Conditions implying regularity of the three dimensional Navier-Stokes equation

0.18 MB·English

by  Stephen Montgomery-Smith

CONDITIONS IMPLYING REGULARITY OF THE
THREE DIMENSIONAL NAVIER-STOKES EQUATION
4
0
0
STEPHEN MONTGOMERY-SMITH
2

n
a Abstract. We obtain logarithmic improvements for conditions
J for regularity of the Navier-Stokes equation, similar to those of

0 Prodi-Serrin or Beale-Kato-Majda. Some of the proofs make use
1 of a stochastic approach involving Feynman-Kac like inequalities.

Aspartoftheourmethods,wegiveadifferentapproachtoapriori
]
P estimates of Foia¸s, Guillop´e and Temam.
A
.
h
t
a
m 1. Introduction
[
TheversionofthethreedimensionalNavier-Stokesequationwestudy

6 is the differential equation in u = u(t) = u(x,t), where t 0, and
v x R3: ≥
7
∈
0 ∂u
= ∆u Ldiv(u u), u(0) = u .
2 0
∂t − ⊗
1
0 Here L denotes the Leray projection. We will not usually be working
3 with classical solutions. We define u(t), 0 t T, to be a solution of
0 ≤ ≤
the Navier-Stokes equation if, whenever u(t ) is sufficiently regular for
/ 0
h
a mild solution
t
a
m t
u(t) = e(t−t0)∆u(t ) e(t−s)∆Ldiv(u(s) u(s))ds
0
: − ⊗
v Zt0
i
X to exist for t [t ,t + τ) for some τ > 0, then u(t) is equal to that
0 0
∈
r mild solution in [t ,t +τ).
a 0 0
We also use other ways to describe the three dimensional Navier-
Stokes equation. First, let us denote the vorticity by w = w(t) =
w(x,t) = curlu. If w is sufficiently smooth then
∂w
= ∆w u w +w u, w(0) = curlu .
0
∂t − ·∇ ·∇
2000 Mathematics Subject Classification. Primary 35Q30, 76D05, Secondary
60H30, 46E30.
Key words and phrases. Navier-Stokes equation, vorticity, Prodi-Serrin condi-
tion, Beale-Kato-Majda condition, Orlicz norm, stochastic methods.
The author was partially supported by an NSF grant.
1
2 STEPHEN MONTGOMERY-SMITH
Another description is given by the so called magnetization variable
[4],[16]. Letm = m(t) = m(x,t)beavectorfieldsatisfyinganequation
∂m
= ∆m u m m ( u)T, m(0) = u + q
0 0
∂t − ·∇ − · ∇ ∇
for some scalar field q = q (x). (Here the superscript T denotes the
0 0
transpose.) Then under sufficient smoothness assumptions we have
that u is the Leray projection of m.
A famous open problem is to prove regularity of the Navier-Stokes
equation, that is, if the initial data u is in L and is regular (which
0 2
in this paper we define to mean that it is in the Sobolev spaces Wn,q
for some 2 q < and all positive integers n), then the solution u(t)
≤ ∞
is regular for all t 0. Such regularity would also imply uniqueness
≥
of the solution u(t). Currently only the existence of weak solutions is
known. Also, it is known that for each regular u that there exists
0
t > 0 such that u(t) is regular for 0 t t . We refer the reader to
0 0
≤ ≤
[3], [6], [7], [14], [21].
In studying this problem, various conditions that imply regularity
have been obtained. For example, the Prodi-Serrin conditions ([17],
[19]) state that for some 2 p < , 3 < q with 2 + 3 1 that
≤ ∞ ≤ ∞ p q ≤
T
u(t) pdt <
k kq ∞
Z0
for all T > 0. If u is a weak solution to the Navier-Stokes equation
satisfying a Prodi-Serrin condition, with regular initial data u , then
0
u is regular (see [20]). (Recently Escauriaza, Seregin and Sver´ak [8]
showed that the condition when q = 3 and p = is also sufficient.)
