Asymptotic Stability and $L^p$ Time-Decay Rates of Rarefaction Waves for Systems of Viscous Conservation Laws

Abstract

We study the asymptotic stability and quantitative time-decay rates of rarefaction waves for one-dimensional $n\times n$ strictly hyperbolic systems of viscous conservation laws. While the nonlinear stability of rarefaction waves for $2\times 2$ systems was established by Xin \cite{Xin1988,Xin1989}, extending such results to general $n\times n$ systems presents additional difficulties arising from the possible coexistence of positive and negative transverse relative characteristic speeds and nonlinear interactions between distinct characteristic families. Moreover, establishing standalone $L^p$ energy estimates and explicit algebraic decay rates for general viscous systems without $L^1$ localization remains a significant challenge. In this paper, we establish both the global $H^2(\R)$ asymptotic stability and, for the first time, standalone physical-space $L^p$ energy estimates and explicit algebraic $L^p$ time-decay rates for weak rarefaction waves. Specifically, for every integer $p\ge 3$, we prove that the perturbation $U(t)-\bar{U}(t)$ around the smooth Burgers rarefaction profile $\bar{U}$ decays in $L^p(\R)$ at the rate \[ \|U(t)-\bar{U}(t)\|_{L^p}\le C_p\bigl(\|U_0-\bar{U}(0)\|_{H^2}^2+δ^3\bigr)^{1/2}(1+t)^{-\frac{1}{2}(\frac{1}{2}-\frac{1}{p})}, \] where $δ$ denotes the wave strength. Our proof combines a diagonal characteristic weight matrix that yields coercive rarefaction damping in all characteristic modes with directed wave-interaction functionals that exploit the strict separation of characteristic speeds to control nonlinear coupling between distinct characteristic families. These estimates lead to a closed $L^p$ Lyapunov inequality, from which the algebraic time-decay rates follow by the one-dimensional Nash inequality.