JFS Journal of Function Spaces 2314-8888 2314-8896 Hindawi 10.1155/2017/3795172 3795172 Research Article A Regularity Criterion for the 3D Incompressible Magnetohydrodynamics Equations in the Multiplier Spaces http://orcid.org/0000-0003-0033-453X Tian Chunhong 1 Zhang Xinguang Nanhang Jincheng College Nanjing 211156 China 2017 2782017 2017 05 03 2017 18 06 2017 2782017 2017 Copyright © 2017 Chunhong Tian. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We are concerned with the regularity criterion for weak solutions to the 3D incompressible MHD equations in this paper. We show that if some partial derivatives of the velocity components and magnetic components belong to the multiplier spaces, then the solution actually is smooth on (0,T).

National Natural Science Foundation of China 11302102 China Postdoctoral Science Foundation 2014M561893
1. Introduction

We consider the 3D incompressible magnetohydrodynamics (MHD) equations.

(MHD) (1) u t - ν Δ u + u · u = - p - 1 2 b 2 + b · b , b t - η Δ b + u · b = b · u , · u = · b = 0 , u 0 , x = u 0 x , b 0 , x = b 0 x . Here u, b describe the flow velocity vector and the magnetic field vector, respectively, p is a scalar pressure, ν>0 is the kinematic viscosity, and η>0 is the magnetic diffusivity, while u0 and b0 are the given initial velocity and initial magnetic field, respectively, with ·u0=·b0=0. If ν=η=0, (1) is called the ideal MHD equations.

Employing the standard energy method, it is easy to prove that, for given initial data (u0,b0)Hs(R3) with s>1/2, there exist a positive time T=T((u0,b0)Hs) and a unique smooth solution (u(t,x),b(t,x)) on [0,T) to the MHD equations satisfying (2)u,bC0,T;HsC10,T;HsC0,T;Hs+2.Whether smooth solutions of (1) on [0,T) will lead to a singularity at t=T is an outstanding open problem; see Sermange and Temam .

However, the solution regularity can be derived when certain growth conditions are satisfied. For regularity of the weak solutions to the 3D MHD equations (1), some numerical experiments [2, 3] seem to indicate that the velocity field plays a more important role than the magnetic field in the regularity theory of solutions to the MHD equations. Recently, inspired by Constantin and Fefferman initial work  where the regularity condition of the direction of vorticity was used to describe the regularity criterion to the Navier-Stokes equations, He and Xin  extended it to the MHD equations and obtained some integrability condition of the magnitude of the only velocity u alone; that is,(3)0Tutpqdt<,2q+3p1,3<p.Later, Zhou and Gala  extended it to the multiplier spaces. Other important studies such as the regularity criteria can be found in  and references therein.

Recently, some regularity criteria in terms of partial velocity components and magnetic components or partial derivative of the velocity components and magnetic components were established . However, the spaces used are not scaling invariant (in other words, not of Serrin’s type). Many researchers were devoted to studying it along this direction. In 2010, Ji and Lee  obtained the following regularity:(4)u~Lp0,T;LqR3,b~Ll0,T;LmR3,2p+3q1,2l+3m1,3<q,m.In 2012, Ni et al.  got some new regularity as follows:(5)~uLp0,T;LqR3,~bLl0,T;LmR3,2p+3q2,2l+3m2,32<q,m.

Our purpose in this paper is to obtain a new regularity criterion of weak solution for the 3D MHD equations in a sense of scaling invariant by employing a different decomposition for nonlinear terms.

Definition 1 (see [<xref ref-type="bibr" rid="B15">1</xref>]).

A weak solution pair (u,b) of the 3D MHD equations (1) is regular in [0,T)×H1R3 provided that (u,b)L(0,T;H1R3).

Notation 1.

Throughout the paper, we use the following notations for the simplicity. The planar components of (u,b) will be denoted by u~=(u1,u2) and b~=(b1,b2) and ~ will be used for ~=(1,2).

Now we state our result as follows.

Theorem 2.

