In this article, we study the regularity criteria for the 3D Navier- Stokes equations involving derivatives of the partial components of the velocity

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

REMARKS ON REGULARITY CRITERIA FOR THE 3D NAVIER-STOKES EQUATIONS

RUIYING WEI, YIN LI

Abstract. In this article, we study the regularity criteria for the 3D Navier- Stokes equations involving derivatives of the partial components of the velocity.

It is proved that if∇_huebelongs to Triebel-Lizorkin space,∇u3 oru3 belongs to Morrey-Campanato space, then the solution remains smooth on [0, T].

1. Introduction

This article is devoted to the Cauchy problem for the following incompressible 3D Navier-Stokes equation:

ut+ (u· ∇)u+∇p=4u, x∈R³, t >0

divu= 0, x∈R³, t >0 (1.1) with initial data

u(x,0) =u₀, x∈R³, (1.2)

where u = (u₁(x, t), u₂(x, t), u₃(x, t)) and p = p(x, t) denote the unknown veloc- ity vector and the unknown scalar pressure, respectively. In the last century, Leray [11] and Hopf [8] proved the global existence of a weak solution u(x, t) ∈ L^∞(0,∞;L²(R³))∩L²(0,∞;H¹(R³)) to (1.1)-(1.3) for any given initial datum u0(x) ∈ L²(R³). However, whether or not such a weak solution is regular and unique is still a challenging open problem. From that time on, different criteria for regularity of the weak solutions has been proposed.

The classical Prodi-Serrin conditions (see [16, 18, 19]) say that if u∈L^t(0, T;L^s(R³)), 2

t +3

s = 1, 3≤s≤ ∞,

then the solution is smooth. Similar results is showed by Beir˜ao da Veiga [1]

involving the velocity gradient growth condition:

∇u∈L^t(0, T;L^s(R³)), 2 t +3

s = 2, 3

2 < s≤ ∞.

Actually, whether the weak solution is smooth when a part of the velocity com- ponents is involved. As for this direction, later on, criteria just for one velocity

2000Mathematics Subject Classification. 35Q35, 76D05.

Key words and phrases. Regularity criteria; Triebel-Lizorkin space; Morrey-Campanato space.

c

2014 Texas State University - San Marcos.

Submitted June 23, 2014. Published October 29, 2014.

1

component appeared. The first result in this direction is due to Neustupa et al [15]

(see also Zhou [21]), where the authors showed that if u3∈L^t(0, T;L^s(R³)), 2

t +3 s =1

2, s∈(6,∞],

then the solution is smooth. A similar result, for the gradient of one velocity component, is independently due to Zhou [22] and Pokorn´y [17]. In [22], Zhou proved that if

∇u3∈L^t(0, T;L^s(R³)), 2 t +3

s = 3

2, 3≤s <∞.

then the solution is smooth on [0, T]. This result is extended by Zhou and Pokorn´y [26]; that is,

∇u3∈L^t(0, T;L^s(R³)), 2 t +3

s = 23

12, 2≤s≤3.

Further criteria, including several components of the velocity gradient, pressure or other quantities, can be found, here we just list some. Zhou and Pokorn´y [25]

proved the regularity condition

u3∈L^t(0, T;L^s(R³)), 2 t +3

s =3 4 + 1

2s, s >10 3 .

And in [10], Jia and Zhou proved that if a weak solution u satisfies one of the following two conditions:

u3∈L^∞(0, T;L¹⁰³(R³)); ∇u3∈L^∞(0, T;L^30/19(R³)),

then u is regular on [0, T]. Dong and Zhang [5] proved that if the horizontal derivatives of the two velocity components

Z T 0

k∇hu(s)k˜ B˙⁰_∞,∞ds <∞,

then the solution keeps smoothness up to timeT, where ˜u= (u1, u2,0), and∇hu˜= (∂1u, ∂˜ 2u,˜ 0). For other kinds of regularity criteria, see [2, 6, 7, 9, 23, 24, 28, 29, 30]

and the references cited therein.

