We consider the regularity of weak solutions to the Navier-Stokes equations inR3

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

TWO COMPONENT REGULARITY FOR THE NAVIER-STOKES EQUATIONS

JISHAN FAN, HONGJUN GAO

Abstract. We consider the regularity of weak solutions to the Navier-Stokes equations inR³. Letu:= (u1, u2, u3) be a weak solution andue:= (u1, u2,0).

We prove thatuis strong solution if∇eusatisfy Serrin’s type criterion.

1. Introduction

In this article we study the regularity of the weak solutions of the Navier-Stokes equations:

ut+u· ∇u+∇p−∆u= 0, (1.1)

divu= 0 in (0,∞)×R³, (1.2)

u|t=0=u0, divu0= 0 inR³ (1.3) whereu:= (u1, u2, u3) represents the velocity andprepresents the pressure.

The existence of global weak solutions for any initial data with finite energy is known since the work of Leray [9]. The smoothness of Leray’s weak solution is not known. While the existence of a regular solution is still an open problem, there are many interesting sufficient conditions which guarantee that a given weak solution is smooth. A well-known condition states that ifu∈L^r(0, T;L^s(R³)) with ²_r+³_s = 1 and s∈ [3,∞], then the solution uis actually regular [4, 5, 6, 12, 13, 14, 15]. A similar condition ω= curlu∈L^r(0, T;L^s(R³)) with ²_r+³_s = 2 where ³₂ ≤s≤ ∞ also implies the regularity as shown by Bei˜ao da Veiga [2]. Chae and Choe [3] proved that ifωe= (ω₁, ω₂,0)∈L^r(0, T;L^s(R³)) with ²_r+³_s = 2 and ³₂ ≤s <∞, then the solution is regular. Kozono and Yatsu [7] showed that ifeω∈L¹(0, T;BM O), then the solution remains smooth. Zhang and Chen [17] proved thatuis regular ifωe∈ L¹(0, T; ˙B⁰_∞,∞). Bae and Choe [1] proved thatuis strong ifeu∈L^r(0, T;L^s(R³)) with ²_r + ³_s = 1 with s > 3. In [3], the authors also proved that u is strong if

∇ue∈L^r(0, T;L^s) with ²_r+³_s ≤1,2≤r≤ ∞and 3≤s≤ ∞, which is not optimal

2000Mathematics Subject Classification. 35Q30, 35K15, 76D03.

Key words and phrases. Navier-Stokes equations; regularity criterion; two component;

multiplier spaces; Besov spaces.

c

2009 Texas State University - San Marcos.

Submitted August 28, 2009. Published September 29, 2009.

Supported by grants 10871097 from the NSFC, 20080441062 from the China Postdoctoral Research Foundation, and 0802020C from Jiangsu Planned Projects for Postdoctoral Research Foundation.

1

from the scaling argument. Here we would like to improve the regularity criterion on∇uein such a way that it undergoes the correct scaling.

There have been many efforts to show that analogous conditions on only one component of the velocity or the gradient of velocity imply the regularity of solutions but all the results are partial, see [8], citez2, z3 and the references there in.

We say that a function belongs to the multiplier spacesM( ˙H^r, L²) if it maps, by point-wise multiplication, ˙H^r inL²:

X˙r:=M( ˙H^r, L²) :={f ∈ S⁰,kf φk_L2 ≤CkφkH˙^r}. (1.4) Similarly we can define ˙Y1+r :=M( ˙H^r,H˙⁻¹) and ˙Y₂⁽⁰⁾ denotes the closure of the Schwartz classS in ˙Y2. We denote Λ := (−∆)¹², then ˙Y2= Λ²BM O[10], ˙Xr and Y˙1+rhave been characterized in [10, 11].

Now we are in a position to state the main result in this paper.

Theorem 1.1. Let u0 ∈ H¹. Assume that one of the following four conditions holds:

∇ue∈L^2−r2 (0, T; ˙Xr) for somer∈[0,1), (1.5)

∇ue∈L^1−r2 (0, T; ˙Y1+r) for somer∈[0,1), (1.6)

∇ue∈C([0, T]; ˙Y₂⁽⁰⁾), (1.7)

∇ue∈L¹(0, T; ˙B_∞,∞⁰ ). (1.8) Then

u∈L^∞(0, T;H¹)∩L²(0, T;H²). (1.9) Here and thereafter, ˙B_p,q^s stands for the homogeneous Besov space, see below for the definition.

