{
  "abstract": "In this paper, the detailed behavior of solutions of both the Stokes and the Navier--Stokes system in the half space is investigated when the initial data belongs to $L^1$. We first give a detailed asymptotic expansion of the solution to the Stokes system supplemented with $L^1$ initial data. With the aid of this expansion, we provide two sufficient conditions on the initial data so that the associated solutions of the Navier--Stokes system do not belong to $L^1$, corresponding to long time and short time behaviors, respectively. Moreover, for the $n$- dimensional Navier--Stokes system, a pointwise spatial lower bound of order $n$ is derived in a conic neighborhood of the $x_n$-axis, provided that the initial data decays faster than order $n$. One of the main difficulties to get precise form of leading order term is the non-commutativity of the Leray projection operator with the Laplacian. Our strategy is to employ the special structure of the Green tensor observed by Han and a cancellation property of the initial data. This also yields the sufficient conditions on the initial data to get precise asymptotic behavior via comparing the linear and nonlinear parts.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.37484",
  "author_total": 3,
  "authors": [
    {
      "name": "Rao Li"
    },
    {
      "name": "Haitao Wang"
    },
    {
      "name": "Chunjing Xie"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2abwryjsj5c95.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.37484",
  "fulltext_url": null,
  "id": "2abwryjsj5c95",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.37484",
  "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": "W7214911118",
  "pdf_url": "https://arxiv.org/pdf/2609.37484",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2abwryjsj5c95.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:35Z",
      "native_id": "2609.37484",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.37484"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "$L^1$ Behavior of the Stokes System and the Navier--Stokes System in the Half Space",
  "title_text": "L^1 Behavior of the Stokes System and the Navier--Stokes System in the Half Space",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:35Z",
  "url": "https://research.draconitic.com/paper/2abwryjsj5c95",
  "venue": "arXiv",
  "versions": []
}