Throughout this paperCwill denote a generic positive constant which can vary from line to line. For simplicity, we shall useR

f(x)dx to denoteR

R³f(x)dx, use k · kp to denotek · kL^p.

The purpose of this article is to improve and extend above known regularity criterion of weak solution for the equations (1.1), (1.2) to the Triebel-Lizorkin space and Morrey-Campanato spaces. The main results of this paper read:

Theorem 1.1. Assume that u0 ∈ H¹(R³) with divu0 = 0. u(x,t) is the corre- sponding weak solution to (1.1)and (1.2)on[0, T). If additionally

Z T 0

k∇heu(·, t)k^p_˙

F⁰

q,2 3q

dt <∞, with 2 p+3

q = 2, 3

2 < q≤ ∞, (1.3) then the solution remains smooth on[0, T].

Theorem 1.2. Assume that u0 ∈ H¹(R³) with divu0 = 0. u(x, t) is the corre- sponding weak solution to (1.1)and (1.2)on[0, T). If additionally

Z T 0

k∇u3(·, t)k

8 3(2−r)

M˙^p,3r

dt <∞, with 0< r≤1, 2≤p≤ 3

r, (1.4)

then the solution remains smooth on[0, T].

Theorem 1.3. Assume that u0 ∈ H¹(R³) with divu0 = 0. u(x,t) is the corre- sponding weak solution to (1.1)and (1.2)on[0, T). If additionally

Z T 0

ku3(·, t)k

8 3−4r

M˙^p,3r

dt <∞, with 0< r < 3

4, 2≤p≤ 3

r, (1.5)

then the solution remains smooth on[0, T].

Remark 1.4. Noticing that the classical Riesz transformation is bounded in ˙B⁰_∞,∞, if we takeq=∞in Theorem 1.1, then the classical Beal-Kato-Majda criterion for the Navier-Stokes equations is obtained; that is, if

Z T 0

k∇hu(·, t)ke B˙_∞,∞⁰ dt <∞ then the solution remains smooth on [0, T].

Remark 1.5. Since (it is proved in [12, 13])

L^q(R³) = ˙M^q,q(R³)⊂M˙^p,q(R³), 1< p≤q <∞, L^3r(R³)⊂M˙^p,3r(R³)⊂X˙r(R³)⊂M˙^2,3r(R³), 2< p≤3

r, 0< r <3 2, the result of Theorem 1.2 is an improved version of [27, Theorem 2]. Also we obtain, if

Z T 0

ku₃(·, t)k

8 3−4r

X˙_r dt <∞, with 0< r < 3 4, then the solution remains smooth on [0, T].

2. Preliminaries

In this section, we shall introduce the Littlewood-Paley decomposition theory, and then give some definitions of the homogeneous Besov space, homogeneous Triebel-Lizorkin space, Morrey-Campanato space and multiplier space as well as some relate spaces used throughout this paper. Before this, let us first recall the weak solutions of (1.1)-(1.3):

Letu0∈L²(R³) with∇ ·u0= 0, a measurableR³-valued vectoruis said to be a weak solution of (1.1)-(1.3) if the following conditions hold:

(1) u(x, t)∈L^∞(0,∞;L²(R³))∩L²(0,∞;H¹(R³));

(2) usolves (1.1)-(1.2) in the sense of distributions;

(3) the energy inequality holds; i.e, kuk²₂+ 2

Z t 0

k4u(·, τ)k²₂dτ ≤ ku0k²₂, 0≤t≤T.

Let us choose a nonnegative radial function ϕ ∈ C^∞(R³) be supported in the annulus {ξ ∈ R³ : ³₄ ≤ |ξ| ≤ ⁸₃}, such that P∞

l=−∞ϕ(2^−lξ) = 1,∀ξ 6= 0. For f ∈S⁰(R³), the frequency projection operators4lis defined as

4_lf =F⁻¹(ϕ(2^−j·))∗f,

whereF⁻¹(g) is the inverse Fourier transform ofg. The formal decomposition f =

∞

X

l=−∞

4lf. (2.1)

is called the homogeneous Littlewood-Paley decomposition. Noticing 4lf =

l+1

X

j=l−1

4j(4lf)

and using the Young inequality, we have the following class Bernstein inequality:

Lemma 2.1 ([3]). Let α∈N, then for all 1 ≤p≤q≤ ∞, sup_|α|=kk∂^α4lfkq ≤ C2^{lk+3l(1p−1q)}k4_lfk_p. andC is a constant independent off, l.

