R E S E A R C H
Open Access
Regularity criterion for a weak solution to the
three-dimensional magneto-micropolar fluid
equations
Yinxia Wang
**Correspondence:
[email protected] School of Mathematics and Information Sciences, North China University of Water Resources and Electric Power, Zhengzhou, 450011, China
Abstract
In this paper, a regularity criterion for the 3D magneto-micropolar fluid equations is investigated. A sufficient condition on the derivative of the velocity field in one direction is obtained. More precisely, we prove that ifux3belongs toL
β(0,T;Lα(R3))
with 3 α+
2
β ≤1 and
α
≥3, then the solution (u,v,b) is regular. MSC: 35K15; 35K45Keywords: magneto-micropolar fluid equations; weak solution; regularity criterion
1 Introduction
In the paper we investigate the initial value problem for magneto-micropolar fluid equa-tions inR
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
∂tu– (μ+χ)u+u· ∇u–b· ∇b+∇(p+|b|) –χ∇ ×v= ,
∂tv–γ v–κ∇∇ ·v+ χv+u· ∇v–χ∇ ×u= ,
∂tb–νb+u· ∇b–b· ∇u= ,
∇ ·u= , ∇ ·b= ,
(.)
with the initial value
t= : u=u(x), v=v(x), b=b(x), (.)
whereu(t,x),v(t,x),b(t,x) andp(t,x) denote the velocity of the fluid, the micro-rotational velocity, magnetic field and hydrostatic pressure, respectively. μ is the kinematic vis-cosity, χ is the vortex viscosity, γ and κ are spin viscosities and ν is the magnetic Reynold.
The incompressible magneto-micropolar fluid equations (.) have been studied exten-sively (see [–] and [–]). The existence and uniqueness of local strong solutions is proved by the Galerkin method in []. In [], the author proved global existence of a strong solution with the small initial data. The existence of weak solutions and the uniqueness of weak solutions in D case were established in []. Yuan [] obtained a Beale-Kato-Majda type blow-up criterion for a smooth solution (u,v,b) to the Cauchy problem for (.) that
relies on the vorticity of velocity∇ ×uonly. Wanget al.[] established a Beale-Kato-Majda blow-up criterion of smooth solutions to the D magneto-micropolar fluid equa-tion with partial viscosity. Fundamental mathematical issues such as the regularity of weak solutions have generated extensive research and many interesting results have been estab-lished (see [, ] and []).
Ifb= , (.) reduces to micropolar fluid equations. The micropolar fluid equations were first proposed by Eringen [] (see also []). The existence of weak and strong solutions for micropolar fluid equations was obtained by Galdi and Rionero [] and Yamaguchi [], respectively. Dong and Chen [] established regularity criteria of weak solutions to the three-dimensional micropolar fluid equations. In [], the authors gave sufficient conditions on the kinematics pressure in order to obtain the regularity and uniqueness of weak solutions to the micropolar fluid equations. For more details on regularity criteria, see [, ] and [].
If bothv= andχ= , then equations (.) reduce to be magneto-hydrodynamic(MHD) equations. Magnetohydrodynamics (MHD), the science of motion of an electrically con-ducting fluid in the presence of a magnetic field, consists essentially of the interaction between the fluid velocity and the magnetic field (see []). Besides their physical ap-plications, the MHD equations are also mathematically significant. The local existence of solutions to the Cauchy problem (.), (.) in the usual Sobolev spacesHs(R) was
established in [] for any given initial datau,B∈Hs(R),s≥. But whether the lo-cal solution can be extended to a global solution is a challenging open problem in the mathematical fluid mechanics. There are numerous important progresses on the fun-damental issue of the regularity for the weak solution to (.), (.) (see [–] and [–]).
The purpose of this paper is to establish the regularity criteria of weak solutions to (.), (.) via the derivative of the velocity in one direction. It is proved that ifTux
β
Lαdt<∞ with α +β ≤ andα≥, then the solution (u,v,b) can be extended smoothly beyond
t=T.
The paper is organized as follows. We first state some important inequalities in Sec-tion . Then we give the definiSec-tion of a weak soluSec-tion and state main results in SecSec-tion , and then we prove the main result in Section .