∞
This is a long way from what is currently known for the so called
Leray-Hopf weak solutions:
T
u(t) pdt <
k kq ∞
Z0
for 2 + 3 3, 2 q 6.
p q ≥ 2 ≤ ≤
Another condition is that of Beale, Kato and Majda [1]. They show
that regularity follows from the condition
T
w(t) dt <
k k∞ ∞
Z0
for all T > 0. (In fact they proved this for the Euler equation, but
the proof works also for the Navier-Stokes equation with only small
modifications.) This was strengthened by Kozono and Taniuchi [12] to
REGULARITY OF NAVIER-STOKES 3
show that regularity follows from the condition
T T
u(t) dt w(t) dt <
k∇ kBMO ≈ k kBMO ∞
Z0 Z0
for all T > 0, where here BMO denotes the space of functions with
bounded mean oscillation.
The purpose of this paper is threefold. First, we would like to pro-
vide some logarithmic improvements to these conditions. Secondly,
we would like to present a stochastic approach to the Navier-Stokes
equation, obtaining our conditions using Feynman-Kac like inequali-
ties. Thirdly, we would like to present a different process for creating
estimates of Foia¸s, Guillop´e and Temam.
To this end, the first result of this paper is the logarithmic improve-
ment to the Prodi-Serrin conditions.
Theorem 1.1. Let 2 < p < , 3 < q < with 2 + 3 = 1. If u is a
∞ ∞ p q
solution to the Navier-Stokes equation satisfying
T u(t) p
k kq dt <
1+log+ u(t) ∞
Z0 k kq
for some T > 0, then u(t) is regular for 0 < t T.
≤
We first present a proof of this result (and indeed of a slightly
stronger result) that uses a standard approach. Then we present a
stochastic approach to the Navier-Stokes equation. This is a kind of
Lagrangian coordinates approach to the Navier-Stokes equation, but
with a probabilistic twist in that we follow the path of each particle
with a stochastic perturbation. A similar approach was adopted by
Busnello, Flandoli and Romito [2].
From this we obtain the following Beale-Kato-Majda type condition.
For 1 q < , define the function on [0, )
≤ ∞ ∞
eλ 1 q
Φ (λ) = − .
q
e 1
(cid:18) − (cid:19)
Define the Φ -Orlicz norm on any space of measurable functions by the
q
formula
f = inf λ > 0 : Φ ( f(x) /λ)dx 1 .
k kΦq q | | ≤
(cid:26) Z (cid:27)
(Thus the triangle inequality is a consequence of the fact that Φ is
q
convex, see [13].)
Theorem 1.2. Let 1 < q < , 3 < r < , and T > 0. Suppose that
∞ ∞
u is a solution to the Navier-Stokes equation satisfying
4 STEPHEN MONTGOMERY-SMITH
(1) for all T (0,T)
0
∈
T
u(t) dt < ,
k∇ kΦq ∞
ZT0
and
(2) either q < 3, or u(t) < for almost every t [0,T].
k kr ∞ ∈
Then u(t) is regular for 0 < t T.
≤
Note that since c for q > q , we may assume without
k·kΦq1 ≤ k·kΦq2 1 2
loss of generality that q > 3/2. Next, if 3/2 < q < 3, since
k·kq ≤
(e 1) ,bytheSobolevinequalityweseethatthesecondhypothesis
− k·kΦq
is automatically satisfied with r = 3q/(3 q). Also, this hypothesis is
−
always satisfied for Leray-Hopf weak solutions with r = 6.
Next we demonstrate how to obtain Theorem 1.1 from Theorem 1.2
using the following result. If u is a solution to the Navier-Stokes equa-
tion, we define the sets
An,q (λ) = t [T ,T ] : nu(t) λ .
