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

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.