We consider the initial value problem of which requires initial conditions u(x,0) =u0(x) and b(x,0) =b0(x) x∈R3 (1.4) The initial conditions satisfy the compatibility condition, i.e

Electronic Journal of Differential Equations, Vol. 2018 (2018), No. 103, pp. 1–12.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

REGULARITY CRITERIA FOR WEAK SOLUTIONS TO 3D INCOMPRESSIBLE MHD EQUATIONS WITH HALL TERM

JAE-MYOUNG KIM

Communicated by Jesus Ildefonso Diaz

Abstract. We study the regularity conditions for a weak solution to the in- compressible 3D magnetohydrodynamic equations with Hall term in the whole spaceR³. In particular, we show the regularity criteria in view of gradient vec- tors in various spaces.

1. Introduction

We consider the incompressible 3D magneto hydro dynamic (MHD) equations with Hall term

∂_tu−∆u+u· ∇u+∇π=b· ∇b, (1.1)

∂tb−∆b+u· ∇b−b· ∇u+∇ ×((∇ ×b)×b) = 0, (1.2)

divu= divb= 0, (1.3)

Here u: QT :=R³×[0, T)→ R³ is the flow velocity vector, b : QT →R³ is the magnetic vector, π = p+ ^|b|₂² : QT → R is the total pressure. We consider the initial value problem of (1.1)–(1.3), which requires initial conditions

u(x,0) =u₀(x) and b(x,0) =b₀(x) x∈R³ (1.4) The initial conditions satisfy the compatibility condition, i.e.

divu0(x) = 0, and divb0(x) = 0.

Definition 1.1. A weak solution pair (u, b) of the incompressible 3D MHD equa- tions with the Hall term (1.1)–(1.4) is regular in QT provided that kuk_L^∞_(QT)+ kbk_L^∞_(QT)<∞.

For a long time, the effects of Hall current on fluids has been a subject of great interest to researchers. A current induced in a direction normal to the electric and magnetic fields is commonly called Hall current [22]. In particular, the effects of Hall current are very important if the strong magnetic field is applied

The mathematical derivations of the incompressible 3D MHD equations with the Hall term could be given in [1] from either two-fluids or kinetic models. It is well- known that the global existence of weak solutions, local existence and uniqueness

2010Mathematics Subject Classification. 35B65, 35Q35, 76W05.

Key words and phrases. Magnetohydrodynamics equation; weak solution;

regularity condition.

2018 Texas State University.c

Submitted July 30, 2017. Published May 7, 2018.

1

of smooth solutions to the system (1.1)–(1.4) were established in [5, 6]. Recently, various results for this equation were proved in view of partial regularity, temporary decay and regularity or blow-up conditions (see [5, 6, 7, 8, 9, 12, 11, 21, 26, 27] and references therein.)

We list only some results relevant to our concerns. In view of the regularity conditions in Lorentz space, He and Wang [13] proved that a weak solution pair (u, b) becomes regular in the presence of a certain type of the integral conditions, typically referred to as Serrin’s condition, namely,

u∈L^q,∞(0, T;L^p,∞(R³)) with 3/p+ 2/q≤1, 3< p≤ ∞, or

∇u∈L^q,∞(0, T;L^p,∞(R³)) with 3/p+ 2/q≤2, 3

2 < p≤ ∞,

(also see [3, 4, 17]). Also, Wang proved in [25] that a weak solution pair (u, b) become regular ifusatisfies

u∈L²(0, T;BM O(R³).

On the other hand, recently, Zhang [27] obtained the regularity criterion

u∈L^1−α2 (0, T; ˙B_∞,∞^−α ), ∇b∈L^1−β2 (0, T; ˙B^−β_∞,∞) (1.5) with −1 < α < 1 and 0 < β < 1. Our study is motivated by these viewpoints, we obtain the regularity conditions for a weak solution to the incompressible 3D MHD equations with the Hall term (1.1)–(1.4) in a whole space. Our proof of main results is based on a priori estimate for the gradient of the velocity field.

