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J. Math. Study doi: 10.4208/jms.v49n2.16.05 Vol. 49, No. 2, pp. 169-194 June 2016 Global Regularity for the 2D Magneto-Micropolar Equations with Partial Dissipation Dipendra Regmi 1 , Jiahong Wu 2 1 Department of Mathematics, Farmingdale State College, SUNY, 2350 Broadhollow Road, Farmingdale, NY 11735 2 Department of Mathematics, Oklahoma State University, 401 Mathematical Sciences, Stillwater, OK 74078 Received 31 March, 2016; Accepted 15 May, 2016 Abstract. This paper studies the global existence and regularity of classical solutions to the 2D incompressible magneto-micropolar equations with partial dissipation. The magneto-micropolar equations model the motion of electrically conducting micropo- lar fluids in the presence of a magnetic field. When there is only partial dissipation, the global regularity problem can be quite difficult. We are able to single out three special partial dissipation cases and establish the global regularity for each case. As special consequences, the 2D Navier-Stokes equations, the 2D magnetohydrodynamic equa- tions, and the 2D micropolar equations with several types of partial dissipation always possess global classical solutions. The proofs of our main results rely on anisotropic Sobolev type inequalities and suitable combination and cancellation of terms. AMS subject classifications: 35Q35, 35B35, 35B65, 76D03 Key words: Global regularity, magneto-micropolar equations, partial dissipation. 1 Introduction This paper aims at the global existence and regularity of classical solutions to the 2D incompressible magneto-micropolar equations with partial dissipation. The standard 3D incompressible magneto-micropolar equations can be written as t u +(u ·∇)u + ( p + 1 2 |b| 2 )=(μ + χ)Δu +(b ·∇)b +2χ∇× ω, t b +(u ·∇)b = νΔb +(b ·∇)u, t ω +(u ·∇)ω +2χω = κ Δω +(α + β)∇∇· ω +2χ∇× u, ∇· u = 0, ∇· b = 0, (1.1) Corresponding author. Email addresses: [email protected] (D. Regmi), [email protected] (J. Wu) http://www.global-sci.org/jms 169 c 2016 Global-Science Press
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Page 1: Global Regularity for the 2D Magneto-Micropolar Equations ...J. Math. Study doi: 10.4208/jms.v49n2.16.05 Vol. 49, No. 2, pp. 169-194 June 2016 Global Regularity for the 2D Magneto-Micropolar

J. Math. Studydoi: 10.4208/jms.v49n2.16.05

Vol. 49, No. 2, pp. 169-194June 2016

Global Regularity for the 2D Magneto-Micropolar

Equations with Partial Dissipation

Dipendra Regmi1, Jiahong Wu2∗

1 Department of Mathematics, Farmingdale State College, SUNY,2350 Broadhollow Road, Farmingdale, NY 117352Department of Mathematics, Oklahoma State University,401 Mathematical Sciences, Stillwater, OK 74078

Received 31 March, 2016; Accepted 15 May, 2016

Abstract. This paper studies the global existence and regularity of classical solutionsto the 2D incompressible magneto-micropolar equations with partial dissipation. Themagneto-micropolar equations model the motion of electrically conducting micropo-lar fluids in the presence of a magnetic field. When there is only partial dissipation, theglobal regularity problem can be quite difficult. We are able to single out three specialpartial dissipation cases and establish the global regularity for each case. As specialconsequences, the 2D Navier-Stokes equations, the 2D magnetohydrodynamic equa-tions, and the 2D micropolar equations with several types of partial dissipation alwayspossess global classical solutions. The proofs of our main results rely on anisotropicSobolev type inequalities and suitable combination and cancellation of terms.

AMS subject classifications: 35Q35, 35B35, 35B65, 76D03

Key words: Global regularity, magneto-micropolar equations, partial dissipation.

1 Introduction

This paper aims at the global existence and regularity of classical solutions to the 2Dincompressible magneto-micropolar equations with partial dissipation. The standard 3Dincompressible magneto-micropolar equations can be written as

∂tu+(u·∇)u+∇(p+ 12 |b|

2)=(µ+χ)∆u+(b·∇)b+2χ∇×ω,

∂tb+(u·∇)b=ν∆b+(b·∇)u,

∂tω+(u·∇)ω+2χω=κ∆ω+(α+β)∇∇·ω+2χ∇×u,

∇·u=0, ∇·b=0,

(1.1)

∗Corresponding author. Email addresses: [email protected] (D. Regmi),

[email protected] (J. Wu)

http://www.global-sci.org/jms 169 c©2016 Global-Science Press

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170 D. Regmi, J. Wu / J. Math. Study, 49 (2016), pp. 169-194

where, for x ∈ R3 and t ≥ 0, u = u(x,t),b = b(x,t),ω = ω(x,t) and p = p(x,t) denote the

velocity field, the magnetic field, the micro-rotation field and the pressure, respectively,and µ denotes the kinematic viscosity, ν the magnetic diffusivity, χ the vortex viscosity,and α, and β and κ the angular viscosities. The 3D magneto-micropolar equations reduceto the 2D magneto-micropolar equations when

u=(u1(x,y,t),u2(x,y,t),0), b=(b1(x,y,t),b2(x,y,t),0),

ω=(0,0,ω(x,y,t)), π=π(x,y,t),

where (x,y) ∈ R2 and we have written π = p+ 1

2 |b|2. More explicitly, the 2D magneto-

micropolar equations can be written as

∂tu+(u·∇)u+∇π=(µ+χ)∆u+(b·∇)b+2χ∇×ω,

∂tb+(u·∇)b=ν∆b+(b·∇)u,

∂tω+(u·∇)ω+2χω=κ∆ω+2χ∇×u,

∇·u=0, ∇·b=0,

(1.2)

where u=(u1,u2), b=(b1,b2), ∇×ω=(−∂yω,∂xω) and ∇×u=∂xu2−∂yu1.The magneto-micropolar equations model the motion of electrically conducting mi-

cropolar fluids in the presence of a magnetic field. Micropolar fluids represent a class offluids with nonsymmetric stress tensor (called polar fluids) such as fluids consisting ofsuspending particles, dumbbell molecules, etc (see, e.g., [6,8–10,17]). A generalization ofthe 2D magneto-micropolar equations is given by

∂tu1+(u·∇)u1+∂xπ=µ11∂xxu1+µ12∂yyu1+(b·∇)b1−2χ∂yω,

∂tu2+(u·∇)u2+∂yπ=µ21∂xxu2+µ22∂yyu2+(b·∇)b2+2χ∂xω,

∂tb+(u·∇)b=ν1∂xxb+ν2∂yyb+(b·∇)u,

∂tω+(u·∇)ω+2χω=κ1∂xxω+κ2∂yyω+2χ∇×u,

∇·u=0, ∇·b=0,

u(x,y,0)=u0(x,y),b(x,y,0)=b0(x,y),ω(x,y,0)=ω0(x,y),

(1.3)

where we have written the velocity equation in its two components. Clearly, if

µ11=µ12=µ21=µ22=µ+χ, ν1=ν2 =ν, κ1 =κ2=κ,

then (1.3) reduces to the standard 2D magneto-micropolar equations in (1.2). This gener-alization is capable of modeling the motion of anisotropic fluids for which the diffusionproperties in different directions are different. In addition, (1.3) allows us to explore thesmoothing effects of various partial dissipations.

The magneto-micropolar equations above are not only important in engineering andphysics, but also mathematically significant. The mathematical study of the magneto-micropolar equations started in the seventies and has been continued by many authors

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D. Regmi, J. Wu / J. Math. Study, 49 (2016), pp. 169-194 171

(see, e.g., [1, 18, 21, 29, 30]). Some of the recent efforts are devoted to the well-posednessproblem and various asymptotic behavior. The focus of this paper will be on the globalexistence and uniqueness problem on the generalized 2D magneto-micropolar equations(1.3) with various partial dissipation. We deal with three main partial dissipation casesand establish the global regularity for each case. For notational convenience, we set χ=1/2 for the rest of the paper.