Remark 1.2. Since L^∞ ( BM O ( B˙⁰_∞,∞, L^3r ⊂ L^3r,∞ ⊂ X˙r and L^1+r3 ⊂ L^{1+r3 ,∞}⊂Y˙1+r, our results improve that given in [3].

2. Preliminaries

We introduce the Littlewood-Paley decomposition. LetS(R³) be the Schwartz class of rapidly decreasing function. Givenf ∈ S(R³), its Fourier transformFf = fˆis defined by

fˆ(ξ) = Z

R³

e^−ixξf(x)dx, and its inverse Fourier transformF⁻¹f =f^∨ is defined by

f^∨(x) = (2π)⁻³ Z

R³

e^ixξf(ξ)dξ.

Let us choose a nonnegative radial functionφ∈ S(R³) such that 0≤φ(ξ)ˆ ≤1, φ(ξ) =ˆ

(1, if|ξ| ≤1, 0, if|ξ| ≥2, and let

ψ(x) =φ(x)−2⁻³φ x 2

, φ_j(x) = 2^3jφ(2^jx), ψ_j(x) = 2^3jψ(2^jx), j∈Z.

Forj ∈Z, the Littlewood-Paley projection operatorsSj and ∆j are, respectively, defined by

S_jf =φ_j∗f, (2.1)

∆jf =ψj∗f. (2.2)

Observe that ∆_j =S_j −S_j−1. Also, if f is an L² function, then S_jf → 0 in L² as j → −∞ and S_jf → f in L² as j → +∞ (this is an easy consequence of Parseval’s theorem). By telescoping the series, we thus have the Littlewood-Paley decomposition

f =

+∞

X

j=−∞

∆jf, (2.3)

for allf ∈L², where the summation is in theL² sense. Notice that

∆jf =

j+2

X

l=j−2

∆l(∆jf) =

j+2

X

l=j−2

ψl∗ψj∗f,

then from the Young inequality, it follows that

k∆jfkL^q ≤C2^3j(1p−1q)k∆jfkL^p, (2.4) where 1≤p≤q≤ ∞,C is a constant independent off, j.

Let s ∈ R, p, q ∈ [1,∞], the homogeneous Besov space ˙B_p,q^s is defined by the full-dyadic decomposition such as

B˙_p,q^s ={f ∈ Z⁰(R³) :kfkB˙_p,q^s <∞}, where

kfkB˙_p,q^s = ^+∞X

j=−∞

2^jsqk∆_jfk^q_Lp

1/q

andZ⁰(R³) denotes the dual space ofZ(R³) ={f ∈ S(R³) :D^αfˆ(0) = 0;∀α∈N³}.

We refer to [16] for more detailed properties.

3. Proof of Theorem 1.1 We set

|∇u|²=X

i,k

|∂kui|², |∇²u|²=X

i,j,k

|∂k∂jui|².

Differentiating both sides of equation (1.1) with respect to xk, taking the scalar product with∂ku, adding overkand, finally, integrating by parts overRⁿ, we show that

1 2

d dt

Z

|∇u|²dx+ Z

|∇²u|²dx=− Z

∇[(u· ∇)u]· ∇udx

=X

i,j,k

Z

∂_ku_i·∂_iu_j·∂_ku_jdx.

Following [1], we consider separately the three cases i 6= 3; i = 3 and j 6= 3;

i=j= 3. We only need to deal with the casei=j= 3. Since∂₃u₃=−∂1u₁−∂2u₂,

it readily follows that Z

∂kui·∂iuj·∂kujdx=− Z

∂ku3·(∂1u1+∂2u2)·∂ku3dx

≤2 Z

|∇eu| · |∇u|²dx.

And thus we get 1 2

d dt

Z

|∇u|²dx+ Z

|∇²u|²dx≤2 Z

|∇u| · |∇u|e ²dx=:I. (3.1) Now we assume that (1.5) holds. Then

I≤2k∇ukL²· k |∇u| · |∇u| ke L²

≤2k∇uk_L2· k∇uke X˙rk∇ukH˙^r

≤Ck∇uk_L2· k∇uke X˙rk∇uk^1−r_L2 k∇²uk^r_L2

by the interpolation inequality

kwkH˙^r ≤Ckwk^1−r_L2 k∇wk^r_L2, (3.2) whence

I≤k∇²uk²_L2+Ck∇euk

2 2−r

X˙r k∇uk²_L2,

for any >0 by Young’s inequality. Inserting the above estimates into (3.1) and takingsmall enough and the Gronwall’s inequality yield (1.9).

