{
  "abstract": "A celebrated conjecture of P.-L. Lions concerns maximal regularity for viscous Hamilton--Jacobi equations. In this paper, we study the endpoint case. We consider normalized strong solutions of $$-Δu+|Du|^γ=f$$ in $\\mathbb T^d$, where $d\\geq 2$, $γ>2$, and $f\\in L^{q_c}(\\mathbb T^d)$ with $q_c=d(γ-1)/γ$. At this critical exponent, the main difficulty is possible concentration under the critical scaling. Assuming that the source terms form a uniformly equi-integrable subset of $L^{q_c}$, we rule out this concentration and prove maximal $L^{q_c}$ regularity for strong solutions. The proof combines a two-stage blow-up argument with a Liouville rigidity theorem. More precisely, the second blow-up yields a uniform local $L^{γq_c}$-bound for the gradients, while the small drift arising from the first blow-up upgrades weak convergence to strong local compactness, ultimately leading to a contradiction with Liouville rigidity. Finally, we apply the maximal regularity theory for Hamilton--Jacobi equations at the endpoint case to establish the existence of ergodic solutions to defocusing second-order mean-field games systems with critical coupling exponents.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.25066",
  "author_total": 1,
  "authors": [
    {
      "name": "Fanze Kong"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/43yf462a8s32a.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.25066",
  "fulltext_url": null,
  "id": "43yf462a8s32a",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.25066",
  "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": "W7204448268",
  "pdf_url": "https://arxiv.org/pdf/2608.25066",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/43yf462a8s32a.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:20Z",
      "native_id": "2608.25066",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.25066"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Endpoint Maximal Regularity for Superquadratic Hamilton-Jacobi Equations and Applications to Ergodic Mean-field Games Systems",
  "title_text": "Endpoint Maximal Regularity for Superquadratic Hamilton-Jacobi Equations and Applications to Ergodic Mean-field Games Systems",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:20Z",
  "url": "https://research.draconitic.com/paper/43yf462a8s32a",
  "venue": "arXiv",
  "versions": []
}