The first partial dissipation case corresponds to (1.3) with

µ11=µ22=0, ν2=0, κ2 =0, µ12=µ21=1, ν1=κ1 =1,

or, more precisely,

∂tu1+(u·∇)u1+∂xπ=∂yyu1+(b·∇)b1−∂yω,

∂tu2+(u·∇)u2+∂yπ=∂xxu2+(b·∇)b2+∂xω,

∂tb+(u·∇)b=∂xxb+b·∇u,

∂tω+(u·∇)ω+ω=∂xxω+∇×u,

∇·u=0, ∇·b=0,

u(x,y,0)=u0(x,y),b(x,y,0)=b0(x,y),ω(x,y,0)=ω0(x,y).

(1.4)

The global existence and regularity result for this case can be stated as follows.

Theorem 1.1. Assume (u0,b0,ω0)∈ H2(R2), and ∇·u0 =∇·b0 = 0. Then (1.4) has a uniqueglobal classical solution (u,b,ω) satisfying, for any T>0,

(u,b,ω)∈L∞([0,T];H2(R2)).

The second partial dissipation case corresponds to (1.3) with

µ11=1, µ22=ν2=κ2=0, µ12=1, µ21=0, ν1=κ1 =1,

or, more precisely,

∂tu1+(u·∇)u1+∂xπ=∆u1+(b·∇)b1−∂yω,

∂tu2+(u·∇)u2+∂yπ=(b·∇)b2+∂xω,

∂tb+(u·∇)b=∂xxb+(b·∇)u,

∂tω+(u·∇)ω+ω=∂xxω+∇×u,

∇·u=0, ∇·b=0,

u(x,y,0)=u0(x,y),b(x,y,0)=b0(x,y),ω(x,y,0)=ω0(x,y).

(1.5)

The global well-posedness for (1.5) is given in the following theorem.

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172 D. Regmi, J. Wu / J. Math. Study, 49 (2016), pp. 169-194

Theorem 1.2. Assume (u0,b0,ω0)∈ H2(R2), and ∇·u0 =∇·b0 = 0. Then (1.5) has a uniqueglobal classical solution (u,b,ω) satisfying, for any T>0,

(u,b,ω)∈L∞([0,T];H2(R2)).

Our third main result establishes the global regularity for the partial dissipation case(1.3) with

µ12=µ22=1, µ11=µ21=ν2 =κ2=0, ν1 =κ1=1,

or, more precisely,

∂tu1+(u·∇)u1+∂xπ=∂yyu1+(b·∇)b1−∂yω,

∂tu2+(u·∇)u2+∂yπ=∂yyu2+(b·∇)b2+∂xω,

∂tb+(u·∇)b=∂xxb+(b·∇)u,

∂tω+(u·∇)ω+ω=∂xxω+∇×u,

∇·u=0, ∇·b=0,

u(x,y,0)=u0(x,y),b(x,y,0)=b0(x,y),ω(x,y,0)=ω0(x,y).

(1.6)

Theorem 1.3. Assume (u0,b0,ω0)∈ H2(R2), and ∇·u0 =∇·b0 = 0. Then (1.6) has a uniqueglobal classical solution (u,b,ω) satisfying, for any T>0,

(u,b,ω)∈L∞([0,T];H2(R2)).

It is worth mentioning some of the special consequences of our theorems. In the spe-cial case when b≡0 and ω≡0, the magneto-micropolar equations become the 2D Navier-Stokes equations and the theorems above assess the global regularity for the Navier-Stokes with various partial dissipation. These results for the Navier-Stokes equationsappear to be new.

Corollary 1.1. Consider (1.4), (1.5) or (1.6) with b≡ 0 and ω ≡ 0. Assume u0 ∈ H2(R2), and∇·u0=0. Then any one of these systems has a unique global solution.

When ω≡0, the magneto-micropolar equations become the magneto-hydrodynamic(MHD) equations. The results in the first two theorems are new for the MHD equationswhile the third one recovers a result in [3].

Corollary 1.2. Consider (1.4), (1.5) or (1.6) with ω≡0. Assume (u0,b0)∈H2(R2), and ∇·u0=∇·b0=0. Then any one of these systems has a unique global solution.

When b≡0, the magneto-micropolar equations become the micropolar equations andthe results in the theorems above reduce to those for the micropolar equations.

Corollary 1.3. Consider (1.4), (1.5) or (1.6) with b≡0. Assume (u0,ω0)∈H2(R2) and ∇·u0=0.Then any one of these systems has a unique global solution.

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D. Regmi, J. Wu / J. Math. Study, 49 (2016), pp. 169-194 173

Now we explain the main difficulties involved in proving the theorems and the meth-ods used here. The general approach to establish the global existence and regularity re-sults consists of two main steps. The first step assesses the local (in time) well-posednesswhile the second extends the local solution into a global one by obtaining global (in time)a priori bounds. For the systems of equations concerned here, the local well-posednessfollows from a standard approach and shall be skipped here. Our main efforts are de-voted to proving the necessary global a priori bounds. More precisely, we show that, forany T>0 and t≤T,

‖(u,b,ω)(·,t)‖H2(R2)≤C, (1.7)

where C denotes a bound that depends on T and the initial data. In general, we rely onthe smoothing effects of the dissipative terms in the systems. When there is no dissipationin (1.3), it is impossible to prove (1.7). Then the issue is how much dissipation we reallyneed in order to prove (1.7). We are able to single out the aforementioned three partialdissipation cases and prove (1.7).

The proof of (1.7) involves three steps. The first step proves the global L2-bound. Thisstep is easy and relies on the divergence-free condition ∇·u=∇·b= 0. The second stepproves the global H1-bound for (u,b,ω). This step is not trivial and fully exploits thepartial dissipation. This step also makes use of the anisotropic Sobolev type inequalities(see Lemma 2.1). This last step is to prove the global H2-bound by using the global H1-bound and various anisotropic inequalities. The whole process involves the estimates ofmany terms and is complex. The details are given in the subsequent sections.

We briefly mention some of closely related results. In [29] Yamazaki obtained theglobal regularity of the 2D magneto-micropolar equation with zero angular viscosity,namely (1.2) with κ=0 and other coefficients being positive. Another partial dissipationcase for the 2D magneto-micropolar equation was studied in [5]. As aforementioned,quite a few global regularity results for the 2D MHD and the 2D micropolar equationswith partial dissipation are available (see, e.g., [2–4, 7, 11–16, 19, 20, 22–28, 31]).

The rest of this paper is divided into three sections with each of them devoted to theproof of one of the theorems stated above. To simplify the notation, we will write ‖ f‖2

for ‖ f‖L2 ,∫

f for∫

R2 f dxdy and write ∂∂x f , ∂x f or fx as the first partial derivative, and

∂2

∂x2 f or ∂xx f as the second partial throughout the rest of this paper.

2 Proof of Theorem 1.1

This section is devoted to the proof of Theorem 1.1. As explained in the introduction,it suffices to establish the global a priori bound for the solution in H2. For the sake ofclarity, we divide this process into two subsections. The first subsection proves the globalH1-bound while the second proves the global H2-bound.

In the proof of Theorem 1.1, the following anisotropic type Sobolev inequality will befrequently used. Its proof can be found in [3].

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174 D. Regmi, J. Wu / J. Math. Study, 49 (2016), pp. 169-194

Lemma 2.1. If f ,g,h,∂yg,∂xh∈L2(R2), then

∫∫

R2| f gh|dxdy≤C‖ f‖2 ‖g‖

122 ‖∂yg‖

122 ‖h‖

122 ‖∂xh‖

122 , (2.1)

where C is a constant.

The following simple fact on the boundedness of Riesz transforms will also be used.