T0,T1 { ∈ 0 1 k∇ kq ≥ }
Theorem 1.3. Given 3 < q q , and a non-negative integer
1 2
≤ ≤ ∞
n, there exists constants c ,c ,c > 0 such that if u(t), 0 t T is
1 2 3 2
≤ ≤
a solution to the Navier-Stokes equation, and if 0 T T , then for
1 2
≤ ≤
all r (0,√T T ) we have
2 1
∈ −
An,q2 (c r3/q2−n−1) c A0,q1 (c r3/q1−1) .
| T1+r2,T2 1 | ≤ 2| T1,T2 3 |
Asimilarresultthatonecanobtain(butwedonotprovehere)isthat
forpositiveintegersnwehave An,2 (c r1/2−n) c A1,2 (c r−1/2) .
| T1+r2,T2 1 | ≤ 2| T1,T2 3 |
Corollary 1.4. Under the hypotheses of Theorem 1.3, there exists a
constant c > 0 with the following properties. If Θ(λ) is a positive
increasing function of λ 0, define
≥
∞
κ = min (cλ−2 T )+,T dΘ(λ).
0 1
{ − }
Z0
Then
T1 T1
Θ( nu(s) 1/(1+n−3/q2))ds cκ+c Θ(c u(s) 1/(1−3/q1))ds.
k∇ kq2 ≤ k kq1
ZT0 Z0
Similarly,
T1 T1
Θ( nu(s) 1/(n−1/2))ds cκ+c Θ(c u(s) 2)ds.
k∇ k2 ≤ k∇ k2
ZT0 Z0
REGULARITY OF NAVIER-STOKES 5
Since the Leray-Hopf weak solution to the Navier-Stokes equation
satisfies T u(t) 2dt < , one can quickly recover the results of
0 k∇ k2 ∞
Foia¸s, Guillop´e and Temam [9] that say that T nu(t) 1/(n−1/2)dt <
R 0 k∇ k2
.
∞ R
2. Theorem 1.1
The hypothesis of Theorem 1.1 imply that, given ǫ (0,T), there
∈
exists T (0,ǫ) with u(T ) L . Let T∗ > T be the first point
0 0 q 0
∈ ∈
of non-regularity for u(t). It is well known that in order to show
that T∗ > T, it is sufficient to show an a priori estimate, that is
sup u(t) < . This is because it is then possible to
T0≤t<min{T∗,T}k kq ∞
extend the regularity beyond T∗ if T∗ T. Without loss of gener-
≤
ality, it is sufficient to consider the case T = T∗ (so as to obtain a
contradiction).
Proof of Theorem 1.1. Weallowallconstantstoimplicitlydependupon
p and q. Let us define quantities
v = u u q/2−1,
| |
3 2
∂u
A = u q/2−1 i ,
| | ∂x
i,j=1(cid:18) j(cid:19)
X
2
3 3
∂u
B = u q/2−3u u k
i k
| | ∂x
j!
i,j=1 k=1
X X
Note that
3 2
∂v
v 2 := i A+B,
|∇ | ∂x ≈
i,j=1(cid:18) j(cid:19)
X
3
∂ ∂u
u q−2u i A+B,
i
∂x | | ∂x ≈
j j
i,j=1
X (cid:0) (cid:1)
3 2
∂
u q−2u c u q−2 v 2.
i
∂x | | ≤ | | |∇ |
i,j=1(cid:18) j (cid:19)
X (cid:0) (cid:1)
We start with the Navier-Stokes equation, take the inner product with
u u q−2, and integrate over R3 to obtain
| |
∂
u q−1 u = u q−2u ∆udx u q−2u Ldiv(u u)dx.
k kq ∂tk kq | | · − | | · ⊗
Z Z
6 STEPHEN MONTGOMERY-SMITH
Integrating by parts, we see that
3
∂ ∂u
u q−2u ∆udx = u q−2u i dx v 2,
| | · − ∂x | | i ∂x ≈ −k∇ k2
j j
Z Z i,j=1
X (cid:0) (cid:1)
and
3
∂
u q−2u Ldiv(u u)dx = u q−2u [L(u u)] dx
i j i
| | · ⊗ ∂x | |
j
Z Z i,j=1
X (cid:0) (cid:1)
c u q/2−1 v L(u u)
≤ k| | ksk∇ k2k ⊗ kr
where r = 1+q/2 and s = (2q+4)/(q 2). Now the Leray projection
−
is a bounded operator on L , and hence L(u u) u 2 . Also
r k ⊗ kr ≈ k k2+q
u q/2−1 u q/2−1. Hence
k| | ks ≈ k k2+q
u q−2u Ldiv(u u)dx c u 1+q/2 v = c v 1+2/q v .
