Electronic Journal of Differential Equations, Vol. 2009(2009), No. 121, pp. 1–6.
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 in R3 . Let u := (u1 , u2 , u3 ) be a weak solution and u
e := (u1 , u2 , 0).
We prove that u is strong solution if ∇e
u satisfy 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,
div u = 0
u|t=0 = u0 ,
(1.1)
3
in (0, ∞) × R ,
div u0 = 0
in R
(1.2)
3
(1.3)
where u := (u1 , u2 , u3 ) represents the velocity and p represents 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 if u ∈ Lr (0, T ; Ls (R3 )) with 2r + 3s = 1
and s ∈ [3, ∞], then the solution u is actually regular [4, 5, 6, 12, 13, 14, 15]. A
similar condition ω = curl u ∈ Lr (0, T ; Ls (R3 )) with 2r + 3s = 2 where 32 ≤ s ≤ ∞
also implies the regularity as shown by Beião da Veiga [2]. Chae and Choe [3] proved
that if ω
e = (ω1 , ω2 , 0) ∈ Lr (0, T ; Ls (R3 )) with 2r + 3s = 2 and 32 ≤ s < ∞, then the
solution is regular. Kozono and Yatsu [7] showed that if ω
e ∈ L1 (0, T ; BM O), then
the solution remains smooth. Zhang and Chen [17] proved that u is regular if ω
e∈
0
L1 (0, T ; Ḃ∞,∞
). Bae and Choe [1] proved that u is strong if u
e ∈ Lr (0, T ; Ls (R3 ))
with 2r + 3s = 1 with s > 3. In [3], the authors also proved that u is strong if
∇e
u ∈ Lr (0, T ; Ls ) with 2r + 3s ≤ 1, 2 ≤ r ≤ ∞ and 3 ≤ s ≤ ∞, which is not optimal
2000 Mathematics 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
2
J. FAN, H. GAO
EJDE-2009/121
from the scaling argument. Here we would like to improve the regularity criterion
on ∇e
u in 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 spaces M (Ḣ r , L2 ) if it maps,
by point-wise multiplication, Ḣ r in L2 :
Ẋr := M (Ḣ r , L2 ) := {f ∈ S 0 , kf φkL2 ≤ CkφkḢ r }.
(1.4)
(0)
Ẏ2
Similarly we can define Ẏ1+r := M (Ḣ r , Ḣ −1 ) and
denotes the closure of the
1
Schwartz class S in Ẏ2 . We denote Λ := (−∆) 2 , then Ẏ2 = Λ2 BM O [10], Ẋr and
Ẏ1+r have been characterized in [10, 11].
Now we are in a position to state the main result in this paper.
Theorem 1.1. Let u0 ∈ H 1 . Assume that one of the following four conditions
holds:
2
∇e
u ∈ L 2−r (0, T ; Ẋr )
∇e
u∈L
2
1−r
(0, T ; Ẏ1+r )
∇e
u∈
for some r ∈ [0, 1),
(1.5)
for some r ∈ [0, 1),
(1.6)
(0)
C([0, T ]; Ẏ2 ),
(1.7)
0
∇e
u ∈ L1 (0, T ; Ḃ∞,∞
).
(1.8)
u ∈ L∞ (0, T ; H 1 ) ∩ L2 (0, T ; H 2 ).
(1.9)
Then
s
stands for the homogeneous Besov space, see below for
Here and thereafter, Ḃp,q
the definition.
3
3
3
0
, L r ⊂ L r ,∞ ⊂ Ẋr and L 1+r ⊂
Remark 1.2. Since L∞ ( BM O ( Ḃ∞,∞
3
L 1+r ,∞ ⊂ Ẏ1+r , our results improve that given in [3].
2. Preliminaries
We introduce the Littlewood-Paley decomposition. Let S(R3 ) be the Schwartz
class of rapidly decreasing function. Given f ∈ S(R3 ), its Fourier transform F f =
fˆ is defined by
Z
fˆ(ξ) =
and its inverse Fourier transform F
e−ixξ f (x)dx,
R3
−1
f = f ∨ is defined by
Z
∨
−3
f (x) = (2π)
eixξ f (ξ)dξ.
