A Class of Large Solutions to the 3D Generalized Hall-MHD Equations
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A Class of Large Solutions to the 3D Generalized Hall-MHD Equations Xing Wu1
· Yanbin Tang2
Received: 6 August 2019 / Accepted: 17 February 2020 © Springer Nature B.V. 2020
Abstract In this paper we derive the global existence of smooth solutions to the 3D generalized Hall-MHD equations for a class of large initial data, whose L∞ norms can be arbitrarily large. In addition, we give an example to show that such a large initial value does exist. Our idea is splitting the generalized heat equations from generalized Hall-MHD system to generate a small quantity for large time t. Keywords Generalized Hall-MHD equations · Large solutions Mathematics Subject Classification 35Q35 · 35A01
1 Introduction We consider a class of large solutions to the Cauchy problem for the following 3D incompressible generalized Hall-MHD equations: ⎧ ut + 2α u + u · ∇u + ∇p = b · ∇b, x ∈ R3 , t > 0, ⎪ ⎪ ⎨ bt + 2β b + u · ∇b + ∇ × ((∇ × b) × b) = b · ∇u, x ∈ R3 , t > 0, (1.1) ⎪ ∇ · u = ∇ · b = 0, x ∈ R3 , t ≥ 0, ⎪ ⎩ 3 x∈R , u(0, x) = u0 (x), b(0, x) = b0 (x), where u, b and p represent the flow velocity vector, the magnetic field vector and scalar 1 pressure, respectively. The fractional Laplacian operator = (−) 2 is defined in terms of the Fourier transform by α f (ξ ) = |ξ |α fˆ(ξ ). When α = β = 1, the system (1.1) is reduced to the standard Hall-MHD system which was derived from either two-fluid or kinetic models by Acheritogaray et al. [1] and has
B X. Wu
[email protected]
1
College of Information and Management Science, Henan Agricultural University, Zhengzhou, Henan 450002, P.R. China
2
School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, Hubei 430074, P.R. China
X. Wu, Y. Tang
received much attention, concerning global weak solutions [2], local and global (small) strong solutions [2–9], and the large time behavior of weak and strong solutions [4, 10–12]. In [2], Chae, Degond and Liu proved the global existence of weak solutions as well as the local well-posedness of smooth solutions in the whole space. Meanwhile, they showed that if u0 H m + b0 H m (m > 52 ) is small enough, the local smooth solution is global in time. Chae and Lee [3] proved two global-in-time existence results of the classical solutions for small initial data in view of (u0 ˙ 3 , b0 ˙ 3 ) or (u0 1 , b0 3 ), this signifH2
H2
2 B˙ 2,1
2 B˙ 2,1
icantly improved the results in [2]. Laterly, Wan and Zhou [4] extended the condition of global existence on the small initial data to the more general norms (u0 ˙ 1 + , b0 ˙ 3 ) H2 H2 or (u0 q3 −1 , b0 q3 −1 q3 ) with 0 < < 1 and 1 < q < ∞, respectively. Recently, Wan B˙ q,1
B˙ q,1 ∩B˙ q,1
and Zhou [7] removed the restriction on in [4], establishing the global existence of strong solution for (1.1) with the Fujita-Kato type initial data. Chae, Wan and Wu [13] proved the local existence of the solutions to the Hall-MHD equations with only a fractional Laplacian magnetic diffusion (−)α b in the space H s (Rd ) for s > 1 + d2 and α >
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