Lemma 2.2. Let f be divergence-free vector field such that ∇ f ∈ Lp for p∈ (1,∞). Then thereexists a pure constant C>0 (independent of p) such that

‖∇ f‖Lp ≤C p2

p−1‖∇× f‖Lp .

The rest of this section is divided into two subsections. The first subsection provesthe global H1-bound while the second proves the global H2-bound.

2.1 H1-Bound

We first state the global L2-bound.

Lemma 2.3. Assume that (u0,b0,ω0) satisfies the condition stated in Theorem 1.1. Let (u,b,ω)be the corresponding solution of (1.4). Then, (u,b,ω) obeys the following global L2-bound,

‖u(t)‖2L2 +‖b(t)‖2

L2 +‖ω(t)‖2L2 +2

∫ t

0‖(∂yu1,∂xu2)‖

2L2 dτ

+2∫ t

0‖∂xb(τ)‖2

L2 dτ+2∫ t

0‖∂xω(τ)‖2

L2 dτ≤C(‖(u0,b0,ω0)‖22)

for any t≥0.

Proof. The proof of the global L2-bound is easy. Taking the L2-inner product of (u,b,ω)with (1.4), respectively, yields

1

2∂t‖u‖2

2+‖(∂yu1,∂xu2)‖22=

(b·∇)b·u+∫

(∇⊥ω)·u,

1

2∂t‖b‖2

2+‖∂xb‖22 =

(b·∇)u·b,

1

2∂t‖ω‖2

2+‖ωx‖22+‖ω‖2

2=∫

(∇×u)ω.

Adding them up and using the fact

(b·∇)b·u+∫

(b·∇)u·b=0,∫

∇⊥ω ·u=∫

(∇×u)ω,

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D. Regmi, J. Wu / J. Math. Study, 49 (2016), pp. 169-194 175

we have

1

2∂t(‖u‖2

2+‖b‖22+‖ω‖2

2)+‖(∂yu1,∂xu2)‖22+‖∂xb‖2

2+‖ωx‖22+‖ω‖2

2=2∫

(∇×u)ω.

To bound the right-hand side, we notice that

2∫

(∇×u)ω=2∫

(∂xu2−∂yu1)ω=−2∫

∂yu1ωdxdy−2∫

∂xωu2.

Applying Holder’s inequality yields

1

2∂t(‖u‖2

2+‖b‖22+‖ω‖2

2)+‖(∂yu1,∂xu2)‖22+‖∂xb‖2

2+‖ωx‖22+‖ω‖2

2

≤1

2(‖∂yu1‖

22+‖∂xω‖2

2)+C(‖u‖22+‖ω‖2

2).

Gronwall’s inequality then implies

‖u‖22+‖b‖2

2+‖ω‖22+

∫ t

0

(

‖(∂yu1,∂xu2)‖22+‖∂xb‖2

2+‖ωx‖22+‖ω‖2

2

)

dτ≤C,

for any 0< t≤T, where C depends only on the initial data.

We next prove the global H1-bound for u,b and ω.

Proposition 2.1. Assume that (u0,b0,ω0) satisfies the condition stated in Theorem 1.1. Let(u,b,ω) be the corresponding solution of (1.4). Then (u,b,ω) satisfies, for any T>0,

u,b,ω∈C([0,T];H1). (2.2)

Proof of Proposition 2.1. To estimate the H1-norm of (u,b,ω), we consider the equations ofΩ=∇×u, ∇ω and of the current density j=∇×b,

Ωt+u·∇Ω=∂xxxu2−∂yyyu1+(b·∇)j−∆ω, (2.3)

∂t∇ω+∇(u·∇ω)+2∇ω=∇ωxx+∇Ω, (2.4)

jt+u·∇j=∂xx j+b·∇Ω+2∂xb1(∂xu2+∂yu1)−2∂xu1(∂xb2+∂yb1), (2.5)

∇·u=0, ∇·b=0.

Dotting (2.3) by Ω, (2.4) by ∇ω and (2.5) by j, we obtain

1

2

d

dt

(

‖Ω‖2L2 +‖j‖2

L2 +‖∇ω‖22

)

+‖∇∂yu1‖22

+‖∇∂xu2‖22+‖∂x j‖2

L2 +‖∇ωx‖22+2‖∇ω‖2

2

= 2∫

[

∂xb1(∂xu2+∂yu1) j−∂xu1(∂xb2+∂yb1)]

jdxdy

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176 D. Regmi, J. Wu / J. Math. Study, 49 (2016), pp. 169-194

−∫

∇ω ·∇u·∇ω+2∫

∇ω ·∇Ω

≡ J1+ J2+ J3+ J4+ J5+ J6.

Invoking the divergence-free condition, we note that

‖∇∂xu1‖22=‖∂xxu1‖

22+‖∂yyu2‖

22,

‖∇∂yu1‖22=‖∂yyu2‖

22+‖∂yyu1‖

22,

‖∇∂xu2‖22=‖∂xxu1‖

22+‖∂xxu2‖

22.

We now estimate the terms on the right. Since j=∂xb2−∂yb1,

J1= 2∫

∂xb1∂xu2∂xb2−2∫

∂xb1∂xu2∂yb1

≡ J11+ J12.

Applying lemma 2.1, Young’s inequality, and the simple fact that

‖∂xb2‖L2 ≤‖j‖L2 , ‖∂xyb1‖L2 ≤‖∂x j‖L2 ,

we have

J11≤ 2

∂xb1∂xu2∂xb2

≤ C‖∂xu2‖2‖∂xb1‖122 ‖∂xyb1‖

122 ‖∂xb2‖

122 ‖∂xxb2‖

122

≤ C‖Ω‖2‖∂xb1‖122 ‖∂x j‖

122 ‖∂xb2‖

122 ‖∂x j‖

122

≤ C‖Ω‖2‖∂xb‖2‖∂x j‖2

≤1

48‖∂x j‖2

2+C‖∂xb‖22‖Ω‖2

2.

Integrating by parts, we have

J12=2∫

∂xb1∂xu2∂yb1=−2∫

u2∂xxb1∂yb1−2∫

u2∂xb1∂xyb1

≡ J121+ J122.

J121≤

−2∫

u2∂xxb1∂yb1

≤C‖∂xxb1‖‖u2‖122 ‖∂yu2‖

122 ‖∂yb1‖

122 ‖∂xyb1‖

122

≤C‖∂x j‖2‖u2‖122 ‖Ω‖

122 ‖j‖

122 ‖∂x j‖

122

≤1

48‖∂x j‖2

2+C‖u2‖2‖j‖2‖Ω‖2.

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D. Regmi, J. Wu / J. Math. Study, 49 (2016), pp. 169-194 177

Similarly,

J122≤

−2∫

u2∂xb1∂xyb1

≤1

48‖∂x j‖2

2+C‖u2‖2‖j‖2‖Ω‖2.

J2, J3, and J4 can be bounded by

J2≤

∂xb1 ∂yu1 j

≤C‖∂xb1‖122 ‖∂xxb1‖

122 ‖∂yu1‖

122 ‖∂yyu1‖

122 ‖j‖2

≤1

48

[

‖∂yyu1‖22+‖∂x j‖2

2

]

+C(‖∂xb1‖22+‖∂yu1‖

22)‖j‖2

2.

J3≤

∂xu1∂xb2 j

≤∫

|(u1∂xxb2 j+u1∂xb2∂x j)|

≤C‖u1‖122 ‖∂yu1‖

122 ‖j‖

122 ‖∂x j‖

122 ‖∂xxb2‖2

+C‖u1‖122 ‖∂yu1‖

122 ‖∂xb2‖

122 ‖∂xxb2‖

122 ‖∂x j‖2

≤1

48‖∂x j‖2

2+C‖u1‖22‖∂yu1‖

22‖j‖2

2.

