Regularizing Model for the 2D MHD Equations with Zero Viscosity

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Volume 2012, Article ID 212786,7pages doi:10.1155/2012/212786

Research Article

Regularizing Model for the 2D MHD Equations with Zero Viscosity

Ensil Kang

¹

and Jihoon Lee

²

1Department of Mathematics, Chosun University, Gwangju 501-759, Republic of Korea

2Department of Mathematics, Sungkyunkwan University, Suwon 440-746, Republic of Korea

Correspondence should be addressed to Jihoon Lee,[email protected] Received 1 March 2012; Accepted 9 April 2012

Academic Editor: Narcisa C. Apreutesei

Copyrightq2012 E. Kang and J. Lee. 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 consider the regularity of two dimensional incompressible magneto-hydrodynamics equations with zero viscosity. We provide an approximating system to the equations and prove global-in- time existence of classical solution to this approximating system. By using approximating system, a priori estimates for the equations can be justified.

1. Introduction

In this paper, we are concerned with regularity problem of solutions to the 2-dimensional incompressible magnetohydrodynamicsMHDequations with zero viscosity

ut u· ∇u∇p b· ∇b, x∈R², t >0, bt−Δb u· ∇b b· ∇u, x∈R², t >0,

∇ ·u∇ ·b0, x∈R², t >0,

1.1

where u u1, u2,b b1, b2, and p are fluid velocity vector field, magnetic field, and pressure function. The underlying idea of MHD is that dynamics of magnetic field induces the force on the fluid and, in turn, the motion of the conducting fluid aﬀects the dynamics of magnetic field. MHD has many applications in electromagnetics, plasma theory, and cosmology. MHD equations describe the dynamics of the conducting fluids and thus, MHD equations are expressed as the combinations of the fluid equations and Maxwell system. We restrict our interest on the incompressible Euler equationwith Lorentz force of magnetic fieldas the fluid equation. This is the special case that the fluid viscosity is quite smaller

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than the magnetic diffusivity. Recall that magnetic Prandtl number is approximately the ratio of viscosity and magnetic diffusivity. So1.1represents the zero magnetic Prandtl number case. In physics, liquid metal usually has the small magnetic Prandtl number. For the two- dimensional incompressible MHD equations with positive viscosity and magnetic diffusivity, it is well known that there exists a unique global classical solution for every initial data u0, b0 ∈ H^m with m ≥ 2 and∇ ·u0 ∇ ·b0 0see1,2 . But if viscosity of the fluid or the magnetic diffusivity is zero, then the global regularity issue for 2D remains as an open problem. For the case that viscosity is positive and the magnetic diffusivity is zero, then there are some studies on the regularity or blow-up criterion. We briefly recall a few of them. In 3 , the authors showed that if_T

0 ∇ ×ωL^∞dt <∞, then the solutionu, bremain smooth on 0, T . In4 , the authors showed that if_T

0 ∇ ×bBMOdt <∞, then the solutionu, bremain smooth on0, T . In5 , it was shown that if_T

0 b×bBMOdt < ∞, then the solutionu, b remain smooth on0, T . For the case that the viscosity is zero and the magnetic diﬀusivity is positive, that is, for the system1.1, we have better a prior estimates. In6–8 , the authors obtained the global existence of more regular weak solution, that is, we haveu∈L^∞0, T;H¹ andb∈L^∞0, T;H¹∩L²0, T;H₀²for anyT >0.

But still the global existence of smooth solution is remained as a challenging open problem. Only some blow-up criterion for the system 1.1is known. In 7 , Cao and Wu showed that if for someT >0,

sup

q≥2

1 q

_T

0

∇u_L^qdt <∞, 1.2

thenu, bremains smooth on0, T . Also Lei and Zhou8 obtained the regularity criterion in terms of theL¹0, T; BMO norm of∇ × ω. In6 , Kozono studied the stability of the solution to1.1 see9 also. And in10 , the authors studied the 2D incompressible MHD equations with horizontal dissipation and horizontal magnetic diﬀusion.

In this paper, we consider approximating system of1.1, which still preserves some properties of1.1. We consider

∂tu u· ∇u∇p −Δ^−αb· ∇b,

∂tb u· ∇b−Δb b· ∇

−Δ^−αu ,

∇ ·u∇ ·b0.

