Global Existence and Exponential Decay of Strong Solutions of Nonhomogeneous Magneto-Micropolar Fluid Equations with Lar
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Journal of Mathematical Fluid Mechanics
Global Existence and Exponential Decay of Strong Solutions of Nonhomogeneous Magneto-Micropolar Fluid Equations with Large Initial Data and Vacuum Xin Zhong Communicated by A. Constantin
Abstract. The present paper concerns an initial boundary value problem of two-dimensional nonhomogeneous magnetomicropolar fluid equations with nonnegative density. We establish the global existence and exponential decay rates of strong solutions. In particular, the initial data can be arbitrarily large. The key idea is to use a lemma of Desjardins (Arch Rational Mech Anal 137:135–158, 1997). Mathematics Subject Classification. 35Q35, 76D03, 76W05. Keywords. Nonhomogeneous magneto-micropolar fluid equations, Global strong solution, Exponential decay, Large initial data.
1. Introduction The three-dimensional (3D for short) nonhomogeneous incompressible magneto-micropolar fluid equations (see [13]) are given by ⎧ ⎪ ρt + div(ρu) = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎨(ρu)t + div(ρu ⊗ u) − (μ1 + ξ)Δu + ∇P = 2ξ∇ × w + b · ∇b, (1.1) bt − νΔb + u · ∇b − b · ∇u = 0, ⎪ ⎪ ⎪ (ρw)t + div(ρu ⊗ w) + 4ξw − μ2 Δw − (μ2 + λ)∇ div w = 2ξ∇ × u, ⎪ ⎪ ⎪ ⎩div u = div b = 0, where ρ, u, w, b, and P denote the density, velocity, micro-rotational velocity, magnetic field, and pressure of the fluid, respectively. The constants μ1 , μ2 , ν, λ, ξ stand for the viscosity coefficients of the fluid satisfying μ1 , μ2 , ν, ξ > 0, μ1 ≥ 2ξ, and 2μ2 + 3λ ≥ 0. In the special case, when ρ = ρ(x1 , x2 , t), u = (u1 (x1 , x2 , t), u2 (x1 , x2 , t), 0), P = P (x1 , x2 , t), b = (b1 (x1 , x2 , t), b2 (x1 , x2 , t), 0), w = (0, 0, w(x1 , x2 , t)),
Supported by National Natural Science Foundation of China (No. 11901474) and the Innovation Support Program for Chongqing Overseas Returnees (No. cx2019130). 0123456789().: V,-vol
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the system (1.1) reduces to the 2D nonhomogeneous magneto-micropolar fluid equations ⎧ ⎪ ρt + div(ρu) = 0, ⎪ ⎪ ⎪ ⊥ ⎪ ⎪ ⎨(ρu)t + div(ρu ⊗ u) − (μ1 + ξ)Δu + ∇P = −2ξ∇ w + b · ∇b, bt − νΔb + u · ∇b − b · ∇u = 0, ⎪ ⎪ ⎪ (ρw)t + div(ρwu) + 4ξw − μ2 Δw = 2ξ∇⊥ · u, ⎪ ⎪ ⎪ ⎩div u = div b = 0,
(1.2)
where u = (u1 , u2 ) is a 2D vector with the corresponding scalar vorticity given by ∇⊥ · u = ∂1 u2 − ∂2 u1 , while w represents a scalar function with ∇⊥ w = (−∂2 w, ∂1 w). Let Ω ⊂ R2 be a bounded smooth domain, and we consider the initial boundary value problem of (1.2) with the initial condition (ρ, ρu, b, ρw)(x, 0) = (ρ0 , ρ0 u0 , b0 , ρ0 w0 )(x), x ∈ Ω,
(1.3)
and the boundary condition u = 0, b = 0, w = 0, x ∈ ∂Ω.
(1.4)
The magneto-micropolar fluid equations introduced by Ahmadi and Shahinpoor [1] is to describe the motion of electrically conducting micropolar fluids in the presence of a magnetic field. Due to the profound physical background and important mathematical significance, a great deal of attention has been focused on studying well-posedness of solutions to the magneto-micropolar fluid equations, both from a pure mathematical point of view and for concrete applications. For more background, we refer to [3
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