{
  "abstract": "We study the one-dimensional viscous Burgers equation on $(0,L)$ with the conservative boundary conditions $u_x=-u^2$. On the half-line, solutions approach a nonlinear self-similar profile, whereas on a bounded interval they converge to a nonconstant equilibrium. We describe explicitly how the dynamics passes between these states at the critical diffusive scale $t=cL^2$. For compactly supported initial data of mass $M$, the rescaled solution converges as $L\\to\\infty$ to an explicit crossover profile $Φ_c$. In similarity variables, $Φ_c$ converges to the half-line profile $f_M$ as $c\\downarrow0$; after rescaling to domain variables, it converges to the interval equilibrium as $c\\to\\infty$. We also determine the sharp onset of confinement: \\[ \\lim_{c\\downarrow0}-c\\log|Φ_c(0)-f_M(0)|=1\\qquad(M\\ne0). \\] Moreover, on compact sets in similarity variables, the interval and half-line solutions differ in $C^k$ by at most $C_{k,\\varepsilon}\\exp(-(1-\\varepsilon)L^2/t)$, uniformly in $L$, and the exponential constant is optimal. Thus we identify not only the transition scale $L^2$, but also the profile governing the crossover and the sharp rate at which the remote boundary becomes visible. We finally discuss the conclusions that persist for space-dependent diffusivity and illustrate the three asymptotic regimes numerically.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.13414",
  "author_total": 2,
  "authors": [
    {
      "name": "Maicon Sonego"
    },
    {
      "name": "Enrique Zuazua"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7gdeq3qxv3dh3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.13414",
  "fulltext_url": null,
  "id": "7gdeq3qxv3dh3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.13414",
  "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",
      "2610|Advanced Mathematical Physics Problems"
    ]
  ],
  "openalex": "W7213353041",
  "pdf_url": "https://arxiv.org/pdf/2609.13414",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7gdeq3qxv3dh3.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.13414",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.13414"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Crossover asymptotics and a sharp confinement rate for the viscous Burgers equation",
  "title_text": "Crossover asymptotics and a sharp confinement rate for the viscous Burgers equation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/7gdeq3qxv3dh3",
  "venue": "arXiv",
  "versions": []
}