2 Preliminaries
In order to prove our main result, we need the following lemma, which may be found in [] (see also [, ] and []).
Lemma . Assume thatθ,λ,ϑ∈Rand satisfy
≤θ, λ<∞,
θ +
λ> , +
ϑ=
θ +
λ.
Assume that f ∈H(R),f
x,fx∈L
λ(R)and f
x∈L
θ(R).Then there exists a positive con-stant such that
fLϑ ≤Cfx
Lλfx
Lλfx
Especially,whenλ= ,there exists a positive constant C=C(θ)such that
fLθ≤Cfx
Lfx
Lfx
Lθ, (.)
which holds for any f ∈H(R)and f
x ∈L
θ(R)with≤μ<∞.
Lemma . Let≤q≤and assume that f ∈H(R).Then there exists a positive con-stant C=C(q)such that
fLq≤Cf
–q
q
L ∂xf
q– q
L ∂xf
q– q
L ∂xf
q– q
L . (.)
Proof It follows from the interpolating inequality that
fLq≤Cf
–q
q
L f q–
q
L . (.)
Using (.) withθ= , we obtain
fL≤C∂xf
L∂xf
L∂xf
L. (.)
Combining (.) and (.) immediately yields (.).
3 Main results
Before stating our main results, we introduce some function spaces. Let
C,σ∞R= ϕ∈C∞R:∇ ·ϕ= ⊂C∞R.
The subspace
Lσ =C,σ∞
R·L = ϕ∈LR:∇ ·ϕ=
is obtained as the closure ofC∞,σwith respect toL-norm · L.Hσr is the closure ofC,σ∞
with respect to theHr-norm
ϕHr=(I–) r
ϕ
L, r≥.
Before stating our main results, we give the definition of a weak solution to (.), (.) (see [, ] and []).
Definition .(Weak solutions) LetT > ,u,b∈Lσ(R),v∈L(R). A measurable
R-valued triple (u,v,b) is said to be a weak solution to (.), (.) on [,T] if the following
conditions hold: .
(u,b)∈L∞,T;Lσ
R∩L,T;H σ
and
v∈L∞,T;LR∩L,T;HR.
. (.), (.) is satisfied in the sense of distributions,i.e., for every (ϕ,ψ)∈H((,T);Hσ)
andφ∈H((,T);H) withϕ(T) =ψ(T) =φ(T) = , the following hold:
T
–u,∂τϕ+u· ∇u,ϕ+ (μ+χ)∇u,∇ϕdτ
– T
b· ∇b,ϕ+χ∇ ×v,ϕdτ=u,ϕ(), T
–v,∂τφ+γ∇v,∇φ+κ∇ ·v,∇φ+ χv,φdτ
+ T
u· ∇v,φ–χ∇ ×u,φdτ=v,φ()
and T
–b,∂τψ+ν∇b,∇ψ+u· ∇b,ψ–b· ∇u,ψdτ=b,ψ().
. The energy inequality, that is,
u(t)L+v(t)
L+b(t)
L+
t
μ∇u(τ)L+γ∇v(τ)
L
dτ
+ t
κ∇ ·v(τ)L+χv(τ)
L+ν∇b(τ)
L
dτ
≤ uL+vL+bL. (.)
Theorem . Let u,b∈Hσ(R)with v∈H(R).Assume that(u,v,b)is a weak solution to(.), (.)on some interval[,T].If
(T)≡ T
ux
β
Lαdt<∞, (.)
where
α +
β ≤, α≥,
then the solution(u,v,b)can be extended smoothly beyond t=T.
4 Proof of Theorem 3.1
Proof Multiplying the first equation of (.) byuand integrating with respect toxonR,
using integration by parts, we obtain
d dtu(t)
L+ (μ+χ)∇u(t)
L=
Rb· ∇b·u dx+ χ
Similarly, we get
d dtv(t)
L+γ∇v(t)
L+κ∇ ·vL+ χvL=χ
R
(∇ ×u)·v dx (.)
and
d dtb(t)
L+ν∇b(t)
L=
Rb· ∇u·b dx. (.)
