Continuous Dependence of Solutions in Low Regularity Spaces for the Hall-MHD Equations
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Continuous Dependence of Solutions in Low Regularity Spaces for the Hall-MHD Equations Xing Wu1
· Wenya Ma1
Received: 9 July 2019 / Revised: 27 January 2020 © Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2020
Abstract In this paper, we consider the strong solutions to the incompressible viscous, resistive Hall-MHD system. Firstly, we show that the solutions to the Hall-MHD system depend continuously on the initial data in H s (R3 ), s > 23 . In addition, we obtain that the solutions to the Hall-MHD system converge to the solutions of the MHD system as the Hall effect coefficient tends to zero. Keywords Hall-MHD equations · Commutator estimates · Continuous dependence Mathematics Subject Classification 76W05 · 35B30 · 35B35
1 Introduction In this paper, we consider the following incompressible viscous resistive Hall-MHD equations: ⎧ u t + u · ∇u − u + ∇ p = B · ∇ B, x ∈ R3 , t > 0, ⎪ ⎪ ⎨ Bt + u · ∇ B − B + η∇ × ((∇ × B) × B) = B · ∇u, x ∈ R3 , t > 0, (1.1) ∇ · u = ∇ · B = 0, x ∈ R3 , t ≥ 0, ⎪ ⎪ ⎩ 3 u(0, x) = u 0 (x), B(0, x) = B0 (x), x ∈ R , where u(t, x) = (u 1 (t, x), u 2 (t, x), u 3 (t, x)) and B(t, x) = (B1 (t, x), B2 (t, x), B3 (t, x)) represent the flow velocity vector and the magnetic field vector, respectively, and p is a scalar pressure. The constant η denotes the Hall effect coefficient. When η = 0, system (1.1) reduces to the classical MHD equation, which has been
Communicated by Yong Zhou.
B 1
Xing Wu [email protected] College of Information and Management Science, Henan Agricultural University, Zhengzhou 450002, Henan, People’s Republic of China
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X. Wu, W. Ma
studied extensively. However, the Hall term ∇ × ((∇ × B) × B) makes the HallMHD system entirely different from the MHD system and prevents straightforward adaptations from arguments used in the mathematical analysis of MHD and related models. Hall-MHD is believed to be an essential feature in the problem of magnetic reconnection. Magnetic reconnection corresponds to changes in the topology of magnetic field lines, which are ubiquitously observed in space. Acheritogaray et al. [1] derived the Hall-MHD system from either two-fluid or kinetic models and proved the global existence of weak solutions in the period setting by using the Galerkin approximation. The global existence of weak solutions as well as the local existence and uniqueness of smooth solutions for (1.1) with −B (with or without −u) in the whole space H s (R3 ) (s > 25 ) went back to Chae et al. [4]. The authors in [8] obtained the global regularity with axisymmetric initial data without swirl. Based on the energy method, Benvenutti and Ferreira [2] established the local existence of strong solutions in H 2 (R3 ). Recently, by making full use of commutator estimates and Sobolev embeddings to control the higher nonlinear term, we [27] obtained the local existence and uniqueness of strong solutions in H s (R3 ) ( 23 < s ≤ 25 ). Just like the 3D Navier–Stokes equations, the global existence for (1.1) evolving from general initial data is
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