Attractors of the 3D Magnetohydrodynamics Equations with Damping
- PDF / 329,031 Bytes
- 15 Pages / 439.37 x 666.142 pts Page_size
- 43 Downloads / 202 Views
Attractors of the 3D Magnetohydrodynamics Equations with Damping Hui Liu1 · Chengfeng Sun2 · Jie Xin1 Received: 15 November 2019 / Revised: 16 March 2020 © Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2020
Abstract The three-dimensional magnetohydrodynamics equation with damping is considered in this paper. Global attractor of the 3D magnetohydrodynamics equations with damping is proved for 4 ≤ β < 5 with any α > 0. Keywords Magnetohydrodynamics equations · Damping · Global attractor Mathematics Subject Classification 76W05 · 35B40 · 35B41
1 Introduction In this paper, we consider the following three-dimensional magnetohydrodynamics (MHD) equations with damping: ⎧ β−1 u + ∇( p + ⎪ ⎪ ∂t u − νu + (u · ∇)u − (b · ∇)b + α|u| ⎪ ⎪ ⎨ ∂t b − κb + (u · ∇)b − (b · ∇)u = f 2 (x), ∇ · u = 0, ∇ · b = 0, ⎪ ⎪ ⎪ ⎪ u|∂ D = b|∂ D = 0, ⎩ u|t=0 = u 0 , b|t=0 = b0 ,
|b|2 2 )
= f 1 (x), (1.1)
Communicated by Yong Zhou. The work is supported by the Natural Science Foundation of Shandong Province under Grant No. ZR2018QA002 and No. ZR2013AM004 and a China NSF Grant No. 11901342, No. 11701269 and No. 11371183, and China Postdoctoral Science Foundation No. 2019M652350.
B
Hui Liu [email protected]
1
School of Mathematical Sciences, Qufu Normal University, Qufu, Shandong 273165, People’s Republic of China
2
School of Applied Mathematics, Nanjing University of Finance and Economics, Nanjing 210023, People’s Republic of China
123
H. Liu et al.
where D ⊆ R3 is a bounded domain with the boundary ∂ D and t > 0. u, b are the fluid velocity and magnetic field, respectively. f 1 (x), f 2 (x) are the external body force. p is the pressure, β ≥ 1 is constant, and α is the damping coefficient. The constants ν, κ ≥ 0 are kinematic viscosity and magnetic resistivity. For simplicity, we set ν = κ = 1. The magnetohydrodynamic model has been investigated by many authors. In [12], Sermange and Temam have proved the well-posedness of solutions for MHD system in a bounded and a periodic domain. At the same time, regularity properties and attractors were also obtained. In [1], the pullback attractors and well-posedness of solutions of 2D MHD system were proved by using the Galerkin method. Based on the well-posedness of strong solutions of 3D MHD system that is difficult problem, many authors in [3,6,18,19,21] have studied the attractors and invariant measures of solutions of 3D-modified MHD system. Our result improves the early results in [21]. The damping term is very important for proving the well-posedness of 3D MHD system. In previous years, well-posedness and regularity of solutions of 3D Navier–Stokes system with damping were proved in [2,20]. In [4,5,7–9,11], the global existence of strong solution for 3D Navier–Stokes system with damping was proved for β > 3 with any α > 0 and α ≥ 41 as β = 3. Moreover, the global well-posedness of the 3D magnetohydrodynamics equations with damping was proved for β ≥ 4 with any α > 0 in [16]. Based on [10], global well-posedness of the 3D magneto–micropolar equat
Data Loading...
