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

TL;DR

This paper establishes a Kolmogorov-type hypothesis for compressible fluids, leading to uniform bounds and strong convergence of Navier-Stokes solutions to Euler solutions in three dimensions, advancing understanding of inviscid limits.

Contribution

It introduces a new Kolmogorov-type hypothesis for compressible flows and demonstrates its implications for uniform bounds and convergence in the inviscid limit.

Findings

Uniform boundedness of fractional derivatives of velocity and sonic speed.

Strong convergence of Navier-Stokes solutions to Euler solutions.

Framework for mathematical existence and numerical interpretation in high Reynolds number limit.

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.