EXISTENCE AND SMOOTHNESS OF THENAVIER–STOKES EQUATION
CHARLES L. FEFFERMAN
The Euler and Navier–Stokes equations describe the motion of a ﬂuid in
R
n
(
n
= 2 or 3). These equations are to be solved for an unknown velocity vector
u
(
x,t
) = (
u
i
(
x,t
))
1
≤
i
≤
n
∈
R
n
and pressure
p
(
x,t
)
∈
R
, deﬁned for position
x
∈
R
n
and time
t
≥
0. We restrict attention here to incompressible ﬂuids ﬁlling all of
R
n
.The
Navier–Stokes
equations are then given by
∂ ∂tu
i
+
n
j
=1
u
j
∂u
i
∂x
j
=
ν
∆
u
i
−
∂p∂x
i
+
f
i
(
x,t
) (
x
∈
R
n
,t
≥
0)
,
(1)div
u
=
n
i
=1
∂u
i
∂x
i
= 0 (
x
∈
R
n
,t
≥
0)(2)with initial conditions(3)
u
(
x,
0) =
u
◦
(
x
) (
x
∈
R
n
)
.
Here,
u
◦
(
x
) is a given,
C
∞
divergencefree vector ﬁeld on
R
n
,f
i
(
x,t
) are the components of a given, externally applied force (e.g. gravity),
ν
is a positive coeﬃcient(the viscosity), and ∆ =
n
i
=1
∂
2
∂x
2
i
is the Laplacian in the space variables. The
Euler equations
are equations (1), (2), (3) with
ν
set equal to zero.Equation (1) is just Newton’s law
f
=
ma
for a ﬂuid element subject to the external force
f
= (
f
i
(
x,t
))
1
≤
i
≤
n
and to the forces arising from pressure and friction.Equation (2) just says that the ﬂuid is incompressible. For physically reasonablesolutions, we want to make sure
u
(
x,t
) does not grow large as

x
→∞
. Hence, wewill restrict attention to forces
f
and initial conditions
u
◦
that satisfy(4)

∂
αx
u
◦
(
x
)
≤
C
αK
(1 +

x

)
−
K
on
R
n
,
for any
α
and
K
and(5)

∂
αx
∂
mt
f
(
x,t
)
≤
C
αmK
(1 +

x

+
t
)
−
K
on
R
n
×
[0
,
∞
)
,
for any
α,m,K.
We accept a solution of (1), (2), (3) as physically reasonable only if it satisﬁes(6)
p,u
∈
C
∞
(
R
n
×
[0
,
∞
))and(7)
R
n

u
(
x,t
)

2
dx < C
for all
t
≥
0 (bounded energy)
.
Alternatively, to rule out problems at inﬁnity, we may look for spatially periodicsolutions of (1), (2), (3). Thus, we assume that
u
◦
(
x
)
,f
(
x,t
) satisfy(8)
u
◦
(
x
+
e
j
) =
u
◦
(
x
)
, f
(
x
+
e
j
,t
) =
f
(
x,t
) for 1
≤
j
≤
n
1
2 CHARLES L. FEFFERMAN
(
e
j
=
j
th
unit vector in
R
n
).In place of (4) and (5), we assume that
u
◦
is smooth and that(9)

