Time-periodic solution to the compressible viscous quantum magnetohydrodynamic model
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Zeitschrift f¨ ur angewandte Mathematik und Physik ZAMP
Time-periodic solution to the compressible viscous quantum magnetohydrodynamic model Ying Yang, Yu Zhou and Qiang Tao Abstract. In this paper, the time-periodic solution to the compressible viscous quantum magnetohydrodynamic model in a periodic domain is studied. Under the boundedness assumption on the external force, we prove the existence of the timeperiodic solution by using the topological degree theory and parabolic regularization method. Furthermore, the uniqueness of the time-periodic solution is shown. Mathematics Subject Classification. 35Q35, 35Q40, 35B10. Keywords. Viscous quantum, Magnetohydrodynamic model, Time-periodic solution, Topological degree theory.
1. Introduction This paper is concerned with the following compressible viscous quantum magnetohydrodynamic (vQMHD) model with friction ⎧ ρt + div(ρu) = 0, ⎪ ⎪ ⎪ √ ⎨ Δ ρ 2 (ρu)t + div(ρu ⊗ u) − μΔu − (μ + λ)divu + ∇P − ρ∇( √ ) = (∇ × B) × B + ρf (x, t), (1.1) ⎪ 2 ρ ⎪ ⎪ ⎩ Bt − ∇ × (u × B) = −∇ × (ν∇ × B), divB = 0. Here u = (u1 , u2 , · · · , uN ) and B = (B1 , B2 , · · · , BN ) are functions with respect to the spatial variable x ∈ Ω = (−L, L)N (N ≥ 1) and the time variable t, which denote the velocity and the magnetic field, respectively. The density of the fluid ρ(x, t) > 0. The pressure P = P (ρ) is supposed to be a smooth ρ) > 0, taking P = αργ for the typical example for some function in a neighborhood of ρ¯ satisfying P (¯ constant α > 0 and γ > 1. The shear and bulk viscosity coefficients μ, λ meet the physical criteria μ > 0 and 2μ + N λ ≥ 0. The constant ν > 0 is the magnetic diffusivity, and represents the Planck constant. The 2 -term can be referred to as the quantum Bohm potential which is a quantum correction to the pressure and satisfies √ Δ ρ |∇ρ|2 ∇ρ ∇ρΔρ ∇ρ · ∇2 ρ 2ρ∇( √ ) = div(ρ∇2 log ρ) = Δ∇ρ + − . (1.2) − ρ ρ2 ρ ρ This term was first introduced for the thermodynamic equilibrium by Wigner [18]. One can find more detailed physical explanation of the model in [5,6]. Moreover, f is a periodic external force term with a period of 2L in space and T in time. The classical compressible MHD has the following basic structure ⎧ ⎪ ⎨ ρt + div(ρu) = 0, (ρu)t + div(ρu ⊗ u) − μΔu − (μ + λ)divu + ∇P = (∇ × B) × B + ρf (x, t), ⎪ ⎩ Bt − ∇ × (u × B) = −∇ × (ν∇ × B), divB = 0. 0123456789().: V,-vol
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It consists of a mass conservation equation, a momentum conservation equation and an induction equation for the electric field. This model has been discussed by many researchers in recent years. Chen et al. [2] investigated the global stability of large solutions to 3D compressible MHD in the whole space. Li and Sun [12] established the existence of global weak solutions of compressible MHD of viscous non-resistive fluids in two-dimensional space. For more results, one can see [3,8,11,13,14,17,24] and the reference therein. Compared with vast literature about classical MHD, there are only a few works about quantum MHD models. Due to strongly nonlinear degenera
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