A New Boundary Condition for the Hall-Magnetohydrodynamics Equation with the Ion-Slip Effect
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Journal of Mathematical Fluid Mechanics
A New Boundary Condition for the Hall-Magnetohydrodynamics Equation with the Ion-Slip Effect Woo Jin Han
and Hyung Ju Hwang
Communicated by G. P. Galdi
Abstract. In this paper, we prove a regularity criteria and the local well-posedness of strong solutions to the magnetohydrodynamics with the Hall and ion-slip effects in a smooth bounded domain. Moreover, for the magnetohydrodynamics with the Hall effect and without ion-slip effect, we prove the global stability of strong large solutions. These results are obtained under an integrable property that depends on a new vorticity boundary condition which is motivated by the generalized Navier-slip boundary condition involving the vorticity. Mathematics Subject Classification. 35Q30, 35Q35, 35Q85, 76W05. Keywords. Magnetohydrodynamics, Hall-MHD, Ion-slip effect, Regularity criteria, Stability.
1. Introduction Let Ω ∈ R3 be a smooth bounded domain. In this paper, We consider the Hall-magnetohydrodynamics (Hall-MHD) equation with or without ion-slip effect in Ω. ∂t u − Δu =Nu (u, B, Π), ∂t B − ΔB =NB (u, B),
∇·u=0
in Ω,
(1.1)
∇·B =0
in Ω,
(1.2)
where Nu (u, B, Π) = −∇ × u × u + ∇ × B × B − ∇Π,
(1.3)
NB (u, B) = ∇ × (u × B) − ∇ × (∇ × B × B) − γ∇ × (B × (∇ × B × B)) ,
(1.4)
with the following boundary conditions u · n = 0,
∇ × u · n = 0,
Δu × n = 0
B · n = 0,
∇ × B · n = 0,
ΔB × n = 0
in ∂Ω, in ∂Ω.
(1.5) (1.6)
and the initial conditions u0 = u(0, x),
B0 = B(0, x)
in Ω.
(1.7)
u, B, Π, n and γ stand for the velocity of the fluid, the magnetic field, the pressure, the unit normal vector of the boundary and a nonnegative ion-slip coefficient respectively. In recent years, MHD has caught a great deal of attention by physicists and mathematicians due to its physical importance, complexity, rich phenomena, and mathematical challenges. The applications of MHD cover a very broad range of problems in geophysics, astrophysics, cosmology, sensors, engineering, and magnetic drug targeting. For instance, MHD is used in devices of electromagnetic stirring, nuclear reactors, and plasma confinement. MHD equations are the mathematical model which is a combination Fully documented templates are available in the elsarticle package on CTAN. 0123456789().: V,-vol
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W. J. Han, H. Ju Hwang
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of the Navier-Stokes equations and the Maxwell equations, for the low-frequency interaction between electrically conducting fluids and electromagnetic fields. There are many introductory books for MHD [10,17,24]. The Hall and ion-slip effect terms are written as −∇ × (∇ × B × B) and −γ∇ × (B × (∇ × B × B)) in (1.4), respectively. These two terms make (1.1)–(1.7) become a quasilinear problem, and make the problem more difficult than the general MHD system or the Navier-Stokes equations which are semilinear. For γ = 0, it recovers the Hall-MHD equations, which was derived in [1] from the two-fluid isothermal Euler-Maxwell system for the electrons and ions and a kinetic equation for the ion distribution function of the plasma. An important w
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