J4≤

∂xu1∂yb1 j

≤∫

∣(u1∂xyb1 j−u1∂yb1 jx)∣

≤C‖u1‖122 ‖∂yu1‖

122 ‖j‖

122 ‖∂x j‖

122 ‖∂xyb1‖2

+C‖u1‖122 ‖∂yu1‖

122 ‖∂yb1‖

122 ‖∂xyb1‖

122 ‖jx‖2

≤1

48‖∂x j‖2

2+C‖u1‖22‖∂yu1‖

22‖j‖2

2.

To bound J5, we use ∇·u=0 and integrate by parts to obtain

J5=−∫

∇ω ·∇u·∇ω

=−2∫

u1ωxx ωx−2∫

u1ωyωxy−∫

(∂xu2+∂yu1)ωxωy.

The terms on the right can be bounded as

u1ωxx ωx

≤C‖ωxx‖2‖u1‖122 ‖∂yu1‖

122 ‖ωx‖

122 ‖ωxx‖

122

≤C‖ωxx‖322 ‖u1‖

122 ‖∂yu1‖

122 ‖ωx‖

122

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178 D. Regmi, J. Wu / J. Math. Study, 49 (2016), pp. 169-194

≤1

48‖ωxx‖

22+C‖u1‖

22‖∂yu1‖

22‖∇ω‖2

2.∣

u1ωyωxy

≤C‖ωxy‖2‖u1‖122 ‖‖∂yu1‖

122 ‖∂xyω‖

122 ‖∂yω‖

122

≤C‖ωxy‖322 ‖u1‖

122 ‖∂yu1‖

122 ‖∇ω‖

122

≤1

48‖∇ωx‖

22+C‖u1‖

22‖∂yu1‖

22‖∇ω‖2

2.∣

(∂xu2+∂yu1)ωxωy

≤C(‖∂xu2‖2+‖∂yu1‖2)‖ωx‖122 ‖ωxy‖

122 ‖ωy‖

122 ‖ωxy‖

122

≤C(‖∂xu2‖2+‖∂yu1‖2)‖∇ω‖2‖∇ωx‖2

≤1

48‖∇ωx‖

22+C(‖∂xu2‖

22+‖∂yu1‖

22)‖∇ω‖2

2.

To estimate J6, we first integrate by part to obtain

J6=2∫

∇ω ·∇Ω=−2∫

ωxx Ω+2∫

ωy Ωy.

The terms on the right can be bounded as follows.∣

ωxx Ω

≤‖ωxx‖2‖Ω‖2 ≤1

2‖∇ωx‖

22+C‖Ω‖2

2,

ωyΩy=∫

(ωy∂xyu2−ωy∂yyu1),∣

ωy∂xyu2

≤C‖∇ω‖2‖∇∂xu2‖2,

ωy∂yyu1

≤C‖∇ω‖2‖∇∂yu1‖2.

Combining the estimates above, together with Gronwall’s inequalities, we obtain

‖Ω‖22+‖j‖2

2+‖∇ω‖22

+∫ t

0

(

‖∇∂yu1‖22+‖∇∂xu2‖

22+‖∂x j‖2

L2 +‖∇ωx‖22+2‖∇ω‖2

2

)

dτ≤C

for any t≤T, where C depends on T and the initial H1-norm. Especially, (2.2) is proven.This completes the proof of Proposition 2.1.

2.2 Global H2 bound and the proof of Theorem 1.1

This subsection proves Theorem 1.1 by establishing the global H2 bound for the solution.

Proof of Theorem 1.1. As we explained before, it suffices to establish the global H2-boundin order to prove Theorem 1.1. The rest of this proof establishes the global H2-bound.

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D. Regmi, J. Wu / J. Math. Study, 49 (2016), pp. 169-194 179

Taking the L2 inner product of (2.3) with ∇Ω and (2.5) with ∇j, and integrating byparts, we obtain

1

2

d

dt(‖∇Ω‖2

2+‖∇j‖22)+‖∆∂yu1‖

22+‖∆∂xu2‖

22+‖∇∂x j‖2

2

= L1+L2+L3+L4+L5+L6, (2.6)

where

L1=−∫

∇Ω·∇u·∇Ω dxdy, L2=−∫

∇j·∇u·∇j dxdy,

L3=2∫

∇Ω·∇b·∇j dxdy, L4=2∫

∇[∂xb1(∂xu2+∂yu1)]·∇j dxdy,

L5=−2∫

∇[∂xu1(∂xb2+∂yb1)]·∇j dxdy, L6=∫

∆Ω∆ω dxdy.

Applying ∇ to (2.4) and taking the L2-inner product with ∆ω, and integrating by parts,we obtain

1

2

d

dt‖∆ω‖2

2+2‖∆ωx‖22+‖∆ω‖2

2 =∫

∆Ω∆ω−∫

∆(u·∇ω)∆ω

≡ L6+L7. (2.7)

Adding (2.6) and (2.7) yields

1

2

d

dt(‖∇Ω‖2

2+‖∇j‖22+‖∆ω‖2

2)+‖∆∂yu1‖22

+‖∆∂xu2‖22+‖∇∂x j‖2

2+2‖∆ωx‖22+‖∆ω‖2

2

= L1+L2+L3+L4+L5+2L6+L7.

We now estimate L1 through L7. We further split L1 into 4 terms.

L1= −∫

∇Ω·∇u·∇Ω dxdy

= −∫

(

∂xu1(∂xΩ)2+∂xu2∂xΩ∂yΩ+∂yu1∂xΩ∂yΩ+∂yu2(∂yΩ)2)

= L11+L12+L13+L14.

Due to Ω=∇×u, we have

∂xxΩ=∆∂xu2, ∂yyΩ=−∆∂yu1, ∂xyΩ=∆∂xu2.

Therefore,

L11=−∫

∂xu1(∂xxu2)2−

∂xu1(∂xyu1)2+2

∂xu1∂xxu2∂xyu1.

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180 D. Regmi, J. Wu / J. Math. Study, 49 (2016), pp. 169-194

Integration by parts yields

∂xu1(∂xxu2)2=−

∂xxu1∂xxu2∂xu2−∫

∂xu1∂xxxu2∂xu2

≡L111+L112,

which can be bounded as

L111≤C‖∂xxu2‖‖∂xxu1‖122 ‖∂xxyu1‖

122 ‖∂xu2‖

122 ‖∂xxu2‖

122

≤C‖∂xxu2‖322 ‖∂xxu1‖

122 ‖∆∂xu2‖

122 ‖∂xu2‖

122

≤1

48‖∆∂xu2‖

22+C‖∂xxu2‖

22‖∂xxu1‖

232 ‖∂xu2‖

232 .

L112≤C‖∂xxxu2‖2‖∂xu1‖122 ‖∂xyu1‖

122 ‖∂xu2‖

122 ‖∂xxu2‖

122

≤C‖∆∂xu2‖2‖Ω‖122 ‖∇Ω‖2‖∂xu2‖

122

≤1

48‖∆∂xu2‖

22+C‖Ω‖2‖∂xu2‖2‖∇Ω‖2

2.

By Lemma 2.1,

L12=∫

∂xu2∂xΩ∂yΩ

≤C‖∂xu2‖2‖∂xΩ‖122 ‖∂xxΩ‖

122 ‖∂yΩ‖

122 ‖∂yyΩ‖

122

≤C‖∂xu2‖2‖∇Ω‖2‖∂xxΩ‖122 ‖∂yyΩ‖

122

≤‖∂xxΩ‖‖∂yyΩ‖+C‖∂xu2‖22‖∇Ω‖2

2

≤1

48(‖∂xxΩ‖2

2+‖∂yyΩ‖22)+C‖∂xu2‖

22‖∇Ω‖2

2.

L13≤

∂yu1∂xΩ∂yΩ

≤C‖∂yu1‖122 ‖∂xyu1‖

122 ‖∂xΩ‖2‖∂yΩ‖

122 ‖∂yyΩ‖

122

≤C‖∂yyΩ‖2‖∂xyu1‖2‖∂yΩ‖2+C‖∂yu1‖2‖∇Ω‖22

≤1

48‖∂yyΩ‖2

2+C(‖∂yu1‖22+‖∂xyu1‖

22)‖∇Ω‖2

2.

