On the Existence of Dissipative Measure-Valued Solutions to the Compressible Micropolar System
- PDF / 385,144 Bytes
- 16 Pages / 547.087 x 737.008 pts Page_size
- 2 Downloads / 182 Views
Journal of Mathematical Fluid Mechanics
On the Existence of Dissipative Measure-Valued Solutions to the Compressible Micropolar System Bingkang Huang Communicated by E. Feireisl
Abstract. Compressible micropolar equations model a class of fluids with microstructure. In this paper we establish the dissipative measure-valued solution to the micropolar fluids. We also give the weak-strong uniqueness principle to this system which means its dissipative measure-valued solution is the same as the classical solution, provided they emanate from the same initial data. Keywords. Micropolar fluid flow, Measure-valued solution, Weak-strong uniqueness.
1. Introduction and Main Result The theory of viscous compressible micropolar system, introduced by [8] to describe the certain microscopic effects in some complex fluids, is an extension of the classical continuum mechanics. Such motion of micropolar system is formulated by mathematical model in Eulerian coordinates as follows: ⎧ ρ + div(ρu) = 0, ⎪ ⎨ t (ρu)t + div(ρu ⊗ u) + ∇p(ρ) = (μ + ξ)Δu + (μ + λ − ξ)∇divu + 2ξ∇ × w, (1.1) ⎪ ⎩ (ρw)t + div(ρu ⊗ w) + 4ξw = μ Δw + (μ + λ )∇divw + 2ξ∇ × u. The initial conditions and boundary conditions are prescribed by (ρ, u, w)|t=0 (t, x) = (ρ0 , u0 , w0 ) (x),
f or
x ∈ Ω ⊂ R3 ,
(u, w)|∂Ω (t, x) = 0.
(1.2)
Here, the unknown functions ρ(t, x), u(t, x), w(t, x) and p(ρ) denote density, velocity, microrotational velocity and pressure, respectively. The coefficients of viscosity μ, λ and the coefficients of microviscosity μ , λ , ξ satisfy μ, μ , ξ > 0,
2μ + 3λ ≥ 0 and
2μ + 3λ ≥ 0.
(1.3)
Different from the classical Navier–Stokes quations, micropolar fluid system exhibits micro-rotational effects and micro-rotational inertia. This model is a significant generalization of the Navier–Stokes equations and has been extensively studied in [13]. There are many meaningful researches on the incompresssible micropolar equations. The incompressible flow has been well studied in [5,24,25]. Particularly, the existence and uniqueness theorems for the incompressible micropolar equations were given by [10]. The existence of weak solutions to the incompressible magneto-micropolar fluid system was proved in [22]. Later on, the existence and uniqueness of strong solutions to the incompressible micropolar fluid system were also given by [20]. Extensive attentions were also paid to the compressible micropolar fluid system. In the one-dimensional setting, the compressible, viscous and heat-conducting micropolar system was considered in [17]. Moreover, the global existence of weak solutions to the three-dimensional compressible micropolar fluids with initial data which may be discontinuous and may contain vacuum 0123456789().: V,-vol
59
Page 2 of 16
B. Huang
JMFM
states was analyzed in [3]. We mention that the existence and large time behavior of strong solutions to a three-dimensional compressible micropolar fluid system was also concerned by [12]. Furthermore, there is a large literature on weak and strong solutions to the micropolar fluid system in [1,6,7,23
Data Loading...
