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628 R. Schnaubelt and M. Veraar

[12] G. Da Prato and J. Zabczyk, Evolution equations with white-noise boundary

conditions, Stochastics Stochastics Rep. 42 (1993), no. 3-4, 167–182.

[13] G. Da Prato and J. Zabczyk, Ergodicity for inﬁnite-dimensional systems, London

Mathematical Society Lecture Note Series, vol. 229, Cambridge University Press,

Cambridge, 1996.

[14] A. Debussche, M. Fuhrman, and G. Tessitore, Optimal control of a stochastic

heat equation with boundary-noise and boundary-control, ESAIM Control Optim.

Calc. Var. 13 (2007), no. 1, 178–205 (electronic).

[15] R.Denk,G.Dore,M.Hieber,J.Pr

uss, and A. Venni, New thoughts on old

resultsofR.T.Seeley, Math. Ann. 328 (2004), no. 4, 545–583.

[16] D. Di Giorgio, A. Lunardi, and R. Schnaubelt, Optimal regularity and Fred-

holm properties of abstract parabolic operators in L

spaces on the real line,Proc.

London Math. Soc. (3) 91 (2005), no. 3, 703–737.

[17] J. Diestel, H. Jarchow, and A. Tonge, Absolutely summing operators,Cam-

bridge Studies in Advanced Mathematics, vol. 43, Cambridge University Press, Cam-

bridge, 1995.

[18] S. Kwapie

n and W.A. Woyczy

nski, Random series and stochastic integrals: single

and multiple, Probability and its Applications, Birkh¨auser Boston Inc., Boston, MA,

1992.

[19] L. Maniar and R. Schnaubelt, The Fredholm alternative for parabolic evolution

equations with inhomogeneous boundary conditions,J.DiﬀerentialEquations235

(2007), no. 1, 308–339.

[20] L. Maniar and R. Schnaubelt, Robustness of Fredholm properties of parabolic

evolution equations under boundary perturbations,J.Lond.Math.Soc.(2)77 (2008),

no. 3, 558–580.

[21] B. Maslowski, Stability of semilinear equations with boundary and pointwise noise,

Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 22 (1995), no. 1, 55–93.

[22] J.M.A.M. van Neerven, M.C. Veraar, and L.W. Weis, Stochastic integration

in UMD Banach spaces, Ann. Probab. 35 (2007), no. 4, 1438–1478.

[23] J.M.A.M. van Neerven, M.C. Veraar, and L.W. Weis, Stochastic evolution

equations in UMD Banach spaces, J. Functional Anal. 255 (2008), 940–993.

[24] G. Pisier, Martingales with values in uniformly convex spaces, Israel J. Math. 20

(1975), no. 3-4, 326–350.

[25] R. Schaubelt and M.C. Veraar, Structurally damped plate and wave equations

with random point force in arbitrary space dimensions, Diﬀerential Integral Equations

23 (2010), 957–988.

[26] R. Schnaubelt, Asymptotic behaviour of parabolic nonautonomous evolution equa-

tions, Functional analytic methods for evolution equations, Lecture Notes in Math.,

vol. 1855, Springer, Berlin, 2004, pp. 401–472.

[27] R. Seeley, Interpolation in L

with boundary conditions, Studia Math. 44 (1972),

47–60.

[28] R.B. Sowers, Multidimensional reaction-diﬀusion equations with white noise bound-

ary perturbations, Ann. Probab. 22 (1994), no. 4, 2071–2121.

Stochastic Equations with Boundary Noise 629

[29] H. Triebel, Interpolation theory, function spaces, diﬀerential operators,seconded.,

Johann Ambrosius Barth, Heidelberg, 1995.

[30] M.C. Veraar, Non-autonomous stochastic evolution equations and applications to

stochastic partial diﬀerential equations,J.Evol.Equ.10 (2010), 85–127.

[31] A. Yagi, Parabolic evolution equations in which the coeﬃcients are the generators

of inﬁnitely diﬀerentiable semigroups. II,Funkcial.Ekvac.33 (1990), no. 1, 139–150.

[32] A. Yagi, Abstract quasilinear evolution equations of parabolic type in Banach spaces,

Boll. Un. Mat. Ital. B (7) 5 (1991), no. 2, 341–368.