Our main results reads as follows.

Theorem 1.2. Suppose that (u, b) is a weak solution of (1.1)–(1.4) with initial conditionu₀, b₀∈H²(R³). If(u, b)satisfies one of the following cases:

Z T 0

k∇uk^q_Lp,∞+k∇bk^m_Ll,∞

dt <∞ (1.6)

with the relations ³_p+²_q = 2, ³₂ < p≤ ∞and ³_l +_m² = 1,3< l≤ ∞. or Z T

0

k∇uk^q_Lp,∞+k∇bk

2 1−β

B˙^−β∞,∞

dt <∞ (1.7)

with the relations ³_p +²_q = 2, ³₂ < p≤ ∞and 0< β < 1, then (u, b)is regular in QT.

Theorem 1.3. Suppose that (u, b) is a weak solution of (1.1)–(1.4) with initial conditionu₀, b₀∈H²(R³). If(u, b)satisfies one of the following two conditions:

Z T 0

k∇uk_{BM O(R}3)+k∇bk²_{BM O(R}3) dt <∞, (1.8) or

Z T 0

k∇uk²_{BM O}−1(R³)+k∇²bk²_{BM O}−1(R³)dt <∞. (1.9) then(u, b)is regular inQ_T.

Theorem 1.2 extends the result by He and Wang [13] with respect to the gradient of the velocity field. Moreover, using the estimate in [24, Lemma A.5], we obtain BM O⁻¹(R³)-regularity condition.

This article is organized as follows: In Section 2 we recall the notion of weak solutions and review some known results. In Section 3, we present the proofs of the Theorem 1.2 and 1.3.

2. Preliminaries

In this section we introduce the notation and definitions to be used in this paper.

We also recall the well-known results for our analysis. For 1≤q≤ ∞, W^k,q(R³) indicates the usual Sobolev space with standard normk · kk,q, i.e.

W^k,q(R³) ={u∈L^q(R³) :D^αu∈L^q(R³),0≤ |α| ≤k}.

Whenq= 2, we denoteW^k,q(R³) by H^k. All generic constants will be denoted by C, which may vary from line to line.

2.1. BMO and Lorentz spaces. The John-Nirenberg space or the Bounded Mean Oscillation space (in short BMO space) [14] consists of all functions f which are integrable on every ballBR(x)⊂R³and satisfy:

kfk²_{BM O}= sup

x∈R³

sup

R>0

1 B(x, R)

Z

B(x,R)

|f(y)−fB_R(y)|dy <∞.

Here, fB_R is the average of f over all ballBR(x) in R³. It will be convenient to define BMO in terms of its dual space,H¹. On the other hand, following [16] letw be the solution to the heat equationwt−∆w= 0 with initial datav. Then

kvk²_{BM O}= sup

x∈R³

sup

R>0

1 B(x, R)

Z

B(x,R)

Z R² 0

|w|²dt dy.

and define theBM O⁻¹-norm by kvk²_{BM O}−1 = sup

x∈R³

sup

R>0

1 B(x, R)

Z

B(x,R)

Z R² 0

|∇w|²dt dy.

We note that ifuis a tempered distribution. Thenu∈BM O⁻¹if and only if there existfⁱ∈BM Owithu=P

∂ifⁱ in [16, Theorem 1].

Letm(ϕ, t) be the Lebesgue measure of the set{x∈R³:|ϕ(x)|> t}, i.e.

m(ϕ, t) :=m{x∈R³:|ϕ(x)|> t}.

We denote by the Lorentz spaceL^p,q(R³) with 1≤p, q≤ ∞with the norm [23]

kϕkL^p,q(R³)=





 R∞

0 t^q(m(ϕ, t))^{q/p dt}_t^1/q

<∞, for 1≤q, sup_t≥0{t(m(ϕ, t))^1/p}, forq=∞.

