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

@misc{he2026asymptotic,
  title = {{Asymptotic Stability and $L^p$ Time-Decay Rates of Rarefaction Waves for Systems of Viscous Conservation Laws}},
  author = {Mutong He and Feimin Huang and Lingjun Liu and Lingda Xu},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.03732},
  eprint = {2611.03732},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.03732},
}

TY  - UNPB
TI  - Asymptotic Stability and L^p Time-Decay Rates of Rarefaction Waves for Systems of Viscous Conservation Laws
AU  - Mutong He
AU  - Feimin Huang
AU  - Lingjun Liu
AU  - Lingda Xu
PY  - 2026
DA  - 2026/11/01
PB  - arXiv
DO  - 10.48550/arxiv.2611.03732
UR  - https://arxiv.org/abs/2611.03732
AB  - We study the asymptotic stability and quantitative time-decay rates of rarefaction waves for one-dimensional n× n strictly hyperbolic systems of viscous conservation laws. While the nonlinear stability of rarefaction waves for 2× 2 systems was established by Xin \cite{Xin1988,Xin1989}, extending such results to general n× 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≥ 3, we prove that the perturbation U(t)−U̅(t) around the smooth Burgers rarefaction profile U̅ decays in L^p(R) at the rate |U(t)−U̅(t)|_L^p≤ C_p(|U_0−U̅(0)|_H^2^2+δ^3)^1/2(1+t)^−1/2(1/2−1/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.
ER  -