Uniform Regularity of the Density-Dependent Incompressible MHD System in a Bounded Domain
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Uniform Regularity of the Density-Dependent Incompressible MHD System in a Bounded Domain Jishan Fan1 · Yong Zhou2,3 Received: 27 April 2020 / Accepted: 5 October 2020 / © Springer Nature B.V. 2020
Abstract In this paper, we prove the uniform-in-η estimates of the local strong solutions of the density-dependent incompressible MHD system in a bounded domain. Here η is the resistivity coefficient. Keywords MHD · Uniform regularity · Bounded domain Mathematics Subject Classification (2010) 35Q30 · 35Q35 · 76D03
1 Introduction Magnetohydrodynamics (MHD) studies the interaction of electromagnetic fields and conducting fluids. In this paper, we consider the following density-dependent incompressible MHD system: ∂t ρ + div (ρu) = 0,
1 2 ∂t (ρu) + div (ρu ⊗ u) + ∇ π + |b| − μu = (b · ∇)b, 2 ∂t b + u · ∇b − b · ∇u = ηb, div u = div b = 0 in × (0, ∞),
Yong Zhou
[email protected] Jishan Fan [email protected] 1
Department of Applied Mathematics, Nanjing Forestry University, Nanjing 210037, People’s Republic of China
2
School of Mathematics (Zhuhai), Sun Yat-sen University, Zhuhai 519082, Guangdong, China
3
Department of Mathematics, Zhejiang Normal University, Jinhua 321004, Zhejiang, China
(1.1) (1.2) (1.3) (1.4)
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Math Phys Anal Geom
(2020) 23:39
u = 0, b · n = 0, rot b × n = 0, on ∂ × (0, ∞), (ρ, u, b)(·, 0) = (ρ0 , u0 , b0 )(·) in ⊂ R3 .
(1.5) (1.6)
Here ρ denotes the density, u the velocity field, π the pressure, and b the magnetic field, respectively. μ is the viscosity coefficient and η is the resistivity coefficient. is a bounded and simply connected domain in R3 with smooth boundary ∂, n is the unit outward normal vector to the boundary ∂. Wu [12] shows the local well-posedness of strong solutions to the problem (1.1)(1.6) with inf ρ0 > 0. Huang and Wang [9] (also see [4]) prove the global wellposedness of the strong solutions. Fan-Li-Nakamura [5] showed a regularity criterion. Fan-Zhou [6] proved the uniform-in-μ(η) local well-posedness of smooth solutions when := Rd . When ρ = 1, Xiao-Xin-Wu [13] studied vanishing viscosity limit when the boundary condition u = 0 is replaced by u · n = 0, rot u × n = 0 on ∂ × (0, ∞). The aim of this paper is to prove some uniform-in-η regularity estimate. We will prove Theorem 1.1 Let 0 < η < 1, 0 < C10 ρ0 C0 , ρ0 ∈ W 1,6 , u0 ∈ H01 ∩ H 2 , b0 ∈ H 2 with b0 · n = 0, rot b0 × n = 0 on ∂ and div u0 = div b0 = 0 in . Then there exist a small time T independent of η > 0 and a unique strong solution (ρ, u, b) to the initial boundary value problem (1.1)-(1.6) such that θ 0, C1 ρ C, ρ ∈ L∞ (0, T ; W 1,6 ), ∂t ρ ∈ L∞ (0, T ; L6 ), u ∈ L∞ (0, T ; H 2 ) ∩ L2 (0, T ; W 2,6 ), ut ∈ L∞ (0, T ; L2 ) ∩ L2 (0, T ; H 1 ), b ∈ L∞ (0, T ; W 1,6 ), bt ∈ L∞ (0, T ; L2 ), ηb ∈ L∞ (0, T ; H 2 ),
(1.7)
with the corresponding norms that are uniformly bounded with respect to η > 0. We will prove Theorem 1.1. by the Banach fixed point theorem. We denote the nonempty set A := {u˜ ∈ A; u(·, ˜ 0) = u0 , div u˜ = 0, u ˜ A A}
with
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