(2.1) Followed in [23], Lorentz spaceL^p,q(R³) may be defined by real interpolation meth- ods

L^p,q(R³) = (L^p1(R³), L^p2(R³))α,q, (2.2) with ¹_p =^1−α_p

1 +_p^α

2, 1≤p1< p < p2≤ ∞. From the interpolation method above, we note that

L^p−12p,2(R³) =

L²(R³), L⁶(R³)

3

2p,2. (2.3)

We also need the H¨older inequality in Lorentz spaces (see [20]) for our proof.

Lemma 2.1. Assume 1 ≤ p1, p2 ≤ ∞, 1 ≤ q1, q2 ≤ ∞ and u ∈ L^p1,q1(R³), v∈L^p2,q2(R³). Thenuv∈L^p3,q3(R³)with _p¹

3 = _p¹

1 +_p¹

2 and _q¹

3 ≤ _q¹

1 +_q¹

2, and kuvkL^p3,q3(R³)≤CkukL^p1,q1(R³)kvkL^p2,q2(R³). (2.4) 2.2. Besov space. Following [23], letB={ξ∈ R^d, |ξ| ≤ ⁴₃} and C ={ξ ∈R^d : 3/4≤ |ξ| ≤8/3}. Choose two nonnegative smooth radial functionχ, ϕsupported, respectively, inBandC such that

χ(ξ) +X

j≥0

ϕ(2^−jξ) = 1, ξ∈R^d, X

j∈Z

ϕ(2^−jξ) = 1, ξ∈R^d\ {0}.

We denote ϕj =ϕ(2^−jξ), h= F⁻¹ϕ and ˜h= F⁻¹χ, where F⁻¹ stands for the inverse Fourier transform. Then the dyadic blocks ∆j and Sj can be defined as follows

∆jf =ϕ(2^−jD)f = 2^jd Z

R^d

h(2^jy)f(x−y)dy, Sjf = X

k≤j−1

∆kf =χ(2^−jD)f = 2^jd Z

R^d

h(2˜ ^jy)f(x−y)dy.

Formally, ∆j =Sj−S_j−1is a frequency projection to annulus{C12^j≤ |ξ| ≤C22^j}, andSj is a frequency projection to the ball{|ξ| ≤C2^j}. One can easily verify that with our choice ofϕ,

∆j∆kf = 0 if|j−k| ≥2 and ∆j(S_k−1f∆kf) = 0 if|j−k| ≥5.

With the introduction of ∆j andSj, let us recall the definition of the Besov space.

Lets∈R,p, q∈[1,∞], the homogeneous space is defined as B˙_p,q^s ={f ∈ S⁰ :kfkB˙^s_p,q <∞}, where

kfkB˙_p,q^s = ( P

j∈Z2^sjqk∆jfk^q_Lp

^1/q

, for 1≤q <∞, sup_j∈Z2^sjk∆jfkL^p, forq=∞,

In particular, whenp=q= 2, the Besov space and Sobolev space are equivalence;

that is

H˙^s≈B˙_2,2^s , H^s≈B_2,2^s . Now we recall first the definition of weak solutions.

Definition 2.2. Let u0, b0 ∈ L²(R³) with the divergence free conditions. We say that (u, b) is a weak solution of Hall-MHD equations (1.1)–(1.3) with initial conditionu0, b0∈L²(R³), ifuandb satisfy the following:

(i) u∈L^∞([0, T);L²(R³))∩L²([0, T);H¹(R³)), andb∈L^∞([0, T);L²(R³))∩ L²([0, T);H¹(R³)).

(ii) (u, b) satisfies (1.1)–(1.2) in the sense of distribution; that is Z T

0

Z

R³

∂φ

∂t + ∆φ+ (u· ∇)φ

u dx dt+ Z

R³

u0φ(x,0)dx= Z T

0

Z

R³

(b· ∇)φ b dx dt

Z T 0

Z

R³

∂φ

∂t + ∆φ+ (u· ∇)φ

b dx dt+ Z

R³

b₀φ(x,0)dx

= Z T

0

Z

R³

(b· ∇)φ u dx dt+ Z T

0

Z

R³

(∇ ×b)×b·(∇ ×φ)dx dt, for allφ∈C₀^∞(R³×[0, T)) with divφ= 0, and

Z

R³

u· ∇ψdx= 0, Z

R³

b· ∇ψdx= 0,

for everyψ∈C₀^∞(R³).