For s ∈ R and (p, q) ∈ [1,∞]×[1,∞], the homogeneous Besov space ˙B_p,q^s is defined by

S˙^s_p,q={f ∈Z⁰(R³) :kfkS˙_p,q^s <∞}, where

kfkB˙_p,q^s =





 P

j∈z2^jsqk4jf(·)k^q_p1/q

, 1≤q <∞, sup_j∈z2^jsk4jf(·)kp, q=∞.

andZ⁰(R³) denote the dual space of

Z⁰(R³) ={f ∈S(R³) :D^αfˆ(0) = 0.∀a∈N³}.

On the other hand, for s∈R,(p, q)∈[1,∞)×[1,∞], and for s∈R, p =q=∞, the homogenous Triebel-Lizorkin space is defined as

F˙_p,q^s ={f ∈Z⁰(R³) :kfkF˙_p,q^s <∞}, kfkF˙_p,q^s =

(k(P

j∈z2^jsq|4_jf(·)|^q)^1/qk_p, 1≤q <∞, ksup_j∈z(2^js|4jf(·)|)kp, q=∞.

Notice that by Minkowski inequality, we have the following two imbedding relations:

B˙_p,q^s ⊂F˙_p,q^s , q≤p;

F˙_p,q^s ⊂B˙_p,q^s , p≤q.

and the following two inclusions:

H˙^s= ˙B_2,2^s = ˙F_2,2^s , L^∞⊂F˙_∞,∞⁰ = ˙B_∞,∞⁰ . We refer to [20] for more properties.

For 1< q ≤p <∞, the homogeneous Morrey-Campanato space inR³ is M˙^p,q=

f ∈L^q_loc(R³);kfkM˙^p,q = sup

x∈R³

sup

R>0

R^p3−3qkfk_q(B(x,R))<∞ , For 1≤p⁰≤q⁰<∞, we define the homogeneous space

N˙^p0,q0 =n

f ∈L^q0|f =X

k∈N

g_k, where g_k∈L^q_comp⁰ (R³) and X

k∈N

d³⁽

1 p0−_q¹0)

k kgkkq⁰ <∞, where dk = diam(suppgk)<∞o .

For 0< α <3/2, we say that a function belongs to the multiplier spacesM( ˙H^α, L²) if it maps, by pointwise multiplication, ˙H^α toL²:

X˙_α:=M( ˙H^α, L²) :=

f ∈S⁰;kf·gkL² ≤CkgkH˙^α,∀g∈H˙^α .

Here, ˙H^α is the homogeneous Sobolev space of orderα, H˙^α=

f ∈L¹_loc;kfkL² ≡Z

R³

|ξ|^2α|ˆu(ξ)|^21/2

<∞ . whereL^p(1≤p≤ ∞) is the Lebsgue space endowed with normk · kp.

Lemma 2.2([4, 12]). Let1≤p⁰≤q⁰<∞, andp, qsuch that ¹_p+_p¹0 = 1,¹_q+_q¹0 = 1.

ThenM˙^p,q is the dual space ofN˙^p0,q0.

Lemma 2.3 ([4, 7, 12]). Let 1 < p⁰ ≤ q⁰ < 2, m≥ 2, and ¹_p +_p¹0 = 1. Denote α=−ⁿ₂ +ⁿ_p +_mⁿ ∈ (0,1] Then there exists a constant C > 0, such that for any u∈L^m(Rⁿ), v∈H˙^α(Rⁿ),

ku·vkN˙^p0,q0 ≤CkukL^mkvkH˙^α.

Lemma 2.4 ([13]). For 0 ≤r ≤ ³₂, let the space M( ˙B₂^r,1 → L²) be the space of functions which are locally square integrable onR³ and such that pointwise multi- plication with these functions maps boundedly the Besov spaceB˙₂^r,1(R³)toL²(R³).