Summing up (.)-(.), we deduce that
d dtu(t)
L+v(t)
L+b(t)
L
+ (μ+χ)∇u(t)L
+γ∇v(t)L+κ∇ ·vL+ χvL+ν∇b(t)
L
=
R
b· ∇b·u dx+χ
R
(∇ ×v)·u dx
+χ
R(∇ ×u)·v dx+
Rb· ∇u·b dx. (.)
By integration by parts and the Cauchy inequality, we obtain
χ
R(∇ ×v)·u dx+ χ
R(∇ ×u)·v dx≤
χ∇uL+χvL. (.)
Using integration by parts, we obtain
Rb· ∇b·u dx+
Rb· ∇u·b dx= . (.)
Combining (.)-(.) yields
d dtu(t)
L+v(t)
L+b(t)
L
+μ∇u(t)L
+γ∇v(t)L+κ∇ ·vL+χv(t)
L+ν∇b(t)
L≤.
Integrating with respect tot, we have
u(t)L+v(t)
L+b(t)
L+
t
μ∇u(τ)L+γ∇v(τ)
L
dτ
+ t
κ∇ ·v(τ)L+χv(τ)
L+ν∇b(τ)
L
dτ
≤ u
L+vL+bL. (.)
Differentiating (.) with respect tox, we obtain
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
∂tux– (μ+χ)ux+ux· ∇u+u· ∇ux–bx· ∇b–b· ∇bx
+∇(p+|b|)x–χ∇ ×vx= ,
∂tvx–γ vx–κ∇ · ∇vx+ χvx+ux· ∇v+u· ∇vx–χ∇ ×ux= , ∂tbx–νbx+ux· ∇b+u· ∇bx–bx· ∇u–b· ∇ux= .
Taking the inner product ofux with the first equation of (.) and using integration by
parts yield
d
dtux(t)
L+ (μ+χ)∇ux(t)
L
= –
R
ux· ∇u·uxdx+
R
bx· ∇b·uxdx
+
Rb· ∇bx·uxdx+ χ
R(∇ ×vx)·uxdx. (.)
Similarly, we get
d
dtvx(t)
L+γ∇vx(t)
L+κ∇ ·vx
L+ χvx
L
= –
R
ux· ∇v·vxdx+χ
R
(∇ ×ux)·vxdx (.)
and
d
dtbx(t)
L+ν∇bx(t)
L
= –
Rux· ∇b·bxdx+
Rbx· ∇u·bxdx+
Rb· ∇ux·bxdx. (.)
Combining (.)-(.) yields
d
dtux(t)
L+vx(t)
L+bx(t)
L
+ (μ+χ)∇ux(t)
L
+γ∇vx(t)
L+κ∇ ·vx
L+ χvx
L+ν∇bx(t)
L
= –
Rux· ∇u·uxdx+
Rbx· ∇b·uxdx+
Rb· ∇bx·uxdx
+χ
R
(∇ ×vx)·uxdx–
R
ux· ∇v·vxdx+χ
R
(∇ ×ux)·vxdx
–
Rux· ∇b·bxdx+
Rbx· ∇u·bxdx+
Rb· ∇ux·bxdx. (.)
Using integration by parts and the Cauchy inequality, we obtain
χ
R
(∇ ×vx)·uxdx+χ
R
(∇ ×ux)·vxdx≤χ∇ux
L+χvx
L. (.)
Using integration by parts, we have
Rb· ∇bx·uxdx+
Rb· ∇ux·bxdx= . (.)
Combining (.)-(.) yields
d
dtux(t)
L+vx(t)
L+bx(t)
L
+μ∇ux(t)
L
+γ∇vx(t)
L+κ∇ ·vx
L+χvx(t)
L+ν∇bx(t)
≤–
Rux· ∇u·uxdx+
Rbx· ∇b·uxdx–
Rux· ∇v·vxdx
–
R
ux· ∇b·bxdx+
R
bx· ∇u·bxdx
I+I+I+I+I. (.)
In what follows, we estimateIj(j= , , . . . , ). By integration by parts and the Hölder
inequality, we obtain
I≤C∇uxLuxLuLα,
where
+
α =
, ≤≤.
It follows from the interpolating inequality that
uxL≤Cux –(–)
L ∇ux (–)
L .
