{
  "abstract": "We establish interior maximal $L^{q_c}$-regularity for bounded strong solutions of $u_t-Δu+|Du|^γ=f$ in $\\mathbb{T}^d\\times(0,T)$, where $d\\geq 2$, $γ>2$, and $q_c=(d+2)(γ-1)/γ$. The estimates are uniform for uniformly bounded families of solutions whose source terms range over a bounded, uniformly equi-integrable subset of $L^{q_c}$. The main difficulty is the possible concentration of the critical gradient energy. We overcome it through a two-scale blow-up argument: the first scale produces a small Hamiltonian coefficient, while energy normalization at the second scale yields strong endpoint compactness, allowing a parabolic Liouville theorem to rule out concentration. The resulting uniform little-Hölder estimate, combined with critical Gagliardo--Nirenberg interpolation, permits absorption of the nonlinear term in the parabolic Calderón--Zygmund estimate. As an application, we construct bounded strong solutions for $f\\in L^{q_c}$ and continuous initial data $u_0$, and prove subsequential strong interior convergence of smooth approximations in $W^{2,1}_{q_c}$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.33798",
  "author_total": 2,
  "authors": [
    {
      "name": "Fanze Kong"
    },
    {
      "name": "Xiaoyu Zeng"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0tqjkkwt09zwy.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.33798",
  "fulltext_url": null,
  "id": "0tqjkkwt09zwy",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.33798",
  "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|Nonlinear Partial Differential Equations"
    ]
  ],
  "openalex": "W7214785371",
  "pdf_url": "https://arxiv.org/pdf/2609.33798",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0tqjkkwt09zwy.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.33798",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.33798"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Maximal Regularity and Existence for Superquadratic Parabolic Hamilton--Jacobi Equations: The Endpoint case",
  "title_text": "Maximal Regularity and Existence for Superquadratic Parabolic Hamilton--Jacobi Equations: The Endpoint case",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/0tqjkkwt09zwy",
  "venue": "arXiv",
  "versions": []
}