讲座题目：On the formation time and regularity analysis of shock curves for 1D conservation laws
主讲人：许刚 教授
主持人：袁海荣 教授
开始时间：2025-11-21 10:00
讲座地址：闵行校区数学楼401报告厅
主办单位：数学科学学院
报告人简介：
许刚，南京师范大学数学科学学院教授，主要从事流体力学的数学理论、双曲型偏微分方程方向研究。许刚教授在Adv. Math, Arch. Ration. Mech. Anal., SIAM JMA等杂志上发表多篇高水平论文。
报告内容：
This talk focuses on shock formation in conservation law ($\partial_t u + \partial_x f(u) = 0$)—a key issue for their physical interpretation and numerical simulation. Existing results cover two extreme initial data cases: Discontinuous data (e.g., Riemann data) may cause instantaneous shocks at $t=0$; $C^1$-smooth data (with $C^2$ flux $f$) leads to finite-time shocks ($t_0 = \frac{-1}{\min_{x\in\mathbb{R}} g'(x)}$, where $g(x)=f'(u_0(x))$) when $\min_{x\in\mathbb{R}} g'(x) < 0$. This shows initial data regularity directly affects shock timing. However, a critical gap exists for "intermediate regularity data" (between discontinuity and $C^1$): No precise criteria distinguish instantaneous vs. finite-time shocks. Our work fills this gap by establishing: (1) Lipschitz-continuous initial data induces finite-time shocks; (2) Unbounded difference quotient data may cause instantaneous shocks at $t=0$. We also analyze shock curve regularity and solution asymptotics near pre-shock points.