From (.), we get
I≤C∇uxLux –(–)
L ∇ux (–)
L ∇u
Lux
Lα
≤C∇ux +(–)
L ux –(–)
L ∇u
Lux
Lα
≤μ
∇ux
L+Cux
L∇u q Lux
q Lα,
where
q=
– (–)= ( –α).
Whenα≥, we have q≤, and the application of the Young inequality yields
I≤μ
∇ux
L+Cux
L
∇u
L+ux δ
Lα
, (.)
where
α +
δ = .
From integration by parts and the Hölder inequality, we obtain
I≤C∇bLbx
Lα–α uxLα≤C∇bLuxLαbx –α
L ∇bx
α
L
≤ν
∇bx
L+∇b α
α–
L ux α
α–
Lα bx α– α–
L
≤ν
∇bx
L+C
∇b
L+ux δ
Lα
bx α– α–
where
α +
δ = .
Similarly,
I≤C∇vLvx
Lα–α uxLα ≤C∇vLuxLαvx
–α
L ∇vx
α
L ≤γ
∇vx
L+∇v α
α–
L ux α
α–
Lα vx α– α–
L
≤γ
∇vx
L+C
∇v
L+ux δ
Lα
vx α– α–
L , (.)
and
I≤C∇bLbx
Lα–α uxL
α
≤C∇bLuxLαbx–
α
L ∇bx
α
L ≤ ν
∇bx
L+∇b α
α–
L ux α
α–
Lα bx α– α–
L
≤ ν
∇bx
L+C
∇bL+ux δ
Lα
bx α– α–
L , (.)
where
α +
δ = .
By integration by parts and the inequality, we have
I≤C∇bxLbxLσuLα
≤C∇bxLbx–( –σ)
L ∇bx (–σ)
L ∇u
Lux
L ≤ν
∇bx
L+Cbx
L∇u q Lux
q Lα,
where
q=
( –α).
Whenα≥, we have q≤, and the application of the Young inequality yields
I≤ν
∇bx
L+Cbx
L
∇u
L+ux δ
Lα
, (.)
where
α +
Combining (.)-(.) yields
d dt
ux
L+bx
L+bx
L
+μ∇ux
L
+γ∇vx
L+κ∇ ·vx
L+χvx
L+ν∇bx
L ≤Cux
L+bx
L
∇uL+ux δ
Lα
+C∇vL+∇bL+ux δ
Lα
vx α– α–
L +bx α– α–
L
.
From the Gronwall inequality, we get
ux
L+bx
L+bx
L+μ
t
∇
ux
Ldτ
+ t
γ∇vx
L+κ∇ ·vx
L+χvx
L+ν∇bx
L
dτ
≤Ce(uL+vL+bL)e(t)u
H+vH+bH
+CuL+vL+bL+(t)
α–
α . (.)
Multiplying the first equation of (.) by –uand integrating with respect toxonR, and
then using integration by parts, we obtain
d
dt∇u(t)
L+ (μ+χ)uL
=
Ru· ∇u·u dx–
Rb· ∇b·u dx– χ
R(∇ ×v)·u dx. (.)
Similarly, we get
d
dt∇v(t)
L+γvL+κ∇∇ ·vL+ χ∇vL
=
Ru· ∇v·v dx– χ
R(∇ ×u)·v dx (.)
and
d
dt∇b(t)
L+νbL
=
Ru· ∇b·b dx–
Rb· ∇u·b dx. (.)
Collecting (.)-(.) yields
d
dt∇u(t)
L+∇v(t)
L+∇b(t)
L
+ (μ+χ)uL
+γvL+κ∇∇ ·vL+ χ∇vL+νbL
=
Ru· ∇u·u dx–
Rb· ∇b·u dx– χ
+
Ru· ∇v·v dx– χ
R(∇ ×u)·v dx
+
R
u· ∇b·b dx–
R
b· ∇u·b dx. (.)
Thanks to integration by parts and the Cauchy inequality, we get
–χ
R(∇ ×v)·u dx– χ
R(∇ ×u)·v dx≤
χuL+χ∇vL. (.)
