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,b∈C0,T;Hs∩C10,T;Hs∩C0,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 [1].
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 [4] where the regularity condition of the direction of vorticity was used to describe the regularity criterion to the NavierStokes equations, He and Xin [5] 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+3p≤1, 3<p≤∞.Later, Zhou and Gala [6] extended it to the multiplier spaces. Other important studies such as the regularity criteria can be found in [7–9] 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 [10–12]. 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 [13] obtained the following regularity:(4)u~∈Lp0,T;LqR3,b~∈Ll0,T;LmR3, 2p+3q≤1, 2l+3m≤1, 3<q,m≤∞.In 2012, Ni et al. [14] got some new regularity as follows:(5)∇~u∈Lp0,T;LqR3,∇~b∈Ll0,T;LmR3, 2p+3q≤2, 2l+3m≤2, 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 reftype="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~∈L2/2r0,T;X˙rR3, with r∈0,1,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 [14].
Remark 4.
Since(7)L3/rR3↪L3/r,∞R3↪X˙rR3as 0≤r≤3/2, where Lp,∞ denotes the weak Lpspace, therefore, our result extends that in [14].
2. Preliminaries
First, we recall the definition and some properties of the multiplier space X˙r introduced recently by LemariéRieusset [15] (see also [16]). The space X˙r of pointwise multipliers which map L2 into H˙r is defined in the following way.
Definition 5.
For 0≤r<3/2, we define the homogeneous space X˙r by (8)X˙r=f∈Lloc2:∀g∈H˙r,fg∈L2,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˙r≤1fgL2<∞.
It is easy to check that (10)fλ·X˙r≤1λrfX˙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 socalled scaling dimension zero property.
Lemma 6.
Assume 0≤r≤1. Then the following inequality holds: (12)uH˙r≤uL21r∇uL2r.
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 reftype="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 (<inlineformula><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></inlineformula>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)12ddt∫u2+b2dx+∫∇u2+∇b2dx=u·∇u+b·∇b,u+u·∇b+b·∇u,b=0,where 〈·,·〉 denotes the innerproduct in L2R3. Integrating from 0 to t for the above inequality, we have(14)u,b∈L∞0,T;L2∩L20,T;H˙1.
Step 2 (<inlineformula><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></inlineformula>estimates).
Differentiating the first equation and the second equation of (1) with respect to xk (1≤k≤3) 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)12ddt∂kuL22+∇∂kuL22=∫∂ku·∇u·∂ku dx∫∂k∇p·∂ku dx+∫∂kb·∇b·∂ku dx,(16)12ddt∂kbL22+∇∂kbL22=∫∂ku·∇b∂kb dx+∫∂kb·∇u∂kb dx.Noting the incompressibility conditions ∇·u=0 and ∇·b=0, since (17)∫∂ku·∇u·∂ku dx=∫∂ku·∇u·∂ku dx,∫∂kb·∇b·∂ku dx=∫∂kb·∇b·∂ku dx+∫b·∇∂kb·∂ku dx,∫∂k∇p·∂ku dx=0,∫∂ku·∇b∂kb dx=∫∂ku·∇b∂kb dx,∫∂kb·∇u∂kb dx=∫∂kb·∇u∂kb