Kolmogorov's Theory of Turbulence and Inviscid Limit of the Navier-Stokes Equations in $\R^3$

TL;DR

This paper establishes conditions under which solutions of the Navier-Stokes equations in three dimensions converge to Euler solutions as viscosity vanishes, using Kolmogorov's turbulence theory and fractional derivatives.

Contribution

It links Kolmogorov's hypothesis to uniform bounds and convergence results for Navier-Stokes solutions in the inviscid limit, providing a new analytical framework.

Findings

Uniform boundedness of fractional derivatives of velocity independent of viscosity

Strong convergence of Navier-Stokes solutions to Euler solutions in D

Weak-star convergence of passive scalars with Young measure representation

Abstract

We are concerned with the inviscid limit of the Navier-Stokes equations to the Euler equations in $\R^3$. We first observe that a pathwise Kolmogorov hypothesis implies the uniform boundedness of the $\alpha^{th}$-order fractional derivative of the velocity for some $\alpha>0$ in the space variables in $L^2$, which is independent of the viscosity $\mu>0$. Then it is shown that this key observation yields the $L^2$-equicontinuity in the time and the uniform bound in $L^q$, for some $q>2$, of the velocity independent of $\mu>0$. These results lead to the strong convergence of solutions of the Navier-Stokes equations to a solution of the Euler equations in $\R^3$. We also consider passive scalars coupled to the incompressible Navier-Stokes equations and, in this case, find the weak-star convergence for the passive scalars with a limit in the form of a Young measure (pdf depending on space and time). Not only do we offer a framework for mathematical existence theories, but also we offer a framework for the interpretation of numerical solutions through the identification of a function space in which convergence should take place, with the bounds that are independent of $\mu>0$, that is in the high Reynolds number limit.