{
  "abstract": "We study a compatible linearization, about a zero-velocity rest equilibrium, of the one-dimensional barotropic compressible Navier-Stokes equations in Lagrangian variables on a finite directed metric graph. The junction closure is an abstract directed common-velocity law selected to preserve weighted-volume and mechanical-power balances. We give a finite-dimensional incidence characterization and a constructive sufficient condition for positive balanced cross-sectional weights. We also provide a common-pressure criterion that produces a nonempty class of compatible constant rest equilibria. For the resulting linear graph system, we prove maximal dissipativity of the homogeneous operator, finite-energy well-posedness for square-integrable boundary tractions, and existence and uniqueness for a linear-quadratic tracking problem with a weak adjoint and an L^2 boundary observation. Positive Lagrangian coordinate parameters are then pulled back to a unit reference graph. Uniform estimates and compactness for varying mass forms yield parameter-to-state continuity, continuity of the optimized L^2 -control value, and existence of an outer minimizer on a compact fixed-topology design set. Under the areal-mass normalization, an edge parameter is the extent of the material coordinate and A_e ℓ_e is the reference mass assigned to the edge. The fixed-reference-profile convention is therefore an abstract reference-mass redistribution model, not an Eulerian pipe-length or geometric shape derivative. A backward-Euler/P 1 section records exact finite-dimensional solvability, energy, envelope, and transpose-adjoint identities on fixed grids. We finally formulate a prospective GFDS program for the full nonlinear network, including the reference-domain sensitivity structure and the analytical results required for a future nonlinear projected gradient flow. The introduction relates this coordinate framework to the geometric gradient-flow analysis of Plotnikov and Sokołowski, with explicit distinctions between curve geometry, network parameters, and conditional descent. No nonlinear state theorem, continuous coordinate derivative, numerical convergence theorem, or continuous-to-discrete gradient claim is made.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "name": "Jan Sokołowski"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/axdka416x5mp2.bib",
  "cited_by": 0,
  "doi": null,
  "fulltext_url": null,
  "id": "axdka416x5mp2",
  "issns": [],
  "landing_url": "https://hal.science/hal-05758667v1",
  "language": "en",
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "hal",
      "0.math"
    ],
    [
      "hal",
      "0.nlin"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://hal.science/hal-05758667/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/axdka416x5mp2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:37Z",
      "native_id": "hal-05758667",
      "source": "hal",
      "url": "https://hal.science/hal-05758667v1"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "hal:0.math",
      "subfield": "general"
    },
    {
      "field": "physics",
      "origin": "hal:0.nlin",
      "subfield": "statistical-nonlinear"
    }
  ],
  "title": "On compressible Navier-Stokes equations on metric graphs",
  "title_text": "On compressible Navier-Stokes equations on metric graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:37Z",
  "url": "https://research.draconitic.com/paper/axdka416x5mp2",
  "venue": "HAL",
  "versions": []
}