∂
αx
∂
mt
f
(
x,t
)
≤
C
αmK
(1 +

t

)
−
K
on
R
3
×
[0
,
∞
)
,
for any
α,m,K.
We then accept a solution of (1), (2), (3) as physically reasonable if it satisﬁes(10)
u
(
x,t
) =
u
(
x
+
e
j
,t
) on
R
3
×
[0
,
∞
) for 1
≤
j
≤
n
and(11)
p,u
∈
C
∞
(
R
n
×
[0
,
∞
))
.
A fundamental problem in analysis is to decide whether such smooth, physicallyreasonable solutions exist for the Navier–Stokes equations. To give reasonable leeway to solvers while retaining the heart of the problem, we ask for a proof of oneof the following four statements.
(A) Existence and smoothness of Navier–Stokes solutions on
R
3
.
Take
ν >
0 and
n
= 3. Let
u
◦
(
x
) be any smooth, divergencefree vector ﬁeld satisfying (4).Take
f
(
x,t
) to be identically zero. Then there exist smooth functions
p
(
x,t
)
,u
i
(
x,t
)on
R
3
×
[0
,
∞
) that satisfy (1), (2), (3), (6), (7).
(B) Existence and smoothness of Navier–Stokes solutions in
R
3
/
Z
3
.
Take
ν >
0 and
n
= 3. Let
u
◦
(
x
) be any smooth, divergencefree vector ﬁeld satisfying(8); we take
f
(
x,t
) to be identically zero. Then there exist smooth functions
p
(
x,t
),
u
i
(
x,t
) on
R
3
×
[0
,
∞
) that satisfy (1), (2), (3), (10), (11).
(C) Breakdown of Navier–Stokes solutions on
R
3
.
Take
ν >
0 and
n
= 3.Then there exist a smooth, divergencefree vector ﬁeld
u
◦
(
x
) on
R
3
and a smooth
f
(
x,t
) on
R
3
×
[0
,
∞
), satisfying (4), (5), for which there exist no solutions (
p,u
)of (1), (2), (3), (6), (7) on
R
3
×
[0
,
∞
).
(D) Breakdown of Navier–Stokes Solutions on
R
3
/
Z
3
.
Take
ν >
0 and
n
= 3. Then there exist a smooth, divergencefree vector ﬁeld
u
◦
(
x
) on
R
3
and asmooth
f
(
x,t
) on
R
3
×
[0
,
∞
), satisfying (8), (9), for which there exist no solutions(
p,u
) of (1), (2), (3), (10), (11) on
R
3
×
[0
,
∞
).These problems are also open and very important for the Euler equations (
ν
= 0),although the Euler equation is not on the Clay Institute’s list of prize problems.Let me sketch the main partial results known regarding the Euler and Navier–Stokes equations, and conclude with a few remarks on the importance of the question.In two dimensions, the analogues of assertions (A) and (B) have been knownfor a long time (Ladyzhenskaya [4]), also for the more diﬃcult case of the Eulerequations. This gives no hint about the threedimensional case, since the maindiﬃculties are absent in two dimensions. In three dimensions, it is known that (A)and (B) hold provided the initial velocity
u
◦
satisﬁes a smallness condition. Forinitial data
u
◦
(
x
) not assumed to be small, it is known that (A) and (B) hold (alsofor
ν
= 0) if the time interval [0
,
∞
) is replaced by a small time interval [0
,T
),with
T
depending on the initial data. For a given initial
u
◦
(
x
), the maximumallowable
T
is called the “blowup time.” Either (A) and (B) hold, or else there isa smooth, divergencefree
u
◦
(
x
) for which (1), (2), (3) have a solution with a ﬁniteblowup time. For the Navier–Stokes equations (
ν >
0), if there is a solution with
EXISTENCE AND SMOOTHNESS OF THE NAVIER–STOKES EQUATION 3
a ﬁnite blowup time
T
, then the velocity (
u
i
(
x,t
))
1
≤
i
≤
3
becomes unbounded nearthe blowup time.Other unpleasant things are known to happen at the blowup time
T
, if
T <
∞
.For the Euler equations (
ν
= 0), if there is a solution (with
f
≡
0, say) with ﬁniteblowup time
T
, then the vorticity
ω
(
x,t
) = curl
x
u
(
x,t
) satisﬁes
T
0
sup
x
∈
R
3

ω
(
x,t
)