The norm inM( ˙B^r,1₂ →L²)is given by the operator norm of pointwise multiplica- tion:

kfk_{M( ˙B}r,1

2 →L²)= sup{kf gk2:kgkB˙^r,1₂ ≤1}.

Then,f belongs toM( ˙B₂^r,1→L²)if and only iff belongs toM˙^2,3r (with equivalence of norms).

3. The proof of main results

Proof of Theorem 1.1. Multiplying (1.1)1 by −4u, integrating by parts, noting that∇ ·u= 0, we have

1 2

d dt

Z

|∇u|²dx+ Z

|4u|²dx= Z

[(u· 5)u]· 4u dx=:I. (3.1) Next we estimate the right-hand side of (3.1), with the help of integration by parts and−∂3u3=∂1u1+∂2u2, one shows that

I=−

3

X

i,j,k=1

Z

∂_ku_i∂_iu_j∂_ku_jdx

=−

2

X

i,j=1 3

X

k=1

Z

∂kui∂iuj∂kujdx−

2

X

i,k=1

Z

∂kui∂iu3∂ku3dx

−

2

X

j,k=1

Z

∂ku3∂3uj∂kujdx−

3

X

k=1

Z

∂ku3∂3u3∂ku3dx

−

2

X

i=1

Z

∂3ui∂iu3∂3u3dx−

2

X

j=1

Z

∂3u3∂3uj∂3ujdx

≤C Z

|∇heu||∇u|²dx.

Thus, the above inequality implies 1

2 d

dtk∇uk²₂+k4uk²₂≤C Z

|∇hu||∇u|e ²dx. (3.2)

Using the Littlewood-Paley decomposition (2.1),∇huecan be written as

∇hue= X

j<−N

4j(∇heu) +

N

X

j=−N

4j(∇hu) +e X

j>N

4j(∇hu).e

where N is a positive integer to be chosen later. Substituting this into (3.2), one obtains

1 2

d

dtk∇uk²₂+k4uk²₂

≤C X

j<−N

Z

|4j(∇hu)||∇u|e ²dx+C

N

X

j=−N

Z

|4j(∇heu)||∇u|²dx +CX

j>N

Z

|4_j(∇_hu)||∇u|e ²dx

=:K1+K2+K3.

(3.3)

ForKi(i= 1,2,3), we now give the estimates one by one. ForK1, using the H¨older inequality, the Young inequality and Lemma 2.1, it follows that

K1≤C X

j<−N

k4j(∇hu)ke ∞k∇uk²₂

≤C X

j<−N

2^3j/2k4j(∇hu)ke 2k∇uk²₂

≤C X

j<−N

2^3j1/2 X

j<−N

k4_j(∇_hu)ke ²₂1/2

k∇uk²₂

≤C2^−3N/2k∇uk³₂.

(3.4)

Where in the last inequality, we use the fact that for alls∈R, ˙H^s= ˙B_2,2^s . ForK2, by the H¨older inequality and the Young inequality, one has

K₂=C Z ^N

X

j=−N

|4j(∇hu)||∇u|e ²dx

≤CN^2q−32q

Z X^N

j=−N

|4_j(∇_hu)|e ^2q/33/(2q)

|∇u|²dx

≤CN^2q−32q k∇heukF˙⁰

q,2q 3

k∇uk²2q q−1

≤CN^2q−32q k∇heukF˙⁰

q,2q 3

k∇uk

2q−3 q

2 k4uk^3/q₂

≤ 1

2k4uk²₂+CNk∇heuk

2q 2q−3

F˙⁰

q,2q 3

k∇uk²₂.

(3.5)

where we used the interpolation inequality

kuks≤Ckuk₂^3s−12kuk³²_˙^−3s

H¹ , for 2≤s≤6.

