U. Le, D.E. Pelinovsky, and P. Poullet
Asymptotic stability of viscous shocks in the modular Burgers equation
Abstract:
Dynamics of viscous shocks is considered in the modular Burgers equation, where
the time evolution becomes complicated due to singularities produced by the modular
nonlinearity. We prove that the viscous shocks are asymptotically stable under odd and general
perturbations. For the odd perturbations, the proof relies on the reduction of the modular
Burgers equation to a linear diffusion equation on a half-line. For the general perturbations,
the proof is developed by converting the time-evolution problem to a system of linear equations
coupled with a nonlinear equation for the interface position. Exponential weights in space are
imposed on the initial data of general perturbations in order to gain the asymptotic decay of
perturbations in time. We give numerical illustrations of asymptotic stability of the viscous
shocks under general perturbations.
Keywords:
modular Burgers equation, viscous fronts, asymptotic stability, linear diffusion equation, convolution estimates,