{
  "abstract": "We study optimal boundary mixing of a nondiffusive scalar transported by a two-dimensional incompressible second-grade fluid. The control is the tangential traction in a Navier-slip boundary condition, and the principal objective is the terminal $(H^1(Ω))'$ mix-norm, supplemented by a quadratic control cost and an optional enstrophy reward. The second-grade constitutive law introduces a spatially filtered acceleration and a generalized vorticity; after lifting the nonhomogeneous boundary data, both $\\mathbf{g}$ and $\\partial_t\\mathbf{g}$ enter the state equation, where $\\mathbf{g}$ is the boundary control variable. We prove global state well-posedness for a passive scalar and well-posedness on a common, control-independent local interval for an active scalar. We then establish existence of an optimal control, directional differentiability of the control-to-state map in the topology required by the terminal objective, and a weak backward adjoint formulation. A duality identity yields the first-order variational inequality. In the absence of the friction coefficient $β$ and enstrophy reward weight $ζ$ (that is, $β=ζ=0$), the additional adjoint regularity implies uniqueness of the optimal control for all sufficiently large control penalties $γ$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.09946",
  "author_total": 1,
  "authors": [
    {
      "name": "Kush Kinra"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9qn53s2r3s34m.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.09946",
  "fulltext_url": null,
  "id": "9qn53s2r3s34m",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.09946",
  "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"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "openalex",
      "2604|Navier-Stokes equation solutions"
    ]
  ],
  "openalex": "W7212110694",
  "pdf_url": "https://arxiv.org/pdf/2609.09946",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9qn53s2r3s34m.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:15Z",
      "native_id": "2609.09946",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.09946"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Boundary control for optimal mixing by two-dimensional second-grade fluids",
  "title_text": "Boundary control for optimal mixing by two-dimensional second-grade fluids",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:15Z",
  "url": "https://research.draconitic.com/paper/9qn53s2r3s34m",
  "venue": "arXiv",
  "versions": []
}