Roland Schnaubelt

Institute of Analysis

Department of Mathematics

Karlsruhe Institute of Technology (KIT)

D-76128 Karlsruhe, Germany

e-mail: roland.schnaubelt@kit.edu

Mark Veraar

Delft Institute of Applied Mathematics

Delft University of Technology

P.O. Box 5031

NL-2600 GA Delft, The Netherlands

e-mail: m.c.veraar@tudelft.nl

mark@profsonline.nl

Progress in Nonlinear Diﬀerential Equations

and Their Applications, Vol. 80, 631–645

 2011 Springer Basel AG

A Note on Necessary Conditions

for Blow-up of Energy Solutions

to the Navier-Stokes Equations

Gregory Seregin

Dedicated to Herbert Amann

Abstract. In the present note, we address the question about behavior of L

norm of the velocity ﬁeld as time t approaches blow-up time T .Itisknown

that the upper limit of the above norm must be equal to inﬁnity. We show

that, for blow-ups of type I, the lower limit of L

-norm equals to inﬁnity as

well.

Mathematics Subject Classiﬁcation (2000). 35K, 76D.

Keywords. Navier-Stokes equations, Cauchy problem, weak Leray-Hopf solu-

tions, local energy solutions, backward uniqueness.

1. Motivation

Consider the Cauchy problem for the classical 3D-Navier-Stokes system

∂

v + v ·∇v −ν Δ v = −∇q

div v =0



in Q

, (1.1)

t=0

= a ∈ C

∞

0,0

). (1.2)

Here, v and q stand for the velocity ﬁeld and for the pressure ﬁeld, respectively,

= R

×]0, +∞[, and

∞

0,0

)={a ∈ C

∞

): diva =0inR

In what follows, we always assume that ν =1.

It is well known due to J. Leray, see [5], the Cauchy problem (1.1), (1.2)

has at least one solution called the weak Leray-Hopf solution. To give its modern

deﬁnition, let us introduce standard energy spaces H and V . H is the closure of

632 G. Seregin

the set C

∞

0,0

)inL

)andV is the closure of the same set with respect to

the norm generated by the Dirichlet integral.

Deﬁnition 1.1. A velocity ﬁeld v ∈ L

∞

(0, +∞; H) ∩L

(0, +∞; V ) is called a weak

Leray-Hopf solution to the Cauchy problem (1.1), (1.2) if the following conditions

hold:



(v · ∂

w + v ⊗ v : ∇w −∇v : ∇w) dxdt =0 (1.3)

for any w ∈ C

∞

) with div w =0inQ

; the function

t →



v(x, t) · w(x)dx (1.4)

is continuous on [0, +∞[ for all w ∈ L

);

v(·,t) −a(·)

→ 0 (1.5)

as t → +0;

v(·,t)



|∇v|

dxdt



≤a

(1.6)

for all t ∈ [0, +∞[.

The deﬁnition does not contain the pressure ﬁeld at all. However, using the

linear theory, we can introduce so-called associated pressure q(·,t), which, for all

t>0, is the Newtonian potential of v

i,j

(·,t)v

j,i

(·,t) and satisﬁes the pressure

equation

−Δ q(·,t)=v

i,j

(·,t)v

j,i

(·,t)=divdivv(·,t) ⊗ v(·,t) (1.7)

in R

.Sincev is known to belong L

), pressure q is in L

). Moreover,

the Navier-Stokes system is satisﬁed in the sense of distributions and even a.e. in

. We refer to the paper [3] for details.

Uniqueness of weak Leray-Hopf solutions is still unknown. However, there is

a simple but deep connection between smoothness and uniqueness. It has been

pointed out by J. Leray in his celebrated paper [5] and reads: any smooth solution

to (1.1), (1.2) is unique in the class of weak Leray-Hopf solutions. The problem of

smoothness of weak Leray-Hopf solutions is actually one of the seven Millennium

problems.

In the paper, we deal with certain necessary conditions for possible blow-ups

of solutions to the Cauchy problem (1.1), (1.2). Suppose that T>0 is the ﬁrst

moment of time when singularities occur. Then, as it has been shown by J. Leray,

given 3 <s≤ +∞, there exists a constant c

such that

v(·,t)

s,R

≥

(T − t)

s−3

(1.8)

for T/2 ≤ t<T.

Blowups for Navier-Stokes Equations 633

However, in the marginal case s =3,wehaveaweakerresult

lim sup

t→T −0

v(·,t)

=+∞, (1.9)

which has been established in [2]. Apparently, a natural question can be raised

whether the statement

lim

t→T −0

v(·,t)

=+∞ (1.10)

is true or not. In [9], there has been proved a weaker version of (1.10), namely,

lim

t→T −0

T − t



v(·,τ)

dτ =+∞. (1.11)

The aim of the present paper is to show validity of (1.10) provided the blow-

up of type I takes place, i.e.,

v(·,t)

∞

≤

∞

√

T − t

(1.12)

for any T/2 ≤ t<T and for some positive constant C

∞

. Our main result can be

formulated as follows.