L14≤

2∫

u2∂yΩ∂yyΩ

≤C‖∂yyΩ‖2‖u2‖122 ‖∂xu2‖

122 ‖∂yΩ‖

122 ‖∂yyΩ‖

122

≤C‖∂yyΩ‖322 ‖u2‖

122 ‖Ω‖

122 ‖∇Ω‖

122

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D. Regmi, J. Wu / J. Math. Study, 49 (2016), pp. 169-194 181

≤1

48‖∂yyΩ‖2

2+C‖u2‖22‖Ω‖2

2‖∇Ω‖22.

To estimate L2, we write it out explicitly as

L2=−∫

∇j·∇u·∇j dxdy

=−∫

(

∂xu1(∂x j)2+∂yu1∂x j∂y j+∂yu2(∂y j)2+∂xu2∂x j∂y j)

=L21+L22+L23+L24.

The terms on the right can be bounded as follows.

L21≤C‖∂x j‖2‖∂xu1‖122 ‖∂xyu1‖

122 ‖∂x j‖

122 ‖∂xx j‖

122

≤C‖Ω‖122 ‖∂xyu1‖

122 ‖∇j‖

32 ‖∇∂x j‖

122

≤1

48‖∇∂x j‖2

2+C‖∂xyu1‖232 ‖Ω‖

232 ‖∇j‖2

2.

L22≤C‖∂yu1‖122 ‖∂xyu1‖

122 ‖∂x j‖

122 ‖∂xy j‖

122 ‖∂y j‖2

≤C‖Ω‖122 ‖∂xyu1‖

122 ‖∇j‖

322 ‖∇∂x j‖

122 .

L23≤C‖∂y j‖2‖∂xu1‖122 ‖∂xyu1‖

122 ‖∂y j‖

122 ‖∂xy j‖

122

≤C‖∇j‖322 ‖∇∂x j‖

122 ‖Ω‖

122 ‖∂xyu1‖

122

≤1

48‖∇∂x j‖2

2+‖Ω‖232 ‖∂xyu1‖

232 ‖∇j‖2

2

L24≤C‖∂xu2‖2‖∂y j‖122 ‖∂xy j‖

122 ‖∂x j‖

122 ‖∂xy j‖

122

≤C‖Ω‖2‖∇j‖2‖∇∂x j‖2.

We now turn to L3. Again we write it out as

L3=∫

∂xΩ∂xb1∂x j+∂xΩ∂xb2∂y j+∂yΩ∂yb1 jx+∂yΩ∂yb2∂y j

≡ L31+L32+L33+L34.

The terms on the right can be bounded as follows.

L31≤

∂xΩ∂xb1∂x j

≤C‖∂xΩ‖2‖∂xb1‖122 ‖∂xxb1‖

122 ‖∂x j‖

122 ‖∂xy j‖

122

≤C‖∂xΩ‖2‖∂xb1‖122 ‖∂x j‖

122 ‖∂x j‖

122 ‖∂xy j‖

122

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182 D. Regmi, J. Wu / J. Math. Study, 49 (2016), pp. 169-194

≤C‖∂x j‖2‖∇∂x j‖2+‖∂x j‖‖∂xb1‖‖∂xΩ‖22

≤1

48‖∇∂x j‖2

2+C(‖∂xb1‖22+‖∂x j‖2

2+1)(‖∇Ω‖22+‖∇j‖2

2).

L32≤1

48‖∇jx‖

22+C(‖∂xb2‖

22+‖∂x j‖2+1)(‖∇Ω‖2

2+‖∇j‖22).

L33≤C‖∂yb1‖122 ‖∂xyb1‖

122 ‖∂x j‖

122 ‖∂xy j‖

122 ‖∇Ω‖2

≤C‖j‖122 ‖∂x j‖

122 ‖∇j‖

122 ‖∇∂x j‖

122 ‖∇Ω‖2

≤1

48‖∇∂x j‖2

2+C‖∂x j‖22‖∇j‖2

2+C‖j‖2‖∇Ω‖22.

L34≤C‖∂xb1‖122 ‖∂xyb1‖

122 ‖∂y j‖

122 ‖∂xy j‖

122 ‖∇Ω‖2

≤C‖j‖122 ‖∂x j‖

122 ‖∇j‖

122 ‖∇∂x j‖

122 ‖∇Ω‖2

≤1

48‖∇∂x j‖2

2+C‖∂x j‖22‖∇j‖2

2+C‖j‖2‖∇Ω‖22.

We now estimate L4.

L4= 2∫

∇[∂xb1(∂xu2+∂yu1)]·∇j dxdy

= 2∫

∂x[∂xb1(∂xu2+∂yu1)]jx+∂y[∂xb1(∂xu2+∂yu1)]jy dxdy

≡ L41+L42.

We bound L41 and L42 as follows.

L41≤

−2∫

∂xb1(∂xu2+∂yu1)∂xx j

≤ C(‖∂xb1‖122 ‖∂xxb1‖

122 ‖∂xu2‖

122 ‖∂xyu2‖

122

+C‖∂xb1‖122 ‖∂xyb1‖

122 ‖∂yu1‖

122 ‖∂xyu1‖

122 )‖∂xx j‖2

≤ C‖j‖122 ‖∇j‖

122 ‖Ω‖

122 ‖Ωy‖

122 ‖∇∂x j‖2

≤1

48‖∇∂x j‖2

2+C‖Ω‖2‖j‖2(‖∇Ω‖22+‖∇j‖2

2).

L42 can be more explicitly written as

L42= 2∫

(∂xyb1∂xu2+∂xb1∂xyu2+∂xyb1∂yu1+∂xb1∂yyu1)∂y j dxdy

≡ L421+L422+L423+L424.

The bounds for the terms on the right are given as follows.

L421≤C‖∂xu2‖2‖∂xyb1‖122 ‖∂xyyb1‖

122 ‖∂y j‖

122 ‖∂xy j‖

122

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D. Regmi, J. Wu / J. Math. Study, 49 (2016), pp. 169-194 183

≤C‖Ω‖2‖∇j‖2‖∇∂x j‖2

≤1

48‖∇∂x j‖2

2+C‖Ω‖22‖∇j‖2

2.

L422≤C‖∂xb1‖122 ‖∂xyb1‖

122 ‖∂xyu2‖2‖∂y j‖

122 ‖∂xy j‖

122

≤C‖j‖122 ‖∂x j‖

122 ‖∂xΩ‖‖∇j‖

122 ‖∇∂x j‖

122

≤1

48‖∇∂x j‖2

2+C‖∂x j‖22‖∇j‖2

2+C‖j‖2‖∇Ω‖22.

L423≤C‖∂xyb1‖122 ‖∂xyyb1‖

122 ‖∂yu1‖

122 ‖∂xyu1‖

122 ‖∂y j‖2

≤C‖∂x j‖122 ‖∇∂x j‖

122 ‖Ω‖

122 ‖∂xΩ‖

122 ‖∇j‖2

≤1

48‖∇∂x j‖2

2+C‖Ω‖2‖∇j‖22+C‖∂x j‖2

2‖∇Ω‖22.

L424≤C‖∂y j‖2‖∂xb1‖122 ‖∂xyb1‖

122 ‖∂yyu1‖

122 ‖∂xyyu1‖

122

≤C‖∇j‖2‖j‖122 ‖∂x j‖

122 ‖∇Ω‖

122 ‖∂yyΩ‖

122

≤1

48‖∂yyΩ‖2

2+C‖∂x j‖22‖∇Ω‖2

2+C‖j‖2‖∇j‖22.

