The Boundary Layer Problem of MHD System with the Non-characteristic Dirichlet Boundary Condition for Velocity
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The Boundary Layer Problem of MHD System with the Non-characteristic Dirichlet Boundary Condition for Velocity Na Wang1 · Shu Wang2
Received: 28 August 2018 / Accepted: 14 October 2019 © Springer Nature B.V. 2019
Abstract In this paper, we study the boundary layer problem for the two-dimensional incompressible MHD system with the non-characteristic Dirichlet boundary condition for the velocity and the perfect conducting wall boundary condition for the magnetic field. Using multiscale analysis and the elaborate energy methods, we rigorously prove that the solutions of the viscous and diffusive MHD system are approximated by the inner solutions with the boundary layers in the sense of L2 norm when the viscosity and diffusion coefficient tend to zero. Keywords Incompressible MHD system · Boundary layer · Non-characteristic Dirichlet boundary condition Mathematics Subject Classification (2010) 35Q35 · 76W05
1 Introduction In this paper, we study the following two-dimensional incompressible MHD system in the domain Ω = {(x, y)| x ∈ T, y ∈ (0, h)}: ⎧ ε ∂t u − εuε + uε · ∇uε − bε · ∇bε + ∇p ε = 0, (x, y, t) ∈ Ω × (0, T ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ∂t bε − εbε + uε · ∇bε − bε · ∇uε = 0, (x, y, t) ∈ Ω × (0, T ) (1.1) ⎪ ∇ · bε = 0, (x, y, t) ∈ Ω × (0, T ) ∇ · uε = 0, ⎪ ⎪ ⎪ ⎪ ⎩ ε = u0 (x, y), bε = b0 (x, y), (x, y) ∈ Ω u t=0
t=0
B N. Wang
[email protected] S. Wang [email protected]
1
College of Mathematics and Information Science, Hebei Normal University, Shijiazhuang, Hebei 050024, P.R. China
2
College of Applied Sciences, Beijing University of Technology, Beijing 100124, P.R. China
N. Wang, S. Wang
with the boundary conditions: ε u = (0, U ), u = (0, U ), ε
∂y b1ε = 0, b2ε = 0, b = 0, ε
y = 0, t ∈ (0, T )
y = h, t ∈ (0, T )
(1.2)
where uε represents the velocity, bε magnetic field, and p ε pressure, respectively. Here, the viscosity coefficient and magnetic diffusion coefficient are assumed to be the same, denoted by ε. u0 and b0 are the initial data independent of ε, and U < 0 is a constant. We assume that all the functions appeared in the paper are L-periodic with respect to the horizontal direction x. Thus we can rewrite Ω = (0, L) × (0, h). In this paper, we consider the case 0 < h < ∞. Of course, our result can be extended to the case h = ∞, where the velocity uε and magnetic field bε will decay to 0 at y → ∞. The well-posedness and regularity of the incompressible viscous and diffuse MHD system (1.1)–(1.2) have been investigated extensively, see [4, 5, 12, 15, 16] etc. In this paper, we focus on studying the zero viscosity-diffusion limit problem of the system (1.1)–(1.3). Setting ε = 0 in (1.1), we can formally obtain the ideal MHD system ⎧ 0 ∂t u + u0 · ∇u0 − b0 · ∇b0 + ∇p 0 = 0, (x, y, t) ∈ Ω × (0, T ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ∂t b0 + u0 · ∇b0 − b0 · ∇u0 = 0, (x, y, t) ∈ Ω × (0, T ) (1.3) ⎪ ⎪ ∇ · u0 = 0, ∇ · b0 = 0, (x, y, t) ∈ Ω × (0, T ) ⎪ ⎪ ⎪ ⎪ ⎩ 0 u = u0 (x, y), b0 = b0 (x, y), (x, y) ∈ Ω t=0
t=0
Since the boundaries are non-characteristic (permeable, i.e. U = 0), in order to keep the well-
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