A H\"older estimate for the trajectories of the Navier-Stokes equations

TL;DR

This paper establishes a H"older estimate for solutions to the Navier-Stokes equations, linking spatial regularity to fluid trajectories and supporting turbulence scaling laws.

Contribution

It proves that certain norms of Navier-Stokes solutions and trajectories can be uniformly estimated, independent of viscosity, in a viscous setting.

Findings

Validated turbulence scaling laws in viscous flows

Established viscosity-independent estimates for solution regularity

Linked spatial regularity to fluid trajectory behavior

Abstract

We study solutions to the Navier-Stokes equations in the class $L^\infty_t C^\alpha_x$. 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 $C^\alpha_{t,x}$-norm of the solution and the $C^{1/(1-\alpha)}$-norm of any fluid trajectory can be estimated by the $L^\infty_tC^\alpha_x$-norm independently of the viscosity parameter $\nu>0$, for times bounded away from zero by a positive power of $\nu$.