We study three-dimensional incompressible Navier-Stokes equa- tions in bounded domains with smooth boundary

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

REGULARITY CRITERIA FOR WEAK SOLUTIONS TO THE 3D NAVIER-STOKES EQUATIONS IN BOUNDED DOMAINS VIA

BMO NORM

JAE-MYOUNG KIM

Communicated by Jesus Ildefonso Diaz

Abstract. We study three-dimensional incompressible Navier-Stokes equa- tions in bounded domains with smooth boundary. We present regularity cri- teria of weak solutions to this equation via the BMO norm.

1. Introduction

We study the three-dimensional Navier-Stokes equation

ut+ (u· ∇)u−∆u+∇π= 0, divu= 0 in QT := Ω×(0, T), (1.1) where Ω is a domain in R³ with smooth boundary ∂Ω ∈C². Here u:QT →R³ is the flow velocity vector and π : QT → R is the pressure. We consider the initial-boundary value problem of (1.1) with initial condition

u(x,0) =u₀(x) x∈Ω (1.2)

together with two types of boundary conditions: Either

u= 0, (1.3)

or

u·n= 0, (∇ ×u)×n= 0, (1.4)

wheren is the outward unit normal vector along boundary∂Ω. The initial condi- tions satisfy the compatibility condition, i.e. ∇ ·u₀(x) = 0. A weak solution uof (1.1)–(1.2) with boundary conditions either

(1.3) or (1.4) is regular in QT provided that kuk_L∞(Q_T) < ∞. The notion of weak solutions will be introduced in Definition 2.1 of Section 2.

The initial conditions hold the compatibility condition, i.e.∇ ·u0(x) = 0. Since Leray [24] proved the existence of weak solutions of the Navier-Stokes equations (see also [16]), regularity question has remained open.

Definition 1.1. A weak solutionuof (1.1)–(1.2) with boundary conditions (1.3) or (1.4) is regular inQT provided thatkuk_L^∞_(QT)<∞.

2010Mathematics Subject Classification. 35Q30, 35B65, 30H35.

Key words and phrases. Navier-Stokes equation; regularity criteria; BMO space.

c

2019 Texas State University.

Submitted March 27, 2018. Published January 17, 2019.

1

It is known that any weak solution becomes unique and regular inQT, provided that the following scaling invariant conditions [3, 7, 27, 30], so called Serrin’s type conditions, are satisfied:

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

∇u∈L^q(0, T;L^p(R³)), 3/p+ 2/q≤2, 3

2 < p≤ ∞, π∈L^q(0, T;L^p(R³)), 3/p+ 2/q≤2, 3

2 < p≤ ∞,

∇π∈L^q(0, T;L^p(R³)), 3/p+ 2/q≤3, 1< p≤ ∞.

In this direction, thee are numerous contributions, see [2, 9, 11, 14, 23, 25, 26, 29].

In view of the regularity conditions in view of the BMO space, Kozono and Taniuchi proved in [20] that a weak solutionubecome regular ifusatisfies

u∈L²(0, T; BMO(R³),

w:=∇ ×u∈L¹(0, T; BMO(R³), T <∞,

which is the result to the space BMO, which is larger thanL^∞(R³). Also, Fan and Ozawa proved in [12] that a weak solutionubecome regular ifusatisfies

∇p∈L^2/3(0, T; BMO(R³)), 0< T <∞.

