{
  "abstract": "We study weak solutions of the incompressible and the compressible Navier--Stokes--Vlasov systems on a bounded domain, where a viscous fluid is coupled to a dispersed particle phase through a drag force; the fluid obeys the no-slip boundary condition and the particles are subject to specular reflection. For solutions with Besov regularity, we show that the Vlasov equation is renormalizable in the sense of DiPerna--Lions, and we give regularity thresholds for the validity of the global energy equality. If the distribution function $f$ belongs to $B_p^{α,\\infty}$ in space and velocity with time integrability $p$, and the fluid velocity $\\mathbf{u}$ belongs to $B_q^{β,\\infty}$ jointly in space and time with integrability $q$, then the energy equality holds provided $α+β>1$ and $β$ exceeds an explicit bound depending on $p$, $q$ and the dimension, under a natural integrability assumption on the pressure; an analogous criterion holds for the compressible system when the density is bounded from above and below. The condition $αp>1$ imposed on $f$ is not merely technical: it is exactly the range in which the macroscopic momentum flux possesses a boundary trace, so that the specular reflection condition passes to the macroscopic energy balance. When the particle phase is absent, our thresholds coincide with the criteria $β>\\frac12+\\frac1q$ and $β>\\frac3q$ obtained by Yu for the incompressible Navier--Stokes equations alone. The proofs combine a boundary cutoff argument with commutator estimates based on a scale-localized Besov modulus, together with a boundary-strip analysis.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.36579",
  "author_total": 1,
  "authors": [
    {
      "name": "Hanyu Teng"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fdpepgk169vdp.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.36579",
  "fulltext_url": null,
  "id": "fdpepgk169vdp",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.36579",
  "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",
      "2604|Navier-Stokes equation solutions"
    ]
  ],
  "openalex": "W7224110995",
  "pdf_url": "https://arxiv.org/pdf/2610.36579",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fdpepgk169vdp.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.36579",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.36579"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Energy equality and renormalization of Navier-Stokes-Vlasov equations",
  "title_text": "Energy equality and renormalization of Navier-Stokes-Vlasov equations",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/fdpepgk169vdp",
  "venue": "arXiv",
  "versions": []
}