Let (u0,b0)H1 with ·u0=·b0=0. If the weak solutions (u,b) to (1) satisfy the following integrability conditions:(6)~u~,~b~L2/2-r0,T;X˙rR3,with r0,1,then the solution (u,b) remains smooth on [0,T].

Remark 3.

Since the components of ~u~ and ~b~ without u3 and b3 are less than those of ~u and ~b, therefore, our result improves that in .

Remark 4.

Since(7)L3/rR3L3/r,R3X˙rR3as 0r3/2, where Lp, denotes the weak Lp-space, therefore, our result extends that in .

2. Preliminaries

First, we recall the definition and some properties of the multiplier space X˙r introduced recently by Lemarié-Rieusset  (see also ). The space X˙r of pointwise multipliers which map L2 into H˙-r is defined in the following way.

Definition 5.

For 0r<3/2, we define the homogeneous space X˙r by (8)X˙r=fLloc2:gH˙r,fgL2,where we denote by H˙r(Rd) the completion of the space D(Rd) with respect to the norm uH˙r=-Δ1/2uL2=ξru^(ξ)L2, where u^(ξ) denotes the Fourier transform of u.

The norm of X˙r is given by the operator norm of pointwise multiplication (9)fX˙r=supgH˙r1fgL2<.

It is easy to check that (10)fλ·X˙r1λrfX˙r,λ>0,f·+x0X˙r=fX˙r.Hence, for any function f(x,t) defined for both spatial and time variables, (11)fλL20,T;X˙r=fL20,T;X˙r,for any λ>0, with fλ(x,t)=λf(λx,λ2t). Therefore, if (u,b) solves the MHD equations, then so does (uλ,bλ). This is the so-called scaling dimension zero property.

Lemma 6.

Assume 0r1. Then the following inequality holds: (12)uH˙ruL21-ruL2r.

The proof of this lemma can be easily obtained by Parseval’s equality and Hölder’s inequality, and we omit it.

3. Proof of Theorem <xref ref-type="statement" rid="thm1.1">2</xref>

By Definition 1, we need to estimate (u,b)L(0,T;H1R3). Thus, we will divide the proof of Theorem 2 into two steps as follows.

Step 1 (<inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M72"><mml:mrow><mml:msup><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mn mathvariant="normal">2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math></inline-formula>-estimates).

Multiplying the first equation and the second equation in (1) by u,  b, respectively, and integrating the resulting equations by parts over R3, we obtain after adding them together from the incompressibility condition ·u=0 and ·b=0(13)12ddtu2+b2dx+u2+b2dx=-u·u+b·b,u+-u·b+b·u,b=0,where ·,· denotes the inner-product in L2R3. Integrating from 0 to t for the above inequality, we have(14)u,bL0,T;L2L20,T;H˙1.

Step 2 (<inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M84"><mml:mrow><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mo>˙</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math></inline-formula>-estimates).