Theorem 1.1. Let T beablow-uptimeandlet,forsome3 <s≤ +∞,thereexist

a positive constant C

such that

v(·,t)

≤

(T − t)

s−3

(1.13)

for any T/2 ≤ t<T.Then(1.10) holds.

Let us outline our proof of Theorem 1.1. Firstly, we reduce the general case

toaparticularones =+∞ showing that if (1.13) is true for some 3 <s<+∞,

then it is true for s =+∞ as well. Secondly, assuming that (1.10) is violated, i.e.,

a sequence t

tending to T exists such that

sup

v(·,t

)

= M<+∞, (1.14)

we may use a blow-up machinery and construct a non-trivial ancient solution

deﬁned in R

×] −∞, 0[ with the following properties. It vanishes at time t =0

and its L

-norm is ﬁnite say at time t = −1. In order to apply backward uniqueness

results, proved in [2], we need to check that the above ancient solution has a certain

behavior at inﬁnity with respect to spatial variables. This can be done with the

help of the conception of so-called local energy solutions to the Cauchy problem,

see [4] and also [7].

Finally, we would like to make the following remark regarding to condition

(1.13). It is interesting to ﬁgure out whether condition (1.14) itself implies regu-

larity. It is worthy to note that the important consequence of (1.14) is that

v(·,T) ∈ L

). (1.15)

634 G. Seregin

Then any reasonable blow up procedure at a singular point gives an ancient solu-

tion vanishing at time t = 0 and a hope to apply backward uniqueness. Actually,

condition (1.13) is one of others that ensure the existence of an ancient solu-

tion with such a property. In principle, we could use the universal scaling as in

[10], which provides the existence of the limit of scaled functions with no addi-

tional assumption. But in such a case, we would loose possibility to use backward

uniqueness results because it is unknown whether or not the corresponding an-

cient solution is zero at t = 0. This also raises the interesting question why we are

still not able to prove smoothness of L

3,∞

-solution with the help of the universal

scaling.

2. Some auxiliary things

In the paper, we are going to use the following notion. B(x

,R) stands for a spatial

ball centered at a point x

and having radius R, B(R)=B(0,R), and B = B(1).

By Q(z

,R), where z

=(x

) is a space-time point, we denote a parabolic ball

B(x

,R)×]t

− R

[, and Q(R)=Q(0,R), Q = Q(1). All constants depending

on non-essential parameters will be denoted simply by c.

Lemma 2.1. Suppose that (1.13) holds for some 3 <s<+∞. Then it is true for

s =+∞.

Proof. From (1.7) and (1.13) it follows that q(·,t) ∈ L

)forT/2 <t<T.

Fix ε>0andz

=(x

)witht

<Tarbitrarily. Applying (1.13) and

H¨older inequality, we ﬁnd



Q(z

,R)

(|v|

+ |q|

)dz ≤ c(s)





Q(z

,R)

(|v|

+ |q|

)dz



5(1−

)

≤ c(s)R

5(1−

)−2





−R

(T − t)

s−3



≤ c(s)C

3−

(T − t

)

s−3

≤ c(s)C



√

T − t



s−3

We let R =



γ(T − t

) and pick up 0 <γ<1sothatc(s)C

s−3

≤ ε/2. Now,

we apply the local regularity theory for suitable weak solutions to the Navier-

Stokes equations, developed in [1], [6], [3], and [2]. It reads that if ε ≤ ε

,where

is a universal constant, then

|v(z

)|≤



γ(T − t

)

for all z

=(x

)withx

∈ R

and T/2 <t

<Tand for some universal

constant c. 

So, we need prove Theorem 1.1 in a particular case s =+∞ only.

Blowups for Navier-Stokes Equations 635

3. Ancient solution

By assumptions of Theorem 1.1, there must be singular points at t = T .Wetake

any of them say x

∈ R

. Then local regularity theory gives the following inequality



Q((x

,T ),R)

(|v|

+ |q|

)dz ≥ ε

> 0 (3.1)

for 0 <R<R

min{1,

√

T } with universal constant ε

. Without loss of

generality, we may assume that x

=0.

Proceeding in the same way as in [10], we can ﬁnd that condition (1.13)

implies the following bound

sup

0<R≤R





Q((0,T ),R)

(|v|

+ |q|

)dz +



Q((0,T ),R)

|∇v|

sup

T −R

<t<T



B(R)

|v(x, t)|



= M

< +∞. (3.2)

Next, we may scale our functions v and q essentially in the same way as it

has been done in [9], namely,

(k)

(y,s)=R

v(R

y, T + R

s),p

(k)

(y,s)=R

q(R

y, T + R

for y ∈ B(R

)andfors ∈] −(R

)

, 0[, where R

√

T − t

Now, let us see what happens if k → +∞. This is more or less well-understood

procedure and the reader can ﬁnd details in [2], [9]–[11]. As a result, we have two

measurable functions u and p deﬁned on Q

−

= R

×] −∞, 0[ with the following

properties:

(k)

→ u in L

(Q(a)),

∇u

(k)

 ∇u in L

(Q(a)),

(k)

p in L

(Q(a)),

(k)

→ u in C([−a

, 0]; L

(B(a)))

(3.3)

for any a>0. The pair u and p satisﬁes the Navier-Stokes equations in Q

−

in the

sense distributions. We call it an ancient solution to the Navier-Stokes equations.

