  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.
