Title: Variational construction of singular characteristics and propagation of singularities Show Chinese title

Title: 变分构造奇异特征和奇异性的传播

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. Show Chinese abstract

Abstract: 在一个光滑的闭流形$M$上，我们为任意一对$(\phi,H)$引入了一种新的最大斜率曲线理论，其中$\phi$是一个半凹函数，$H$是一个哈密顿量。 通过使用梯度流理论中的最大斜率曲线概念，[Cannarsa, P.; Cheng, W.,\textit{广义特征和Lax-Oleinik算子：全局理论}. Calc. Var. Partial Differential Equations 56 (2017), no. 5, 56:12] 中构建的内在奇异特性，[Cannarsa, P.; Yu, Y.,\textit{半凸函数的奇异动力学}. J. Eur. Math. Soc. 11 (2009), no. 5, 999--1024] 中开发的光滑逼近方法，以及 [Khanin, K.; Sobolevski, A.,\textit{关于哈密顿-雅可比方程的拉格朗日轨迹的动力学}. Arch. Ration. Mech. Anal. 219 (2016), no. 2, 861--885] 中研究的断裂特征，我们证明了此类最大斜率曲线的存在性和稳定性，并讨论了一些新的弱KAM特性。 我们还证明了对于任何一对$(\phi,H)$，极大斜率曲线正好是处处具有右导数的断裂特征。 应用这个理论，我们建立了严格奇异特征和断裂特征的全局变分构造。 此外，我们证明了一个关于广义特征上截断点全局传播的结果，以及关于严格奇异特征上奇点传播的结果，针对弱KAM解。 我们还得到了严格奇异特征上的连续性方程，这阐明了奇点传播问题中的质量传输性质。