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A Hölder estimate for the trajectories of the Navier-Stokes equations

  • Ming-Yuan Chang
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Abstract

We study solutions to the Navier-Stokes equations in the class L∞t Cαx. Landau and Lifshitz [LL87] predicted that the Eulerian and Lagrangian temporal structure functions for turbulence exhibit 1/3 and 1/2 scaling laws, respectively. These laws were justified for the Euler equations in [Ise23,Ise25], assuming the spatial structure functions satisfies a 1/3 scaling law. We demonstrate them in a viscous setting by proving that the Cαt,x-norm of the solution and the C1/(1−α)-norm of any fluid trajectory can be estimated by the L∞tCαx-norm independently of the viscosity parameter ν>0, for times bounded away from zero by a positive power of ν.

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Cite

BibTeXRIS
BibTeX
@misc{chang2026hölder,
  title = {{A Hölder estimate for the trajectories of the Navier-Stokes equations}},
  author = {Ming-Yuan Chang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2605.22006},
  eprint = {2605.22006},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2605.22006},
}
RIS
TY  - UNPB
TI  - A Hölder estimate for the trajectories of the Navier-Stokes equations
AU  - Ming-Yuan Chang
PY  - 2026
DA  - 2026/09/14
PB  - arXiv
DO  - 10.48550/arxiv.2605.22006
UR  - https://arxiv.org/abs/2605.22006
AB  - We study solutions to the Navier-Stokes equations in the class L^∞_t C^α_x. Landau and Lifshitz [LL87] predicted that the Eulerian and Lagrangian temporal structure functions for turbulence exhibit 1/3 and 1/2 scaling laws, respectively. These laws were justified for the Euler equations in [Ise23,Ise25], assuming the spatial structure functions satisfies a 1/3 scaling law. We demonstrate them in a viscous setting by proving that the C^α_t,x-norm of the solution and the C^1/(1−α)-norm of any fluid trajectory can be estimated by the L^∞_tC^α_x-norm independently of the viscosity parameter ν>0, for times bounded away from zero by a positive power of ν.
ER  -