It follows from (.)-(.) and integration by parts that
d
dt∇u(t)
L+∇v(t)
L+∇b(t)
L
+μuL
+γvL+κ∇∇ ·vL+χ∇vL+νbL
≤–
R∇u· ∇u· ∇u dx+
R∇b· ∇b· ∇u dx–
R∇u· ∇v· ∇v dx
–
R∇
u· ∇b· ∇b dx+
R∇
b· ∇u· ∇b dx
J+J+J+J+J. (.)
In what follows, we estimateJi(i= , . . . , ).
By (.) and the Young inequality, we deduce that
J≤C∇uL≤C∇u
L∇x˜∇uL∇ux
L ≤ μ
∇x˜∇u
L+C∇uL∇uxL ≤ μ
∇x˜∇u
L+C
∇uL+∇ux
L
∇uL. (.)
By (.) and the Young inequality, we have
J≤ ∇uL∇b
L ≤C∇u
L∇x˜∇u
L∇ux
L∇bL∇x˜∇b
L∇bx
L ≤ μ
∇x˜∇u
L+C∇u
L∇ux
L∇b
L∇x˜∇b
L∇bx
L
≤ μ
∇x˜∇u
L+ ν
∇˜x∇b
L+C∇uL∇ux
L∇bL∇bx
L
≤ μ
∇x˜∇u
L+ ν
∇˜x∇b
L
+C∇bL
∇uL+∇ux
L+∇bx
L
. (.)
Similarly, we obtain
J≤ ∇uL∇v
L ≤ μ
∇x˜∇u
L+ γ
∇x˜∇v
L
+C∇vL
∇u
L+∇ux
L+∇vx
L
J≤ ∇uL∇b
L ≤ μ
∇x˜∇u
L+ ν
∇x˜∇b
L
+C∇bL
∇u
L+∇ux
L+∇bx
L
(.)
and
J≤ ∇uL∇b
L ≤ μ
∇x˜∇u
L+ ν
∇x˜∇b
L
+C∇bL
∇u
L+∇ux
L+∇bx
L
. (.)
Combining (.)-(.) yields
d
dt∇u(t)
L+∇v(t)
L+∇b(t)
L
+μuL
+γvL+κ∇∇ ·vL+χ∇vL+νbL ≤C∇uL+∇vL+∇bL
×∇uL+∇ux
L+∇vx
L+∇bx
L
. (.)
From (.), the Gronwall inequality, (.) and (.), we know that (u,v,b)∈L∞(,T;
H(R)). Thus, (u,v,b) can be extended smoothly beyondt=T. We have completed the
proof of Theorem ..
Competing interests
The author declares that she has no competing interests.
Author’s contributions
The author completed the paper herself. The author read and approved the final manuscript.
Received: 30 January 2013 Accepted: 4 March 2013 Published: 25 March 2013 References
1. Gala, S: Regularity criteria for the 3D magneto-micropolar fluid equations in the Morrey-Campanato space. Nonlinear Differ. Equ. Appl.17, 181-194 (2010)
2. Ortega-Torres, EE, Rojas-Medar, MA: On the uniqueness and regularity of the weak solution for magneto-micropolar fluid equations. Rev. Mat. Apl.17, 75-90 (1996)
3. Ortega-Torres, EE, Rojas-Medar, MA: On the regularity for solutions of the micropolar fluid equations. Rend. Semin. Mat. Univ. Padova122, 27-37 (2009)
4. Ortega-Torres, EE, Rojas-Medar, MA: Magneto-micropolar fluid motion: global existence of strong solutions. Abstr. Appl. Anal.4, 109-125 (1999)
5. Rojas-Medar, MA: Magneto-micropolar fluid motion: existence and uniqueness of strong solutions. Math. Nachr.188, 301-319 (1997)
6. Rojas-Medar, MA, Boldrini, JL: Magneto-micropolar fluid motion: existence of weak solutions. Rev. Mat. Complut.11, 443-460 (1998)
7. Yuan, B: Regularity of weak solutions to magneto-micropolar fluid equations. Acta Math. Sci.30, 1469-1480 (2010) 8. Yuan, J: Existence theorem and blow-up criterion of the strong solutions to the magneto-micropolar fluid equations.
