On the interior regularity criterion and the number of singular points to the Navier-Stokes equations

0.23 MB·English

by  Wendong Wang

On the interior regularity criterion and the number of singular
points to the Navier-Stokes equations
Wendong Wang and Zhifei Zhang
School of Mathematical Sciences and BICMR, Peking University, Beijing 100871,P.R. China
2
1 E-mail: [email protected] and [email protected]
0
2 January 3, 2012

n
a
J
Abstract
5
We establish some interior regularity criterions of suitable weak solutions for the 3-D

] Navier-Stokes equations, which allow the vertical part of the velocity to be large under
P
the local scaling invariant norm. As an application, we improve Ladyzhenskaya-Prodi-
A
Serrin’s criterion and Escauriza-Seregin-Sˇver´ak’s criterion. We also show that if weak
.
h solution u satisfies
at u(,t) Lp C( t)32−pp
m k · k ≤ −
for some 3<p< , then the number of singular points is finite.
[
∞

1
v 1 Introduction
0
0
1 We consider the three dimensional incompressible Navier-Stokes equations
1
1. ∂tu−∆u+u·∇u+∇π = 0, (1)
(cid:26) divu= 0,
0
2
where u(x,t) = (u (x,t),u (x,t),u (x,t)) denotes the unknown velocity of the fluid, and the
1 1 2 3
: scalar function π(x,t) denotes the unknown pressure.
v
i In a seminal paper [12], Leray proved the global existence of weak solution with finite
X
energy. It is well known that weak solution is unique and regular in two spatial dimensions.
r
a In three dimensions, however, the question of regularity and uniqueness of weak solution is
an outstanding open problem in mathematical fluid mechanics.
Inafundamentalpaper[1], Caffarelli-Kohn-Nirenbergproved thatone-dimensionalHaus-
dorff measure of the possible singular points of suitable weak solution u is zero (see also
[13, 22, 14, 23]). The proof is based on the following ε-regularity criterion: there exists an
ε > 0 such that if u satisfies
limsupr−1 u(y,s)2dyds ε, (2)
r→0 ZQr(z0)|∇ | ≤
then u is regular at z . The same result remains true if (2) is replaced by
0
limsupr−2 u(y,s)3dyds ε. (3)
r→0 ZQr(z0)| | ≤
1
Thequantities on theleft handsideof (2)and (3)are scaling invariant. More general interior
regularity criterions were obtained by Gustafson-Kang-Tsai [7] in terms of scaling invariant
quantities (see Proposition 2.5). In the firstpart of this paper, we will establish some interior
regularity criterions, which allow the vertical part of the velocity to be large under the local
scaling invariant norm. The proof is based on the blow-up argument and an observation that
if the horizontal part of the velocity is small, then the blow-up limit satisfies u = 0, hence
h
∂ u = 0 and
3 3
∂ u ∆u +∂ π = 0, ∆π = 0.
t 3 3 3
−
Using new interior regularity criterions, we improve Ladyzhenskaya-Prodi-Serrin regularity
criterions, which state if the weak solution u satisfies
2 3
u Lq(0,T;Lp(R3)) with + 1, p 3,
∈ q p ≤ ≥
then it is regular in (0,T) R3, see [19, 5, 21, 4]. It should be pointed out that the regularity
×
in the class L∞(0,T;L3(R3)) is highly nontrivial, since it does not fall in the framework
of small energy regularity. This case was solved by Escauriza-Seregin-Sˇvera´k [4] by using
blow-up analysis and the backward uniqueness for the parabolic equation.
