Hamilton-Jacobi方程（组）的粘性解从不同角度体现了Hamilton动力学复杂性决定的不可逆现象。本项目将就粘性解的奇性传播、Lax-Oleinik半群的紧性估计以及Hamilton-Jacobi方程组的内蕴耦合系统，深入研究其与动力学复杂性之间的关系。尤其是建立Hamilton动力学中的大范围轨道连接机制与这些不可逆性的体现数学上深刻的联系。以期推动相关领域的突破性进展。

The irreversibility for the viscosity solutions of Hamilton-Jacobi equations embodies the dynamical complexity in the study of Hamiltonian dynamical systems. The main purpose of this project will be contributed to the study between dynamical complexity and the irreversibility associated to the propagation of singularities, compactness estimate of Lax-Oleinik semigroups and the intrinsic coupling of Hamilton-Jacobi systems, especially to establish the deep relations between the mechanism for the global connecting orbits in Hamiltonian systems and the the irreversibility phenomenon. We expect the breakthrough in the relevant fields.