3. Proof of main results

Proof of Theorem 1.2. (L²-estimate or energy estimate): By the standard energy estimate, we obtain

1 2

d dt

Z

(|u|²+|b|²)dx+ Z

(|∇u|²+|∇b|²)dx= 0. (3.1)

•(H¹-estimate): Testing−∆uand−∆bto the fluid equation and by the magnetic equation of (1.1) and (1.2), respectively, using the integrating by parts, integrating on domain, we have

1 2

d

dt(k∇u(τ)k²_L2(R³)+k∇b(τ)k²_L2(R³)) + Z

R³

(|∆u|²+|∆b|²)dx

≤ − Z

R³

∇[(u· ∇)u] :∇udx+ Z

R³

∇[(b· ∇)b] :∇u dx

− Z

R³

∇[(u· ∇)b]· ∇b dx+ Z

R³

∇[(b· ∇)u] :∇bdx +

Z

∇((∇ ×b)×b)∇∇ ×b dx :=I1+I2+I3+I4+I5.

(3.2)

We estimate separately the terms in the right hand side of (3.2). The first termI₁ is computed as follows:

|I1| ≤ k∇uk³_L3, (3.3) where the divergence free condition ofuis used.

On the other hand, we observe that I2+I4≤

Z

R³

|∇u||∇b|². since

Z

R³

(b· ∇)∇b· ∇u dx+ Z

R³

(b· ∇)∇u· ∇b dx

=

3

X

j=1

Z

R³

bj

∂∇b

∂xj

∇u dx+∂∇u

∂xj

∇b dx

=−

3

X

j=1

Z

R³

bj

∂(∇b∇u)

∂xj

dx= 0, where we use the product rule and divb= 0.

Note that

k∇fk²_L4≤Ck∇fk_B˙−β

∞,∞kfkH˙^1+β with 0< β <1, (3.4) kfkH˙^1+β ≤Ck∇fk^1−β_L2 k∆fk^β_L2 with 0< β <1, (3.5) (e.g. see [18, 19] and [2, Theorem 2.42]).

First of all, using the interpolation (2.2), Lemma 2.1, H¨older and Young’s in- equalities, we estimateI3 as follows:

|I3| ≤ Z

R³

|∇b|²|∇u|dx≤ k∇ukL^p,∞k|∇b|²k

L

p p−1,1

=k∇ukL^p,∞k∇bk²

L

2p p−1,1

≤Ck∇ukL^p,∞k∇bk^2θ_L2k∇²bk^2(1−θ)_L2

≤Ck∇uk

2p 2p−3

L^p,∞k∇bk²_L2+ 1

16k∇²bk²_L2,

(3.6)

whereθ= 1−_2p³. Similarly, from (3.3), we have

|I1| ≤ k∇uk³_L3≤Ck∇uk

2p 2p−3

L^p,∞k∇uk²_L2+ 1

16k∇²uk²_L2.

Case 1: Again, using the interpolation (2.2), Lemma 2.1, H¨older and Young’s inequalities, we boundI5 as follows.

|I5| ≤Ck∇bk_Ll,∞k∇bk

L^l−22l,2k∆bk_L2,2

≤Ck∇bk_Ll,∞k∇bk

l−3 l

L² k∆bk_L^l+3l2

≤Ck∇bk

2l l−3

L^l,∞k∇bk²_L2+ 1

16k∆bk²_L2. Summing the termsI1–I5, inequality (3.2) becomes

1 2

d

dt(k∇uk²_L2+k∇bk²_L2) +1 2 Z

R³

(|∇²u|²+|∇²b|²)dx

≤C k∇uk

2p 2p−3

L^p,∞+k∇bk

2l l−3

L^l,∞

(k∇uk²_L2+k∇bk²_L2).

