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

Dixi Wang (Department of Mathematics, University of Florida, Gainesville, Fl., U.S.A.)

Cheng Yu (Department of Mathematics, University of Florida, Gainesville, Fl., U.S.A.)

Xinhua Zhao (School of Mathematics and Systems Science, Guangdong Polytechnic Normal University, Guangzhou, China)

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

Received 10 November 2021

Published 23 May 2022