Convergence rates of vanishing diffusion limit on nonlinear hyperbolic system with damping and diffusion

Abstract

The aim of this paper is to study the global unique solvability on C∞-solution to the Cauchy problem of a special 2 × 2 nonlinear hyperbolic system with damping and diffusion. Furthermore, we also investigate the convergence rates as the diffusion parameter β goes to zero. It is shown that the convergence rates in C∞-norm is of the order O(β). The novelty of analysis taken in the present article is to obtain the uniform-in-β a priori estimates on Hs-norm for any \documentclass[12pt]{minimal}\begin{document}$s\in \mathbb {N}$\end{document}s∈N, which are implemented by introducing induction. We focus on two cases in this paper: both α = β and α ≠ β since the uniform-in-β a prioriestimates obtained are distinctly different for different cases.