{
  "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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.03732",
  "author_total": 4,
  "authors": [
    {
      "name": "Mutong He"
    },
    {
      "name": "Feimin Huang"
    },
    {
      "name": "Lingjun Liu"
    },
    {
      "name": "Lingda Xu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cyw5e5ezmbvrh.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.03732",
  "fulltext_url": null,
  "id": "cyw5e5ezmbvrh",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.03732",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.03732",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cyw5e5ezmbvrh.ris",
  "sources": [
    {
      "first_seen": "2026-11-04T15:21:29Z",
      "native_id": "2611.03732",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.03732"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Asymptotic Stability and $L^p$ Time-Decay Rates of Rarefaction Waves for Systems of Viscous Conservation Laws",
  "title_text": "Asymptotic Stability and L^p Time-Decay Rates of Rarefaction Waves for Systems of Viscous Conservation Laws",
  "type": "preprint",
  "updated_at": "2026-11-06T18:21:01Z",
  "url": "https://research.draconitic.com/paper/cyw5e5ezmbvrh",
  "venue": "arXiv",
  "versions": []
}