{
  "abstract": "We prove the existence of unique solutions ${u\\in H^1(0,T;L^2_γ(\\mathbb{R}^n))\\cap L^2(0,T;H^1_γ(\\mathbb{R}^n))}$ of the transport equation $u_t=F\\cdot\\nabla u+f$ and the orthogonal dynamics equation ${u_t= Q_\\text{Z} (F\\cdot\\nabla) Q_\\text{Z} u+f}$, where $F$ is a time-independent vector field, $f$ is a forcing term, $P_\\text{Z}=1-Q_\\text{Z}$ is a Zwanzig projection (conditional expectation) on $L^2_γ(\\mathbb{R}^n)$ defined by $P_\\text{Z}h \\propto \\int h(\\cdot ,y) ρ(\\cdot,y)\\rm{d}y$, and $γ$ is the standard Gaussian measure with density $ρ$. To this end, we develop a vanishing viscosity method based on the maximal regularity theory for the Ornstein-Uhlenbeck operator on $L^2_γ(\\mathbb{R}^n)$. The following assumptions guarantee the existence and uniqueness of strong solutions: ${f\\in H^1(0,T;L^2_γ(\\mathbb{R}^n))\\cap L^2(0,T;H^1_γ(\\mathbb{R}^n))}$, ${g\\in H^1_γ(\\mathbb{R}^n)}$, $F\\in W^{1,\\infty}(\\mathbb{R}^n,\\mathbb{R}^n)$ and $\\rm{ess\\,sup}_{\\mathbb{R}^n}\\left(x\\cdot F-\\rm{div}F\\right) < \\infty$, where $g$ denotes the initial value function.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.18647",
  "author_total": 1,
  "authors": [
    {
      "name": "Christoph Widder"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/94e68x956szsb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.18647",
  "fulltext_url": null,
  "id": "94e68x956szsb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.18647",
  "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.DS"
    ],
    [
      "openalex",
      "2604|Gas Dynamics and Kinetic Theory"
    ]
  ],
  "openalex": "W7221753690",
  "pdf_url": "https://arxiv.org/pdf/2610.18647",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/94e68x956szsb.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.18647",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.18647"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "Existence and Uniqueness Proof for the Transport Equation and the Orthogonal Dynamics Equation for Zwanzig's Projection on Gaussian Spaces: A Vanishing Viscosity Method",
  "title_text": "Existence and Uniqueness Proof for the Transport Equation and the Orthogonal Dynamics Equation for Zwanzig's Projection on Gaussian Spaces: A Vanishing Viscosity Method",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/94e68x956szsb",
  "venue": "arXiv",
  "versions": []
}