{
  "abstract": "We prove non-uniqueness for the advection-diffusion equation with a drift in a logarithmically near-critical class below $L_t^2L_x^\\infty$. The first construction is a diffusion-assisted inverse cascade of scalar energy that diverges as $t \\to 0$. The second uses compressed inverse mixing, with diffusion acting perturbatively, and produces a bounded parabolic solution with the energy jump at the initial time. In both constructions the drift and scalar are smooth for positive time. We also consider the drift in $L^{2,\\infty}_tL^\\infty_x$ and solutions satisfying the energy inequality starting from almost every time. In this setting, we prove uniqueness of solutions with bounded energy, and construct non-unique solution with energy blowing up as time goes to zero.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.01779",
  "author_total": 2,
  "authors": [
    {
      "name": "Alexey Cheskidov"
    },
    {
      "name": "Xiaoran Liu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b58gnf8m3zrne.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.01779",
  "fulltext_url": null,
  "id": "b58gnf8m3zrne",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.01779",
  "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": "W7216038925",
  "pdf_url": "https://arxiv.org/pdf/2610.01779",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/b58gnf8m3zrne.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:07Z",
      "native_id": "2610.01779",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.01779"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Inverse Energy Cascade and Non-uniqueness for the Advection-Diffusion Equation",
  "title_text": "Inverse Energy Cascade and Non-uniqueness for the Advection-Diffusion Equation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:07Z",
  "url": "https://research.draconitic.com/paper/b58gnf8m3zrne",
  "venue": "arXiv",
  "versions": []
}