In Leary’s paper [12], he also proved that if [ T,0) is the maximal existence interval of
−
smooth solution, then for p > 3, there exits c > 0 such that
p
3−p
u(,t) Lp cp( t) 2p .
k · k ≥ −
In general, if u satisfies
3−p
u(,t) Lp C( t) 2p , (4)
k · k ≤ −
the regularity of the solution at t = 0 remains unknown except p = 3. Recently, for the ax-
isymmetric Navier-Stokes equations, important progress has been madeby Chen-Strain-Yau-
Tsai [2, 3] and Koch-Nadirashvili-Segegin-Sˇvera´k [10], where they showed that the solution
does not develop Type I singularity (i.e, u(,t) L∞ C( t)−21) by using De-Giorgi-Nash
k · k ≤ −
method and Liouville theorem respectively. However, the case without the axisymmetric
assumption is still open. The second part of this paper will be devoted to show that the
number of singular points is finite if the solution satisfies (4) for 3 < p < . The proof is
∞
based on an improved ε-regularity criterion: if the suitable weak solution (u,π) satisfies
t0 1
sup u(x,t)2dx+ u(x,t)4dx 2dt
Z | | Z Z | |
t∈[−1+t0,t0] B1(x0) −1+t0(cid:0) B1(x0) (cid:1)
t0 1
+ π(x,t)2dx 2dt ε ,
6
Z Z | | ≤
−1+t0(cid:0) B1(x0) (cid:1)
then u is regular in Q (z ), see Proposition 5.1.
1 0
2
This paper is organized as follows. In section 2, we introduce some definitions and no-
tations. In section 3, we establish some new interior regularity criterions of suitable weak
solutions. In section 4, we apply them to improve Ladyzhenskaya-Prodi-Serrin’s criterion
and Escauriza-Seregin-Sˇvera´k’s criterion. Section 4 is devoted to the proof of the number of
singular points under the condition (4). In the appendix, we present the estimates of the
pressure and some scaling invariant quantities.
2
2 Definitions and notations
Let us first introduce the definition of weak solution.
Definition 2.1 Let Ω R3 and T > 0. We say that u is a Leray-Hopf weak solution of (1)
⊂
in Ω = Ω ( T,0) if
T
× −
1. u L∞( T,0;L2(Ω)) L2( T,0;H1(Ω));
∈ − ∩ −
2. u satisfies (1) in the sense of distribution;
3. u satisfies the energy inequality: for a.e. t [ T,0],
∈ −
t
u(x,t)2dx+2 u2dxds u(x, T)2dx.
Z | | Z Z |∇ | ≤ Z | − |
Ω −T Ω Ω
Furthermore, the pair (u,π) is called a suitable weak solution if π L3/2(Ω ) and the
T
∈
energy inequality is replaced by the following local energy inequality: for any nonnegative
φ C∞(R3 R) vanishing in a neighborhood of the parabolic boundary of Ω ,
∈ c × T
t
u(x,t)2φdx+2 u2φdxds
Z | | Z Z |∇ |
Ω −T Ω
t
u2(∂ φ+ φ)+u φ(u2 +2π)dxds, for a.e. t [ T,0].
s
≤ Z Z | | △ ·∇ | | ∈ −
−T Ω
Remark 2.2 In general, we don’t know whether a Leray-Hopf weak solution is a suitable
weak solution. However, if u is a Leray-Hopf weak solution and u L4(Ω ), then it is also
T
∈
a suitable weak solution, which can be verified by using a standard mollification procedure.