Our study is motivated by the works above, that is, we obtain the regularity condi- tions for a weak solution to 3D Naiver-Stokes equations (1.1)–(1.2) with the bound- ary conditions (1.3) or (1.4) in bounded domains. In particular, for bounded do- mains, the difficulty lies in treating the pressure. To be more precise, in the case that Ω =R³, using the equation of pressure, we observe that the pressureπsatisfies kπk_Lp(Rⁿ)≤Ckuk²_L2p(R³), 1< p <∞. (1.5) However, it is not known yet whether or not the estimate above (1.5) holds for domains with the boundary condition. Thus, the methods of proof in a whole spaceR³ do not seem to be applicable to our case. To overcome these difficulties, we use the maximal estimates of Stokes system for both cases of slip and no-slip boundary conditions, regarding the nonlinear term as an external force (see Lemma 2.2 in section 2). Since such estimates of the Stokes system are also available for domain with boundaries, this approach allows for control of pressure and is useful for our analysis. On the other hand, to obtain the regularity condition for a vorticity vector, we consider the vorticity equations for Navier-Stokes equations to avoid the estimate of terms containing the pressure term. In this case, our proof is based on a priori estimate for the vorticity. At last, we give regularity criteria for the pressure to this equations using the maximal regularity theorem (see Lemma 2.2 in section 2). Our main results read as follows.

Theorem 1.2. Suppose thatuis a weak solution to (1.1)–(1.2)with initial condi- tionsu₀∈H²(Ω)∩W^1,q(Ω),q >3and boundary conditions (1.3)or (1.4). Assume further that usatisfies

kukL²(0,T;BMO(Ω))<∞ Then, ubecomes regular in QT.

Theorem 1.3. Suppose thatuis a weak solution to (1.1)–(1.2)with initial condi- tionsu0∈H²(Ω)∩W^1,q(Ω),q >3and boundary conditions (1.3)or (1.4). Assume further that w:=∇ ×usatisfies

kwk_L1(0,T;BMO(Ω))<∞ Then, ubecomes regular in QT.

Theorem 1.4. Suppose thatuis a weak solution to (1.1)–(1.2)with initial condi- tionsu₀∈H²(Ω)∩W^1,q(Ω),q >3 and boundary condition (1.3). Assume further that usatisfies

kπk_L2(0,T;BMO(Ω))<∞ Then, ubecome regular in Q_T.

Theorem 1.5. Suppose thatuis a weak solution to (1.1)–(1.2)with initial condi- tionsu0∈H²(Ω)∩W^1,q(Ω),q >3and boundary conditions (1.3)or (1.4). Assume further that usatisfies

k∇πk_L2/3(0,T;BMO(Ω))<∞ Then, ubecomes regular in QT.

Remark 1.6. Theorem 1.4 can be extended to any dimension Ω⊂R^N, because we do not deal with the terms related to pressure. On the other hand, Theorems 1.2, 1.3 and 1.5 can be restricted to the case n= 3,4 in view of [17, Remark 3.2]

or [19].

Remark 1.7. Theorems 1.5 is given in [13] under the boundary condition (1.4).

For the convenience of readers, we give a sketch of the proof.

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 Theorems 1.2–1.5.

2. Preliminaries

In this section, we introduce the notation and definitions used throughout this paper. We also recall some lemmas which are useful for our analysis. For 1≤q≤ ∞ and a nonnegative integer k, W^k,q(Ω) indicates the standard Sobolev space with normk · kk,q, i.e.,W^k,q(Ω) ={u∈L^q(Ω) :D^αu∈L^q(Ω),0≤ |α| ≤k}. As usual, W₀^k,q(Ω) is defined as the completion ofC₀^∞(Ω) inW^k,q(Ω). Whenq= 2, we write W^k,q(Ω) asH^k(Ω). LetI be a finite time interval. For a functionf(x, t), Ω⊂R³, we denotekfk_L^p,q

x,t(Ω×I)=kfk_L^q

t(I;L^p_x(Ω))=

kfk_L^p_{x(Ω) L}q

t(I). All generic constants will be denoted byC, which may vary from line to line. We recall first the definition of weak solutions.

Definition 2.1. Letu0∈L²(Ω) with divu0= 0. We say thatuis a distributional solution (or weak solution) of (1.1)–(1.2) ifusatisfies the following:

(1) u∈L^∞(0, T;L²(Ω))∩L²(0, T;H¹(Ω)) and usatisfies Z T

0

Z

Ω

∂φ

∂t + (u· ∇)φ

u dx dt+ Z

Ω

u0φ(x,0)dx= Z T

0

Z

Ω

∇u:∇φ dx dt for allφ∈ C₀^∞(Ω×[0, T)) with divφ= 0.

