{
  "id": "2506.17801",
  "version": "v1",
  "published": "2025-06-21T19:59:57.000Z",
  "updated": "2025-06-21T19:59:57.000Z",
  "title": "Low regularity well-posedness of nonlocal dispersive perturbations of Burgers' equation",
  "authors": [
    "Luc Molinet",
    "Didier Pilod",
    "Stéphane Vento"
  ],
  "comment": "48 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "We consider the Cauchy problem associated to a class of dispersive perturbations of Burgers' equations, which contains the low dispersion Benjamin-Ono equation, (also known as low dispersion fractional KdV equation), $$ \\partial_tu-D_x^{\\alpha}\\partial_xu=\\partial_x(u^2) \\, ,$$ and prove that it is locally well-posed in $H^s(\\mathbb K)$, $\\mathbb K=\\mathbb R$ or $\\mathbb T$, for $s>s_{\\alpha}$, where \\begin{equation*} s_\\alpha=\\begin{cases} 1-\\frac{3\\alpha}4 & \\text{for} \\quad \\frac23 \\le \\alpha \\le 1; \\frac 32(1-\\alpha) & \\text{for} \\quad \\frac13 \\le \\alpha \\le \\frac23; \\frac 32-\\frac{\\alpha}{1-\\alpha} & \\text{for} \\quad 0 < \\alpha \\le \\frac13 . \\end{cases} \\end{equation*} The uniqueness is unconditional in $H^s(\\mathbb K)$ for $s>\\max\\{\\frac12,s_{\\alpha}\\}$. Moreover, we obtain \\emph{a priori} estimates for the solutions at the lower regularity threshold $s>\\widetilde{s}_\\alpha$ where \\begin{equation*} \\widetilde{s}_\\alpha=\\begin{cases} \\frac 12-\\frac \\alpha 4 & \\text{for} \\quad \\frac23 \\le \\alpha \\le 1; 1-\\alpha & \\text{for} \\quad \\frac12 \\le \\alpha \\le \\frac23; \\frac 32-\\frac{\\alpha}{1-\\alpha} & \\text{for} \\quad 0 < \\alpha \\le \\frac12 . \\end{cases} \\end{equation*} As a consequence of these results and of the Hamiltonian structure of the equation, we deduce global well-posedness in $H^s(\\mathbb K)$ for $s>s_{\\alpha}$ when $\\alpha>\\frac23$, and in the energy space $H^{\\frac{\\alpha}2}(\\mathbb K)$ when $\\alpha>\\frac45$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-06-21T19:59:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35A01",
      "35A02",
      "35B45",
      "35E15",
      "35Q53"
    ],
    "keywords": [
      "low regularity well-posedness",
      "nonlocal dispersive perturbations",
      "low dispersion fractional kdv equation",
      "low dispersion benjamin-ono equation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 48,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}