Let (u,π) be a solution of (1) and introduce the following scaling
uλ(x,t) = λu(λx,λ2t), πλ(x,t) =λ2π(λx,λ2t), (5)
for any λ > 0, then the family (uλ,πλ) is also a solution of (1). We introduce some invariant
quantities under the scaling (5):
A(u,r,z )= sup r−1 u(y,t)2dy,
0
Z | |
−r2+t0≤t<t0 Br(x0)
C(u,r,z ) = r−2 u(y,s)3dyds,
0
Z | |
Qr(z0)
E(u,r,z ) = r−1 u(y,s)2dyds,
0
Z |∇ |
Qr(z0)
D(π,r,z0) = r−2 π(y,s) 32dyds,
Z | |
Qr(z0)
3
where z = (x ,t),Q (z ) = ( r2+t ,t ) B (x ), and B (x ) is a ball of radius r centered
0 0 r 0 0 0 r 0 r 0
− ×
at x . We also denote Q by Q (0) and B by B (0). We also denote
0 r r r r
1−3−2
G(f,p,q;r,z0)= r p qkfkLp,q(Qr(z0)),
2−3−2
H(f,p,q;r,z0) = r p qkfkLp,q(Qr(z0)),
1−3−2
G(f,p,q;r,z0)= r p qkf −(f)Br(x0)kLp,q(Qr(z0)),
He(f,p,q;r,z0) = r2−p3−2qkf −(f)Br(x0)kLp,q(Qr(z0)),
where the mixed space-etime norm is defined by
k·kLp,q(Qr(z0))
def t0 q 1
f = f(x,t)pdx pdt q,
k kLp,q(Qr(z0)) (cid:16)Zt0−r2(cid:16)ZBr(x0)| | (cid:17) (cid:17)
and(f) is theaverage of f intheball B (x ). For the simplicity of notations, we denote
Br(x0) r 0
A(u,r,(0,0)) = A(u,r), C˜(u,r) =C(u (u) ,r), G(f,p,q;r,(0,0)) = G(f,p,q;r)
− Br
and so on. These scaling invariant quantities will play an important role in the interior
regularity theory.
Now we recall the definitions of Lorentz space and BMO space [6].
Definition 2.3 Let Ω Rn and 1 p,ℓ . We say that a measurable function f
⊂ ≤ ≤ ∞ ∈
Lp,ℓ(Ω) if f < + , where
k kLp,ℓ(Ω) ∞
1
kfkLp,ℓ(Ω) d=ef  (cid:16)suRp0∞σσℓx−1|{Ωx;∈fΩ>;|fσ|>1pσ}f|opℓrdℓσ(cid:17)=ℓ+ fo.r ℓ < +∞,
|{ ∈ | | }| ∞
σ>0


Moreover, f(x,t) Lq,s( T,0;Lp,ℓ(Ω)) if f(,t) Lq,s( T,0).
∈ − k · kLp,ℓ(Ω) ∈ −
The following facts will be used frequently: for any R > 0,
f f , if ℓ ℓ ; (6)
k kLp,ℓ1 ≤ k kLp,ℓ2 1 ≥ 2
f p1 C Rp1 Ω +Rp1−p f p , if p > p . (7)
k kLp1(Ω) ≤ | | k kLp,∞(Ω) 1
(cid:0) (cid:1)
Recall that a local integrable function f BMO(Rn) if it satisfies
∈
1
sup f(x) f dx < .
R>0,x0∈Rn |BR(x0)|ZBR(x0)| − BR(x0)| ∞
Moreover, f(x) VMO(Rn) if f(x) BMO(Rn) and for any x Rn,
0
∈ ∈ ∈
1
limsup f(x) f dx = 0.
B (x ) Z | − BR(x0)|
R↓0 | R 0 | BR(x0)
We say that a function u BMO−1(Rn) if there exist U BMO(Rn) such that u =
j
n ∂ U . VMO−1(Rn) is∈defined similarly. A remarkable pr∈operty of BMO function is
j=1 j j
P
1
sup f(x) f qdx < .
R>0,x0∈Rn |BR(x0)|ZBR(x0)| − BR(x0)| ∞
4
for any 1 q < .
≤ ∞
Let us conclude this section by recalling the following ε-regularity results. Here and what
follows, we defineasolution uto beregular at z = (x ,t ) if u L∞(Q (z )) for some r > 0.
0 0 0 r 0
∈
Proposition 2.4 [1, 14] Let (u,π) be a suitable weak solution of (1) in Q (z ). There exists
1 0
an ε > 0 such that if
0
u(x,t)3 + π(x,t)3/2dxdt ε ,
0
Z | | | | ≤
Q1(z0)
then u is regular in Q (z ). Moreover, π can be replaced by π (π) in the integral.
