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 $\alpha^{ th}$-order fractional 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 strong convergence of $\sqrt{ \rho^{ \mu}}\bf u^{ \mu}$ in $L^2$ space. This shows that the inviscid limit is a weak solution to the corresponding Euler equations.
Categories: math.AP