{
  "abstract": "We consider second-order strictly hyperbolic operators whose coefficients are Lipschitz continuous in space and may oscillate near the initial time. Their first two time derivatives are bounded by the corresponding powers of $\\frac{1}{t}(\\log\\frac{1}{t})^α$ with $0\\leqslantα<1$, and the first time derivative satisfies the corresponding spatial Lipschitz bound. By combining Tarama's energy with frequency-dependent weights, we prove an energy estimate without loss for $α=0$ and with a stretched logarithmic upper bound for the loss when $0<α<1$, which implies an arbitrarily small loss in the Sobolev scale. These estimates extend the analysis of Colombini, Del Santo and Fanelli [\\emph{Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5)} \\textbf{26} (2025)] for spatially Lipschitz coefficients to time-derivative bounds with an additional logarithmic factor.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.13343",
  "author_total": 2,
  "authors": [
    {
      "name": "Wenhui Chen"
    },
    {
      "name": "Michael Reissig"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9f9r6jg6td8q9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.13343",
  "fulltext_url": null,
  "id": "9f9r6jg6td8q9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.13343",
  "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",
      "2610|Advanced Mathematical Physics Problems"
    ]
  ],
  "openalex": "W7221549235",
  "pdf_url": "https://arxiv.org/pdf/2610.13343",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9f9r6jg6td8q9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:28:40Z",
      "native_id": "2610.13343",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.13343"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Loss of derivatives for hyperbolic operators with coefficients oscillating in time",
  "title_text": "Loss of derivatives for hyperbolic operators with coefficients oscillating in time",
  "type": "preprint",
  "updated_at": "2026-10-30T12:28:40Z",
  "url": "https://research.draconitic.com/paper/9f9r6jg6td8q9",
  "venue": "arXiv",
  "versions": []
}