Vanishing viscosity limit of the 2D micropolar equations for planar rarefaction wave to a Riemann problem
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Zeitschrift f¨ ur angewandte Mathematik und Physik ZAMP
Vanishing viscosity limit of the 2D micropolar equations for planar rarefaction wave to a Riemann problem Guiqiong Gong and Lan Zhang
Abstract. We are concerned with the vanishing viscosity limit of the 2D compressible micropolar equations to the Riemann solution of the 2D Euler equations which admit a planar rarefaction wave. In this article, the key point of the analysis is to introduce the hyperbolic wave, which helps us obtain the desired uniform estimates with respect to the viscosities. Moreover, the proper combining of rotation terms and damping term is also important, which contributes to closing the basic energy estimates. Finally, a family of smooth solutions for the 2D micropolar equations converging to the corresponding planar rarefaction wave solution with arbitrary strength is pursued. Mathematics Subject Classification. 35L65 and 35Q35 and 76N10. Keywords. Micropolar system, Planar rarefaction wave, Vanishing viscosity limit.
Contents 1. 2.
Introduction and main result Construction of the solution profile 2.1. Smooth approximate rarefaction wave 2.2. Hyperbolic wave 2.3. Approximate solution profile 3. Reformulation of the problem 4. A priori estimates Acknowledgements References
1. Introduction and main result In this article, we consider the compressible, viscous, micropolar fluid model which describes the microrotational and spin inertia. This model is a significant generalization of the Navier–Stokes equations and has been extensively studied. The 3D micropolar equations are described as follows: ⎧ ⎨ ρt + div(ρu) = 0, (ρu)t + div(ρu ⊗ u) + ∇p(ρ) = (μ1 + ζ1 )Δu + (μ1 + λ1 − ζ1 )∇divu + 2ζ1 rotw, (1.1) ⎩ (ρw)t + div(ρu ⊗ w) + 4ζ1 w = μ2 Δw + (μ2 + λ2 )∇divw + 2ζ1 rotu, where, the unknowns are ρ(t, x), u(t, x), w(t, x), and P (ρ) for t ≥ 0, x = (x1 , x2 , x3 ) ∈ Ω ⊂ R3 , denoting, respectively, the fluid density, velocity, microrotational velocity, and pressure, respectively. The 0123456789().: V,-vol
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G. Gong and L. Zhang
ZAMP
shear viscosity μ1 , bulk viscosity λ1 , microrotation viscosity ζ1 and angular viscosities μ2 , λ2 are positive constants satisfying (1.2) 2μ1 + 3λ1 − 4ζ1 ≥ 0, 2μ2 + 3λ2 ≥ 0. Here we consider the ideal polytropic gas, that is, the pressure p is given by p = aργ ,
(1.3)
where a > 0 is the gas constant and γ > 1 is the adiabatic exponent. The system (1.1) can reduce to the 2D micropolar equations when we let ρ = ρ(x, t),
u = (u1 (x, t), u2 (x, t), 0),
w = (0, 0, w3 (x, t)),
x = (x1 , x2 ).
More precisely, the two-dimensional micropolar equations can be written as follows ⎧ ⎨ ρt + div(ρu) = 0, (ρu)t + div(ρu ⊗ u) + ∇p(ρ) = (μ1 + ζ1 )Δu + (μ1 + λ1 − ζ1 )∇divu + 2ζ1 rotw, ⎩ (ρw)t + div(ρu ⊗ w) + 4ζ1 w = μ2 Δw + 2ζ1 rotu,
(1.4)
where rotw = (∂2 w, −∂1 w) and Ω = rotu = ∂1 u2 − ∂2 u1 . Moreover, we consider the spatial domain as (x1 , x2 ) ∈ R × T. Then, we define the vanishing parameter ε > 0 such that μ1 = με,
λ1 = λε,
μ2 = μ ε,
ζ1 = ζε,
λ2 = λ ε.
(1.5)
The system (1.4) is supplemented wi
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