Differentiating the first equation and the second equation of (1) with respect to xk  (1k3) and multiplying the first and the second equations of (1) by u/xk=ku and b/xk=kb, respectively, and then, by integrating by parts over R3, we get(15)12ddtkuL22+kuL22=-ku·u·kudx-kp·kudx+kb·b·kudx,(16)12ddtkbL22+kbL22=-ku·bkbdx+kb·ukbdx.Noting the incompressibility conditions ·u=0 and ·b=0, since (17)ku·u·kudx=ku·u·kudx,kb·b·kudx=kb·b·kudx+b·kb·kudx,kp·kudx=0,ku·bkbdx=ku·bkbdx,kb·ukbdx=kb·ukbdx+b·kukbdx,then (15) and (16) can be rewritten as (18)12ddtkuL22+kuL22=-ku·u·kudx+kb·b·kudx+b·kb·kudx,12ddtkbL22+kbL22=-ku·bkbdx+kb·ukbdx+b·kukbdx.Since (b·)kukbdx+(b·)kb·kudx=0, adding up (18), we have by summing up over k  (1k3)(19)12ddtuL22+bL22+2uL22+2bL22=-k=13ku·u·kudx+k=13kb·b·kudx-k=13ku·bkbdx+k=13kb·ukbdxI1+I2+I3+I4.First, we rewrite I1 into seven parts as follows: (20)I1=-i,j,k=13kuiiujkujdx=-i,j,k=12kuiiujkujdx-j,k=12ku33ujkujdx-i,k=12kuiiu3ku3dx-i,j=123uiiuj3ujdx-k=13ku33u3kujdx-i=123uiiu33u3dx-j=123u33uj3ujdxm=17I1m.According to the definition of ~u~, the former four terms of the right-hand side of (20) are bounded by (21)m=14I1mC~u~uudx.And the latter three terms can be estimated, noting that 3u3=-1u1-2u2 from the incompressible condition (22)m=57I1mC-1u1-2u2uudxC~u~uudx.Thus, using Hölder’s inequality and Young’s inequality, we have (23)I1C~u~uudxC~u~uL2uL2C~u~X˙ruH˙ruL2C~u~X˙ruL21-r2uL2ruL2C~u~X˙r2/2-ruL22+122uL22.Next, we will bound the term I2. We can get, by splitting I2 into seven parts, (24)I2=k=13kb·b·kudx=i,j,k=13kbiibjkujdx=i,j,k=12kbiibjkujdx+j,k=12kb33bjkujdx+i,k=12kbiib3ku3dx+i,j=123biibj3ujdx+k=13kb33b3kujdx+i=123biib33u3dx+j=123b33bj3ujdxm=17I2m.Obviously, the former four terms of the right-hand side of (24) are bounded by (25)m=14I2mC~u~ubdx+~u~bbdx+~b~ubdx.Noting that 3u3=-1u1-2u2 and 3b3=-1b1-2b2 from the incompressible conditions, the latter three terms can be estimated as follows: (26)m=57I2mC-1u1-2u2bbdx+-1b1-2b2ubdxC~u~bbdx+~b~ubdx.So, by Hölder’s inequality and Young’s inequality, we obtain(27)I2C~u~ubdx+~u~bbdx+~b~ubdxC~u~uL2bL2+~u~bL2bL2+~b~bL2uL2C~u~X˙ruH˙rbL2+~u~X˙rbH˙rbL2+~b~X˙ruH˙rbL2C~u~X˙ruL21-r2uL2rbL2+~u~X˙rbL21-r2bL2rbL2+~b~X˙ruL21-r2uL2rbL2C~u~X˙r2/2-r+~b~X˙r2/2-ruL22+bL22+122uL22+2bL22.Similar to the estimate of I2, for I3-I4, we have (28)I3+I4C~u~X˙r2/2-r+~b~X˙r2/2-ruL22+bL22+122uL22+2bL22.Combining the above estimates (23), (27), and (28) into (19), we deduce (29)12ddtuL22+bL22+2uL22+2bL22C~u~X˙r2/2-r+~b~X˙r2/2-ruL22+bL22+122uL22+2bL22.Therefore, we have (30)ddtuL22+bL22+2uL22+2bL22C~u~X˙r2/2-r+~b~X˙r2/2-ruL22+bL22.Due to Gronwall’s inequality, it follows from (30) that (31)supt0,TuL22+bL22+0T2uL22+2bL22dtu0L22+b0L22expC0T~u~X˙r2/2-r+~b~X˙r2/2-rdt.Thanks to ~u~,~b~L2/2-r([0,T);X˙r(R3)), we have(32)u,bL0,T;H˙1L20,T;H2.Combining (14) with (32), we have(33)u,bL0,T;H1.By the standard arguments of continuation of local solutions , we can draw the conclusion that the smooth solution (u,b) remains smooth at t=T. This completes the proof of Theorem 2.

Conflicts of Interest

The author declares that they have no conflicts of interest.

Acknowledgments

The work was supported by the National Natural Science Foundation of China (11302102) and China Postdoctoral Science Foundation funded project (2014M561893).

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