Finally, using H¨older inequality and Lemma 2.1,K3 can be estimated as K3=C X

j>N

Z

|4j(∇heu)||∇u|²dx

≤C X

j>N

k4j(∇heu)k3k∇uk²₃

≤C X

j>N

2^j2k4j(∇heu)k2k∇uk²₃

≤C(X

j>N

2^−j)^1/2(X

j>N

2^2jk4j(∇hu)ke ²₂)^1/2k∇uk2k4uk2

≤C2^−N/2k∇uk2k4uk²₂.

(3.6)

Substituting (3.4), (3.5) and (3.6) in (3.3), we obtain d

dtk∇uk²₂+k4uk²₂

≤C2^−32Nk∇uk³₂+CNk∇huke

2q 2q−3

F˙⁰

q,2q 3

k∇uk²₂+C2^−N/2k∇uk2k4uk²₂.

(3.7)

Now we chooseN such thatC2^−N/2k∇uk₂≤ ¹₂; that is N ≥ln(k∇uk²₂+e) + lnC

ln 2 + 2.

Thus (3.7) implies d

dtk∇uk²₂≤C+Ck∇huke ^p_˙

F⁰

q,2q 3

ln(k∇uk²₂+e)k∇uk²₂. Taking the Gronwall inequality into consideration, we obtain

ln(k∇uk²₂+e)≤Ch 1 +

Z T 0

k∇heuk^p_˙

F⁰

q,2q 3

(τ)dτ ·e

RT 0 k∇huke ^p_˙

F0 q,2q

3

(τ)dτ

i . The proof of Theorem 1.1 is complete under the condition (1.3).

Proof of Theorem 1.2. Multiplying (1.1)1 by −4hu, integrating by parts, noting that∇ ·u= 0, we have

1 2

d dt

Z

|∇_hu|²dx+ Z

|∇∇_hu|²dx= Z

[(u· 5)u]· 4_hu dx=:J. (3.8) Next we estimate the right-hand side of (3.8), with the help of integration by parts and−∂3u₃=∂₁u₁+∂₂u₂, one shows that

J =−

3

X

i,j=1 2

X

k=1

Z

∂_ku_i∂_iu_j∂_ku_jdx

=−

2

X

i,j,k=1

Z

∂_ku_i∂_iu_j∂_ku_jdx−

2

X

i,k=1

Z

∂_ku_i∂_iu₃∂_ku₃dx

−

3

X

j=1 2

X

k=1

Z

∂ku3∂3uj∂kujdx

=:J₁+J₂+J₃.

(3.9)

ForJ2 andJ3, we obtain

|J₂+J₃| ≤C Z

|∇u₃||∇_hu||∇u|dx. (3.10) J1 is a sum of eight terms, using the fact−∂3u3=∂1u1+∂2u2, we can estimate it as

J₁=− Z

(∂₁u₁+∂₂u₂)[(∂₁u₁)²−∂₁u₁∂₂u₂+ (∂₂u₂)²]dx

− Z

(∂₁u₁+∂₂u₂)[(∂₂u₁)²+∂₁u₂∂₂u₁+ (∂₁u₂)²]dx

= Z

∂3u3[(∂1u1)²−∂1u1∂2u2+ (∂2u2)²+ (∂2u1)²+∂1u2∂2u1+ (∂1u2)²]dx

≤C Z

|∇u3||∇hu||∇u|dx.

(3.11) Substituting the estimates (3.9)-(3.11) in (3.8), we obtain

1 2

d

dtk∇huk²₂+k∇∇huk²₂≤C Z

|∇u3||∇hu||∇u|dx=:L. (3.12) when 2< p≤ ³_r, using Lemmas 2.2 and 2.3, and the Young inequality, we obtain

L≤Ck∇u3k _˙

M^p,3rk|∇u| · |∇hu|k

N˙^p0,3−r3

≤Ck∇u₃k _˙

M^p,3rk∇_hukH˙^rk∇uk₂

≤Ck∇u3k _˙

M^p,3rk∇uk_L2k∇huk^1−r₂ k∇∇huk^r₂

≤1

2k∇∇huk²₂+Ck∇u3k

2 2−r

M˙^p,r³

k∇uk²₂.

