{
  "abstract": "While the asymptotic stability of rarefaction waves for the compressible Navier--Stokes equations has been well established since the pioneering works of Matsumura--Nishihara \\cite{MatsumuraNishihara1986} and Liu--Xin \\cite{LiuXin1988}, the classical $L^2$ energy method yields only qualitative convergence in $L^\\infty(\\R)$ as $t\\to\\infty$ without any explicit decay rate. In this paper, we establish for the first time a standalone \\(L^4\\) energy estimate for perturbations of a weak rarefaction wave in the one-dimensional isentropic compressible Navier--Stokes system, thereby obtaining the first explicit algebraic time-decay rates for this model. Specifically, for small initial perturbations in $H^2(\\R)$ and small wave strength $δ$, we prove that $(v-V,u-U)$ decays in $L^4(\\R)$ at the rate \\[ \\|(v-V,u-U)(t)\\|_{L^4}\\le C\\bigl(\\|(v_0-V(0),u_0-U(0))\\|_{H^2}+\\sqrtδ\\bigr)(1+t)^{-1/8}, \\] where \\((V,U)\\) denotes the smooth rarefaction wave, while the relative Riemann invariants \\(z_\\pm\\) of the perturbed system decay in \\(L^\\infty(\\R)\\) at the rate \\((1+t)^{-1/6}\\). The proof relies on a new local quartic energy estimate for \\(z_\\pm\\) and \\(z_\\pm-Q\\), where \\(Q=μ(v)v_x-μ(V)V_x\\), as well as a nonlocal half-space wave-interaction functional for corrected characteristic modes and a weighted higher-order derivative energy estimate coupled at the scale of the one-dimensional Nash inequality.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.01956",
  "author_total": 4,
  "authors": [
    {
      "name": "Mutong He"
    },
    {
      "name": "Feimin Huang"
    },
    {
      "name": "Lingjun Liu"
    },
    {
      "name": "Lingda Xu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/apkv2afk26p3x.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.01956",
  "fulltext_url": null,
  "id": "apkv2afk26p3x",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.01956",
  "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/2611.01956",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/apkv2afk26p3x.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.01956",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.01956"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Time-Decay Rates toward Rarefaction Waves for the One-Dimensional Isentropic Compressible Navier--Stokes Equations",
  "title_text": "Time-Decay Rates toward Rarefaction Waves for the One-Dimensional Isentropic Compressible Navier--Stokes Equations",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/apkv2afk26p3x",
  "venue": "arXiv",
  "versions": []
}