{
  "abstract": "We study the stability and instability of stratified steady states $ρ_s=ρ_s(y)$ for the two-dimensional incompressible porous media equation on $\\mathbb{T}\\times(-1,1)$ and $\\mathbb{T}\\times\\mathbb{R}$. On the periodic channel, uniformly decreasing steady states satisfying natural boundary-compatibility conditions are nonlinearly stable under small $H^m$ perturbations, for every integer $m>2$. The solutions converge in $L^2$ to the measure-preserving stratification of the initial density at the rate $t^{-m/2}$. Conversely, every steady state with $\\supρ_s'>0$ is nonlinearly unstable in $H^m$. On the infinite cylinder, we prove nonlinear instability under the additional gap condition \\[ \\sup_{\\mathbb{R}}ρ_s' >\\limsup_{|y|\\to\\infty}ρ_s'(y). \\] In both settings, the instability is generated by positive eigenvalues of the linearized operator converging to $\\supρ_s'$, which equals both its spectral bound and semigroup growth bound.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.37397",
  "author_total": 3,
  "authors": [
    {
      "name": "Seyed Abdolhamid Banihashemi"
    },
    {
      "name": "Sepehr Mohammadkhani"
    },
    {
      "name": "Huy Q. Nguyen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8ey0mh3jgjd07.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.37397",
  "fulltext_url": null,
  "id": "8ey0mh3jgjd07",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.37397",
  "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-ph"
    ],
    [
      "arxiv",
      "math.AP"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "openalex",
      "2604|Navier-Stokes equation solutions"
    ]
  ],
  "openalex": "W7214934851",
  "pdf_url": "https://arxiv.org/pdf/2609.37397",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8ey0mh3jgjd07.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:31Z",
      "native_id": "2609.37397",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.37397"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    }
  ],
  "title": "Sharp stability and instability of stratified steady states for the incompressible porous media equation",
  "title_text": "Sharp stability and instability of stratified steady states for the incompressible porous media equation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:35Z",
  "url": "https://research.draconitic.com/paper/8ey0mh3jgjd07",
  "venue": "arXiv",
  "versions": []
}