(3.13)

where we used the inequality kfkH˙^r =k|ξ|^rfkˆ 2= (

Z

|ξ|^2r|fˆ|^2r|fˆ|^2−2rdξ)^1/2≤ kfk^1−r₂ k∇fk^r₂, with 0< r≤1.

In the casep= 2, using H¨olders inequality, Lemma 2.4, and the Young inequality, we can estimateLas

L≤Ck|∇u3| · |∇hu|k2k∇uk2

≤Ck∇u3k _˙

M^2,3rk∇hukB˙₂^r,1k∇uk2

≤Ck∇u3k _˙

M^2,3rk∇huk^1−r₂ k∇∇huk^r₂k∇uk2

≤ 1

2k∇∇huk²₂+Ck∇u3k

2 2−r

M˙^2,3r

k∇uk²₂.

(3.14)

where we used the following interpolation inequality [14]: for 0≤r≤1,kfkB˙^r,1₂ ≤ kfk^1−r₂ k∇fk^r₂.

Now, gathering (3.13) and (3.14) together and substituting into (3.12), we obtain d

dtk∇_huk²₂+k∇∇_huk²₂≤Ck∇u₃k

2 2−r

M˙^p,3r

k∇uk²₂. (3.15)

Multiplying (1.1)1 by−4u, integrating by parts, noting that∇ ·u= 0, we have (see [25])

1 2

d

dtk∇uk²₂+k4uk²₂= Z

[(u· 5)u]· 4u dx

≤C Z

|∇hu||∇u|²dx

≤Ck∇huk2k∇uk²₄

≤Ck∇_huk₂k∇uk^1/2₂ k∇uk₆³²

≤Ck∇huk2k∇uk^1/2₂ k∇∇huk2k4uk^1/2₂ . Integrating, with respect tot, yields

1

2k∇u(t)k²₂+ Z t

0

k4u(τ)k²₂dτ

≤1

2k∇u0k²₂+C sup

0≤τ≤t

k∇hu(τ)k2( Z t

0

k∇u(τ)k²₂dτ)^1/4

×Z t 0

k∇∇hu(τ)k²₂dτ^1/2Z t 0

k4u(τ)k²₂dτ^1/4 .

(3.16)

Substituting (3.15) in (3.16), using H¨olders inequality and the Young inequality, we obtain

1

2k∇u(t)k²₂+ Z t

0

k4u(τ)k²₂dτ

≤1

2k∇u0k²₂+ (C+C Z t

0

k∇u3k

2 2−r

M˙^p,3r

k∇u(τ)k²₂dτ)Z t 0

k4u(τ)k²₂dτ1/4

≤C+CZ t 0

k∇u3k

2 2−r

M˙^p,3r

k∇u(τ)k₂³²k∇u(τ)k^1/2₂ dτ^4/3 +1

2 Z t

0

k4u(τ)k²₂dτ

≤C+CZ t 0

k∇u3k

8 3(2−r)

M˙^p,3r

k∇u(τ)k²₂dτZ t 0

k∇u(τ)k²₂dτ1/3

+1 2

Z t 0

k4u(τ)k²₂dτ

≤C+C Z t

0

k∇u3k

8 3(2−r)

M˙^p,3r k∇u(τ)k²₂dτ +1 2

Z t 0

k4u(τ)k²₂dτ.

(3.17)

Absorbing the last term into the left hand side, applying the Gronwall inequality and combining with the standard continuation argument, we conclude that the solu- tionsucan be extended beyondt=Tprovided that∇u3∈L^3(2−r)8 (0, T; ˙M^p,3r(R³)).

This completes the proof of Theorem 1.2.

Proof of Theorem 1.3. We start from (3.9), we can estimateJ2 andJ3 as

|J2+J3| ≤C Z

|u3||∇u||∇∇hu|dx. (3.18)

From (3.11), we find that

J1≤C Z

|u3||∇u||∇∇hu|dx. (3.19) Combining (3.9),(3.17), (3.18) with (3.8), we obtain

1 2

d

dtk∇huk²₂+k∇∇huk²₂≤C Z

|u3||∇u||∇∇hu|dx=:V. (3.20) When 2< p≤³_r, similarly as in the proof ofLin (3.13), we obtain

V ≤Cku3k _˙

M^p,3rk|∇u| · |∇∇hu|k_˙

N^p0,3−r3

≤Cku3k _˙

M^p,3rk∇ukH˙^rk∇∇huk2

≤Cku3k _˙

M^p,3rk∇∇hukL²k∇uk^1−r₂ k4uk^r₂

≤ 1

2k∇∇huk²₂+Cku3k²

M˙^p,3rk∇uk^2(1−r)₂ k4uk^2r₂ .

