Decay Rates of the Compressible Hall-MHD Equations for Quantum Plasmas
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Decay Rates of the Compressible Hall-MHD Equations for Quantum Plasmas Xiaoyu Xi1
· Xueke Pu1 · Boling Guo2
Received: 2 January 2020 / Accepted: 17 June 2020 © Springer Nature B.V. 2020
Abstract In this paper, decay rates of the compressible Hall-MHD equations for quantum plasmas in three-dimensional whole space are studied. By using a general energy method, the time decay rates for higher-order spatial derivatives of density, velocity and magnetic field are established when the initial perturbation belongs to H˙ −s with 0 ≤ s < 32 . Keywords Decay rates · Hall-MHD equations · Quantum plasmas · Energy method Mathematics Subject Classification 35M31 · 35Q35
1 Introduction The compressible viscous Hall-MHD equations for quantum plasmas can be written as ⎧ ρt + div(ρu) = 0, ⎪ ⎪ ⎪ √ ⎪ ⎪ ρ 2 ⎪ ⎪ ⎪ ⎨ (ρu)t + div(ρu ⊗ u) − μu − (μ + λ)∇divu + ∇P − 2 ρ∇ √ρ = (∇ × B) × B, ⎪ (∇ × B) × B ⎪ ⎪ Bt − ∇ × (u × B) + ∇ × = −∇ × (ν∇ × B), divB = 0, ⎪ ⎪ ⎪ ρ ⎪ ⎪ ⎩ (ρ, u, B)(x, t)|t=0 = (ρ0 (x), u0 (x), B0 (x)), (1)
B X. Xi
[email protected] X. Pu [email protected] B. Guo [email protected]
1
School of Mathematics and Information Science, Guangzhou University, Guangzhou, 510006, P.R. China
2
Institute of Applied Physics and Computational Mathematics, P. O. Box 8009, Beijing, 100088, P.R. China
X. Xi et al.
where (x, t) ∈ R3 × R+ denotes the space-time position. The unknown functions ρ, u and B represent the density, velocity, and magnetic field, respectively. The pressure P (ρ) is a smooth function in a neighborhood of 1. ≥ 0 denotes the Planck constant and ν is the magnetic diffusivity. The constants μ > 0 and λ are the viscosity coefficients with the usual phys , ical condition 3λ + 2μ ≥ 0. ≥ 0 is called Hall coefficient. The Hall term ∇ × (∇×B)×B ρ which is the key to understand the problem of magnetic reconnection in space plasmas, star formulation, neutron stars, and geo-dynamo, due to the Ohm’s law (see [1, 8, 18, 29, 36, 41] √ ρ and the references therein). The expression √ρ can be interpreted as a quantum potential, i.e. Bohm potential, which satisfies 2ρ∇
√ ρ |∇ρ|2 ∇ρ ∇ρρ ∇ρ · ∇ 2 ρ − . − = div(ρ∇ 2 logρ) = ∇ρ + √ ρ ρ2 ρ ρ
The quantum correction terms, which are closely related to Bohm potential, can be tracked back to Wigner [42] for the thermodynamic equilibrium. For more physical interpretations of the model, one may refer to Hass [13, 14]. Furthermore, we assume ¯ ¯ u0 − u, ¯ B0 − B)(x) = 0, lim (ρ0 − ρ,
(2)
|x|→∞
¯ = (1, 0, 0) is the constant state. In particular, when = 0, (1) become the where (ρ, ¯ u, ¯ B) classical compressible quantum magnetohydrodynamic equations. There is much literature concerned with the decay of the solutions for the fluid dynamics. For the convergence rates for the compressible Navier-Stokes equations with or without external forces, see, for example, the works in [3–6, 12, 16–22, 25–30]. For higher-order spatial derivatives of solutions, Guo and Wang [12] employed a general energy method to build the time convergence rates by assuming the init
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