On the global well-posedness and exponential stability of 3D heat conducting incompressible Navier-Stokes equations with temperature-dependent coefficients and vacuum

Jianxia He; Qingyan Li; Jianxia He; Qingyan Li

doi:10.3934/era.2024253

Electronic Research Archive

2024, Volume 32, Issue 9: 5451-5477. doi: 10.3934/era.2024253

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Research article

On the global well-posedness and exponential stability of 3D heat conducting incompressible Navier-Stokes equations with temperature-dependent coefficients and vacuum

Jianxia He ^{1
,
,},
Qingyan Li ²

1.
Center for Nonlinear Studies, School of Mathematics, Northwest University, Xi'an 710127, China
2.
School of Sciences, Chang'an University, Xi'an 710064, China

Received: 01 July 2024 Revised: 05 September 2024 Accepted: 09 September 2024 Published: 26 September 2024

This paper focuses on investigating the initial-boundary value problem of incompressible heat conducting Navier-Stokes equations with variable coefficients over bounded domains in $ \mathbb{R}^3 $, where the viscosity coefficient and heat conduction coefficient are powers of temperature. We obtain the global well-posedness of a strong solution under the assumption that the initial data and the measure of the initial vacuum region are sufficiently small. It is worth mentioning that the initial density is allowed to contain vacuum, and there are no restrictions on the power index of the temperature-dependent viscosity coefficient and heat conductivity coefficient. At the same time, the exponential decay-in-time results are also obtained.
- Navier-Stokes equations,
- degenerate and temperature-dependent transport coefficients,
- strong solution,
- vacuum
Citation: Jianxia He, Qingyan Li. On the global well-posedness and exponential stability of 3D heat conducting incompressible Navier-Stokes equations with temperature-dependent coefficients and vacuum[J]. Electronic Research Archive, 2024, 32(9): 5451-5477. doi: 10.3934/era.2024253

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Abstract

This paper focuses on investigating the initial-boundary value problem of incompressible heat conducting Navier-Stokes equations with variable coefficients over bounded domains in $ \mathbb{R}^3 $, where the viscosity coefficient and heat conduction coefficient are powers of temperature. We obtain the global well-posedness of a strong solution under the assumption that the initial data and the measure of the initial vacuum region are sufficiently small. It is worth mentioning that the initial density is allowed to contain vacuum, and there are no restrictions on the power index of the temperature-dependent viscosity coefficient and heat conductivity coefficient. At the same time, the exponential decay-in-time results are also obtained.

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