| | · ⊗ ≤ k k2+q k∇ k2 k k2+4/qk∇ k2
Z
From the Sobolev and interpolation inequalities
v c 3/(q+2)v c v (q−1)/(q+2) v 3/(q+2),
k k2+4/q ≤ k|∇| k2 ≤ k k2 k∇ k2
and hence
u q−2u Ldiv(u u)dx c v 1−1/q v 1+3/q.
| | · ⊗ ≤ k k2 k∇ k2
Z
NowapplyYoung’sinequality ab ((q 3)a2q/(q−3)+(q+3)b2q/(q+3))/2q
≤ −
for a,b 0, to obtain
≥
u q−2u Ldiv(u u)dx c v 2 +c v 2(q−1)/(q−3),
| | · ⊗ ≤ 1k∇ k2 2k k2
Z
where c may be made as small as required by making c larger. Hence
1 2
∂
u q−1 u c v 2(q−1)/(q−3),
k kq ∂tk kq ≤ k k2
that is,
∂
u c u p+1,
∂tk kq ≤ k kq
and so
∂ c u p
log(1+log+ u ) k kq .
∂t k kq ≤ 1+log+ u
k kq
Integrating, we see that for T t < T
0
≤
T u(s) p
log(1+log+ u(t) ) log(1+log+ u(T ) )+c k kq ds,
k kq ≤ k 0 kq 1+log+ u(s)
ZT0 k kq
which provides a uniform bound for u(t) . (cid:3)
k kq
REGULARITY OF NAVIER-STOKES 7
Remark 2.1. Note that this proof can easily be adapted to show that
a sufficient condition for regularity is that
T u(s) p
k kq ds < ,
Θ( u(s) ) ∞
Z0 k kq
where Θ is any increasing function for which
∞ 1
dx = .
xΘ(x) ∞
Z1
3. A Priori Estimates
This section is devoted to theproof ofTheorem 1.3 andCorollary 1.4
TheproofisverysimilartotheproofScheffer’sTheorem[18]thatstates
that the Hausdorff dimension of the set of t for which the solution u(t)
is not regular is 1/2. The main tool is the following result is due to
Gruji´c and Kukavica [10] (see also [15]).
Theorem 3.1. There exist constants a,c > 0 and a function T :
(0, ) (0, ), with T(λ) as λ 0, with the following proper-
ties∞. If→u ∞L (R3), then th→ere∞is a so→lution u(t) (0 t T( u ))
0 ∈ q ≤ ≤ k 0kq
to the Navier-Stokes equation, with u(0) = u , and u(x,t) is the re-
0
striction of an analytic function u(x+iy,t)+iv(x+iy,t) in the region
x+ iy C3 : y a√t , and u( +iy,t)+iv( +iy,t) c u
{ ∈ | | ≤ } k · · kq ≤ k 0kq
for y a√t.
| | ≤
Proof of Theorem 1.3. First let us show that there exists a constants
c ,c ,c > 0suchthatifu(t),t r2 t t isasolutiontotheNavier-
1 3 4 0 0
− ≤ ≤
Stokes equation, and A0,q1 (c r3/q1−1) < c r2, then nu(t ) <
| t0−r2,t0 3 | 4 k∇ 0 kq2
c r3/q2−n−1.
1
To see this, Let us first consider the case when t = 0 and r = 1. By
0
hypothesis, we see that there exists t [ 1, 1+c ]with u(t) < c .
∈ − − 4 k kq1 3
By Theorem 3.1 and the appropriate Cauchy integrals, if c is small
4
enough, then there exists a constant c > 0 such that nu(0) < c .
