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Loss of derivatives for hyperbolic operators with coefficients oscillating in time

  • Wenhui Chen
  • Michael Reissig
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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 1/t(log1/t)α with 0leqslantα<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.

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Cite

BibTeXRIS
BibTeX
@misc{chen2026loss,
  title = {{Loss of derivatives for hyperbolic operators with coefficients oscillating in time}},
  author = {Wenhui Chen and Michael Reissig},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.13343},
  eprint = {2610.13343},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.13343},
}
RIS
TY  - UNPB
TI  - Loss of derivatives for hyperbolic operators with coefficients oscillating in time
AU  - Wenhui Chen
AU  - Michael Reissig
PY  - 2026
DA  - 2026/10/09
PB  - arXiv
DO  - 10.48550/arxiv.2610.13343
UR  - https://arxiv.org/abs/2610.13343
AB  - 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 1/t(log1/t)^α with 0leqslantα<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.
ER  -