(2) usatisfies divergence free condition; that is, R

Ωu· ∇ψdx= 0 for any ψ∈ C^∞( ¯Ω).

We consider the following Stokes system which is the linearized Navier-Stokes equations,

vt−∆v+∇p=f, divv= 0 inQT := Ω×(0, T) (2.1) with initial data v(x,0) = v0(x). As in (1.3) and (1.4), boundary data of v are again assumed to be either no-slip or slip conditions, namely

v(x, t) = 0, x∈∂Ω or (2.2)

v·n= 0, (∇ ×v)×n= 0, x∈∂Ω. (2.3) Next, we recall maximal estimates of the Stokes system in terms of mixed norms (see [15, Theorem 5.1] and [28, Theorem 1.2] for no-slip and slip boundary cases, respectively).

Lemma 2.2. Let 1 < l, m <∞. Suppose that f ∈ L^l,m_x,t(QT)and v0 ∈ D¹⁻

1 m,m

l .

If (v, p) is the solution of the Stokes system (2.1) satisfying one of the boundary conditions (2.2)or (2.3), then the following estimate is satisfied:

kvtk_Ll,m

x,t(QT)+k∇²vk_Ll,m

x,t(QT)+k∇pk_Ll,m x,t(QT)

≤Ckfk_Ll,m

x,t(Q_T)+kv₀k

D¹⁻

1 m,m

l (Ω). (2.4)

We note thatkv0k

D¹⁻

1 m,m

l (Ω)≤ kv0k_W1,l(Ω)because D¹⁻

1 m,m

l (Ω) := [Ll(Ω), W^1,l((Ω))]₁₋¹

m,m

(see e.g., [1, Chapter 7]) and, therefore,kv0k

D¹⁻

m1,m

l (Ω)in (2.4) can be replaced by kv₀k_W1,l(Ω).

The John-Nirenberg space or the space of the Bounded Mean Oscillation (in short BMO space) [22] consists of all functionsf which are integrable on every ball B_R(x)⊂R³ and satisfy

kfk²_BMO= sup

x∈R³

sup

R>0

1 B(x, R)

Z

B(x,R)

|f(y)−f_BR(y)|dy <∞.

Here,f_BR is the average off over all ballB_R(x) inR³. Next we recall a Gagliardo- Nirenberg inequality using BMO-norm (See [8, Theorem 2.3] and [21, Theorem 2.2]).

Lemma 2.3. Suppose that 1≤p < r <∞andf ∈L^p(Ω)∩BMO(Ω). Then there exists a constantC=C(n, p, r,Ω)such that

kfk_Lr(Ω)≤Ckfk^p/r_Lp(Ω)kfk¹⁻_BMO(Ω)^pr .

Also, we recall estimates with respect to smooth vector field under the slip boundary condition. (See [4, Lemma 2.2],[5, Theorem 2.1] and [6, Lemma 2.1-2.2]).

Lemma 2.4. Let Ω be a smooth domain in R³. Then, for each q > 1, regular smooth vector fieldsf,

(a)

− Z

Ω

∆f·f|f|^q−2dx=1 2

Z

Ω

|f|^q−2|∇f|²dx+4(q−2) q²

Z

Ω

f|^q/2

2dx

− Z

∂Ω

|f|^q−2(n· ∇)f·f dS.

(b) Moreover, using the vector identity,

(n· ∇)f·f = (f· ∇)f·n+ ((∇ ×f)×n)·f, we deduce that

− Z

Ω

∆f·f|f|^q−2dx

= 1 2 Z

Ω

|f|^q−2|∇f|²dx+4(q−2) p²

Z

Ω

∇|f|^q/2

2dx

− Z

∂Ω

|f|^p−2(f· ∇)f·ndS− Z

∂Ω

|f|^p−2((∇ ×f)×n)·f dS.

