characteristics and propagation of singularities

TL;DR

This paper develops a new variational framework for analyzing singular characteristics and their propagation in Hamilton-Jacobi equations on manifolds, unifying and extending previous theories with stability and propagation results.

Contribution

It introduces a variational construction of maximal slope curves and broken characteristics, establishing their existence, stability, and propagation properties for weak KAM solutions.

Findings

Maximal slope curves are exactly broken characteristics with right derivatives everywhere.

Global propagation of cut points and singular points along generalized and strict singular characteristics.

Continuity equation along strict singular characteristics clarifies mass transport in singularity propagation.

Abstract

On a smooth closed manifold $M$, we introduce a novel theory of maximal slope curves for any pair $(\phi,H)$ with $\phi$ a semiconcave function and $H$ a Hamiltonian. By using the notion of maximal slope curve from gradient flow theory, the intrinsic singular characteristics constructed in [Cannarsa, P.; Cheng, W., \textit{Generalized characteristics and Lax-Oleinik operators: global theory}. Calc. Var. Partial Differential Equations 56 (2017), no. 5, 56:12], the smooth approximation method developed in [Cannarsa, P.; Yu, Y. \textit{Singular dynamics for semiconcave functions}. J. Eur. Math. Soc. 11 (2009), no. 5, 999--1024], and the broken characteristics studied in [Khanin, K.; Sobolevski, A., \textit{On dynamics of Lagrangian trajectories for Hamilton-Jacobi equations}. Arch. Ration. Mech. Anal. 219 (2016), no. 2, 861--885], we prove the existence and stability of such maximal slope curves and discuss certain new weak KAM features. We also prove that maximal slope curves for any pair $(\phi,H)$ are exactly broken characteristics which have right derivatives everywhere. Applying this theory, we establish a global variational construction of strict singular characteristics and broken characteristics. Moreover, we prove a result on the global propagation of cut points along generalized characteristics, as well as a result on the propagation of singular points along strict singular characteristics, for weak KAM solutions. We also obtain the continuity equation along strict singular characteristics which clarifies the mass transport nature in the problem of propagation of singularities.