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In this paper, we consider the inviscid limit of inhomogeneous incompressibleNavier-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 $\alpha^{th}$-orderfractional derivatives of $\sqrt{\rho^\mu}{\bf u}^\mu $ in $L^2_x$ for some$\alpha>0$, independent of the viscosity. The uniform bounds can provide strongconvergence of $\sqrt{\rho^{\mu}}\bf u^{\mu}$ in $L^2$ space. This shows thatthe inviscid limit is a weak solution to the corresponding Euler equations.

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