1.3

We study the regularity issue for1.3. We can rewrite1.3into the equations of the vorticity ω∂1u2−∂2u1and the current densityj∂1b2−∂2b1as follows:

ωt u· ∇ω −Δ^−α

b· ∇j

, x∈R², t >0, jt u· ∇j−Δj b· ∇−Δ^−αω−∂2b1

1 −Δ^−α

∂1u1−∂2b2

1 −Δ^−α

∂1u2

∂1b1

1 −Δ^−α

∂2u1∂1b2

1 −Δ^−α

∂2u2, x∈R², t >0.

1.4

We state our main results.

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Theorem 1.1. Assumeu0, b0 ∈ H³ and ∇ ·u0 ∇ ·b0 0. Then for any T > 0 andα ∈ 0,1/2, there exists a unique solutionu, b∈L^∞0, T;H³×L^∞0, T;H³∩L²0, T;H₀⁴of1.3.

Furthermore, there exist constantsCandDsatisfying

u_L∞0,T;H³b_L∞0,T;H³b_L20,T;H⁴₀≤De^CT, 1.5

whereCandDdepend onu0H³,b0H³, andα.

2. Global-in-Time Existence of Smooth Solution

To proveTheorem 1.1, we present some regularity criterion for the solution to1.3. The proof is standard and very similar to the criterion in7,8 . But for the readers’ sake, we provide the sketch of the proof.

Proposition 2.1. Assume the initial datau0, b0∈H³,∇ ·u0 0 and∇ ·b0 0. Letu, bbe the corresponding solution of 1.3. If, for someT >0,

_T

0

ωt_L∞dt <∞, 2.1

thenu, bis regular on0, T , namely,u, b∈C0, T ;H³.

Proof. We provide brief sketch of proof. If we take∇³ operator on the fluid equations and magnetic field equations and take inner product with∇³uand ∇³b, respectively, then we obtain

1 2

d dt

∇³u²

L²∇³b²

L²

∇⁴b²

L²

−

R²∇³u· ∇u∇³u dx

R²∇³−Δ^−αb· ∇b∇³u dx

−

R²∇³u· ∇b∇³b dx

R²∇³

b· ∇−Δ^−αu

∇³b dx.

2.2

There are some cancellation properties, that is,

−

R²u· ∇∇³u· ∇³u dx0,

R²−Δ^−α b· ∇∇³b

∇³u dx

R² b· ∇∇³b

∇³−Δ^−αu dx −

R² b· ∇∇³−Δ^−αu

∇³b dx.

2.3

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Using the previous cancellation, we have 1

2 d dt

∇³u²

L²∇³b²

L²

∇⁴b²

L²≤C∇u_L∞

∇³u²

L²∇²b²

L²

. 2.4

Then recall the following Beale-Kato-Majda’s logarithmic inequality11 ,

∇u_L^∞ ≤Cω_L^∞1ln1u_H³. 2.5

We have the conclusion via Gronwall type inequality.

We provide the Proof ofTheorem 1.1.

Proof ofTheorem 1.1. For simplicity of the exposition, the calculations are presented on 0 smooth solutions. All the calculations can be justified by using continuation method of the local solutions. If we multiply both sides of the first and second equations of1.4byωand j, respectively, and integrate overR², then we have

1 2

d

dtω²_L2≤

R²b· ∇j·−Δ^−αω dx, 1

2 d dtj²

L²∇j²

L² ≤

R²b· ∇−Δ^−αω·j dxC

R²|∇u||∇B|jdx

≤

R²b· ∇−Δ^−αω·j dxC∇u_L²j²

L⁴.

2.6

Since we have

R²b· ∇−Δ^−αω·j dx−

R²b· ∇−Δ^−αj·ω dx, 2.7 we obtain the following by adding the above inequalities:

1 2

d

dt ω²_L2j²

L²

∇j²

L²≤C∇u_L2j²

L⁴

≤Cω_L2j

L²∇j

L²

≤Cω²_L2j²

L² 1 2∇j²

L².

