Variational construction of singular characteristics and propagation of singularities

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., $\mathit{\text{Generalized}}\mathit{\text{characteristics}}\mathit{\text{and}}\mathit{\text{Lax-Oleinik}}\mathit{\text{operators:}}\mathit{\text{global}}\mathit{\text{theory}}$. Calc. Var. Partial Differential Equations 56 (2017), no. 5, 56:12], the smooth approximation method developed in [Cannarsa, P.; Yu, Y. $\mathit{\text{Singular}}\mathit{\text{dynamics}}\mathit{\text{for}}\mathit{\text{semiconcave}}\mathit{\text{functions}}$. J. Eur. Math. Soc. 11 (2009), no. 5, 999–1024], and the broken characteristics studied in [Khanin, K.; Sobolevski, A., $\mathit{\text{On}}\mathit{\text{dynamics}}\mathit{\text{of}}\mathit{\text{Lagrangian}}\mathit{\text{trajectories}}\mathit{\text{for}}\mathit{\text{Hamilton-Jacobi}}\mathit{\text{equations}}$. 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.