Uniform Stability of Oscillatory Shocks for KdV-Burgers Equation

We study viscous-dispersive shock waves with infinite oscillations of the Korteweg-de Vries-Burgers (KdVB) equation. First, we establish detail structures of the shock waves, including the rates at which the local extrema converge to the left end state towards the left far field. Then, by exploiting the structural properties of the shocks, we show the $L^2$-contraction property of the shock profiles under arbitrarily large perturbations, up to time-dependent shifts. This property implies both time-asymptotic stability and uniform stability with respect to the viscosity and dispersion coefficients. This uniformity yields the existence of zero viscosity-dispersion limits, on which Riemann shocks are orbitally stable.