Low Regularity Well-Posedness for the 3D Generalized Hall-MHD System
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Low Regularity Well-Posedness for the 3D Generalized Hall-MHD System Renhui Wan1 · Yong Zhou2,3
Received: 25 May 2015 / Accepted: 9 September 2016 © Springer Science+Business Media Dordrecht 2016
Abstract In this paper, we obtain the local well-posedness for the 3D incompressible Hallmagnetohydrodynamics (Hall-MHD) system with Λ2α u and Λ2β B, 0 < α ≤ 1, 12 < β ≤ 1. Our results improve regularity conditions on the initial data of previous works. Keywords Generalized Hall-MHD system · Local well-posedness · Low regularity Mathematics Subject Classification 35R11 · 35Q35 · 76B03
1 Introduction In this paper, we consider the 3D generalized Hall-MHD system given by ⎧ ∂t u + u · ∇u + Λ2α u + ∇p = B · ∇B, ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ∂t B + u · ∇B − B · ∇u + Λ2β B = −∇ × (∇ × B) × B , ⎪ ⎪ div u = div B = 0, ⎪ ⎪ ⎪ ⎩ u(x, 0) = u0 (x), B(x, 0) = B0 (x),
(1.1)
here t ≥ 0, x ∈ R3 , p, u, B stand for scalar pressure, velocity vector and magnetic vector, 1 respectively, div u0 = div B0 = 0, α, β ≥ 0 and Λ = (−Δ) 2 , which is defined by α f (ξ ) = |ξ |α f (ξ ). Λ
B Y. Zhou
[email protected] R. Wan [email protected]; [email protected]
1
Department of Mathematics, Zhejiang University, Hanzhou 310027, China
2
School of Mathematics, Shanghai University of Finance and Economics, Shanghai, 200433, China
3
Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, 21589, Saudi Arabia
R. Wan, Y. Zhou
The Hall term ∇ × ((∇ × B) × B) makes Hall-MHD system entirely different from MHD system. When α = β = 1, (1.1) reduces the standard Hall-MHD system which was derived in [1] from kinetic models. Then extensive studies have been made for the Hall-MHD system. Local classical solutions for the Hall-MHD system with −ΔB (with or without −Δu) and global weak solutions for that with −Δu and −ΔB were obtained in [4]. In [3], local well-posedness was also obtained with −Δu and −ΔB only assuming initial data in H 2 (R3 ). The author in [22] obtained the global regularity for (1.1) with α ≥ 54 and β ≥ 74 , which was optimal by combining the scaling invariance with energy estimates for generalized Navier-Stokes system (B = 0) and simple Hall problem (u = 0). For some other results on regularity criterions, global small solutions, large time behavior, weak-strong uniqueness and magneto-helicity identity, we refer to [5, 6, 17] and [10]. For the Hall-MHD system with α = β = 1, [15] obtained the global regularity with axisymmetric initial data without swirl, namely, u0 = ur0 (r, z)er + uz0 (r, z)ez ,
B0 = B0θ (r, z)eθ .
(1.2)
But provided without −ΔB (β = 0), called non-resistive Hall-MHD system, the authors in [8] showed it generated the singularity with initial data (1.2). Very recently, based on Besov space techniques, the authors in [7] obtained the local well-posedness for the Hall-MHD system with α = 0 and β > 12 . So far, for the case with α = 0 and β ≤ 12 , the corresponding result is empty. For other related results, we refer to [11–14, 16, 19, 23] and references therein. It is worth pointing out that all t
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