Lemma 2.5. Assume thatuis a regular enough satisfying the boundary condition (1.4)on∂Ω. Then, the for w=∇ ×uwe have

−∂w

∂n ·w= (1jk1βγ+2jk2βγ+3jk3βγ)wjwβ∂knγ on∂Ω,

where ijk denotes the totally anti-symmetric tensor such that (a×b) = 1jkajbk. In particular,

Z

Ω

∆w·wdx≤ − Z

Ω

|∇w|²dx+C Z

∂Ω

|w|²dx.

3. Proof of main results

Proof of Theorem 1.2. Following the argument in [18, 19], it is sufficient to show theL⁴-estimate ofu. Suppose thatT^∗ be the first time of singularity. Thenumust satisfies for anyδ >0,

t%Tlim^∗

ku(·, t)k⁴_L4+ Z t

T^∗−δ

|∇u(·, τ)| |u(·, τ)|

2 L²

+

∇|u(·, τ)|²

2 L²

dτ

=∞.

(3.1)

In the proof, we consider only the boundary condition (1.4), since the case of (1.3) is much simpler. Multiplying the first equation of (1.1) with|u|²u, and integrating over Ω, we have

1 4

d dt

Z

Ω

|u|⁴+ Z

Ω

|∇u|²|u|²+1 2

Z

Ω

|∇|u|²|²

=− Z

Ω

∇π|u|²u+

3

X

i,j=1

Z

∂Ω

u_j,xiu_j|u|²n_i,

(3.2)

where we used integration by parts, divergence-free conditions ofuand trace the- orem. Let be a sufficiently small positive number, which will be specified later.

Integrating (3.2) in time over (T^∗−, τ) for anyτwithT^∗− < τ < T^∗, we observe that

1 4

Z

Ω

|u(·, τ)|⁴dx−1 4 Z

Ω

|u(·, T^∗−)|⁴dx +

Z τ

T^∗−

Z

Ω

|∇u|²|u|²dx dt+1 2

Z τ

T^∗−

Z

Ω

∇|u|²

2dx dt

≤ Z τ

T^∗−

Z

Ω

|∇π||u|²|u|dx dt+ Z τ

T^∗−

Z

Ω

|u|³|∇u|dx dt:=I1+I2.

(3.3)

For convenience, we denoteQτ := Ω×(T^∗−, τ). Using H¨older’s inequality, the first termI1 can be estimated as follows:

I₁≤ Z τ

T^∗−

k∇πk_L2kuk³_L6≤C Z τ

T^∗−

k∇πk_L2kuk²_L4kuk_BMO

≤Ck∇πk_L2(Qτ)kuk_L2(0,τ;BMO(Ω)) sup

T^∗−<t<τ

ku(·, t)k²_L4,

For convenience of computations, we denoteC:=ku(·, T^∗−)kW^1,2(Ω). Using the estimate (2.4), we continue to estimateI1 as

I1≤C

k(u· ∇)uk_L2(Qτ)+C

kuk_L2((T^∗−,τ);BMO(Ω)) sup

T^∗−<t<τ

ku(·, t)k²_L4

≤Ck(u· ∇)uk_L2(Qτ)kuk_L2((T^∗−,τ);BMO(Ω)) sup

T^∗−<t<τ

ku(·, t)k²_L4 (3.4) +CCkuk_L2(0,τ;BMO(Ω)) sup

T^∗−<t<τ

ku(·, t)k²_L4. On the other hand, by direct calculations,I2is bounded by

C^1/2

|u||∇u|

_L2(Q

τ) sup

T^∗−<t<τ

ku(·, t)k²_L4.

