{
  "abstract": "We study the Cauchy problem for the one-dimensional isothermal compressible Navier--Stokes system with the van der Waals equation of state, and analyze the well-posedness and asymptotic stability of steady-state phase-transition solutions at subcritical temperatures. Working in Lagrangian coordinates, we use variational techniques to construct piecewise smooth steady-state solutions corresponding to two-phase equilibrium states under finite-mass perturbations. Introducing artificial-viscosity regularization, we first establish existence of the global minimizer of the regularized variational problem, unique within each fixed symmetry class; passing to the vanishing artificial-viscosity limit then yields a physically admissible steady-state solution with two phase interfaces. For initial perturbations that are small with respect to energy, piecewise-gradient, total-variation, and integral norms but may exhibit large pointwise jumps across phase boundaries, we derive uniform a priori bounds for weak solutions near the two-phase equilibrium profile. We further prove that these global weak solutions converge to the steady-state two-phase equilibrium in $L^\\infty(\\mathbb{R})$ as $t\\to\\infty$, thereby characterizing their large-time asymptotic behavior. Physically, finite mass perturbations break the uniqueness of the uniform steady state and generate infinitely many piecewise constant two-phase equilibria satisfying the Maxwell equal-area rule. Artificial viscosity acts as a selection principle that singles out the physically relevant two-interface steady-state configuration, and this two-phase equilibrium is asymptotically stable under small integral perturbations that allow large-amplitude discontinuities at phase interfaces.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.05965",
  "author_total": 4,
  "authors": [
    {
      "name": "Yazhou Chen"
    },
    {
      "name": "Yi Peng"
    },
    {
      "name": "Xiaoding Shi"
    },
    {
      "name": "Xiaoping Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/e4e9w2czhxpvq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.05965",
  "fulltext_url": null,
  "id": "e4e9w2czhxpvq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.05965",
  "language": null,
  "licence": "CC-BY-NC-SA-4.0",
  "licence_label": "CC BY-NC-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "openalex",
      "2604|Navier-Stokes equation solutions"
    ]
  ],
  "openalex": "W7220728311",
  "pdf_url": "https://arxiv.org/pdf/2610.05965",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/e4e9w2czhxpvq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.05965",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.05965"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Steady-State Phase Transition States and Large-Time Asymptotics for the Cauchy Problem of Compressible Navier-Stokes Equations with van der Waals Equation of State",
  "title_text": "Steady-State Phase Transition States and Large-Time Asymptotics for the Cauchy Problem of Compressible Navier-Stokes Equations with van der Waals Equation of State",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/e4e9w2czhxpvq",
  "venue": "arXiv",
  "versions": []
}