ABSTRACT

In this talk, I will discuss the onset and early development of shocks in scalar conservation laws, with main emphasis on the one-dimensional case. For one-dimensional scalar conservation laws, I will present a sharp regularity criterion describing when shocks form instantaneously and when they appear only after a positive time. More precisely, Lipschitz continuous initial data lead to shock formation in finite time, while continuous but non-Lipschitz initial data with unbounded difference quotients may generate shocks immediately. This shows that Lipschitz regularity is the exact threshold between instantaneous and delayed shock formation.  I will also briefly describe the local structure of the solution near pre-shock points and the regularity of the emerging shock curves. At the end, I will mention some related results in two dimensions, where weak singularities in the initial gradient along a curve can also lead to shock formation and introduce new geometric features. Overall, the talk aims to give a unified picture of how the regularity of the initial data governs shock formation and development in scalar conservation laws.