J. Lin, D. E. Pelinovsky, and B. de Rijk
On the extinction of multiple shocks in scalar viscous conservation laws
SIAM J. Applied Dynamical Systems 23 (2024) 2323-2363
Abstract:
We are interested in the dynamics of interfaces, or zeros, of shock waves in
general scalar viscous conservation laws with a locally Lipschitz continuous flux function,
such as the modular Burgers' equation. We prove that all interfaces coalesce within finite
time, leaving behind either a single interface or no interface at all. Our proof relies on
mass and energy estimates, regularization of the flux function, and an application of the
Sturm theorems on the number of zeros of solutions of parabolic problems. Our analysis
yields an explicit upper bound on the time of extinction in terms of the initial condition
and the flux function. Moreover, in the case of a smooth flux function, we characterize the
generic bifurcations arising at a coalescence event with and without the presence of odd
symmetry. We identify associated scaling laws describing the local interface dynamics
near collision. Finally, we present an extension of these results to the case of anti-shock
waves converging to asymptotic limits of opposite signs. Our analysis is corroborated by
numerical simulations in the modular Burgers' equation and its regularizations.
Keywords:
modular Burgers equation; viscous shocks; finite-time extinction; traveling waves; energy estimates;