Global strong solutions to the Cauchy problem of the 3D heat-conducting fluids

Abstract

In this paper, we study the global strong solutions to the three-dimensional (3D) heat-conducting incompressible Navier–Stokes equations with density-temperature-dependent viscosity and heat-conducting coefficients in ${\mathbb{R}}^3$. By using the t-weighted a priori estimates, we prove the global existence and exponential decay-in-time rates of strong solutions to the Cauchy problem when the $L^{3/2}$-norm of the initial density is suitably small. It should be noted that the velocity and absolute temperature can be large initially, and the initial density contains vacuum case.

Keywords:

• Exponential decay-in-time
• density-temperature-dependent viscosity and heat-conductivity
• vacuum
• global strong solution

Maths:

• 35B40
• 35B45
• 76N10

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The work was supported by Xinjiang University of Finance and Economics Research Fund Project [grant number 2022XYB006].