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arXiv · 2605.22006

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 $L^\infty_t C^α_x$. 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 $L^\infty_tC^α_x$-norm independently of the viscosity parameter $ν>0$, for times bounded away from zero by a positive power of $ν$.

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Ming-Yuan Chang. 2026-09-14. A Hölder estimate for the trajectories of the Navier-Stokes equations. https://arxiv.org/abs/2605.22006

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