We now estimate L5. More explicitly, L5 is written as

L5= −2∫

∇[∂xu1(∂xb2+∂yb1)]·∇j dxdy

= −2∫

∂x[∂xu1(∂xb2+∂yb1)]∂x j+∂y[(∂xu1(∂xb2+∂yb1)]∂y j dxdy

≡ L51+L52.

L51 is bounded as follows.

L51≤ C‖∂xu1‖122 ‖∂xyu1‖

122 ‖∂xb2‖

122 ‖∂xxb2‖

122 ‖∂xx j‖2

+C‖∂xu1‖122 ‖∂xyu1‖

122 ‖∂yb1‖

122 ‖∂xyb1‖

122 ‖∂xx j‖2

≤ C‖Ω‖122 ‖∇Ω‖

122 ‖j‖

122 ‖∇j‖

122 ‖∇∂x j‖2

≤1

48‖∇∂x j‖2

2+C‖Ω‖2‖j‖2(‖∇Ω‖22+‖∇j‖2

2).

L52 contains four terms.

L52= −2∫

(∂xyu1∂xb2+∂xu1∂xyb2+∂xyu1∂yb1+∂xu1∂yyb1)∂y j dxdy

≡ L521+L522+L523+L524.

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184 D. Regmi, J. Wu / J. Math. Study, 49 (2016), pp. 169-194

These terms are estimated as follows.

L521≤1

48‖∇∂x j‖2

2+C(‖∂x j‖22+‖j‖2

2)(‖∇j‖22+‖∇Ω‖2

2).

L522≤1

48‖∇∂x j‖2

2+C‖Ω‖232 ‖∂xyu1‖

232 ‖∇j‖2

2.

L523≤C‖∂xyu1‖122 ‖∂xyyu1‖

122 ‖∂yb1‖

122 ‖∂xyb1‖

122 ‖∂y j‖2

≤C‖Ωyy‖122 ‖∂x j‖

122 ‖∇Ω‖

122 ‖j‖

122 ‖∇j‖2

≤1

48‖∂yyΩ‖2

2+C‖∂x j‖22‖∇Ω‖2

2+C‖j‖2‖∇j‖22.

L524≤C‖∂y j‖2‖∂xu1‖122 ‖∂xyu1‖

122 ‖∂yyb1‖

122 ‖∂xyyb1‖

122

≤C‖∇j‖2‖Ω‖122 ‖∂xyu1‖

122 ‖∇j‖

122 ‖∇∂x j‖

122

≤1

48‖∇∂x j‖2

2+C‖∂xyu1‖22‖∇j‖2

2+‖Ω‖2‖∇j‖22.

L6 can be easily bounded.

L6=∫

∆Ω∆ω=∫

Ωxx∆ω+∫

Ωyy∆ω

with

Ωxx∆ω=−∫

Ωx∆ωx ≤‖∇Ω‖2‖∆ωx‖2,

Ωyy∆ω

≤‖Ωyy‖2‖∆ω‖2.

We now estimate the last term L7.

L7 =−∫

∆(u·∇ω)∆ω=−∫

∆(u1∂1ω+u2∂2ω)∆ω≡ L71+L72.

We first split L71 and L72 each into two terms.

L71 =−∫

∂xx(u1∂xω+u2∂yω)∆ω= L711+L712.

L72 =−∫

∂yy(u1∂xω+u2∂yω)∂xxω−∫

∂yy(u1∂xω+u2∂yω)∂yyω= L721+L722.

These terms are bounded as follows.

|L711|=

−∫

∂x(u1∂xω)∆ωx

−∫

∂xu1∂xω∆ωx

+

u1∂xxω∆ωx

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D. Regmi, J. Wu / J. Math. Study, 49 (2016), pp. 169-194 185

≤C‖∆ωx‖2‖∂xu1‖122 ‖∂xyu1‖

122 ‖∂xω‖

122 ‖∂xxω‖

122

+C‖∆ωx‖2‖∂xxω‖122 ‖∂xxxω‖

122 ‖u1‖

122 ‖∂yu1‖

122

≤C‖∆ωx‖2‖∆ω‖122 ‖Ω‖

122 ‖∇Ω‖

122 ‖∇ω‖

122 +C‖∆ωx‖

322 ‖∇ωx‖

122 ‖u1‖

122 ‖Ω‖

122

≤1

48‖∆ωx‖

22+C‖Ω‖2

2‖∇Ω‖22+‖∇ω‖2

2‖∆ω‖22+C‖u1‖

22‖∇ωx‖

22‖Ω‖2

2.

|L712|=

−∫

∂xu2∂yω∆ωx−∫

u2∂xyω∆ωx

≤C‖∆ωx‖2‖∂xu2‖122 ‖∂xyu2‖

122 ‖∂yω‖

122 ‖∂xyω‖

122

+C‖∆ωx‖2‖u2‖122 ‖∂xu2‖

122 ‖∂xyω‖

122 ‖∂xyyω‖

122 .

L721=∫

∂yy(u1∂xω+u2∂yω)∂xxω=∫

∂xx(u1∂xω+u2∂yω)∂yyω.

Obviously L721 admits the same bound as that for L711,

|L721|≤1

48‖∆ωx‖

22+C‖Ω‖2

2‖∇Ω‖22+‖∇ω‖2

2‖∆ω‖22+C‖u1‖

22‖∇ωx‖

22‖Ω‖2

2.

To estimate L722, we write it out explicitly and integrate by parts,

L722 =∫

∂yy(u1∂xω+u2∂yω)∂yyω

=∫

∂y(∂yu1∂xω+u1∂xyω)∂yyω+∫

∂y(∂yu2∂2ω+u2∂yyω)∂yyω

=∫

[∂yyu1∂xω+2∂yu1∂xyω+u1∂xyyω]∂yyω

+∫

[∂yyu2∂yω+2∂yu2∂yyω+u2∂yyyω]∂yyω.

The terms on the right can be bounded as follows.∣

∂yyu1∂xω∂yyω

≤ C‖∂xω‖2‖∂yyu1‖122 ‖∂yyyu1‖

121 ‖∂yyω‖

122 ‖∂xyyω‖

122

≤ C‖∂xω‖2‖∇∂yu1‖122 ‖∆∂yu1‖

122 ‖∆ω‖

122 ‖∆∂xω‖

122

≤1

48‖∆∂yu1‖

22+

1

48‖∆ωx‖

22+C‖ωx‖

22[‖∇∂yu1‖

22+‖∆ω‖2

2)

≤1

48‖∆∂yu1‖

22+

1

48‖∆ωx‖

22+C‖ωx‖

22(‖∇Ω‖2

2+‖∆ω‖22).

∂yu1∂xyω∂yyω

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186 D. Regmi, J. Wu / J. Math. Study, 49 (2016), pp. 169-194

≤ C‖∂yyω‖2‖∂xyω‖122 ‖∂xyyω‖

122 ‖∂yu1‖

122 ‖∂xyu1‖

122

≤ C‖∆ω‖2‖∇ωx‖122 ‖∆ωx‖

122 ‖∂yu1‖

122 ‖∇Ω‖

122

≤1

48‖∆ωx‖

22+C‖∇ωx‖

22‖∇Ω‖2

2+C‖∂yu1‖2‖∆ω‖22

u1∂xyyω∂yyω

≤ C‖∂xyyω‖‖u1‖122 ‖∂yu1‖

122 ‖∂yyω‖

122 ‖∂xyyω‖

122

≤ C‖∆ωx‖322 ‖u1‖

122 ‖∂yu1‖

122 ‖∆ω‖

122

≤1

48‖∆ωx‖

22+C‖u1‖

22‖∂yu1‖

22‖∆ω‖2

2.

∂yyu2∂yω∂yyω

≤ C‖∂yyu2‖2‖∂yω‖122 ‖ωyy‖2‖ωxyy‖

122

≤ ‖ωy‖2‖∆ωx‖2+‖∇∂yu1‖22‖∆ω‖2

2

≤1

48‖∆ωx‖

22+C‖∇ω‖2

2+C‖∇∂yu1‖22‖∆ω‖2

2.

