Global Existence of Heat-Conductive Incompressible Viscous Fluids
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Global Existence of Heat-Conductive Incompressible Viscous Fluids Xia Ye1
Received: 30 April 2015 / Accepted: 23 October 2016 © Springer Science+Business Media Dordrecht 2016
Abstract In this paper, we consider the Cauchy problem of non-stationary motion of heatconducting incompressible viscous fluids in R2 , where the viscosity and heat-conductivity coefficient vary with the temperature. It is shown that the Cauchy problem has a unique global-in-time strong solution (u, θ )(x, t) on R2 × (0, ∞), provided the initial norm ∇u0 L2 is suitably small, or the lower-bound of the coefficient of heat conductivity (i.e. κ) is large enough, or the derivative of viscosity (i.e. |μ (θ )|) is small enough. Keywords Heat-conducting fluids · local existence · global existence
1 Introduction We consider the Cauchy problem for the system of PDE’s modeling the motion of a heatconducting incompressible viscous fluid: ⎧ ut + (u · ∇)u − div(μ(θ )∇u) + ∇P = 0, ⎪ ⎪ ⎪ ⎨θ + (u · ∇)θ − div(κ(θ )∇θ ) − μ(θ )|∇u|2 = 0, t (1.1) ⎪divu = 0, ⎪ ⎪ ⎩ u|t=0 = u0 , θ |t=0 = θ0 , with x ∈ R2 and t > 0. Here, the unknowns u, θ , P denote the velocity, temperature and pressure of the fluid, respectively. μ(θ ) is the coefficient of viscosity, κ(θ ) is the coefficient of heat conductivity, and they are functions of the temperature, which are assumed to satisfy μ(·), κ(·) ∈ C 3 [0, ∞ ), 0 < μ ≤ μ(θ ) ≤ μ, ¯ |μ (θ )| ≤ μ1 , 0 < κ ≤ κ(θ ) ≤ κ, ¯
|κ (θ )| ≤ κ1 ,
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