(3.7)

Case 2: Using (3.4) and (3.5), we boundI5 as follows:

I5=−X

i

Z

(∇ ×b×∂ib)∂i∇ ×bdx≤Ck∇bk²_L4k∆bkL²

≤Ck∇bk_B˙−β

∞,∞kbkH˙^1+βk∆bk_L2 ≤Ck∇bk_B˙−β

∞,∞k∇bk^1−β_L2 k∆bk^1+β_L2

≤ 1

16k∆bk²_L2+Ck∇bk

2 1−β

B˙^−β∞,∞

k∇bk²_L2. Summing the termsI1–I5, the (3.2) becomes

1 2

d

dt(k∇uk²_L2+k∇bk²_L2) +1 2 Z

R³

(|∇²u|²+|∇²b|²)dx

≤C k∇uk

2p 2p−3

L^p,∞+k∇bk

2 1−β

B˙_∞,∞^−β

(k∇uk²_L2+k∇bk²_L2).

(3.8)

For Cases 1 and 2 with the given conditions, we apply the Grownwall’s inequality to estimates (3.7) and (3.8), respectively, to find

sup

0<τ≤T

(k∇u(τ)k²_L2+k∇b(τ)k²_L2) + Z T

0

Z

R³

(|∇²u|²+|∇²b|²)dx dt

≤C(k∇u(0)k²_L2+k∇b(0)k²_L2).

•(H²-estimate) Applying the operator ∆ to (1.1)–(1.2), then multiplying it by ∆u and ∆b, respectively, and integrating on domain, we obtain

1 2

d

dt(k∆uk²+k∆bk²)dx+ (k∇∆u|²_L2+k∇∆b|²_L2)

=− Z

R³

∆(u· ∇u)·∆u dx+ Z

R³

∆(b· ∇b)·∆u dx

− Z

R³

∆(u· ∇b)·∆b dx +

Z

R³

∆(b· ∇u)·∆b dx− Z

R³

∆((∇ ×b)×b)·∆∇ ×b dx :=J₁+J₂+J₃+J₄+J₅

By the commutator estimate in [10, Theorem 2.1 or Corollary 2.1] or [15], we note that

Z

R³

∆[(u· ∇)u],∆u dx

≤Ck∇ukH²kuk²_H2,

Z

R³

∆[(u· ∇)b],∆b dx

≤Ck∇ukH²kbk²_H2,

Z

R³

∆[(b· ∇)u],∆b dx

≤Ck∇ukH²kbk²_H2.

Also, integrating by parts we obtain the estimate for the remaining convection term follows as:

Z

R³

∆[(b· ∇)b],∆u dx

≤C|h∆[b⊗b],∆∇u dx| ≤Ck∇ukH²kbk²_H2. Thus

|J₁+J₂+J₃+J₄| ≤Ck∇uk_H2(kuk²_H2+kbk²_H2)

≤C(kuk⁴_H2+kbk⁴_H2) + 1

128k∇uk²_H2

≤C(kuk²_H2+kbk²_H2)(kuk²_H2+kbk²_H2) + 1

128k∇uk²_H2

(3.9)

Case 1: For the termJ5, using the chain rule, we note that J5=

Z

R³

(∇ ×b×∆b+ 2∂i(∇ ×b)×∂ib)∇∆b dx (3.10) And thus, we have

|J₅| ≤ k∇bk_Ll,∞k∆bk

L

2l

l−2,2k∇∆bk_L2 ≤Ck∇bk

2l l−3

L^l,∞k∆bk²_L2+ 1

128k∇∆bk²_L2

Summing the estimate of termsJ1–J5with the energy estimate andH¹-estimates, we obtain

1 2

d

dt(kuk²_H2+kbk²_H2) +1

2(ku|²_H3+kb|²_H3)

