Inviscid limit of the inhomogeneous incompressible Navier-Stokes equations under the weak Kolmogorov hypothesis in ${\mathbb{R}}^3$

Abstract

In this paper, we consider the inviscid limit of inhomogeneous incompressible Navier-Stokes equations under the weak Kolmogorov hypothesis in ${\mathbb{R}}^3$. In particular, we first deduce the Kolmogorov-type hypothesis in ${\mathbb{R}}^3$, which yields the uniform bounds of $\hat{I}{\pm }^{th}$-order fractional derivatives of $\sqrt{\ddot{I}{}^{\hat{I}}¼}{\mathbf{u}}^{\hat{I}}¼$ in $L_x^2$ for some $\hat{I}\pm >0$, independent of the viscosity. The uniform bounds can provide strong convergence of $\sqrt{\ddot{I}{}^{\hat{I}}¼}{\mathbf{u}}^{\hat{\mathbf{I}}}¼$ in $L^2$ space. This shows that the inviscid limit is a weak solution to the corresponding Euler equations.