dx+∫b·∇∂ku∂kb dx,then (15) and (16) can be rewritten as (18)12ddt∂kuL22+∇∂kuL22=∫∂ku·∇u·∂ku dx+∫∂kb·∇b·∂ku dx+∫b·∇∂kb·∂ku dx,12ddt∂kbL22+∇∂kbL22=∫∂ku·∇b∂kb dx+∫∂kb·∇u∂kb dx+∫b·∇∂ku∂kb dx.Since ∫(b·∇)∂ku∂kb dx+∫(b·∇)∂kb·∂ku dx=0, adding up (18), we have by summing up over k (1≤k≤3)(19)12ddt∇uL22+∇bL22+∇2uL22+∇2bL22=∑k=13∫∂ku·∇u·∂ku dx+∑k=13∫∂kb·∇b·∂ku dx∑k=13∫∂ku·∇b∂kb dx+∑k=13∫∂kb·∇u∂kb dx≜I1+I2+I3+I4.First, we rewrite I1 into seven parts as follows: (20)I1=∑i,j,k=13∫∂kui∂iuj∂kujdx=∑i,j,k=12∫∂kui∂iuj∂kujdx∑j,k=12∫∂ku3∂3uj∂kujdx∑i,k=12∫∂kui∂iu3∂ku3dx∑i,j=12∫∂3ui∂iuj∂3ujdx∑k=13∫∂ku3∂3u3∂kujdx∑i=12∫∂3ui∂iu3∂3u3dx∑j=12∫∂3u3∂3uj∂3ujdx≜∑m=17I1m.According to the definition of ∇~u~, the former four terms of the righthand side of (20) are bounded by (21)∑m=14I1m≤C∫∇~u~∇u∇udx.And the latter three terms can be estimated, noting that ∂3u3=∂1u1∂2u2 from the incompressible condition (22)∑m=57I1m≤C∫∂1u1∂2u2∇u∇udx≤C∫∇~u~∇u∇udx.Thus, using Hölder’s inequality and Young’s inequality, we have (23)I1≤C∫∇~u~∇u∇udx≤C∇~u~∇uL2∇uL2≤C∇~u~X˙r∇uH˙r∇uL2≤C∇~u~X˙r∇uL21r∇2uL2r∇uL2≤C∇~u~X˙r2/2r∇uL22+12∇2uL22.Next, we will bound the term I2. We can get, by splitting I2 into seven parts, (24)I2=∑k=13∫∂kb·∇b·∂ku dx=∑i,j,k=13∫∂kbi∂ibj∂kujdx=∑i,j,k=12∫∂kbi∂ibj∂kujdx+∑j,k=12∫∂kb3∂3bj∂kujdx+∑i,k=12∫∂kbi∂ib3∂ku3dx+∑i,j=12∫∂3bi∂ibj∂3ujdx+∑k=13∫∂kb3∂3b3∂kujdx+∑i=12∫∂3bi∂ib3∂3u3dx+∑j=12∫∂3b3∂3bj∂3ujdx≜∑m=17I2m.Obviously, the former four terms of the righthand side of (24) are bounded by (25)∑m=14I2m≤C∫∇~u~∇u∇bdx+∫∇~u~∇b∇bdx+∫∇~b~∇u∇bdx.Noting that ∂3u3=∂1u1∂2u2 and ∂3b3=∂1b1∂2b2 from the incompressible conditions, the latter three terms can be estimated as follows: (26)∑m=57I2m≤C∫∂1u1∂2u2∇b∇bdx+∫∂1b1∂2b2∇u∇bdx≤C∫∇~u~∇b∇bdx+∫∇~b~∇u∇bdx.So, by Hölder’s inequality and Young’s inequality, we obtain(27)I2≤C∫∇~u~∇u∇bdx+∫∇~u~∇b∇bdx+∫∇~b~∇u∇bdx≤C∇~u~∇uL2∇bL2+∇~u~∇bL2∇bL2+∇~b~∇bL2∇uL2≤C∇~u~X˙r∇uH˙r∇bL2+∇~u~X˙r∇bH˙r∇bL2+∇~b~X˙r∇uH˙r∇bL2≤C∇~u~X˙r∇uL21r∇2uL2r∇bL2+∇~u~X˙r∇bL21r∇2bL2r∇bL2+∇~b~X˙r∇uL21r∇2uL2r∇bL2≤C∇~u~X˙r2/2r+∇~b~X˙r2/2r∇uL22+∇bL22+12∇2uL22+∇2bL22.Similar to the estimate of I2, for I3I4, we have (28)I3+I4≤C∇~u~X˙r2/2r+∇~b~X˙r2/2r∇uL22+∇bL22+12∇2uL22+∇2bL22.Combining the above estimates (23), (27), and (28) into (19), we deduce (29)12ddt∇uL22+∇bL22+∇2uL22+∇2bL22≤C∇~u~X˙r2/2r+∇~b~X˙r2/2r∇uL22+∇bL22+12∇2uL22+∇2bL22.Therefore, we have (30)ddt∇uL22+∇bL22+∇2uL22+∇2bL22≤C∇~u~X˙r2/2r+∇~b~X˙r2/2r∇uL22+∇bL22.Due to Gronwall’s inequality, it follows from (30) that (31)supt∈0,T∇uL22+∇bL22+∫0T∇2uL22+∇2bL22dt≤∇u0L22+∇b0L22expC∫0T∇~u~X˙r2/2r+∇~b~X˙r2/2rdt.Thanks to ∇~u~,∇~b~∈L2/2r([0,T);X˙r(R3)), we have(32)u,b∈L∞0,T;H˙1∩L20,T;H2.Combining (14) with (32), we have(33)u,b∈L∞0,T;H1.By the standard arguments of continuation of local solutions [1], we can draw the conclusion that the smooth solution (u,b) remains smooth at t=T. This completes the proof of Theorem 2.