R3
Let us choose a nonnegative radial function φ ∈ S(R3 ) such that
(
1, if |ξ| ≤ 1,
0 ≤ φ̂(ξ) ≤ 1, φ̂(ξ) =
0, if |ξ| ≥ 2,
and let
ψ(x) = φ(x) − 2−3 φ
x
,
2
φj (x) = 23j φ(2j x),
ψj (x) = 23j ψ(2j x),
j ∈ Z.
EJDE-2009/121
REGULARITY FOR THE NAVIER-STOKES EQUATIONS
3
For j ∈ Z, the Littlewood-Paley projection operators Sj and ∆j are, respectively,
defined by
Sj f = φj ∗ f,
(2.1)
∆j f = ψj ∗ f.
(2.2)
Observe that ∆j = Sj − Sj−1 . Also, if f is an L2 function, then Sj f → 0 in
L2 as j → −∞ and Sj f → 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
+∞
X
f=
∆j f,
(2.3)
j=−∞
2
for all f ∈ L , where the summation is in the L2 sense. Notice that
∆j f =
j+2
X
∆l (∆j f ) =
l=j−2
j+2
X
ψl ∗ ψj ∗ f,
l=j−2
then from the Young inequality, it follows that
1
1
k∆j f kLq ≤ C23j( p − q ) k∆j f kLp ,
(2.4)
where 1 ≤ p ≤ q ≤ ∞, C is a constant independent of f, j.
s
is defined by the
Let s ∈ R, p, q ∈ [1, ∞], the homogeneous Besov space Ḃp,q
full-dyadic decomposition such as
s
Ḃp,q
= {f ∈ Z 0 (R3 ) : kf kḂ s < ∞},
p,q
where
kf kḂ s =
+∞
X
2jsq k∆j f kqLp
p,q
1/q
j=−∞
and Z (R ) denotes the dual space of Z(R3 ) = {f ∈ S(R3 ) : Dα fˆ(0) = 0; ∀α ∈ N3 }.
We refer to [16] for more detailed properties.
0
3
3. Proof of Theorem 1.1
We set
|∇u|2 =
X
|∂k ui |2 ,
|∇2 u|2 =
i,k
X
|∂k ∂j ui |2 .
i,j,k
Differentiating both sides of equation (1.1) with respect to xk , taking the scalar
product with ∂k u, adding over k and, finally, integrating by parts over Rn , we show
that
Z
Z
Z
1 d
|∇u|2 dx + |∇2 u|2 dx = − ∇[(u · ∇)u] · ∇udx
2 dt
XZ
=
∂k ui · ∂i uj · ∂k uj dx.
i,j,k
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 case i = j = 3. Since ∂3 u3 = −∂1 u1 −∂2 u2 ,
4
J. FAN, H. GAO
EJDE-2009/121
it readily follows that
Z
Z
∂k ui · ∂i uj · ∂k uj dx = − ∂k u3 · (∂1 u1 + ∂2 u2 ) · ∂k u3 dx
Z
≤ 2 |∇e
u| · |∇u|2 dx.
And thus we get
1 d
2 dt
Z
|∇u|2 dx +
Z
|∇2 u|2 dx ≤ 2
Z
|∇e
u| · |∇u|2 dx =: I.
(3.1)
Now we assume that (1.5) holds. Then
I ≤ 2k∇ukL2 · k |∇e
u| · |∇u| kL2
≤ 2k∇ukL2 · k∇e
ukẊr k∇ukḢ r
2
r
≤ Ck∇ukL2 · k∇e
ukẊr k∇uk1−r
L2 k∇ ukL2
by the interpolation inequality
r
kwkḢ r ≤ Ckwk1−r
L2 k∇wkL2 ,
(3.2)
whence
2
2−r
k∇uk2L2 ,
I ≤ k∇2 uk2L2 + Ck∇e
ukẊ
r
for any > 0 by Young’s inequality. Inserting the above estimates into (3.1) and
taking small enough and the Gronwall’s inequality yield (1.9).
