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.