Math. Methods Appl. Sci.31, 1113-1130 (2008)
9. Zhang, Z, Yao, A, Wang, X: A regularity criterion for the 3D magneto-micropolar fluid equations in Triebel-Lizorkin spaces. Nonlinear Anal.74, 2220-2225 (2011)
10. Wang, Y, Hu, L, Wang, Y: A Beale-Kato-Madja criterion for magneto-micropolar fluid equations with partial viscosity. Bound. Value Probl.2011, Article ID 128614 (2011)
11. Eringen, AC: Theory of micropolar fluids. J. Math. Mech.16, 1-18 (1966)
13. Galdi, GP, Rionero, S: A note on the existence and uniqueness of solutions of the micropolar fluid equations. Int. J. Eng. Sci.15, 105-108 (1977)
14. Yamaguchi, N: Existence of global strong solution to the micropolar fluid system in a bounded domain. Math. Methods Appl. Sci.28, 1507-1526 (2005)
15. Dong, B, Chen, Z: Regularity criteria of weak solutions to the three-dimensional micropolar flows. J. Math. Phys.50, Article ID 103525 (2009)
16. Wang, Y, Chen, Z: Regularity criterion for weak solution to the 3D micropolar fluid equations. J. Appl. Math.2011, Article ID 456547 (2011)
17. Dong, B, Jia, Y, Chen, Z: Pressure regularity criteria of the three-dimensional micropolar fluid flows. Math. Methods Appl. Sci.34, 595-606 (2011)
18. Wang, Y, Yuan, H: A logarithmically improved blow-up criterion for smooth solutions to the 3D micropolar fluid equations. Nonlinear Anal., Real World Appl.13, 1904-1912 (2012)
19. Lifschitz, AE: Magnetohydrodynamics and spectral theory. In: Developments in Electromagnetic Theory and Applications, vol. 4. Kluwer Academic, Dordrecht (1989)
20. Sermange, M, Temam, R: Some mathematical questions related to the MHD equations. Commun. Pure Appl. Math. 36, 635-666 (1983)
21. Cao, C, Wu, J: Two regularity criteria for the 3D equations. J. Differ. Equ.248, 2263-2274 (2010)
22. Gala, S: Extension criterion on regularity for weak solutions to the 3D MHD equations. Math. Methods Appl. Sci.33, 1496-1503 (2010)
23. He, C, Xin, Z: On the regularity of solutions to the magnetohydrodynamic equations. J. Differ. Equ.213, 235-254 (2005)
24. He, C, Wang, Y: On the regularity for weak solutions to the magnetohydrodynamic equations. J. Differ. Equ.238, 1-17 (2007)
25. He, C, Wang, Y: Remark on the regularity for weak solutions to the magnetohydrodynamic equations. Math. Methods Appl. Sci.31, 1667-1684 (2008)
26. Lei, Z, Zhou, Y: BKM criterion and global weak solutions for magnetohydrodynamics with zero viscosity. Discrete Contin. Dyn. Syst., Ser. A25, 575-583 (2009)
27. Wang, Y, Zhao, H, Wang, Y: A logarithmally improved blow up criterion of smooth solutions for the three-dimensional MHD equations. Int. J. Math.23, Article ID 1250027 (2012)
28. Wu, J: Viscous and inviscid magneto-hydrodynamics equations. J. Anal. Math. (Jerus.)73, 251-265 (1997) 29. Zhou, Y: Remarks on regularities for the 3D MHD equations. Discrete Contin. Dyn. Syst.12, 881-886 (2005)
30. Zhou, Y, Gala, S: Regularity criteria for the solutions to the 3D MHD equations in the multiplier space. Z. Angew. Math. Phys.61, 193-199 (2010)
31. Zhou, Y, Gala, S: A new regularity criterion for weak solutions to the viscous MHD equations in terms of the vorticity field. Nonlinear Anal.72, 3643-3648 (2010)
32. Zhou, Y, Gala, S: Regularity criteria for the solutions to the 3D MHD equations in the multiplier space. Z. Angew. Math. Phys.61, 193-199 (2010)
33. Admas, RA: Sobolev Spaces. Academic Press, New York (1975)
34. Galdi, GP: An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Vols I, II. Springer, New York (1994)
35. Ladyzhenskaya, OA: Mathematical Theory of Viscous Incompressible Flow, 2nd edn. Gordon & Breach, New York (1969). English translation
doi:10.1186/1687-2770-2013-58