Next we assume that (1.6) holds. Then
I ≤ 2k∇ukḢ 1 k |∇e
u| · |∇u| kḢ −1
≤ Ck∇ukḢ 1 k∇e
ukẎ1+r k∇ukḢ r
2
r
≤ Ck∇2 ukL2 k∇e
ukẎ1+r k∇uk1−r
L2 k∇ ukL2
2
1−r
≤ k∇uk2L2 + Ck∇e
ukẎ
1+r
(by (3.2))
k∇uk2L2
for any > 0 by Young’s inequality. Inserting the above estimates into (3.1) and
taking small enough and the Gronwall’s inequality give (1.9).
Now we assume that (1.7) holds. For any > 0, then there exist α and β such
that ∇e
u = α + β, kαkL∞ (0,T ;Y2 ) ≤ and β ∈ L∞ ((0, T ) × R3 ),
Z
I ≤ 2 |α| · |∇u|2 dx + 2kβkL∞ k∇uk2L2
≤ 2k∇ukḢ 1 k| α| · ∇ukḢ −1 + Ck∇uk2L2
≤ 2k∇ukḢ 1 kαkẎ2 k∇ukḢ 1 + Ck∇uk2L2
≤ 2k∇2 uk2L2 + Ck∇uk2L2 .
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 decomposition (2.3), we decompose ∇e
u as follows:
∇e
u=
+∞
X
`=−∞
∆` (∇e
u) =
X
`<−N
∆` (∇e
u) +
N
X
`=−N
∆` (∇e
u) +
X
`>N
∆` (∇e
u).
EJDE-2009/121
REGULARITY FOR THE NAVIER-STOKES EQUATIONS
5
Here N is a positive integer to be chosen later. Substituting this into I, we have
N Z
X Z
X
|∆` (∇e
u)| |∇u|2 dx + 2
I≤2
|∆` (∇e
u)| |∇u|2 dx
`<−N
+2
`=−N
XZ
(3.3)
|∆` (∇e
u)| |∇u|2 dx
`>N
=: I1 + I2 + I3 .
For I1 , from the Hölder inequality and (2.4), it follows that
X
I1 ≤ 2
k∆` (∇e
u)kL∞ k∇uk2L2
`<−N
3
X
≤ 2k∇uk2L2
u)kL2
2 2 ` k∆` (∇e
`<−N
− 23 N
≤ C2
3
k∇uk2L2 k∇e
ukL2 ≤ C2− 2 N k∇uk3L2 .
For I2 , from the Hölder inequality, it follows that
N
X
I2 ≤ 2k∇uk2L2
k∆` (∇e
u)kL∞
`=−N
≤ Ck∇uk2L2 · N k∇e
ukḂ 0
∞,∞
= CN k∇uk2L2 k∇e
ukḂ 0
∞,∞
.
For I3 , from the Hölder inequality and (2.4), it follows that
X
I3 ≤ 2k∇uk2L3
k∆` (∇e
u)kL3
`>N
≤
Ck∇uk2L3
X
`
u)kL2
2 2 k∆` (∇e
`>N
≤ Ck∇uk2L3
X
2−`
1/2 X
`>N
≤
N
Ck∇uk2L3 2− 2
`>N
k∇ u
ekL2
k∇uk2L3 k∇2 ukL2
−N
2
k∇ukL2 k∇2 uk2L2
≤ C2
1/2
2
−N
2
≤ C2
22` k∆` (∇e
u)k2L2
by the Gagliardo-Nirenberg inequality
k∇uk2L3 ≤ Ck∇ukL2 k∇2 ukL2 .
Inserting the above estimates into (3.3), we find that
3
I ≤ C2− 2 N k∇uk3L2 + CN k∇e
ukḂ 0
∞,∞
N
k∇uk2L2 + C2− 2 k∇ukL2 k∇2 uk2L2 .