Integration by parts yields

∂yu2∂yyω∂yyω=−∫

∂xu1∂yyω∂yyω=2∫

u1∂yyω∂xyyω

and

u2∂yyyω∂yyω=1

2

u2∂y[∂yyω]2=∫

u1∂xyyω∂yyω,

which can be bounded as∣

u1∂xyyω∂yyω

≤ C‖∂xyyω‖2‖∂yyω‖122 ‖∂xyyω‖

122 ‖u1‖

122 ‖∂yu1‖

122

≤ ‖∆ωx‖322 ‖∆ω‖

122 ‖u1‖

122 ‖∂yu1‖

122

≤ ‖∆ωx‖22+C‖u1‖

22‖∂yu1‖

22‖∆ω‖2

2.

Collecting the estimates above and applying Gronwall’s inequality, we obtain the desiredglobal H2-bound. This completes the proof for the global H2-bound and thus the proofof Theorem 1.1.

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D. Regmi, J. Wu / J. Math. Study, 49 (2016), pp. 169-194 187

3 Proof of Theorem 1.2

This section proves Theorem 1.2, which assesses the global existence and regularity ofsolutions to (1.5). Again the main task is to prove the global H2-bound for the solution.

First of all, we can easily prove the following global L2-bound.

Lemma 3.1. Assume that (u0,b0,ω0) satisfies the condition stated in Theorem 1.2. Let (u,b,ω)be the corresponding solution of (1.5). Then, (u,b,ω) obeys the following global L2-bound,

‖u(t)‖2L2 +‖b(t)‖2

L2 +‖ω(t)‖2L2 +2

∫ t

0‖∂xu1,∂yu1‖

2L2 dτ+2

∫ t

0‖∂xb(τ)‖2

L2 dτ

+2∫ t

0‖∂xω(τ)‖2

L2 dτ≤C(‖u0‖2L2 ,‖b0‖

2L2 ,‖ω0‖

22)

for any t≥0.

The rest of this section is divided into two subsections. The first subsection proves theglobal H1-bound while the second proves the global H2-bound, which leads to the proofof Theorem 1.2.

3.1 Global H1 bound

This subsection proves that the solution of (1.5) is globally bounded in H1-norm. Moreprecisely, we prove the following proposition.

Proposition 3.1. Assume that (u0,b0,ω0) satisfies the condition stated in Theorem 1.2. Let(u,b,ω) be the corresponding solution of (1.5). Then (u,b,ω) satisfies, for any T>0,

u,b,ω∈C([0,T];H1).

Proof. To prove the global H1-bound, we start with the equations for Ω=∇×u, ∇ω andj=∇×b,

Ωt+u·∇Ω=−∂xxyu1−∂yyyu1+(b·∇)j−∆ω, (3.1)

∂t∇ω+∇(u·∇ω)+2∇ω=∇ωxx+∇Ω, (3.2)

jt+u·∇j=∂xx j+b·∇Ω+2∂xb1(∂xu2+∂yu1)−2∂xu1(∂xb2+∂yb1), (3.3)

∇·u=0, ∇·b=0.

Taking the inner product of these equations with (Ω,∇ω, j), integrating by parts andapplying ∇·u=0 and ∇·b=0, we obtain

1

2

d

dt

(

‖Ω‖2L2 +‖j‖2

L2 +‖∇ω‖22

)

+‖(∇∂xu1,∇∂yu1)‖22

+‖∂x j‖2L2 +‖∇ωx‖

22+2‖∇ω‖2

2

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188 D. Regmi, J. Wu / J. Math. Study, 49 (2016), pp. 169-194

= 2∫

[

∂xb1(∂xu2+∂yu1) j−∂xu1(∂xb2+∂yb1)]

jdxdy

−∫

∇ω ·∇u·∇ω+2∫

∇ω ·∇Ω.

= J1+ J2+ J3+ J4+ J5+ J6.

The terms J1, J2, J3, J4 can be estimated in a similar fashion as in the previous section andtheir corresponding bounds are

J1≤1

48‖∂x j‖2

2+C‖∂xb‖22‖Ω‖2

2+C‖u2‖‖j‖2‖Ω‖2,

J2≤1

48

[

‖∂yyu1‖22+‖∂x j‖2

2

]

+C(‖∂xb1‖22+‖∂yu1‖

22)‖j‖2

2,

J3≤1

48‖∂x j‖2

2+C‖u1‖22‖∂yu1‖

22‖j‖2

2,

J4≤1

48‖∂x j‖2

2+C‖u1‖22‖∂yu1‖

22‖j‖2

2.

We focus on the bounds for J5 and J6. Writing out the terms in J5 explicitly and applying∇·u=0, we obtain

J5 =−∫

∇ω ·∇u·∇ω=2∫

u1ωxx ωx−2∫

u1ωyωxy−∫

(∂xu2+∂yu1)ωxωy.

The terms on the right are bounded as follows.

u1 ωxxωx

≤1

48‖ωxx‖

22+C‖u1‖

22‖∂yu1‖

22‖∇ω‖2

2.

u1ωyωxy

≤1

48‖∇ωx‖

22+C‖u1‖

22‖∂yu1‖

22‖∇ω‖2

2.

∂yu1ωxωy

≤1

48‖∇ωx‖

22+C‖∂yu1‖

22‖∇ω‖2

2.

By integration by parts,

∂xu2ωxωy=−∫

∂xyu2ωxω−∫

∂xu2ωxyω.

The two terms on the right admit the following bounds.

−∫

∂xyu2ωxω

≤C‖∂xyu2‖2‖ωx‖122 ‖ωxy‖

122 ‖ω‖

122 ‖ωx‖

122

≤C‖∇∂xu1‖2‖ωx‖2‖∇ωx‖122 ‖ω‖

122

≤C‖∇∂xu1‖22+‖∇ωx‖2‖ω‖2‖∂xω‖2‖∇ω‖

≤C‖∇∂xu1‖22+‖∇ωx‖

22+‖ω‖2

2‖ωx‖22‖∇ω‖2

2.

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D. Regmi, J. Wu / J. Math. Study, 49 (2016), pp. 169-194 189

−∫

∂xu2ωxyω

≤C‖ωxy‖2‖∂xu2‖122 ‖∂xyu2‖

122 ‖ω‖

122 ‖ωx‖

122

≤C‖∇ωx‖‖Ω‖122 ‖∇∂xu1‖

122 ‖ω‖

122 ‖ωx‖

122

≤‖∇ωx‖22+C‖Ω‖2‖∇∂xu1‖2‖ω‖2‖ωx‖2

≤‖∇ωx‖22+‖∇∂xu1‖

22+C‖Ω‖2

2‖ωx‖22‖ω‖2

2.

Finally, we deal with J6. There are two terms in J6,

J6=2∫

∇ω ·∇Ω=−∫

ωxx Ω+2∫

ωyΩy.

They can be bounded as follows.

ωxx Ω

≤‖ωxx‖2‖Ω‖2≤1

2‖∇ωx‖

22+C‖Ω‖2

2,

ωy Ωy=∫

(ωy∂xyu2−ωy∂yyu1),∣

ωy∂xyu2

≤C‖∇ω‖2‖∇∂yu1‖2,

ωy∂yyu1

≤C‖∇ω‖2‖∇∂yu1‖2.

Collecting the estimates above and applying Gronwall’s inequality allows us to concludethe global H1-bound. This completes the proof of Proposition 3.1.

3.2 Global H2 Bound and the proof of Theorem 1.2

This subsection proves Theorem 1.2. As we explained before, it suffices to prove theglobal H2-bound. This is given in the following proof.

