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# Dynamics of Partial Differential Equations

## Volume 19 (2022)

### Number 3

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

Pages: 191 – 206

DOI: https://dx.doi.org/10.4310/DPDE.2022.v19.n3.a2

#### Authors

#### 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, this limit is a weak solution of the corresponding Euler equations. We first deduce the Kolmogorov-type hypothesis in $\mathbb{R}^3$, which yields the uniform bounds of $\alpha^\mathit{th}$‑order fractional derivatives of $\sqrt{\rho^\mu} \mathbf{u}^\mu$ in $L^2_x$ for some $\alpha \gt 0$, independent of the viscosity. The uniform bounds can provide strong convergence of $\sqrt{\rho^\mu} \mathbf{u}^\mu$ in $L^2$ space. This shows that the inviscid limit is a weak solution to the corresponding Euler equations.

#### Keywords

inviscid limit, Kolmogorov hypothesis, inhomogeneous Navier–Stokes equations, Euler equations

#### 2010 Mathematics Subject Classification

35D30, 35Q31, 76D05

Cheng Yu is partially supported by Collaboration Grants for Mathematicians from Simons Foundation with award Number: 637792.

Xinhua Zhao is supported by the National Natural Science Foundation of China #12101140 and the Talent Special Project of Guangdong Polytechnic Normal University #99166030406.

Received 10 November 2021

Published 23 May 2022