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

We study solutions to the Navier-Stokes equations in the class $L_t^{\mathrm{\infty }}{C}_x^{\alpha }$. Landau and Lifshitz [LL87] predicted that the Eulerian and Lagrangian temporal structure functions for turbulence exhibit $1\mathrm{/}3$ and $1\mathrm{/}2$ scaling laws, respectively. These laws were justified for the Euler equations in [Ise23,Ise25], assuming the spatial structure functions satisfies a $1\mathrm{/}3$ scaling law. We demonstrate them in a viscous setting by proving that the $C_{t,x}^{\alpha }$-norm of the solution and the $C^{1\mathrm{/}(1-\alpha )}$-norm of any fluid trajectory can be estimated by the $L_t^{\mathrm{\infty }}{C}_x^{\alpha }$-norm independently of the viscosity parameter $\nu >0$, for times bounded away from zero by a positive power of $\nu$.