三维粘性系数依赖于密度的不可压缩热传导Navier-Stokes方程的全局强解

Global Strong Solution for 3D Viscous Incompressible Heat Conducting Navier-Stokes Equations with Density-Dependent Viscosity

摘要: 本文研究了三维粘性系数依赖于密度的非齐次不可压缩热传导Navier-Stokes方程。首先，当粘性系数的梯度的范数满足${\Vert \nabla \mu \left(\rho \right)\Vert }_{L^{\infty }\left(0,T;L^p\right)}<\infty$ 时，存在一个整体强解，此外，如果初始能量适当小，证明了三维粘性非齐次热传导变粘性Navier-Stokes方程整体强解的唯一性。

Abstract: In this paper, we investigate an 3D viscosity incompressible heat conducting Navier-Stokes equations with density-dependent viscosity. First, we obtain that there exists a global strong solution provided the norm of the gradient of viscosity satisfies ${\Vert \nabla \mu \left(\rho \right)\Vert }_{L^{\infty }\left(0,T;L^p\right)}<\infty$ . Moreover, if energy is suitably small, we show the uniqueness of the global strong solution to the three-dimensional viscous non-homogeneous heat conducting Navier-Stokes equations with variable viscosity.