N
Now we choose N so that C2− 2 k∇ukL2 ≤ 21 ; i.e.,
N ≥2+
Then
2 log+ (2Ck∇ukL2 )
.
log 2
1
I ≤ C + Ck∇uk2L2 + Ck∇uk2L2 log(e + k∇ukL2 ) + k∇2 uk2L2 .
2
6
J. FAN, H. GAO
EJDE-2009/121
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].
References
[1] H. O. Bae and H. J. Choe; A regularity criterion for the Navier-Stokes equations, (Preprint
2005).
[2] H. Beirão da Veiga; A new regularity class for the Navier-Stokes equations in Rn , Chinese
Ann. Math. 16 (1995), 407-412.
[3] D. Chae and H. J. Choe; Regularity of solutions to the Navier-Stokes equations, Electronic
J. Differential Equations, Vol. 1999 (1999), No. 5, pp. 1-7.
[4] L. Escauriaza, G. A. Seregin and V.Šverák; L3,∞ -solutions of the Navier-Stokes equations
and backward uniqueness, Russ. Math. Surv. 58(2) (2003), 211-248.
[5] E. Fabes, B. Jones and N. M. Riviere; The initial value problem for the Navier-Stokes equations with data in Lp , Arch. Rat. Mech. Anal. 45 (1972), 222-248.
[6] Y. Giga; Solutions for semilinear parabolic equations in Lp and regularity of weak solutions
of the Navier-Stokes equations, J. Diff. Eqns. 62 (1986), 186-212.
[7] H. Kozono and N. Yatsu; Extension criterion via two-Components of vorticity on strong
solutions to the 3-D Navier-Stokes equations, Math. Z., 246 (2004), 55-68.
[8] I. Kukavica and M. Ziane; One component regularity for the Navier-Stokes equations, Nonlinearity 19 (2006), 453-469.
[9] J. Leray; Sur le mouvement d’un liquide visqueux emplissant l’espace, Acta Math. 63 (1934),
193-248.
[10] V. G. Maz’ya and T. O. Shaposhnikova; Theory of Multipliers in Spaces of Differentiable
Functions. Monographs and Studies in Mathematics 23, Pitman, Boston, MA, 1985.
[11] V. G. Maz’ya; On the theory of the n-dimensional Schrödinger operator, Izv. Akad. Nauk
SSSR (Ser. Mat.) 28 (1964), 1145-1172.
[12] T. Ohyama; Interior regularity of weak solutions to the Navier-Stokes equations, Proc. Japan
Acad. 36 (1960), 273-277.
[13] J. Serrin; On the interior regularity of weak solutions of the Navier-Stokes equations, Arch.
Rat. Mech. Anal. 9 (1962), 197-195.
[14] H. Sohr and W. Von Wahl; On the regularity of the pressure of weak solutions of NavierStokes equations, Archiv Math. 46 (1986), 428-439.
[15] M. Struwe; On partial regularity results for the Navier-Stokes equations, Comm. Pure Appl.
Math. 41 (1988), 437-458.
[16] H. Triebel; Interpolation Theory, Function Spaces, Differential Operators, North Holland,
Amsterdam-New York-Oxford, 1978.
[17] Z. Zhang and Q. Chen; Regularity criterion via two components of vorticity on weak solutions
to the Navier-Stokes equations in R3 , J. Differential Equations 216 (2005), 470-481.
[18] Y. Zhou, A new regularity result for the Navier-Stokes equations in terms of the gradient of
one velocity component, Methods Appl. Anal. 9 (2002), 563–578.
[19] Y. Zhou, A new regularity criterion for weak solutions to the Navier- Stokes equations, J.
Math. Pures Appl. 84 (2005), 1496–1514.
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: fanjishan@njfu.com.cn
Hongjun Gao
School of Mathematical Science, Nanjing Normal University, Nanjing, 210097, China
E-mail address: gaohj@njnu.edu.cn