Summing the estimates ofI1andI2 with using Young’s inequality, we obtain 1

4 Z

Ω

|u(·, τ)|⁴dx−1 4

Z

Ω

|u(·, T^∗−)|⁴dx +

Z τ

T^∗−

Z

Ω

|∇u|²|u|²dx dt+1 2

Z τ

T^∗−

Z

Ω

∇|u|²

2dx dt

≤Ck |u||∇u| k_L2(Qτ)kuk_L2(0,τ;BMO(Ω)) sup

T^∗−<t<τ

ku(·, t)k²_L4

+CCkuk_L2(0,τ;BMO(Ω)) sup

T^∗−<t<τ

ku(·, t)k²_L4

≤ 1

2k |u||∇u| k²_L2(Q_τ)+CC²+C(kuk²_L2(0,τ;BMO(Ω))+) sup

T^∗−<t<τ

ku(·, t)k⁴_L4. Since the above estimate holds for alltwithT^∗− < t < τ, we obtain

sup

T^∗−<t<τ

ku(·, t)k⁴_L4+ Z τ

T^∗−

Z

Ω

|∇u|²|u|²|dx dt+1 2

Z τ

T^∗−

Z

Ω

∇|u|²

2dx dt

≤ Z

Ω

|u(·, T^∗−)|⁴dx+CC² +C

kuk²_L2((T^∗−,τ);BMO(Ω))+ sup

T^∗−<t<τ

ku(·, t)k⁴_L4. With sufficiently smallso that kuk²_L2((T^∗−,τ);BMO(Ω))+

≤ _2C¹ with a constant C >0 in the above estimate, we have

ku(·, t)k⁴_L4,∞

x,t(Q_τ)+1

2k |∇u||u| k²_L2(Q_τ)+1

2k∇|u|²k²_L2(Q_τ)

≤2ku(·, T −)k⁴_L4

x(Ω)+CC².

For simplicity, we denoteQ= Ω×(T^∗−, T^∗). Sinceτ is arbitrary withτ < T^∗, we obtain

ku(·, t)k⁴_L4,∞

x,t (Q)+1 2

|∇u||u|

2

L²(Q)+1

2k∇|u|²k²_L2(Q)≤C,

where C is a constant depending onku(·, T^∗−)kW^1,2(Ω). This is contrary to the hypothesis of (3.1). Therefore,T^∗cannot be a maximal time of existence less than

or equal toT. This completes the proof.

Proof of Theorem 1.3. First, we consider the vorticity equation

wt−∆w+ (u· ∇)w−(u· ∇)w= 0. (3.5) Multiplying the first equation of (3.5) byw, integrating over Ω, and adding them, we have

1 2

d dt

Z

Ω

|w|²+ Z

Ω

|∇w|²≤ Z

Ω

|w||∇u||w|+ Z

∂Ω

∂w

∂n ·w

:=II1+II2, where we use Lemmas 2.4 and 2.5. Using H¨older inequality and Lemma 2.3, the termII₁ is estimated as follows:

II₁≤ k∇uk_L3(Ω)kwk²_L3(Ω)≤Ckwk³_L3(Ω)≤Ckwk²_L2(Ω)kwk_BMO(Ω),

Next, we can easily estimate II2. Indeed, we use the Trace theorem (see e.g., [10, pp 257-258]) and smoothness of boundary to find

II₂≤ Z

∂Ω

∂w

∂n ·w ≤C

Z

Ω

|w|², Summing the estimatesII1 andII2, we obtain

d

dtkwk²_L2+k∇wk²_L2≤C(1 +kwkBMO(Ω))kwk²_L2. (3.6) Applying the Gronwall’s inequality to (3.6),

sup

0<t<T

kw(t)k²_L2+ Z T

0

k∇wk²_L2 ≤Ckw0k²_L2,

which is the desired result.

Proof of Theorem 1.4. First, we note that, without loss of generality, the mean value of the pressureπis assumed to be zero, namelyR

Ωπ(·, t)dx= 0 for each time t∈[0, T). We getπsatisfies

kπk_L2(Ω)≤Ck∇πk_L2(Ω).

