Kolmogorov’s Theory of Turbulence and Inviscid Limit of the Navier-stokes Equations in R3

We are concerned with the inviscid limit of the Navier-Stokes equations to the Euler equations in R. We first observe that a pathwise Kolmogorov hypothesis implies the uniform boundedness of the α-order fractional derivatives of the velocity for some α > 0 in the space variables in L, which is independent of the viscosity μ > 0. Then it is shown that this key observation yields the Lequicontinuity in the time variable and the uniform bound in L, for some q > 2, of the velocity independent of μ > 0. These results lead to the strong convergence of solutions of the Navier-Stokes equations to a solution of the Euler equations in R . 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 μ > 0, that is in the high Reynolds number limit.