{
  "abstract": "In this paper, we will construct periodic solutions to the viscous steady Burgers equation with an external force $f(x)$, based on the following formal expansion $u^\\varepsilon(x)=u_0(x)+\\varepsilon u_1(x)+\\cdots+\\varepsilon^n u_n(x)+\\cdots$, where $\\varepsilon \\ge 0$ represents the viscosity and $u_0(x)$ is a solution to non-viscous steady Burgers equation with the external force $f(x)$. We will focus on the solutions which are uniformly bounded with respect to the viscosity. In our previous work, starting from $u_0=-(2+\\cos x)$, the authors constructed the periodic solutions to the viscous steady Burgers equation with the external force $f=u_0(x)u_{0x}=-2\\sin x-\\sin x\\cos x$. In this paper, we will extend the main result obtained in our previous work and construct the solutions starting from general $u_0$ and $f$ satisfying the non-viscous steady Burgers equation $u_0(x)u_{0x}=f$. It will be shown that there exists a $\\varepsilon_0>0$, which depends on $n$, such that for any $0<\\varepsilon<\\varepsilon_0$, the viscous steady Burgers equation with the external force $f$ has a periodic solution $u^\\varepsilon(x) \\in C^2([0,2π])$, satisfying $|u^\\varepsilon(x)-u_0(x)-\\varepsilon u_1(x)-\\cdots-\\varepsilon^n u_n(x)| \\leq C\\varepsilon^{n+1}$, where $C>0$ is a constant which depends on $n$, but is independent of $\\varepsilon$. Compared with Jauslin-Kreiss-Moser's result, we present a new approach to construct periodic solutions to the viscous steady Burgers with a general external force. The constructed solutions will tend to the ones of the non-viscous Burgers equation with a sharper convergence rate (up to higher order) when the viscosity vanishes.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.29704",
  "author_total": 2,
  "authors": [
    {
      "name": "Yuhan Cao"
    },
    {
      "name": "Quansen Jiu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/athdhm5j4hfs1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.29704",
  "fulltext_url": null,
  "id": "athdhm5j4hfs1",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.29704",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "openalex",
      "2604|Navier-Stokes equation solutions"
    ]
  ],
  "openalex": "W7204869564",
  "pdf_url": "https://arxiv.org/pdf/2608.29704",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/athdhm5j4hfs1.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2608.29704",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.29704"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Existence of Periodic Solutions to Steady Viscous Burgers Equation with a General Force",
  "title_text": "Existence of Periodic Solutions to Steady Viscous Burgers Equation with a General Force",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/athdhm5j4hfs1",
  "venue": "arXiv",
  "versions": []
}