The proof of Theorem 1.4 is similar to that of Theorem 1.2. Indeed, from (3.3), we note that

1 4

Z

Ω

|u(·, τ)|⁴dx−1 4

Z

Ω

|u(·, T^∗−)|⁴dx +

Z τ

T^∗−

Z

Ω

|∇u|²|u|²dx dt+1 2

Z τ

T^∗−

Z

Ω

∇|u|²

2dx dt

≤ Z τ

T^∗−

Z

Ω

|π| |u| |u||∇u|dx dt+ Z τ

T^∗−

Z

Ω

|u|³|∇u|dx dt:=III₁+III₂. Using H¨older’s inequality, the first termIII1 can be estimated as

III1≤ Z τ

T^∗−

kπkL⁴kukL⁴

|u||∇u|

_L2≤C Z τ

T^∗−

kπkL⁴kukL⁴

|u||∇u|

_L2

≤ Z τ

T^∗−

Ckπk^1/2_L2kπk^1/2_BMOkukL⁴

|u||∇u|

_L2

≤Ck∇πk^1/2_L2(Q_τ)

Z τ

T^∗−

kπk²_BMOkuk⁴_L4dt1/4

|u||∇u|

_L2(Q

τ)

For convenience of computations, we denoteC:=ku(·, T^∗−)k_W1,2(Ω). Using the estimate (2.4), we obtain

III1≤C

k(u· ∇)uk^1/2_L2(Q_τ)+C

Z τ

T^∗−

kπk²_BMOkuk⁴_L4dt1/4

|u||∇u|

_L2(Q

τ),

≤Ck(u· ∇)uk^1/2_L2(Q_τ)

Z τ

T^∗−

kπk²_BMOkuk⁴_L4dt1/4

|u||∇u|

_L2(Q

τ)

+CCZ τ T^∗−

kπk²_BMOkuk⁴_L4dt1/4

|u||∇u|

_L2(Q

τ).

(3.7) Following similar computations as inI2, we obtain

III2≤C

|u||∇u|

_L2(Q

τ) sup

T^∗−<t<τ

ku(·, t)k²_L4. (3.8) Summing (3.7)-(3.8) and using Young’s inequality, we obtain

1 4

Z

Ω

|u(·, τ)|⁴dx−1 4 Z

Ω

|u(·, T^∗−)|⁴dx +

Z τ

T^∗−

Z

Ω

|∇u|²|u|²dx dt+1 2

Z τ

T^∗−

Z

Ω

∇|u|²

2dx dt

≤Ck(u· ∇)uk^1/2_L2(Q_τ)

Z τ

T^∗−

kπk²_BMOkuk⁴_L4dt1/4

|u||∇u|

_L2(Q

τ)

+CC

Z τ

T^∗−

kπk²_BMOkuk⁴_L4dt^1/4

|u||∇u|

_L2(Q

τ)

+C¹²

|u||∇u|

_L2(Q

τ) sup

T^∗−<t<τ

ku(·, t)k²_L4

≤1 2

|u||∇u|

2

L²(Qτ)+CC⁴+ChZ τ T^∗−

kπ(·, t)k²_BMOdt+i sup

T^∗−<t<τ

ku(·, t)k⁴_L4. With sufficiently smallso that

Z τ

T^∗−

kπ(·, t)k²_BMOdt+

≤ 1 2C with a constantC in the above estimate, we have

ku(·, t)k⁴_L4,∞

x,t(Qτ)+1

2k |∇u||u|k²_L2(Qτ)+1

2k∇|u|²k²_L2(Qτ)

≤2ku(·, T −)k⁴_L4

x(Ω)+CC⁴.

By the same argument as in the proof of Theorem 1.2, we finally obtain ku(·, t)k⁴_L4,∞

x,t (Q)+1 2

|∇u||u|

2

L²(Q)+1

2k∇|u|²k²_L2(Q)≤C,

whereC is a constant depending onku(·, T^∗−)kW^1,2(Ω). Proof of Theorem 1.5. This proof is similar to that of Theorem 1.2. Indeed, from From (3.3), we note that