21 0 − Br
Proposition 2.5 [7] Let (u,π) be a suitable weak solution of (1) in Q (z ) and w = u.
1 0
∇×
There exists an ε > 0 such that if one of the following two conditions holds,
1
1. G(u,p,q;r,z ) ε for any 0 < r < 1, where 1 3 + 2 2;
0 ≤ 1 2 ≤ p q ≤
2. H(w,p,q;r,z ) ε for any 0 < r < 1, where 2 3 + 2 3 and (p,q) = (1, );
0 ≤ 1 2 ≤ p q ≤ 6 ∞
then u is regular at z .
0
3 Interior regularity criterions of suitable weak solution
The purpose of this section is to establish some interior regularity criterions, which allow
the vertical part of the velocity to be large under the local scaling invariant norm. These
results improve some classical results and Gustafson-Kang-Tsai’s result (Proposition 2.5).
Set u= (u ,u ). Let us state our main results.
h 3
Theorem 3.1 Let (u,π) be a suitable weak solution of (1) in Q and satisfy
1
C(u,1)+D(π,1) M.
≤
Then there exists a positive constant ε depending on M such that if
2
C(u ,1) ε ,
h 2
≤
then u is regular at (0,0).
Theorem 3.2 Let (u,π) be a suitable weak solution of (1) in Q and satisfy
1
G(u,p,q;r) M for any 0 < r < 1,
≤
where 1 3 + 2 < 2, 1 < q . There exists a positive constant ε depending on p,q,M
≤ p q ≤ ∞ 3
such that (0,0) is a regular point if
G(u ,p,q;r∗) ε
h 3
≤
for some r∗ with 0 < r∗ < min 1,(C(u,1)+D(π,1))−2 .
{2 }
5
Theorem 3.3 Let (u,π) be a suitable weak solution of (1) in Q and satisfy
1
H( u,p,q;r) M for any 0 < r < 1,
∇ ≤
where 2 3 + 2 < 3,1 < p . There exists a positive constant ε depending on p,q,M
≤ p q ≤ ∞ 4
such that (0,0) is a regular point if
H( u ,p,q;r∗) ε (8)
h 4
∇ ≤
for some r∗ with 0 < r∗ < min 1,(C(u,1)+D(p,1))−2 .
{2 }
Remark 3.4 As a special case of Theorem 3.2, it follows that u is regular if
M ε
3
u , u ,
3 h
| |≤ √T t | | ≤ √T t
− −
which improves Leray’s result [12]. And from Theorem 3.3, it follows that u is regular at
(0,0) if for any 0 < r < 1,
r−1 u 2dxdt M2, r−1 u 2dxdt ε2,
Z |∇ 3| ≤ Z |∇ h| ≤ 4
Qr Qr
which improves Caffarelli-Kohn-Nirenberg’s result [1].
The proof of Theorem 3.1 is based on compactness argument and the following lemma.
Lemma 3.5 Let (u,π) be a suitable weak solution of (1) in Q and D(π,1) M. Then u
1
≤
is regular at (0,0) if
e
9/5 8/5
C(u,r ) cε r for some 0 < r 1.
0 ≤ 0 0 0 ≤
Here c is a small constant depending on M.
Proof. By (27) and H¨older inequality, for 0 < r < r /4 we have
0
r2 r r2
C(u,r)+D(π,r) 0C(u,r )+C( )5/2D(π,r )+C 0C(u,r )
≤ r2 0 r 0 r2 0
0
e r5/2 r2 e
CM +C 0C(u,r ).
≤ r9/2 r2 0
0
Choosing r = ( ε0 )2/5r9/5 and by assumption, we infer that
2CM 0
C(u,r)+D(π,r) < ε ,
0
which implies that (0,0) is a regular pointeby Proposition 2.4. (cid:3)
6
Now let us turn to the proof of Theorem 3.1.