2.8

By using Gronwall’s inequality, we have sup

0≤t≤T ωt²_L2jt²

L²

_T

0

∇jt²

L²dt

≤ ω0²_L2j0²

L²

exp

C

_T

0

j²

L²dt

.

2.9

Then we obtainω∈L^∞0, T;L²andj∈L^∞0, T;L²∩L²0, T;H¹.

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By standardL^p,qestimate of the Stokes system or heat equationsee12,13 , we have _t

0

∇j^q_L_pdt≤C _t

0

b· ∇

−Δ^−αu^q

L^pdtCj0

H², 2.10

for anyp, q∈1,∞. We also recall the following inequality:

b· ∇

−Δ^−αu

L^p ≤ bL^2pu_W^1−2α,2p

0 . 2.11

Sincej ∈L^∞0, T;L²∩L²0, T;H¹,b∈L^∞0, T;L^2pforp <∞. Then we have _t

0

∇j^q

L^pdt≤C _t

0

u^q

W₀^1−2α,2pdtC. 2.12

If we consider a usual trajectory map Xx, t such that dX/dtx, t uXx, t, t and Xx,0 x, then we can rewrite the first equation of1.4as

d

dtωXx, t, t −Δ^−α∇ ·

bjXx, t, t

. 2.13

Thus we have

sup

0≤t≤Tω_Lr ≤ ω0_Lr _t

0

bj

W^1−2α,rdt, 2.14

for anyr∈1,∞ .

From the calculus inequality, we have bj

W^1−2α,r ≤C

b_L2rj

W^1−2α,2rb_W1−2α,2rj

L^2r

. 2.15

Forr∈2,∞andp≥2r/1αr, by Sobolev inequality, we have j

W^1−2α,2r ≤Cj

W^1,p₀ Cj

L². 2.16

For the caser∞, we choosepsuch thatαp >1, then we have j

W^1−2α,∞ ≤Cj

W₀^1,pCj

L². 2.17

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Hence for anyr <∞andp≥2r/1αr, we have

sup

0≤t≤Tω_Lr ≤ ω0_Lr C _T

0

b_L2rj

W^1−2α,2rb_W1−2α,2rj

L^2rdt

≤ ω0_Lr C _T

0

j

W₀^1,pdtC

.

2.18

Sincej ∈L^∞0, T;L²∩L²0, T;H₀¹, we have the finiteness ofbL^∞0,T;L^2rbL²0,T;W^1−2α,r j_L²_0,T;H¹

0in the above inequality.

If we use2.12, then for anyq∈1,∞, we have

sup

0≤t≤Tω_Lr ≤CCT^q−1/q _T

0

u^q

W₀^1−2α,2pdt 1/q

. 2.19

If we choosepsuch that 4r/1αr≤2p≤r/1−αrwe can choose suchpifr≥3/5α, then for anyq∈1,∞, we have

sup

0≤t≤Tω_Lr ≤CCT^q−1/q _T

0

ω^q_Lrdt _1/q

. 2.20

By using Gronwall’s inequality, we have sup

0≤t≤Tω_L^r ≤C, 2.21

for anyr∈3/5α,∞. In turn, it gives the bound of∇jL^q0,t;L^pfor allp, q∈1,∞.

Forr∞andp >1/α, we have

sup

0≤t≤Tω_L∞≤ ω0_L∞C _T

0

b_L∞j

W^1−2α,∞b_W1−2α,∞j

L^∞dt

≤ ω0_L^rC T

0

j

W₀^1,pdtC

.

2.22

We already have the finiteness of∇jL¹0,T;L^p; this gives our conclusion byProposition 2.1.

Remark 2.2. With the similar arguments in the proof ofTheorem 1.1and this approximating system, we can prove rigorously that the solutionu, bto1.1satisfiesu ∈ L^∞0, T;W^1,p andb∈L^∞0, T;W^1,p∩L^q0, T;W₀^2,pfor anyT >0 andp, q∈2,∞×1,∞as the remark in8 .

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Acknowledgment

J. Lee’s work was partly supported by the National Research Foundation of Korea NRF- 2009-0072320.

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Regularizing Model for the 2D MHD Equations with Zero Viscosity

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