Instantaneous versus Finite-Time Shock Formation and Development in Scalar Conservation Laws
数学专题报告
报告题目(Title)：Instantaneous versus Finite-Time Shock Formation and Development in Scalar Conservation Laws
报告人(Speaker)：许刚 教授（南京师范大学）
地点(Place)：教七406
时间(Time)：2025年12月4日（周四）14:00-15:00
邀请人(Inviter)：许孝精
报告摘要
We investigate the precise relationship between the initial regularity of data and the onset of shock formation in the one-dimensional scalar conservation law
$$ \partial_t u + \partial_x f(u) = 0, \qquad u(x,0) = u_0(x) \in L^\infty(\mathbb{R}). $$
It is well known that discontinuous initial data, such as Riemann-type profiles, produce shocks instantaneously at t = 0, while sufficiently smooth (C^1) data lead to finite-time gradient blow-up determined by the minimal characteristic derivative of f'(u_0).
This talk focus on sharp regularity criteria that delineate these two regimes. Specifically, we prove that for Lipschitz continuous initial data, shocks form only after a strictly positive time, whereas for continuous but non-Lipschitz data with unbounded difference quotient, shock formation occurs instantaneously.
These results provide a complete classification of shock initiation scenarios according to the local regularity of the initial data and identify the Lipschitz threshold as the exact borderline between immediate and delayed shock formation.
In addition, we characterize the regularity of emerging shock curves and describe the asymptotic behavior of the solution near pre-shock points.
Our analysis unifies and extends classical smooth-to-shock transition theories, offering a refined understanding of how small-scale singularities in the initial profile govern the temporal onset of shocks.