Proof of Theorem 3.1. Assume that the statement of the proposition is false, then there
exist a constant M and a sequence (uk,πk), which are suitable weak solutions of (1) in Q
1
and singular at (0,0), and satisfies
1
C(uk,1)+D(πk,1) M, C(uk,1) .
≤ h ≤ k
Then by the local energy inequality, it is easy to get
A(uk,3/4)+E(uk,3/4) C(M),
≤
hence by using Lions-Aubin’s lemma, there exists a suitable weak solution (v,π′) of (1) such
that (at most up to subsequence),
uk → v, ukh → 0 in L3(Q21), πk ⇀ π′ in L32(Q12),
as k + . That is, v = 0, which gives ∂ v = 0 by v = 0, and hence,
h 3 3
→ ∞ ∇·
∂ v +∂ π′ v =0, π′ = 0, or
t 3 3 3
−△ −△
∂ v+ π′ v = 0,
t
∇ −△
which implies that v C(M) in Q by the classical result of linear Stokes equation(see
1/4
| | ≤
[2] for example). However, (0,0) is a singular point of uk, hence by Lemma 3.5, for any
0 < r < 1/4,
cε9/5r8/5 lim r−2 uk 3dxdt
0 ≤ k→∞ ZQr | |
lim C(v,r) C(M)r3,
≤ k→∞ ≤
which is a contradiction by letting r 0. (cid:3)
→
The proof of Theorem 3.2 is motivated by [18] and based on the blow-up argument.
Proof of Theorem 3.2. Assume that the statement of the proposition is false, then there
exist constants p,q,M and a sequence (uk,πk), which are suitable weak solutions of (1) in
Q and singular at (0,0), and satisfy
1
G(uk,p,q;r) M for all 0 <r < 1,
≤
1
G(uk,p,q;r ) ,
h k ≤ k
where 0 < r < min 1,(C(uk,1)+D(πk,1))−2 . Then it follows from Lemma 6.2 that
k {2 }
A(uk,r)+E(uk,r)+D(πk,r) C(M,p,q)
≤
for any 0 < r < r .
k
Set vk(x,t) = r uk(r x,r2t),qk(x,t) =r2πk(r x,r2t). Then
k k k k k k
A(vk,r)+E(vk,r)+D(qk,r) C(M,p,q)
≤
7
for any 0 < r < 1. Lions-Aubin’s lemma ensures that there exists a suitable weak solution
(v¯,π¯) of (1) such that (at most up to subsequence),
vk v¯ in L3(Q1), qk ⇀ q¯ in L32(Q1),
→ 2 2
1
vk ⇀ 0 in Lq(( ,0);Lp(B )),
h −4 12
as k + . Then we have v¯ = 0 and
h
→ ∞
∂ v¯ +∂ q¯ v¯ = 0,
t 3 3 3
−△
which implies that v¯ C(M) in Q . However, (0,0) is a singular point of vk, hence by
3 1
| | ≤ 4
Proposition 2.4 and (26), for any 0< r < 1/4,
ε liminfr−2 vk 3+ qk 3/2dxdt
0
≤ k→∞ ZQr| | | |
r ρ
Climinf C(v¯,r)+ D(qk,ρ)+( )2C(vk,ρ)
≤ k→∞ (cid:16) ρ r (cid:17)
r ρ
C C(v¯,r)+ +( )2C(v¯,ρ)
≤ ρ r
(cid:16) (cid:17)
Cr1/2 (by choosing ρ = r12),
≤
which is a contradiction if we take r small enough. (cid:3)
Proof of Theorem 3.3. Without loss of generality, let us assume that
8 3 2
< + < 3.
