On Liouville-type theorem for the stationary compressible Navier–Stokes equations in $  \mathbb{R}^{3}  $

Caifeng Liu; Pan Liu; Caifeng Liu; Pan Liu

doi:10.3934/era.2024019

Electronic Research Archive

2024, Volume 32, Issue 1: 386-404. doi: 10.3934/era.2024019

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Research article

On Liouville-type theorem for the stationary compressible Navier–Stokes equations in $ \mathbb{R}^{3} $

Caifeng Liu ¹,
Pan Liu ^{2
,
,}

1.
School of Mathematics, Northwest University, Xi'an 710127, Shaanxi, China
2.
School of Mathematics and Statistics, Yulin University, Yulin 719000, Shaanxi, China

Received: 15 October 2023 Revised: 06 December 2023 Accepted: 18 December 2023 Published: 28 December 2023

In this paper, we study the Liouville-type theorem for the stationary barotropic compressible Navier–Stokes equations in $ \mathbb{R}^{3} $. Based on a fairly general framework of a kind of local mean oscillations integral and Morrey spaces, we prove that the velocity and the density of the flow are trivial without any integrability assumption on the gradient of the velocity.
- Liouville-type theorem,
- compressible Navier–Stokes equaitons,
- barotropic,
- local mean oscillations integral,
- Morrey spaces
Citation: Caifeng Liu, Pan Liu. On Liouville-type theorem for the stationary compressible Navier–Stokes equations in $ \mathbb{R}^{3} $[J]. Electronic Research Archive, 2024, 32(1): 386-404. doi: 10.3934/era.2024019

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Abstract

In this paper, we study the Liouville-type theorem for the stationary barotropic compressible Navier–Stokes equations in $ \mathbb{R}^{3} $. Based on a fairly general framework of a kind of local mean oscillations integral and Morrey spaces, we prove that the velocity and the density of the flow are trivial without any integrability assumption on the gradient of the velocity.

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