7 k∇ kq2 1
Now, by replacing u(x,t) by r−1u(r−1x,r−2(t t )), we can relax the
0
−
restriction r = 1 and t = 0, and we obtain the statement we asserted.
0
Next, given ǫ > 0, it is trivial to find a finite collection t ,...,t in
1 N
A = An,q2 (c r3/q2−n−1) such that the sets [t r2,t ] are disjoint,
T1+r2,T2 1 n − n
but the sets [t r2 ǫ,t + ǫ] cover A. By the above observation,
n n
− −
A0,q1 (c r3/q1−1) c r2.
| t0−r2,t0 3 | ≥ 4
Hence
r2 N
A Nr2 < c−1 A0,q1 (c r3/q1−1) c−1 A0,q1 (c r3/q1−1) .
r2 +2ǫ| | ≤ 4 | tn−r2,tn 3 | ≤ 4 | T1,T2 3 |
n=1
X
8 STEPHEN MONTGOMERY-SMITH
Since ǫ is arbitrary, the result follows. (cid:3)
Proof of Corollary 1.4. We only prove the first inequality. By Theo-
rem 1.3, there exist constants c ,c ,c > 0 such that
1 2 3
T1
Θ( nu(s) 1/(1+n−3/q2))ds
k∇ kq2
ZT0
∞
= s [T ,T ] : nu(s) 1/(1+n−3/q2) > λ dΘ(λ)
|{ ∈ 0 1 k∇ kq2 }|
Z0
∞
c κ+ s [c λ−2,T ] : nu(s) 1/(1+n−3/q2) > λ dΘ(λ)
≤ 1 |{ ∈ 2 1 k∇ kq2 }|
Z0
∞
c κ+c s [0,T ] : u(s) 1/(1−3/q1) > c λ dΘ(λ)
≤ 1 1 |{ ∈ 1 k kq1 3 }|
Z0
T1
= c κ+c Θ(c−1 u(s) 1/(1−3/q1))ds.
1 1 3 k kq1
Z0
(cid:3)
4. A Stochastic Description
Let us give a little motivation. Suppose that we defined ϕ (x) to
t0,t1
be X(t ), where X satisfies the equation
0
dX(t) = u(X(t),t)dt, X(t ) = x,
1
then ϕ would be the “back to coordinates map” that takes a point
t0,t1
at t = t to where it was carried from by the flow of the fluid at time
1
t = t . For the Euler equation, this provides a very effective way to
0
describe the solution, for example, the equation for vorticity can be
rewritten in a Lagrangian form:
t
w(x,t) = w(ϕ (x),0)+ w(ϕ (x),s) u(ϕ (x),s)ds.
0,t s,t s,t
·∇
Z0
Similarly, for the magnetization variable we have
t
m(x,t) = m(ϕ (x),0) m(ϕ (x),s) ( u(ϕ (x),s))T ds.
0,t s,t s,t
− · ∇
Z0
For the Navier-Stokes equation this formula is not true, and the Lapla-
cian term can make things complicated. One approach to dealing with
this is described in the paper by Constantin [5]. However, we take a
different approach using Brownian motion, using a kind of “randomly
perturbed back to coordinates map.” Such a method was already dis-
cussed in the paper [16], here we make the discussion more rigorous.
The author recently found out that a similar approach was followed by
Busnello, Flandoli and Romito in [2].
REGULARITY OF NAVIER-STOKES 9
The hypothesis of Theorem 1.2 imply that, given ǫ (0,T), there
∈
exists t′ (0,ǫ) with u(t′) L . Then by known results (for example
r
∈ ∈
Theorem 3.1), it follows that there exists 0 < T < ǫ such that u(T )
0 0
Wn,r′ for all r′ [r, ] and positive integers n. Furthermore, arguin∈g
∈ ∞
as in Section 2, we only need to prove sup u(t) <
T0≤t<min{T∗,T}k kr ∞
underthea priori assumptionthatthesolutionisregularfort [T ,T].
