Kolmogorov-type theory of compressible 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 for compressible fluids in R3. Motivated by the Kolmogorov hypothesis (1941) for incompressible flow, we introduce a Kolmogorov-type hypothesis for barotropic flows, in which the density and the sonic speed normally vary significantly. We then observe that the compressible Kolmogorov-type hypothesis implies the uniform boundedness of some fractional derivatives of the weighted velocity and sonic speed in the space variables in L2, which is independent of the viscosity coefficient µ > 0. It is shown that this key observation yields the equicontinuity in both space and time of the density in Lγ and the momentum in L2, as well as the uniform bound of the density in Lq1 and the velocity in Lq2 for some fixed q1 > γ and q2 > 2, independent of µ > 0, where γ > 1 is the adiabatic exponent. These results lead to the strong convergence of solutions of the Navier–Stokes equations to a solution of the Euler equations for barotropic fluids in R3. 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.