3 p q
The other case can be reduced to it by H¨older inequality. By Lemma 6.2, we have
A(u,r)+E(u,r)+D(π,r) C(M) r1/2 C(u,1)+D(π,1) +1 C(M),
≤ ≤
(cid:0) (cid:0) (cid:1) (cid:1)
for any 0 < r r , min 1, C(u,1) + D(p,1) −2 . This together with interpolation
≤ 1 {2 }
inequality gives (cid:0) (cid:1)
C(u,r) C(M) for any 0 < r r . (9)
1
≤ ≤
We get by Poinca´re inequality that
G(u ,p ,q ;r) CH( u ,p,q;r),
h 1 1 h
≤ ∇
where p = 3p ,q = q, henece it follows from (e28) and (8) that
1 3−p 1
C(uh,r∗) C(M) A(uh,r∗)+E(uh,r∗) 11−−32δδG(uh,p1,q1;r∗)1−12δ,
≤
e ≤ C(M)G(cid:0) (uh,p1,q1;r∗)1−12δ ≤(cid:1)C(M)eε41−12δ,
where δ = 2 3 2 (0, 1), heence by (9) for 0< r < r∗
− p1 − q1 ∈ 3
r r∗ r r∗ 1
C(u ,r) C( )C(u ,r∗)+C( )2C(u ,r∗) C(M) +( )2ε1−2δ .
h ≤ r∗ h r h ≤ r∗ r 4
(cid:0) (cid:1)
Taking r small enough, and then ε small eneough such that
4
C(u ,r) ε3.
h ≤ 3
Then the result follows from Theorem 3.2. (cid:3)
8
4 Applications of interior regularity criterions
4.1 Ladyzhenskaya-Prodi-Serrin’s criterion
Using the interior regularity criterions established in Section 3, we present Ladyzhenskaya-
Prodi-Serrin’s type criterions in Lorentz spaces.
Theorem 4.1 Let u be a Leray-Hopf weak solution of (1) in R3 ( 1,0). Assume that u
× −
satisfies
kukLq,∞((−1,0);Lp,∞(R3)) ≤ M, kuhkLq,ℓ((−1,0);Lp,∞(R3)) < ∞, (10)
where 3 + 2 = 1, 3 < p , and 1 ℓ < . Then u is regular in R3 ( 1,0]. For ℓ =
p q ≤ ∞ ≤ ∞ × − ∞
or p = 3, the same result holds if the second condition of (10) is replaced by
uh Lq,∞((−1,0);Lp,∞(R3)) ε5,
k k ≤
where ε is a small constant depending on M.
5
Remark 4.2 For ℓ = , we improve Kim-Kozono’s result [9] and He-Wang’s result [8],
∞
where the smallness of all components of the velocity is imposed. In general case, we improve
Sohr’s result [20] by allowing the vertical part of the velocity to fall in weak Lp space.
Remark 4.3 Under the condition (10), it can be verified that Leray-Hopf weak solution is
suitable weak solution. We left it to the interested readers.
The proof is based on the following lemma.
Lemma 4.4 Assume that u satisfies
u Lq,∞((−1,0);Lp,∞(R3)) m,
k k ≤
where 3 + 2 = 1, 3 p . Then for any 0< r < 1 and 0< ǫ < 1, there hold
p q ≤ ≤ ∞
G(u, 9 p,4q;r) Cǫ45q +Cǫ−5qmq, 3 < p < ,
10 5 ≤ ∞
A(u,r) Cǫ2+Cǫ−1m3, p = 3,
≤
3
G(u, , ;r) Cǫ3/2+Cǫ−1/2m2, p= .