0
If f: R3 R is regular, and T t t < T, define A ∈f(x) =
→ 0 ≤ 0 ≤ 1 t0,t1
α(x,t ), where α satisfies the transport equation
1
∂α
= ∆α u α, α(x,t ) = f(x).
0
∂t − ·∇
Since div(u) = 0, an easy integration by parts argument shows that
∂
α(x,t)dx = 0,
∂t
Z
and hence if f is also in L , then
1
A f(x)dx = f(x)dx.
t0,t1
Z Z
Since stochastic differential equations traditionally move forwards in
time, it will be convenient to consider a time reversed equation. Let
b(t) be three dimensional Brownian motion. For T t t < T ,
0 0 1 1
define the random function ϕ : R3 R3 by ϕ ≤(x) ≤= X( t ),
t0,t1 → t0,t1 − 0
where X satisfies the stochastic differential equation:
dX(t) = u(X(t),t)dt+√2db(t), X( t ) = x.
1
− −
It follows by the Ito Calculus [11] that if T t t < T, then
0 0 1
≤ ≤
A f(x) = Ef(ϕ (x)).
t0,t1 t0,t1
(Here as in the rest of the paper, E denotes expected value.) Note that
if f is also in L , then
1
Ef(ϕ (x))dx = f(x)dx.
t0,t1
Z Z
Applying the usual dominated and monotone convergence theorems, it
quickly follows that the last equality is also true if f is any function in
L , or if f is any positive function.
1
Now let us develop the equations for the magnetization variable.
(The same approach will also work for the vorticity.) If we set m(T ) =
0
u(T ), then we note that m is the unique solution to the integral equa-
0
tion
t
m(t) = A u(T ) A (m(s) ( u(s))T)ds (T t < T).
T0,t 0 − s,t · ∇ 0 ≤
ZT0
10 STEPHEN MONTGOMERY-SMITH
Uniqueness followsquickly bytheusualfixedpointargumentover short
intervals, remembering that u(t) is regular for T t < T.
0
≤
Consider also the random quantity m˜ = m˜(x,t) as the solution to
the integral equation for T t < T
0
≤
t
m˜(x,t) = u(ϕ (x),T ) m˜(ϕ (x),s) ( u(ϕ (x),s))T ds.
T0,t 0 − s,t · ∇ s,t
ZT0
Again, it is very easy to show that a solution exists by using a fixed
point argument over short time intervals. It is seen that Em˜ satisfies
the same equation as m, and hence Em˜ = m.
Next, ϕ (ϕ (x)) = ϕ (x), since both are Y(t ) where Y(t) is
t0,t1 t1,t2 t0,t2 0
the solution to the integral equation
t
Y(t) = ϕ (x)+ u(Y(s),s)ds+√2(b b ).
t1,t2 −t − −t1
Zt1
Hence
s2
m˜(ϕ (x),s ) m˜(ϕ (x),s ) = m˜(ϕ (x),s) ( u(ϕ (x),s))T ds.
s1,t 1 − s2,t 2 s,t · ∇ s,t
Zs1
Thus, by Gronwall’s inequality, if T t < T
0
≤
t
m˜(x,t) exp u(ϕ (x),s) ds u(ϕ (x),T ) .
| | ≤ |∇ s,t | | T0,t 0 |
(cid:18)ZT0 (cid:19)
(This is essentially the Feynman-Kac formula.) The goal, then, is to
find uniform estimates on the quantity
t
exp u(ϕ (x),s) ds .
s,t
|∇ |
(cid:18)ZT0 (cid:19)
This we proceed to do in the next section.
5. Theorem 1.2
Let us fix q and r satisfying the hypothesis of Theorem 1.2, and
allow all constants to implicitly depend upon q and r. We retain the
notation from the previous section, in particular the definitions of T ,
0
T∗ and T.
Proof of Theorem 1.2. Since u(t) < for almost every t [0,T],
k kr ∞ ∈
by Theorem 1.3, we see that u(t) < for almost every t [0,T].
k∇ k∞ ∞ ∈
Hence, there exists λ > T−1 such that
0
1
u(t) dt ,
k∇ kΦq ≤ q
ZB