Kolmogorov-type theory of compressible turbulence and inviscid limit of the Navier-Stokes equations in R3

[摘要] We are concerned with the inviscid limit of the Navier-Stokes equations to the Euler equations for compressible fluids in 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-q1 and the velocity in L-q2 for some fixed q(1) > gamma and q(2) > 2, independent of mu > 0, 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 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. (C) 2019 Elsevier B.V. Ail rights reserved.

[发布日期] 2019-12-15

[关键词] Inviscid limit;Navier-Stokes equations;Compressible Kolmogorov-type hypotheses;Compressible turbulence;Existence;Euler equations