{
  "abstract": "Previous work strengthened the time integrability of integral-invariant weak solutions from L^1 to L^\\infty within a spatial L^2 framework.This paper studies the propagation of Sobolev regularity for integral-invariant weak solutions of the conservative linear scalar transport equation.We first derive a representation formula along the flow and a distributional equation from integral identities against adjoint-transported test functions on domains moving with the same characteristic flow,and relate the weak formulation in time, including an endpoint term for the initial data, to representatives satisfying the pointwise-in-time integral definition.We then give a unified proof for k=1,2:under u_0\\in H^k(Ω_0)and Assumption \\mathsf A_{k+1},the pullback of the solution belongs to C^1([0,T];H^k(Ω_0)),the material derivative has spatial $H^k$ regularity,and the Eulerian time derivative has spatial H^{k-1} regularity.The corresponding time continuity is understood after pullback to the fixed domain by the flow.In both cases, U\\in H^1(Q);when k=2, we further obtain \\nabla_xU\\in H^1(Q;\\R^d).The pullback differentiability and the spatial estimates for the material derivativeprovide information beyond first-order space-time Sobolev regularity alone.The proof combines estimates for the characteristic flow with Sobolev product and composition rules.Under the additional assumption that the zero extension of the initial data belongs to H^2(\\R^d),we also establish $H^1(0,T;H^1(\\R^d))$ regularity in the fixed ambient space.Counterexamples show that these results concern regularity propagation under additional assumptions: general L^2 initial data do not automatically yield spatial smoothing,and spatial regularity of the material derivative cannot replace the corresponding regularity of the Eulerian time derivative.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.39527",
  "author_total": 1,
  "authors": [
    {
      "name": "Zhengrong Xie"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1tv630hkjvews.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.39527",
  "fulltext_url": null,
  "id": "1tv630hkjvews",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.39527",
  "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/2610.39527",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1tv630hkjvews.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.39527",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.39527"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Propagation of Sobolev Regularity for Integral-Invariant Weak Solutions of the Linear Transport Equation",
  "title_text": "Propagation of Sobolev Regularity for Integral-Invariant Weak Solutions of the Linear Transport Equation",
  "type": "preprint",
  "updated_at": "2026-11-06T18:21:01Z",
  "url": "https://research.draconitic.com/paper/1tv630hkjvews",
  "venue": "arXiv",
  "versions": []
}