The Decay Estimates for Magnetohydrodynamic Equations with Coulomb Force
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Wuhan Institute Physics and Mathematics, Chinese Academy of Sciences, 2020
http://actams.wipm.ac.cn
THE DECAY ESTIMATES FOR MAGNETOHYDRODYNAMIC EQUATIONS WITH COULOMB FORCE∗
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Wenxuan ZHENG (
School of Mathematical Sciences, Xiamen University, Xiamen 361005, China School of Mechanical and Electronic Engineering, Tarim University, Alar 843300, China E-mail : [email protected]
Zhong TAN (
§)
School of Mathematical Sciences and Fujian Provincial Key Laboratory on Mathematical Modeling and Scientific Computing, Xiamen University, Xiamen 361005, China E-mail : [email protected] Abstract In this article we consider the compressible viscous magnetohydrodynamic equations with Coulomb force. By spectral analysis and energy methods, we obtain the optimal time decay estimate of the solution. We show that the global classical solution converges to its equilibrium state at the same decay rate as the solution of the linearized equations. Key words
lower convergence rates; upper decay rates; spectral analysis; energy method
2010 MR Subject Classification
1
76W05; 35Q35; 76X05
Introduction
The compressible viscous magnetohydrodynamic equations with Coulomb force in the isentropic case of the form ∂t ρ˜ + divm ˜ = 0, m ˜ ⊗m ˜ m ˜ m ˜ ˜ ˜ ×B ˜ + ρ˜∇Φ = 0, ˜ + div + ∇P − µ∆ − (µ + λ)∇div − curlB ∂t m ρ˜ ρ˜ ρ˜ ! ˜ m ˜ × B ∂ B ˜ − ν∆B ˜ − curl ˜ = 0, = 0, divB t ρ˜ −∆Φ = ρ˜ − ρ¯. (1.1) describes some physically interesting and important phenomena, such as the dynamics of a charge transport where the compressible electron fluid interacts with its own electric field against ˜ and Φ = a charged ion background under the influence of the magnetic field. Here ρ˜, m, ˜ B Received April 30, 2019; revised July 22, 2020. The first author was supported by the National Natural Science Foundation of China (11271305, 11531010).
No.6
W.X. Zheng & Z. Tan: THE DECAY ESTIMATES FOR MHD EQUATIONS
1929
Φ(x, t) are the density, momentum, magnetic field, and electric potential. The pressure P = P (ρ) is a smooth function with P ′ (ρ) > 0 for ρ > 0. µ, λ are the viscosity coefficients of the flow satisfying µ > 0 and 2µ + 3λ ≥ 0. The constant ν > 0 is the magnetic diffusivity acting as the magnetic diffusion coefficient of the magnetic field. ρ¯ is the positive constant background ionic density. We complement (1.1) with the Cauchy data ˜ ∇Φ)(x, 0) = (˜ ˜0 (x), ∇Φ0 (x)) → (¯ (˜ ρ, m, ˜ B, ρ0 (x), m ˜ 0 (x), B ρ, 0, 0, 0) as |x| → ∞.
(1.2)
When we neglect the viscous terms, that is, λ = ν = µ = 0, (1.1) will become the compressible ideal MHD-Poisson equations, which can describe some important phenomena in astrophysics; e.g. solar flares (cf. [3, 27, 35]). For ideal MHD-Poisson equations, Federbush, Luo and Smoller [11] first proved the existence of axi-symmetric stationary solutions when γ > 2 by a variational method. Wang and Liu [31] proved the existence of stationary solutions when γ > 43 . Jang, Strauss, and Wu [17] proved the existence of axi-symmetric stationary solutions with a
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