Kolmogorov-Type Theory of Compressible Turbulence and Inviscid Limit of the Navier-Stokes Equations in ${\mathbb{R}}^3$

Abstract

We are concerned with the inviscid limit of the Navier-Stokes equations to the Euler equations for compressible fluids in ${\mathbb{R}}^3$. 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 $L^2$, which is independent of the viscosity coefficient $\mu >0$. It is shown that this key observation yields the equicontinuity in both space and time of the density in $L^{\gamma }$ and the momentum in $L^2$, as well as the uniform bound of the density in $L^{q_1}$ and the velocity in $L^{q_2}$ independent of $\mu >0$, for some fixed $q_1>\gamma$ and $q_2>2$, where $\gamma >1$ 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 ${\mathbb{R}}^3$. 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.

20 pages. arXiv admin note: text overlap with arXiv:1008.1546

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