dt
=
∞
(Beale–Kato–Majda)
,
so that the vorticity blows up rapidly.Many numerical computations appear to exhibit blowup for solutions of theEuler equations, but the extreme numerical instability of the equations makes itvery hard to draw reliable conclusions.The above results are covered very well in the book of Bertozzi and Majda [1].Starting with Leray [5], important progress has been made in understanding
weak solutions
of the Navier–Stokes equations. To arrive at the idea of a weak solution of a PDE, one integrates the equation against a test function, and then integrates byparts (formally) to make the derivatives fall on the test function. For instance, if (1)and (2) hold, then, for any smooth vector ﬁeld
θ
(
x,t
) = (
θ
i
(
x,t
))
1
≤
i
≤
n
compactlysupported in
R
3
×
(0
,
∞
), a formal integration by parts yields(12)
R
3
×
R
u
·
∂θ∂tdxdt
−
ij
R
3
×
R
u
i
u
j
∂θ
i
∂x
j
dxdt
=
ν
R
3
×
R
u
·
∆
θdxdt
+
R
3
×
R
f
·
θdxdt
−
R
3
×
R
p
·
(div
θ
)
dxdt.
Note that (12) makes sense for
u
∈
L
2
,
f
∈
L
1
,
p
∈
L
1
, whereas (1) makes senseonly if
u
(
x,t
) is twice diﬀerentiable in
x
. Similarly, if
ϕ
(
x,t
) is a smooth function,compactly supported in
R
3
×
(0
,
∞
), then a formal integration by parts and (2)imply(13)
R
3
×
R
u
·
x
ϕdxdt
= 0
.
A solution of (12), (13) is called a
weak solution
of the Navier–Stokes equations.A longestablished idea in analysis is to prove existence and regularity of solutionsof a PDE by ﬁrst constructing a weak solution, then showing that any weak solutionis smooth. This program has been tried for Navier–Stokes with partial success.Leray in [5] showed that the Navier–Stokes equations (1), (2), (3) in three spacedimensions always have a weak solution (
p,u
) with suitable growth properties.Uniqueness of weak solutions of the Navier–Stokes equation is
not
known. For theEuler equation, uniqueness of weak solutions is strikingly false. Scheﬀer [8], and,later, Schnirelman [9] exhibited weak solutions of the Euler equations on
R
2
×
R
with compact support in spacetime. This corresponds to a ﬂuid that starts fromrest at time
t
= 0, begins to move at time
t
= 1 with no outside stimulus, andreturns to rest at time
t
= 2, with its motion always conﬁned to a ball
B
⊂
R
3
.Scheﬀer [7] applied ideas from geometric measure theory to prove a partialregularity theorem for suitable weak solutions of the Navier–Stokes equations.
4 CHARLES L. FEFFERMAN
Caﬀarelli–Kohn–Nirenberg [2] improved Scheﬀer’s results, and F.H. Lin [6] simpliﬁed the proofs of the results in Caﬀarelli–Kohn–Nirenberg [2]. The partial regularity theorem of [2], [6] concerns a parabolic analogue of the Hausdorﬀ dimensionof the singular set of a suitable weak solution of Navier–Stokes. Here, the
singular set
of a weak solution
u
consists of all points (
x
◦
,t
◦
)
∈
R
3
×
R
such that
u
is unbounded in every neighborhood of (
x
◦
,t
◦
). (If the force
f
is smooth, and if (
x
◦
,t
◦
) doesn’t belong to the singular set, then it’s not hard to show that
u
can becorrected on a set of measure zero to become smooth in a neighborhood of (
x
◦
,t
◦
).)To deﬁne the parabolic analogue of Hausdorﬀ dimension, we use
parabolic cylinders
Q
r
=
B
r
×
I
r
⊂
R
3
×
R
, where
B
r
⊂
R
3
is a ball of radius
r
, and
I
r
⊂
R
is aninterval of length
r
2
. Given
E
⊂
R
3
×
R
and
δ >
0, we set
P
K,δ
(
E
) = inf
∞
i
=1
r
Ki
:
Q
r
1
,Q
r
2
,
···
cover
E,
and each
r
i
< δ
and then deﬁne
P
K
(
E
) = lim
δ
→
0+
P
K,δ
(
E
)
.
The main results of [2], [6] may be stated roughly as follows.
Theorem.
(A) Let
u
be a weak solution of the Navier–Stokes equations, satisfying suitable growth conditions. Let
E
be the singular set of
u
. Then
P
1
(
E
) = 0
.(B) Given a divergencefree vector ﬁeld
u
◦
(
x
)
and a force
f
(
x,t
)
satisfying (4)and (5), there exists a weak solution of Navier–Stokes (1), (2), (3) satisfying thegrowth conditions in (A).
In particular, the singular set of
u
cannot contain a spacetime curve of the form
{
(
x,t
)
∈
R
3
×
R
:
x
=
φ
(
t
)
}
. This is the best partial regularity theorem known sofar for the Navier–Stokes equation. It appears to be very hard to go further.Let me end with a few words about the signiﬁcance of the problems posed here.Fluids are important and hard to understand. There are many fascinating problems and conjectures about the behavior of solutions of the Euler and Navier–Stokesequations. (See, for instance, Bertozzi–Majda [1] or Constantin [3].) Since we don’teven know whether these solutions exist, our understanding is at a very primitivelevel. Standard methods from PDE appear inadequate to settle the problem. Instead, we probably need some deep, new ideas.
References
[1] A. Bertozzi and A. Majda,
Vorticity and Incompressible Flows
, Cambridge U. Press, Cambridge, 2002.[2] L. Caﬀarelli, R. Kohn, and L. Nirenberg,
Partial regularity of suitable weak solutions of theNavier–Stokes equations
, Comm. Pure & Appl. Math.
35
(1982), 771–831.[3] P. Constantin,
Some open problems and research directions in the mathematical study of ﬂuid dynamics
, in Mathematics Unlimited–2001 and Beyond, Springer Verlag, Berlin, 2001,353–360.[4] O. Ladyzhenskaya,
The Mathematical Theory of Viscous Incompressible Flows
(2nd edition),Gordon and Breach, New York, 1969.[5] J. Leray,
Sur le mouvement d’un liquide visquex emplissent l’espace
, Acta Math. J.
63
(1934),193–248.[6] F.H. Lin,
A new proof of the Caﬀarelli–Kohn–Nirenberg theorem
, Comm. Pure. & Appl.Math.
51
(1998), 241–257.[7] V. Scheﬀer,
Turbulence and Hausdorﬀ dimension
, in Turbulence and the Navier–Stokes Equations, Lecture Notes in Math.
565
, Springer Verlag, Berlin, 1976, 94–112.