Pointwise stability estimates for periodic traveling wave solutions of systems of viscous conservation laws

In the previous paper [9], we showed time asymptotic behavior with detailed decaying rates of perturbations of periodic traveling reaction–diffusion waves under small initial perturbations with a Gaussian rate and an algebraic rate. Here, we establish pointwise nonlinear stability up to an appropriate modulation of periodic traveling waves of systems of viscous conservation laws under small algebraic decaying initial data. Similar to the reaction–diffusion equations, by using Bloch decomposition, we start with pointwise bounds on the Green function of the linearized operator about underlying solutions.