Global Dynamics of 3-D Compressible Micropolar Fluids with Vacuum and Large Oscillations
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Journal of Mathematical Fluid Mechanics
Global Dynamics of 3-D Compressible Micropolar Fluids with Vacuum and Large Oscillations Bingkang Huang, Lvqiao Liu and Lan Zhang Communicated by E. Feireisl
Abstract. This paper deals with the Cauchy problem to three-dimensional compressible viscous and heat-conducting micropolar fluid model. We establish the global existence and uniqueness of classical solution to this specific system with small initial energy. In particular, the small initial energy allows the solution to have large oscillations and initial density may contain a vacuum. Mathematics Subject Classification. 76N10, 35Q35, 35B40, 35D35. Keywords. Compressible micropolar fluids, Vacuum, Large oscillations, Cauchy problem.
1. Introduction The compressible micropolar fluids can describe many phenomena of interactions between macro and micro motions, which extend classical continuum mechanics by taking into account the effects of microstructure present in the medium. Particularly, this system can be used in modeling of clouds with dust and some biological fluids in [11,25]. In this article we are concerned with the three dimensional compressible viscous and heat-conducting micropolar system, which is in the thermodynamical sense perfect and polytropic. The mathematical model of this fluids is formulated by the following equations ⎧ ρt + div(ρu) = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ρ(ut + u · ∇u) = (λ + μ − μr )∇divu + (μ + μr )Δu − ∇P + 2μr rotw, ⎪ ⎪ ⎨ jI ρ(wt + u · ∇w) = 2μr (rotu − 2w) + (c0 + cd − ca )∇divw + (ca + cd )Δw, (1.1) 2 ⎪ 1 ⎪ 2 ⎪ ⎪ rotu − w ρ(et + u · ∇e) = −P divu + λ(divu) + 2μD : D + 4μr ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎩ + c0 (divw)2 + (ca + cd )∇w : ∇wT + (cd − ca )∇w : ∇w + κΔθ, where ρ := ρ(t, x), u := (u1 , u2 , u3 )(t, x), w := (w1 , w2 , w3 )(t, x), e := e(t, x), D := 12 (∇u + ∇uT ) are mass density, velocity, micro-rotation velocity, internal energy, deformation tensor, respectively. Moreover, we Rθ in which θ := θ(t, x) is absolute temperature. assume that pressure P := Rρθ and internal energy e := γ−1 jI is a positive constant which represents micro-inertia density. R is a positive constant and γ > 1 is the adiabatic constant. κ > 0 denotes the heat conduction coefficient. λ and μ are the coefficients of viscosity. μr , c0 , cd and ca are the coefficients of micro-viscosity. These coefficients also satisfy μ > 0, 3λ + 2μ ≥ 0, μr > 0, (1.2) cd > 0, 3c0 + 2cd ≥ 0, ca > 0. Here we study the classical solution to the system (1.1) with the far-field behavior (ρ, u, w, θ)(t, x) → (1, 0, 0, 1), 0123456789().: V,-vol
as
|x| → +∞,
t > 0,
(1.3)
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and initial data (ρ, ρu, ρw, ρθ)(t = 0, x) = (ρ0 , ρ0 u0 , ρ0 w0 , ρ0 θ0 )(x),
x ∈ R3 .
(1.4)
There are a lot of literatures on the mathematical theory of the compressible micropolar fluids. Since the pioneering works on the compressible micropolar fluids in [11,25,26], the micro-continuum model recently has received considerable attention in terms of mathematical analysis. Noticing that when we ignore the effect of the micro-rotation velocity, the system (1.1
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