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

TL;DR

This paper investigates the inviscid limit of inhomogeneous incompressible Navier-Stokes equations in three dimensions under a weak Kolmogorov hypothesis, establishing uniform bounds and strong convergence to Euler solutions.

Contribution

It derives a Kolmogorov-type hypothesis in $R^3$ and proves the strong convergence of solutions to the Euler equations as viscosity vanishes.

Findings

Uniform bounds of fractional derivatives of velocity and density.

Strong convergence of velocity in $L^2$ space.

Inviscid limit solutions are weak solutions to Euler equations.

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 $\alpha^{th}$-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.