REMARKS ON THE REGULARITY CRITERIA OF THE SOLUTIONS OF THE 3D MICROPOLAR FLUID EQUATIONS

Qiao Liu

doi:10.11948/2014019

2014 Volume 4 Issue 4

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Qiao Liu. REMARKS ON THE REGULARITY CRITERIA OF THE SOLUTIONS OF THE 3D MICROPOLAR FLUID EQUATIONS[J]. Journal of Applied Analysis & Computation, 2014, 4(4): 355-365. doi: 10.11948/2014019

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Qiao Liu. REMARKS ON THE REGULARITY CRITERIA OF THE SOLUTIONS OF THE 3D MICROPOLAR FLUID EQUATIONS[J]. Journal of Applied Analysis & Computation, 2014, 4(4): 355-365. doi: 10.11948/2014019

REMARKS ON THE REGULARITY CRITERIA OF THE SOLUTIONS OF THE 3D MICROPOLAR FLUID EQUATIONS

Qiao Liu

Department of Mathematics, Hunan Normal University, Changsha, Hunan 410081, China

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Abstract

Abstract

We provide two regularity criteria for the weak solutions of the 3D micropolar fluid equations, the first one in terms of one directional derivative of the velocity, i.e.,∂₃u, while the second one is is in terms of the behavior of the direction of the velocity u/|u|. More precisely, we prove that if ∂₃u ∈ L^β(0, T; L^α(R³)) with 2/β + 3/α ≤ 1 + 1/α, 2 < α ≤ ∞, 2 ≤ β < ∞; or div (u/|u|) ∈ L^4/(1-2r) (0, T; Ẋ_r(R³)) with 0 ≤ r < 1/2, then the weak solution (u(x, t), ω(x, t)) is regular on R³×[0, T]. Here Ẋ_r(R³) is the multiplier space.
- Micropolar fluid equations /
- regularity criteria /
- Navier-Stokes equations /
- weak solution
MSC: 35B65;35Q35

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References

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