Generalized MHD System with Velocity Gradient in Besov Spaces of Negative Order
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Generalized MHD System with Velocity Gradient in Besov Spaces of Negative Order Zujin Zhang1
Received: 22 February 2016 / Accepted: 20 December 2016 / Published online: 28 December 2016 © Springer Science+Business Media Dordrecht 2016
Abstract This paper studies the 3D generalized MHD system with fractional diffusion terms (−)α u and (−)β b with 0 < α < 54 ≤ β, and establishes a regularity criterion involving the velocity gradient in Besov spaces of negative order. This improves Fan et al. (Math. Phys. Anal. Geom. 17:333–340, 2014) a lot. Keywords Regularity criteria · Generalized MHD system · Fractional diffusion Mathematics Subject Classification (2000) 35B65 · 35Q30 · 76D03
1 Introduction In the present paper, we consider the 3D generalized MHD system: ∂t u + (u · ∇)u − (b · ∇)b + (−)α u + ∇π = 0, ∂t b + (u · ∇)b − (b · ∇)u + (−)β b = 0, ∇ · u = ∇ · b = 0,
(1)
b(0) = b0 ,
u(0) = u0 ,
where u = (u1 , u2 , u3 ) is the fluid velocity field, b = (b1 , b2 , b3 ) is the magnetic field; π is the pressure; α and β are two positive constants. The fractional Laplacian operator defined through the Fourier transform
F (−)α f (ξ ) = |ξ |2α F f (ξ )
B Z. Zhang
[email protected]
1
School of Mathematics and Computer Sciences, Gannan Normal University, Ganzhou 341000, Jiangxi, P.R. China
(2)
140
Z. Zhang
appears in a wide class of physical systems and engineering problems, including Lévy flights, stochastic interfaces and anomalous diffusion problems. For simplicity, we write 1 Λ = (−) 2 . When b = 0, (1) reduces to the generalized Navier-Stokes equations, which was first studied by J.L. Lions (see [2, p. 334]), and Ladyzhenskaya showed its global regularity in case α ≥ 54 [6, p. 286], but we were unable to locate the references. Motivated by this, we want to study how the parameters α and β affects the regularity of the solutions of (1). Wu [6] generalized the results of Ladyzhenskaya, and established the global-in-time regularity of n-dimensional version of (1) under the assumption α≥
1 n + , 2 4
β≥
1 n + . 2 4
(3)
n α+β ≥1+ . 2
(4)
Later in 2011, Wu [7] relaxed (3) to be α≥
1 n + , 2 4
β > 0,
For system (1) with 0 < α < 54 or 0 < β < 54 , it is open for its global smooth solutions. And some regularity criteria were derived in [2, 3, 8, 9]. In particular, when 0 < α < 54 ≤ β, Fan et al. [2] showed the following regularity criterion ∇u ∈ Lp 0, T ; Lq R3 ,
2α 3 + = 2α, p q
3 < q ≤ ∞. 2α
(5)
The purpose of this paper is to improve (5) from classical Lebesgue spaces to larger Besov spaces of negative order (see [1, Propositions 2.20, 2.21, 2.39]): 3 3 −α B˙ ∞,∞ R ⊃ B˙ 03 ,∞ R3 ⊃ L α R3 .
(6)
α
One is referred to [1, Chap. 2] for more details on Besov spaces and [5] for applications in Navier-Stokes equations. Precisely, our result reads as Theorem 1 Let 0 < α < ∇ · b0 = 0. If
5 4
≤ β. Assume that u0 , b0 ∈ H m (R3 ) with m > 3 2α −γ ∇u ∈ L 2α−γ 0, T ; B˙ ∞,∞ R
5 2
and ∇ · u0 = (7)
for some 0 < γ < 2α, then the solution (u, b) of (1) can be extended smoothl
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