≤C k∇uk

2q 2q−3

L^q,∞+k∇bk

2l l−3

L^l,∞+kuk²_H2+kbk²_H2

(kuk²_H2+kbk²_H2)

(3.11)

Case 2: Using (3.4) and (3.5), we boundJ₅ as follows:

|J₅| ≤Ck∆bk_B˙−β

∞,∞k∇bkH˙^1+βk∇∆bk_L2

≤Ck∇bkB˙∞,∞^−β k∇²bk^1−β_L2 k∇∆bk^1+β_L2

≤Ck∇bk

2 1−β

B˙_∞,∞^−β kbk²_H2+ 1

128k∇∆bk²_L2.

As in case 1, summing the estimate of termsJ1–J5 with the energy estimate and H¹-estimates, we obtain

1 2

d

dt(kuk²_H2+kbk²_H2) +1

2(ku|²_H3+kb|²_H3)

≤C k∇uk

2p 2p−3

L^p,∞+k∇bk

2 1−β

B˙^−β∞,∞

+kuk²_H2+kbk²_H2

(kuk²_H2+kbk²_H2)

(3.12)

Under the assumption, we apply Grown’s inequality to the estimates (3.11) and (3.12), respectively, we finally obtain

sup

0≤τ≤T

(ku(τ)k²_H2+kb(τ)k²_H2) +kuk²_H3+kbk²_H3 ≤(ku0k²_H2+kb0k²_H2)

Th proof is complete.

Proof of Theorem 1.3. (H¹-estimate): Testing−∆uand−∆bto the fluid equation and the magnetic equation of (1.1) and (1.2), respectively, using the integrating by parts, integrating on domain, we have

1 2

d

dt(k∇u(τ)k²_L2(R³)+k∇b(τ)k²_L2(R³)) + Z

R³

(|∆u|²+|∆b|²)dx

≤ − Z

R³

∇[(u· ∇)u] :∇udx+ Z

R³

∇[(b· ∇)b] :∇udx

− Z

R³

∇[(u· ∇)b] :∇b dx+ Z

R³

∇[(b· ∇)u] :∇bdx +

Z

∇((∇ ×b)×b) :∇∇ ×b dx :=I1+I2+I3+I4+I5.

(3.13)

Case 1: By the H¨older, Young inequalities and the space duality BM O-H¹, we have

Z

R³

|∇b|²|∇u|dx≤ k∇ukBM Ok|∇b|²k_H¹ ≤ k∇ukBM Ok∇bkL²k∇bkL²

=k∇ukBM Ok∇bk²_L2. Similarly, we obtain

Z

R³

|∇u|³dx≤Ck∇ukBM Ok∇uk²_L2.

Again, by the vector identity, the H¨older and Young inequalities, we have I₅=

Z

R³

∇[(∇ ×b)×b]· ∇(∇ ×b)dx

= Z

R³

(∇ ×b)× ∇b− ∇(∇ ×b)×b

· ∇(∇ ×b)dx

≤Ck∇bkBM Ok|∇b||∇²b|k_H¹

≤Ck∇bk²_{BM O}k∇bk²_L2+1

8k∇²bk²_L2.

(3.14)

Summing the estimates above, the (3.13) becomes d

dt(k∇uk²_L2+k∇bk²_L2) + Z

(|∇²u|²+|∇²b|²)dx

≤C(k∇ukBM O+k∇bk²_{BM O})(k∇uk²_L2+k∇bk²_L2).