Moreover, since inequalities (1.12) and (3.2) are invariant with respect to the

Navier-Stokes scaling, we can show that

sup

0<a<+∞





Q(a)

(|u|

+ |p|

)de +



Q(a)

|∇u|

sup

−a

<s<0



B(a)

|u(y, s)|



≤ M

< +∞ (3.4)

636 G. Seregin

and

|u(y, s)|≤

∞

√

−s

(3.5)

for all e =(y, s) ∈ Q

−

The important consequence of (1.15) and the last line in (3.3), is the following

fact

u(·, 0) = 0 (3.6)

in R

, see [9] in a similar situation.

Now, our goal is to show that the above ancient solution is non-trivial. Un-

fortunately, we cannot get this by direct passing to the limit in the formula



Q(a)

(|u

(k)

+ |p

(k)

)de =



Q(aR

)

(|v|

+ |q|

)dz ≥ ε

> 0 (3.7)

for aR

< 3/4. The reason is simple: there is no hope to prove strong convergence

of the pressure. However, we still have local strong convergence of u

(k)

so that



Q(a)

(k)

de →



Q(a)

|u|

de (3.8)

for any 0 <a≤ 3/4.

To prove that our ancient solution is non-trivial, let us ﬁrst note that accord-

ing to (3.2)



Q(a)

(|u

(k)

+ |p

(k)

)de ≤ M

(3.9)

for suﬃcient large k and for all a ∈]0, 3/4].

The second observation is quite typical when treating the pressure. In the

ball B(3/4), the pressure can be split into two parts

(k)

= p

(k)

+ p

(k)

where the ﬁrst term is deﬁned by the variational identity



B(3/4)

(k)

(y,s)Δ ϕ(y)dy = −



B(3/4)

(k)

(y,s) ⊗u

(k)

(y,s):∇

ϕ(y)dy

being valid for any ϕ ∈ W

(B(3/4)) with ϕ =0on∂B(3/4). It is not diﬃcult to

show that the ﬁrst counter-part of the pressure satisﬁes the estimate

p

(k)

(·,s)

,B(3/4)

≤ cu

(k)

(·,s)

3,B(3/4)

(3.10)

Blowups for Navier-Stokes Equations 637

for all −∞ <s<0 while the second one is a harmonic function in B(3/4) for the

same s.Sincep

(k)

(·,s) is harmonic, we have

sup

y∈B(1/2)

(k)

(y, s)|

≤ c



B(3/4)

(k)

(y, s)|

dy (3.11)

≤ c



B(3/4)

(k)

(y, s)|

dy + c



B(3/4)

(k)

(y, s)|

dy.

Then, for any 0 <a<1/2,

≤



Q(a)

(|u

(k)

+ |p

(k)

)de ≤ c



Q(a)

(|u

(k)

+ |p

(k)

+ |p

(k)

)de

≤ c



Q(a)

(|u

(k)

+ |p

(k)

)de + ca



−a

sup

y∈B(1/2)

(k)

(y, s)|

ds.

Combining (3.9)–(3.11), we ﬁnd

≤ c



Q(3/4)

(k)

de + ca



−a



B(3/4)



(k)

(y,s)|

+ |u

(k)

(y,s)|



≤ c



Q(3/4)

(k)

de + ca



Q(3/4)



(k)

+ |u

(k)



≤ c



Q(3/4)

(k)

de + cM

for the same a. Passing to the limit and choosing suﬃciently small a, we show that

0 <cε

≤



Q(3/4)

|u|

de (3.12)

for some positive 0 <a<1/2. So, our ancient solution u is non-trivial.

If would show that for some positive R

∗

|u| + |∇u|∈L

∞

((R

\ B(R

∗

))×] − (5/6)

, 0[),

we could use arguments from [2] and conclude that, by (3.6), ∇∧u ≡ 0inR

×] −

(3/4)

, 0[ which, together with the incompressibility condition, means that u(·,t)

is harmonic in R

. And it is bounded there. So, u must be a function of t only. But

estimate (3.4) says that such a function must be zero in ] − (3/4)

, 0[. The latter

contradicts (3.12).