Proof of Theorem 1.2. Dotting (3.1) with ∇Ω, applying ∇ to (3.2) and then dotting with∆ω, and dotting (3.3) with ∆j, we obtain

1

2

d

dt(‖∇Ω‖2

2+‖∇j‖22+‖∆ω‖2

2)+‖∆∂xu1‖22

+‖∆∂yu1‖22+‖∇∂x j‖2

2+2‖∆ωx‖22+‖∆ω‖2

2

= L1+L2+L3+L4+L5+2L6+L7,

where

L1=−∫

∇Ω·∇u·∇Ω dxdy, L2=−∫

∇j·∇u·∇j dxdy,

L3=2∫

∇Ω·∇b·∇j dxdy, L4=2∫

∇[∂xb1(∂xu2+∂yu1)]·∇j dxdy,

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190 D. Regmi, J. Wu / J. Math. Study, 49 (2016), pp. 169-194

L5=−2∫

∇[∂xu1(∂xb2+∂yb1)]·∇j dxdy, L6=∫

∆Ω∆ω dxdy,

L7=−∫

∆(u·∇ω)∆ω dxdy.

We write the four terms in L1 explicitly,

L1= −∫

(∇Ω·∇u·∇Ω)dxdy

= −∫

(

∂xu1(∂xΩ)2+∂xu2∂xΩ∂yΩ+∂yu1∂xΩ∂yΩ+∂yu2(∂yΩ)2)

≡ L11+L12+L13+L14.

These terms can be bounded as follows.

L11≤C‖∂xΩ‖322 ‖∂xu1‖

122 ‖∂xxu1‖

122 ‖∂xyΩ‖

122

≤C‖∇Ω‖322 ‖∂xu1‖

122 ‖∇∂xu1‖

122 ‖∆∂xu1‖

122

≤1

48‖∆∂xu1‖

22+C‖∂xu1‖

232 ‖∇∂xu1‖

232 ‖∇Ω‖2

2.

L12=∫

∂xu2∂xΩ∂yΩ

≤C‖∂xu2‖2‖∂xΩ‖122 ‖∂xyΩ‖

122 ‖∂yΩ‖

122 ‖∂xyΩ‖

122

≤C‖∂xu2‖2‖∇Ω‖2‖∂xyΩ‖2

≤1

48‖∆∂xu1‖

22+C‖∂xu2‖

22‖∇Ω‖2

2.

We note that ∂xyΩ=∂xy(∂xu2−∂yu1)=(−∂xxx−∂xyy)u1, and thus

‖∂xyΩ‖22≤C‖∆∂xu1‖

22.

Therefore,

L13≤1

48‖∂yyΩ‖2

2+C(‖∂yu1‖22+‖∂xyu1‖

22)‖∇Ω‖2

2.

L14≤1

48‖∂yyΩ‖2

2+C‖u2‖22‖Ω‖2

2‖∇Ω‖22.

The rest of the terms are bounded as follows.

L2≤1

48‖∇∂x j‖2

2+C‖∂xyu1‖232 ‖Ω‖

232 ‖∇j‖2

2+‖Ω‖22‖∇j‖2

2.

L3≤1

48‖∇∂x j‖2

2+C(‖j‖22+‖∂x j‖2

2)(‖∇Ω‖22+‖∇j‖2

2).

L4≤1

48‖∇∂x j‖2

2+1

48‖Ωyy‖

22+C(‖Ω‖2

2+‖j‖22+‖∂x j‖2

2)(‖∇ω‖22+‖∇j‖2

2).

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D. Regmi, J. Wu / J. Math. Study, 49 (2016), pp. 169-194 191

L5 admits a similar bound as L4.

L6≤1

48‖∆ωx‖

22+

1

48‖∆∂yu1‖

22+C(‖∇Ω‖2

2+‖∆ω‖22

L7≤1

48‖∆ωx‖

22+

1

48‖∆∂yu1‖

22+C(‖∇ω‖2

2+‖∇ωx‖22+‖Ω‖2

2)(‖∇Ω‖22+‖∆ω‖2

2).

Combining the bounds above and then applying Gronwall’s inequality lead to the de-sired global H2-bound. This completes the proof of Theorem 1.2.

4 Proof of Theorem 1.3

This section proves Theorem 1.3. Due to the similarity to the proofs in the previoustwo sections, we shall omit most of the details and provide only the estimates that aresignificantly different from the previous ones.

First, the following global L2-bound holds.

Lemma 4.1. Assume that (u0,b0,ω0) satisfies the conditions in Theorem 1.3. Let (u,b,ω) be thecorresponding solution of (1.6). Then, (u,b,ω) obeys the following global L2-bound,

‖u(t)‖2L2 +‖b(t)‖2

L2 +‖ω(t)‖2L2

+2∫ t

0‖∂xu1,∂yu2‖

22dτ+2

∫ t

0‖∂xb(τ)‖2

L2 dτ+2∫ t

0‖∂xω(τ)‖2

L2 dτ

≤C(‖u0‖2L2 ,‖b0‖

2L2 ,‖ω0‖

22)

for any t≥0.

The following global H1-bound can also be established.

Proposition 4.1. Assume that (u0,b0,ω0) satisfies the conditions in Theorem 1.3. Let(u,b,ω) be the corresponding solution of (1.6). Then (u,b,ω) satisfies, for any T>0,

u,b,ω∈C([0,T];H1).

We briefly indicate the proof of Proposition 4.1. Again we invoke the equations ofΩ=∇×u, ∇ω and j=∇×b

Ωt+u·∇Ω=−∆∂yu1+(b·∇)j−∆ω,∂t∇ω+∇(u·ω)+2∇ω=∇ωxx+∇Ω,jt+u·∇j=∂xx j+b·∇Ω+2∂xb1(∂xu2+∂yu1)−2∂xu1(∂xb2+∂yb1),∇·u=0, ∇·b=0.

Dotting the equations above with Ω, ∇ω and j, we obtain

1

2

d

dt

(

‖Ω‖2L2 +‖j‖2

L2 +‖∇ω‖22

)

+‖(∇∂xu1,∇∂yu1)‖22

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192 D. Regmi, J. Wu / J. Math. Study, 49 (2016), pp. 169-194

+‖∂x j‖2L2 +‖∇ωx‖

22+2‖∇ω‖2

2

= 2∫

[

∂xb1(∂xu2+∂yu1) j−∂xu1(∂xb2+∂yb1)]

jdxdy

−∫

∇ω ·∇u·∇ω+2∫

∇ω ·∇Ω

≡ J1+ J2++J3+ J4+ J5+ J6.

All these term can be bounded as in the previous cases except the term∫

∂xu2ωxωy, whichcan be handled as follows. We first integrate by parts to obtain

∂xu2ωxωy=−∫

u2∂xxωωy−∫

u2ωx∂xyω.

The two terms on the right are bounded by

−∫

u2∂xxωωy

≤C‖∂xxω‖2‖u2‖122 ‖∂yu2‖

122 ‖ωy‖

122 ‖∂xyω‖

122

≤C‖∇∂xω‖322 ‖u2‖

122 ‖∂yu2‖

122 ‖∇ω‖

122 .

≤‖∇ωx‖22+C‖u2‖

22‖∂yu1‖

22‖∇ω‖2

2∣

−∫

u2ωx∂xyω

≤C‖∂xyω‖2‖u2‖122 ‖∂yu2‖

122 ‖ωx‖

122 ‖∂xxω‖

122

≤‖∇ωx‖22+C‖u2‖

22‖∂yu1‖

22‖∇ω‖2

2.

Collecting the estimates would yield the desired global H1-bound.As in the previous two cases, we can prove Theorem 1.3 by establishing the global

H2-bound. Since the process of proving the global H2-bound is similar to the previoustwo cases, we shall omit the details.

Acknowledgments

D. Regmi thanks the Department of Mathematics at Farmingdale State College (SUNY)for its summer research support. J. Wu was partially supported by NSF grant DMS1209153,by the AT&T Foundation at Oklahoma State University and by NSFC (No. 11471103, agrant awarded to Professor Baoquan Yuan).

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