Existence and uniqueness of time periodic solutions to the compressible magneto-micropolar fluids in a periodic domain
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Zeitschrift f¨ ur angewandte Mathematik und Physik ZAMP
Existence and uniqueness of time periodic solutions to the compressible magnetomicropolar fluids in a periodic domain Xinli Zhang and Hong Cai
Abstract. We consider the existence and uniqueness of a time periodic solution for the compressible magneto-micropolar fluids with time periodic forces in a periodic domain. More precisely, under some smallness and symmetry assumptions on the external forces, we prove the existence of the periodic solution by a regularized approximation scheme and the topological degree theory. The uniqueness of the periodic solution is obtained by energy estimates. Mathematics Subject Classification. 35Q30, 35Q35, 35B10, 76N10. Keywords. Time periodic solution, Compressible magneto-micropolar fluids, Topological degree theory, Energy estimates.
1. Introduction In this paper, we investigate the time periodic solution to the following three-dimensional compressible magneto-micropolar fluids ⎧ ρt + ∇ · (ρu) = 0, ⎪ ⎪ ⎪ ⎪ (ρu) ⎪ t + ∇ · (ρu ⊗ u) + ∇P (ρ) = (μ + ν)u + (μ + λ − ν)∇divu + 2ν∇ × ω ⎪ ⎨ +(∇ × H) × H + ρf, (1.1) + ∇ · (ρu ⊗ ω) + 4νω = μ ω + (μ + λ )∇divω + 2ν∇ × u + ρg, (ρω) ⎪ t ⎪ ⎪ ⎪ Ht − ∇ × (u × H) = −∇ × (σ∇ × H), ⎪ ⎪ ⎩ ∇ · H = 0. Here, the fluid density ρ(x, t) > 0, the fluid velocity field u(x, t) = (u1 , u2 , u3 )(x, t) ∈ R3 , the microrotational velocity ω(x, t) = (ω1 , ω2 , ω3 )(x, t) ∈ R3 , the magnetic field H(x, t) = (H1 , H2 , H3 )(x, t) ∈ R3 are unknown functions for x ∈ Ω := (−L, L)3 . The pressure P (ρ) is a smooth function in a neighborhood of ρ¯ satisfying P (¯ ρ) > 0 with a given constant ρ¯ > 0. Moreover, the constants μ > 0 and λ are the shear and bulk viscosity coefficients of the flow, and the constant ν > 0 is the dynamics microrotation viscosity satisfying the physical condition 2μ + 3λ − 4ν ≥ 0. The two constants μ > 0 and λ are the angular viscosity coefficients satisfying 2μ + 3λ ≥ 0. The constant σ > 0 is the resistivity coefficient, which is inversely proportional to the electrical conductivity constant. In addition, the given external forces f (x, t) = (f1 , f2 , f3 )(x, t) and g(x, t) = (g1 , g2 , g3 )(x, t) are periodic both in space and time with period 2L and T > 0 satisfying f (x, t) = −f (−x, t), 0123456789().: V,-vol
g(x, t) = −g(−x, t).
(1.2)
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X. Zhang and H. Cai
ZAMP
1.1. Known results Many practical applications, which have an inherent interest of physical and mathematical nature, involve the hydrodynamic flow in the presence of a magnetic field. Compressible magnetic-micropolar fluids is a mathematical model which can be used to describe the motion of aggregates of small solid ferromagnetic particles relative to viscous magnetic fluids, such as water, hydrocarbon, ester, fluorocarbon, etc., in which they are immersed, covers a wide range of heat and mass transfer phenomena, under the action of magnetic fields, and are of great importance in practical and mathematical applications [2,10,19,20]. Because of its mathematical challenge and physical importance, many research
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