∞ 2 ≤ ∞
Proof. First we consider the case of 3 < p < . Using the definition of Lorentz space, we
∞
infer that
0 8q
r(54−38p)q−2 u 190pdx 9pdt
Z−r2(cid:16)ZBr | | (cid:17)
0 ∞ 8q
Cr(54−38p)q−2 σ190p−1 x Br; u(x,t) > σ dσ 9pdt
≤ Z−r2(cid:16)Z0 |{ ∈ | | }| (cid:17)
0 ∞ 8q
Cr(54−38p)q−2 r2(r3R190p)98pq + σ190p−1 x Br; u(x,t) > σ dσ 9pdt
≤ (cid:16) Z−r2(cid:16)ZR |{ ∈ | | }| (cid:17) (cid:17)
≤ Cr(54−38p)q−2(cid:16)r2(r3R190p)98pq +R−44q5 Z−0r2ku(·,t)kL89qp,∞dt(cid:17)
Cr(54−38p)q−2 r2(r3R190p)98pq +R−44q5r−(1−p3)89q+2I(r)
≤
(cid:16) (cid:17)
Cǫ45q +Cǫ−445qI(r),
≤
9
where we take R = ǫr−1 and the estimate of I(r) is given by
(1−3)8q−2 0 8q
I(r) r p 9 u(,t) 9 dt
≡ Z−r2k · kLp,∞(B1)
∞
Cr(1−p3)89q−2 σ89q−1 t ( r2,0); u(,t) Lp,∞ > σ dσ
≤ Z |{ ∈ − k · k }|
0
∞
Cr(1−p3)89q−2 R89qr2+ σ89q−1 t ( r2,0); u(,t) Lp,∞ > σ dσ
≤ (cid:16) ZR |{ ∈ − k · k }| (cid:17)
≤ Cr(1−p3)89q−2(cid:16)R89qr2+R−q9kukqLq,∞(−1,0;Lp,∞(B1))(cid:17)
Cr(1−p3)89q−2 R89qr2+R−q9mq
≤
(cid:16) (cid:17)
Cǫ89q +Cǫ−9qmq (R = ǫrp3−1).
≤
This gives the first inequality. For p = 3, we consider
∞
sup r−1 u2dx C sup r−1 σ x B ; u(x,t) > σ dσ
r
Z | | ≤ Z |{ ∈ | | }|
−r2<t<0 Br −r2<t<0 0
∞
C sup r−1 R2r3+ σ x B ; u(x,t) > σ dσ
r
≤ −r2<t<0 (cid:16) ZR |{ ∈ | | }| (cid:17)
C sup r−1 R2r3+R−1 u(,t) 3 ,
≤ k · kL3,∞
−r2<t<0 (cid:16) (cid:17)
which gives the second inequality by taking R = ǫr. Let g(t) = u(,t) . Then we
k · kL∞(B1)
have
0 ∞
r−1/2 g(t)3/2dt Cr−1/2 σ12 t ( r2,0); g(t) > σ dσ
Z ≤ Z |{ ∈ − | | }|
−r2 0
∞
Cr−1/2 R23r2+ σ12 t ( r2,0); g(t) > σ dσ
≤ (cid:16) ZR |{ ∈ − | | }| (cid:17)
Cr−21 R23r2+R−21m2 ,
≤
(cid:16) (cid:17)
which gives the third inequality by taking R = ǫr. (cid:3)
Proof of Theorem 4.1. By translation invariance and Theorem 3.2, it suffices to show that
G(u,p ,q ;r) M, G(u ,p ,q ;r) ε , (11)
1 1 h 1 1 3
≤ ≤
for any 0 <r < 1/2 and some (p ,q ) with 1 3 + 2 < 2. For 3 < p < , let p = 9 p and
1 1 ≤ p1 q1 ∞ 1 10
q = 4q, then 3 + 2 < 5 < 2. For ℓ < , we have u 0 as r 0.
1 5 p1 q1 4 ∞ k hkLq,ℓ(−r2,0;Lp,∞(Br)) → →
Hence by Lemma 4.4, the condition (11) holds if we take ǫ small enough, and then take r
small enough. The proof of the other cases is similar. We omit the details. (cid:3)
4.2 Escauriza-Seregin-Sˇver´ak’s criterion
The following theorem improves Escauriza-Seregin-Sˇvera´k’s criterion by noting the inclusion
L3(R3) L3,ℓ(R3) for ℓ > 3 and L3(R3) VMO−1(R3).
⊂ ⊂
10