1 4

Z

Ω

|u(·, τ)|⁴dx−1 4 Z

Ω

|u(·, T^∗−)|⁴dx

+ Z τ

T^∗−

Z

Ω

|∇u|²|u|²dx dt+1 2

Z τ

T^∗−

Z

Ω

∇|u|²

2dx dt

≤ Z τ

T^∗−

Z

Ω

|∇π| |u|²|u|dx dt+ Z τ

T^∗−

Z

Ω

|u|³|∇u|dx dt:=IV1+IV2. Using H¨older’s inequality, the first termIV₁can be estimated as

IV₁≤ Z τ

T^∗−

k∇πkL⁴kuk³_L4 ≤C Z τ

T^∗−

k∇πk^1/2_L2k∇πk^1/2_BMOkuk³_L4

≤Ck∇πk^1/2_L2(Qτ)

Z τ

T^∗−

k∇π(·, t)k^2/3_BMOdt^3/4 sup

T^∗−<t<τ

ku(·, t)k³_L4, For convenience of computations, we denoteC:=ku(·, T^∗−)k_W1,2(Ω). Using the estimate (2.4), we obtain

IV₁≤C

k(u· ∇)uk^1/2_L2(Q

τ)+CZ τ T^∗−

k∇π(·, t)k^2/3_BMOdt^3/4

× sup

T^∗−<t<τ

ku(·, t)k³_L4,

≤Ck(u· ∇)uk^1/2_L2(Qτ)

Z τ

T^∗−

k∇π(·, t)k^2/3_BMOdt^3/4 sup

T^∗−<t<τ

ku(·, t)k³_L4

+CCZ τ T^∗−

k∇π(·, t)k^2/3_BMOdt3/4

sup

T^∗−<t<τ

ku(·, t)k³_L4.

(3.9)

Following similar computations as inI2, we obtain IV2≤C^1/2

|u||∇u|

_L2(Q

τ) sup

T^∗−<t<τ

ku(·, t)k²_L4. (3.10) Summing (3.7)-(3.8) and using Young’s inequality, we obtain

1 4

Z

Ω

|u(·, τ)|⁴dx−1 4

Z

Ω

|u(·, T^∗−)|⁴dx +

Z τ

T^∗−

Z

Ω

|∇u|²|u|²dx dt+1 2

Z τ

T^∗−

Z

Ω

∇|u|²

2dx dt

≤Ck(u· ∇)uk^1/2_L2(Q_τ)

Z τ

T^∗−

k∇π(·, t)k^2/3_BMOdt^3/4 sup

T^∗−<t<τ

ku(·, t)k³_L4

+CC

Z τ

T^∗−

k∇π(·, t)k^2/3_BMOdt^3/4 sup

T^∗−<t<τ

ku(·, t)k³_L4

+C^1/2

|u||∇u|

_L2(Q

τ) sup

T^∗−<t<τ

ku(·, t)k²_L4

≤ 1

2k |u||∇u| k²_L2(Q_τ)+CC⁴+CZ τ T^∗−

k∇π(·, t)k^2/3_BMOdt+

× sup

T^∗−<t<τ

ku(·, t)k⁴_L4. With sufficiently smallso that

Z τ

T^∗−

k∇π(·, t)k^2/3_BMOdt+

≤ 1 2C with a constantC in the above estimate, we have

ku(·, t)k⁴_L4,∞

x,t(Q_τ)+1

2k |∇u||u| k²_L2(Q_τ)+1

2k∇|u|²k²_L2(Q_τ)

≤2ku(·, T −)k⁴_L4

x(Ω)+CC⁴.

By the same argument from the proof of Theorem 1.2, we finally obtain the desired

result.

Remark 3.1. The arguments of Theorems 1.2–1.4 also hold for a whole spaceRⁿ because Lemma 2.2 also established for these cases.

Acknowledgments. I would like to thank the anonymous referee for the useful comments. Jae-Myoung Kim was supported by grants NRF-20151009350 and NRF- 2016R1D1A1B03930422.

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Jae-Myoung Kim

Center for Mathematical Analysis & Computation, Yonsei University, Seoul, Korea E-mail address:[email protected]