Global strong solutions to the 3D inhomogeneous heat-conducting incompressible fluids

ABSTRACT

This paper concerns the global regularity to an initial boundary value problem for the three-dimensional (3D) inhomogeneous heat-conducting fluids with density and absolute temperature-dependent viscosity. Let $u_0$ be the initial velocity. Through some time-weighted a priori estimates, we establish global existence of strong solutions for positive initial density under assumption that $\Vert u_0{\Vert }_{L^2}$ is small. For the case when initial vacuum is allowed, we show that strong solutions globally exist provided $\Vert \mathrm{\nabla }{u}_0{\Vert }_{L^2}$ small. It is worth pointing out that the initial temperature can be arbitrarily large.

KEYWORDS:

• Global regularity
• heat-conducting fluid
• density and temperature-dependent viscosity
• non-vacuum
• vacuum

AMS SUBJECT CLASSIFICATIONS:

• 35B40
• 35B45
• 76D03
• 76D05

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work is supported by Natural Science Foundation of Fujian Province of China [grant number 2015J01582], [grant number JAT160026].