Speaker: 许刚教授，南京师范大学

Inviter: 流体力学方程研究中心

Title: Instantaneous versus Finite-Time Shock Formation and Development in Scalar Conservation Laws

Language: Chinese

Time & Venue: 2025.12.04 10:30-11:30 南楼N204

Abstract: We investigate the precise relationship between the initial regularity of data and the onset of shock formation in the one-dimensional scalar conservation law.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 the flux. 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.