(3.21)

Whenp= 2, similarly as in the proof ofL in (3.14), we have V ≤Ck|u3| · |∇u|k2k∇∇huk2

≤Cku3k _˙

M^2,3rk∇ukB˙₂^r,1k∇∇huk2

≤Cku3k _˙

M^2,3rk∇uk^1−r₂ k4uk^r₂k∇∇huk2

≤ 1

2k∇∇huk²₂+Cku3k²_˙

M^2,3rk∇uk^2(1−r)₂ k4uk^2r₂ .

(3.22)

Substituting (3.21) and (3.22) in (3.20), we find that d

dtk∇huk²₂+k∇∇huk²₂≤Cku3k²

M˙^2,3rk∇uk^2(1−r)₂ k4uk^2r₂ . (3.23) Substituting (3.23) in (3.16), we obtain

1

2k∇u(t)k²₂+ Z t

0

k4u(τ)k²₂dτ

≤ 1

2k∇u0k²₂+ C+C

Z t 0

ku3k²_˙

M^p,3rk∇u(τ)k^2(1−r)₂ k4u(τ)k^2r₂ dτ

×Z t 0

k4u(τ)k²₂dτ1/4

≤C+1 4

Z t 0

k4u(τ)k²₂dτ +CZ t 0

ku3k²_˙

M^p,3rk∇u(τ)k^2(1−r)₂ k4u(τ)k^2r₂ dτ4/3

≤C+1 4

Z t 0

k4u(τ)k²₂dτ +CZ t 0

k4u(τ)k²₂dτ^4r/3

×Z t 0

ku3k

2 1−r

M˙^p,3r

k∇u(τ)k²₂dτ^4(1−r)₃

≤C+1 2

Z t 0

k4u(τ)k²₂dτ +CZ t 0

ku3k

2 1−r

M˙^p,3r

k∇u(τ)k

3−4r 2(1−r)

2 k∇u(τ)k

1 2(1−r)

2 dτ

4(1−r) 3−4r

≤C+1 2

Z t 0

k4u(τ)k²₂dτ +CZ t 0

ku3k

8 3−4r

M˙^p,3r

k∇u(τ)k²₂dτZ t 0

k∇u(τ)k²₂dτ_3−4r¹

≤C+1 2

Z t 0

k4u(τ)k²₂dτ +C Z t

0

ku3k

8 3−4r

M˙^p,3r

k∇u(τ)k²₂dτ.

By a similar argument as in the proof of Theorem 1.2, provided that u3 ∈ L^3−4r8 (0, T; ˙M^p,3r(R³)), we complete the proof of Theorem 1.3.

Acknowledgements. Ruiying Wei and Yin Li would like to express sincere grat- itude to Professor Zheng-an Yao of Sun Yat-sen University for enthusiastic guid- ance and constant encouragement. This work is partially supported by Guang- dong Provincial culture of seedling of China (no. 2013LYM0081), and Guangdong Provincial NSF of China (no. S2012010010069), the Shaoguan Science and Technol- ogy Foundation (no. 313140546), and Science Foundation of Shaoguan University.

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Ruiying Wei

School of Mathematics and Information Science, Shaoguan University, Shaoguan, Guangdong 512005, China.

Department of Mathematics, Sun Yat-sen University , Guangzhou, Guangdong 510275, China

E-mail address:weiruiying521@163.com

Yin Li

School of Mathematics and Information Science, Shaoguan University, Shaoguan, Guangdong 512005, China.

Department of Mathematics, Sun Yat-sen University , Guangzhou, Guangdong 510275, China

E-mail address:liyin2009521@163.com