(3.15)

Case 2: Following [24, Lemma A.5], we note that

kuk²_L4 =kuukL² ≤Ck∇ukL²kukBM O⁻¹. Using this estimate, we have

Z

R³

|∇b|²|∇u|dx≤ k∇uk_L2k∇bk²_L4 ≤Ck∇uk_L2k∇²bk_L2k∇bk_{BM O}⁻¹

≤Ck∇uk²_L2k∇bk²_{BM O}−1+1

8k∇²bk²_L2. Similarly, we obtain

Z

R³

|∇u|³dx≤Ck∇uk²_L2k∇uk²_{BM O}−1+1

8k∇²uk²_L2. By the vector identity, the H¨older and Young inequalities, we have

I₅= Z

R³

∇[(∇ ×b)×b]· ∇(∇ ×b)dx

= Z

R³

(∇ ×b)× ∇b− ∇(∇ ×b)×b

· ∇(∇ ×b)dx

≤Ck(∇ ×b)× ∇bkL²k∇²bkL²

≤Ck∇²bk²_BMO−1k∇bk²_L2+ 1

258k∇²bk²_L2

(3.16)

Using the estimates above, (3.13) becomes d

dt(k∇uk²_L2+k∇bk²_L2) + Z

(|∇²u|²+|∇²b|²)dx

≤C(k∇uk²_{BM O}−1+k∇bk²_{BM O}−1)(k∇uk²_L2+k∇bk²_L2).

(3.17)

• (H²-estimate) Taking ∆ to (1.1)–(1.2), then multiplying it by ∆u and ∆b, re- spectively, and integrating on domain, we derive

1 2

d

dt(k∇²uk²+k∇²bk²)dx+ (k∇∆u|²_L2+k∇∆b|²_L2)

=− Z

R³

∆(u· ∇u)·∆u dx+ Z

R³

∆(b· ∇b)·∆u dx− Z

R³

∆(u· ∇b)·∆b dx +

Z

R³

∆(b· ∇u)·∆b dx− Z

R³

∆((∇ ×b)×b) : ∆∇ ×b dx :=J1+J2+J3+J4+J5

(3.18)

As in the proof of Theorem 1.2, namely (3.9), we note that

|J1+J2+J3+J4| ≤C(kuk²_H2+kbk²_H2)(kuk²_H2+kbk²_H2) + 1

128k∇uk²_H2. Case 1. From (3.10) with the space dualityBM O-H¹, we have

|J5| ≤ k∇bkBMOk∇²bkL²k∇∆bkL² ≤Ck∇bk²_BMOk∇²bk²_L2+ 1

128k∇∆bk²_L2. Summing the estimates J₁–J₅ with the energy estimate and H¹-estimates, the (3.18) becomes

d

dt(kuk²_H2+kbk²_H2) + (ku|²_H3+kb|²_H3)

≤C

k∇uk_{BM O}+k∇bk²_BMO+kuk²_H2+kbk²_H2

(kuk²_H2+kbk²_H2).

(3.19)

Case 2. From (3.9), we note that

|J₁+J₂+J₃+J₄| ≤C(kuk²_H2+kbk²_H2)(kuk²_H2+kbk²_H2) + 1

128k∇uk²_H2. Following [24, Lemma A.5], we note that

kuuk_L2≤Ck∇uk_L2kuk_{BM O}−1.

As in the previous proof, namely, from (3.10) with the space dualityBM O-H¹, we have

|J5| ≤Ck∇²bk_BMO−1k∇²bk_L2+ 1

128k∇∆bk²_L2.

SummingJ1–J5 with the energy estimate andH¹-estimate, the (3.18) becomes d

dt(kuk²_H2+kbk²_H2) + (ku|²_H3+kb|²_H3)

≤C(k∇uk²_{BM O}−1+k∇²bk²_{BM O}−1+kuk²_H2+kbk²_H2)(kuk²_H2+kbk²_H2).

(3.20) Under the assumption, we apply Gronwall’s inequality to the estimates (3.19) and (3.20), respectively, we finally obtain

sup

0≤τ≤T

(ku(τ)k²_H2+kb(τ)k²_H2) +kuk²_H3+kbk²_H3 ≤(ku₀k²_H2+kb₀k²_H2).

The proof is complete.

Acknowledgments. J.-M. Kim was supported by grants NRF-2015R1A5A1009350 and NRF-2016R1D1A1B03930422.

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Jae-Myoung Kim

Department of Mathematical Sciences, Seoul National University, Seoul, Korea E-mail address:[email protected]