Global Regularity of Three-Dimensional Incompressible Magneto-Micropolar Fluid Equations with Damping
- PDF / 262,537 Bytes
- 7 Pages / 439.37 x 666.142 pts Page_size
- 76 Downloads / 182 Views
Global Regularity of Three-Dimensional Incompressible Magneto-Micropolar Fluid Equations with Damping Yuze Deng1 · Ling Zhou2 Received: 14 July 2020 / Revised: 20 August 2020 / Accepted: 1 September 2020 © Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2020
Abstract We deal with the Cauchy problem of three-dimensional incompressible magnetomicropolar fluid equations with a nonlinear damping term α|u|β−1 u (α > 0 and β ≥ 1) in the momentum equations. By cancelation properties of the system under study, we show that there exists a unique global strong solution for any β ≥ 4. Our work extends previous results. Keywords Incompressible magneto-micropolar fluid equations · Global strong solution · Damping Mathematics Subject Classification 35Q35 · 35B65 · 76N10
1 Introduction Consider the following incompressible magneto-micropolar fluid equations with damping in R3 :
Communicated by Yong Zhou.
B
Ling Zhou [email protected] Yuze Deng [email protected]
1
Institute of Innovation and Entrepreneurship, Southwest University, Chongqing 400715, People’s Republic of China
2
School of Mathematics and Statistics, Southwest University, Chongqing 400715, People’s Republic of China
123
Y. Deng, L. Zhou
⎧ β−1 u + ∇ p = χ ∇ × w + b · ∇b, ⎪ ⎪∂t u − (μ + χ )u + u · ∇u + α|u| ⎪ ⎪ ⎪ ∂t w − γ w + u · ∇w + 2χ w − κ∇ div w = χ ∇ × u, ⎪ ⎪ ⎪ ⎪ ⎨∂t b − νb + u · ∇b − b · ∇u = 0, div u = div b = 0, ⎪ ⎪ ⎪ ⎪ ⎪ u(0, x) = u0 (x), b(0, x) = b0 (x), w(0, x) = w0 (x), ⎪ ⎪ ⎪ ⎪ ⎩ lim |u(t, x)| = lim |b(t, x)| = lim |w(t, x)| = 0. |x|→∞
|x|→∞
(1.1)
|x|→∞
Here, u is the velocity field, w is the micro-rotational velocity, b is the magnetic field, and p is the pressure. In the damping term, α > 0 and β ≥ 1 are constants. μ is the kinematic viscosity, χ is the vortex viscosity, ν1 is the magnetic Reynolds number. κ and γ denote angular viscosities. The prescribed functions u0 (x), b0 (x) are the initial data with div u0 = div b0 = 0. The system (1.1) is related to the following Navier-Stokes equations with damping: ⎧ β−1 u + ∇ p = 0, ⎪ ⎪∂t u − μu + u · ∇u + α|u| ⎪ ⎪ ⎨div u = 0, u(0, x) = u0 (x), ⎪ ⎪ ⎪ ⎪ ⎩ lim |u(t, x)| = 0.
(1.2)
|x|→∞
Eq. (1.2) is studied firstly by Cai and Jiu [1], they showed the existence of a global weak solution for any β ≥ 1 and global strong solutions for β ≥ 27 . Moreover, the uniqueness is shown for any 27 ≤ β ≤ 5. In [8], Zhang et al. proved for β > 3 and u0 ∈ H 1 ∩ L β+1 that the system (1.1) has a global strong solution and the strong solution is unique when 3 < β ≤ 5. Later, Zhou [10] improved the results obtained in [1,8]. He proved that the strong solution exists globally for β ≥ 3 and u0 ∈ H 1 . Moreover, regularity criteria of (1.2) are also established for 1 ≤ β < 3 as follows: if u(t, x) satisfies u ∈ L s (0, T ; L γ ) with
3 2 + ≤ 1, 3 < γ < ∞, s γ
or ∇u ∈ L s˜ (0, T ; L γ˜ ) with
3 2 + ≤ 1, 3 < γ˜ < ∞, s˜ γ˜
then the solution remains smooth on [0, T ]. Recently, Zhong [9] showed the global existence of strong solution to the problem (1.2) under some smalln
Data Loading...