Next we assume that (1.6) holds. Then I≤2k∇ukH˙¹k |∇u| · |∇u| ke H˙⁻¹

≤Ck∇ukH˙¹k∇eukY˙_1+rk∇ukH˙^r

≤Ck∇²uk_L2k∇eukY˙_1+rk∇uk^1−r_L2 k∇²uk^r_L2 (by (3.2))

≤k∇uk²_L2+Ck∇uke

2 1−r

Y˙1+rk∇uk²_L2

for any >0 by Young’s inequality. Inserting the above estimates into (3.1) and takingsmall enough and the Gronwall’s inequality give (1.9).

Now we assume that (1.7) holds. For any > 0, then there existαand β such that∇eu=α+β,kαkL^∞(0,T;Y₂)≤andβ ∈L^∞((0, T)×R³),

I≤2 Z

|α| · |∇u|²dx+ 2kβkL^∞k∇uk²_L2

≤2k∇ukH˙¹k|α| · ∇ukH˙⁻¹+Ck∇uk²_L2

≤2k∇ukH˙¹kαkY˙₂k∇ukH˙¹+Ck∇uk²_L2

≤2k∇²uk²_L2+Ck∇uk²_L2.

Inserting the above estimates into (3.1) and taking small enough and then the Gronwall’s inequality show (1.9).

Finally we assume that (1.8) holds. Then using the Littlewood-Paley decompo- sition (2.3), we decompose∇ueas follows:

∇eu=

+∞

X

`=−∞

∆`(∇eu) = X

`<−N

∆`(∇u) +e

N

X

`=−N

∆`(∇eu) +X

`>N

∆`(∇u).e

HereN is a positive integer to be chosen later. Substituting this intoI, we have I≤2 X

`<−N

Z

|∆`(∇u)| |∇u|e ²dx+ 2

N

X

`=−N

Z

|∆`(∇eu)| |∇u|²dx + 2X

`>N

Z

|∆`(∇u)| |∇u|e ²dx

=:I1+I2+I3.

(3.3)

ForI1, from the H¨older inequality and (2.4), it follows that I1≤2 X

`<−N

k∆`(∇eu)kL^∞k∇uk²_L2

≤2k∇uk²_L2

X

`<−N

2<sup>32`</sup>k∆`(∇u)ke _L2

≤C2^−32Nk∇uk²_L2k∇uke L² ≤C2^−32Nk∇uk³_L2. ForI₂, from the H¨older inequality, it follows that

I2≤2k∇uk²_L2

N

X

`=−N

k∆`(∇u)ke L^∞

≤Ck∇uk²_L2·Nk∇uke B˙_∞,∞⁰

=CNk∇uk²_L2k∇uke B˙_∞,∞⁰ . ForI₃, from the H¨older inequality and (2.4), it follows that

I3≤2k∇uk²_L3

X

`>N

k∆`(∇eu)k_L3

≤Ck∇uk²_L3

X

`>N

2<sup>2`</sup>k∆`(∇eu)k_L2

≤Ck∇uk²_L3

X

`>N

2^−`1/2 X

`>N

2<sup>2`</sup>k∆`(∇u)ke ²_L2

^1/2

≤Ck∇uk²_L32^−N2k∇²euk_L2

≤C2^−N2k∇uk²_L3k∇²ukL²

≤C2^−N2k∇uk_L2k∇²uk²_L2

by the Gagliardo-Nirenberg inequality

k∇uk²_L3 ≤Ck∇uk_L2k∇²uk_L2. Inserting the above estimates into (3.3), we find that

I≤C2^−32Nk∇uk³_L2+CNk∇eukB˙⁰_∞,∞k∇uk²_L2+C2^−N2k∇ukL²k∇²uk²_L2. Now we chooseN so thatC2^−N2k∇ukL²≤ ¹₂; i.e.,

N ≥2 +2 log⁺(2Ck∇uk_L2)

log 2 .

Then

I≤C+Ck∇uk²_L2+Ck∇uk²_L2log(e+k∇uk_L2) +1

2k∇²uk²_L2.

Insetting the above estimates into (3.1) and the Gronwall’s inequality give (1.9).

This completes the proof.

Acknowledgements. The authors are indebted to Professor H. O. Bae who kindly sent us the preprint [1].

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Jishan Fan

School of Mathematical Science, Nanjing Normal University, Nanjing, 210097, China Department of Applied Mathematics, Nanjing Forestry University, Nanjing 210037, China

E-mail address:[email protected]

Hongjun Gao

School of Mathematical Science, Nanjing Normal University, Nanjing, 210097, China E-mail address:[email protected]