similar solutions and backward uniqueness
Julio Andr´
es Montero Rosero
Universidad de los Andes
Facultad de Ciencias
Departamento de Matem´
aticas
Bogot´
a
Navier-Stokes equations: possible blow-up, self
similar solutions and backward uniqueness
Julio Andr´
es Montero Rosero
Disertaci´
on presentada para optar al t´ıtulo de
Doctor en Ciencias - Matem´
aticas
Advisor
Jean Carlos Cortissoz Iriarte
Universidad de los Andes
Facultad de Ciencias
Departamento de Matem´
aticas
Bogot´
a
Title in English
Navier-Stokes equations: possible blow-up, self similar solutions and backward
uniqueness
Abstract: In this thesis we consider two problems for the Navier-Stokes equations. One problem is blow-up rates to solutions of the Navier-Stokes equation if a
sin-gularity exists at time T in the homogeneous periodic Sobolev spaces ˙Hs(T3), in particular in the cases s= 3
2 and s= 5
2 where we prove the blow-up rates
ku(t)k˙
H32(T3) ≥
c3 2
(T −t)12
and ku(t)k˙
H52(T3) ≥
c5 2
(T −t)|log(T −t)|.
The techniques employed in the proofs are used to give a simpler proof of Chen
and Pavlovic’s results on the blow-up of Euler equation as given in [7]. Another
problem we are concerned with regards the existence of self-similar solutions. In
the final chapter, we prove, under a growth condition, that a backward self-similar
mild solution in the space ˙Hs(R3) with 1
2 ≤ s < 3
2 has to be identically equal to zero. A more general result has been proved by Neˇcas, R˚uˇziˇcka and ˇSver´ak in [32]
and Tsai [39]. However, our techniques allow an unified treatment of Tsai’s result
in the case of homogeneous Sobolev spaces. We also obtain some results related to
the backward uniqueness of the Navier-Stokes equation using elementary techniques.
Keywords: Blow-up, Navier-Stokes, Sobolev Spaces, Self-similar Solutions, Back-ward uniqueness
Para la culminaci´on de mis estudios doctorales hubo muchos que me brindaron apoyo
y ayuda, a todos ellos gracias.
Para llevar a buen t´ermino mis estudios de posgrado dos personas fueron
fun-damentales: los profesores Leonardo Rend´on y Jean Carlos Cortissoz. El profesor
Leonardo en la maestr´ıa y el profesor Jean en el doctorado. Ellos con su paciencia,
comprensi´on y conocimiento me guiaron para que esa etapa de mi vida estuviera
llena de aprendizaje y satisfaciones.
He de mencionar tambi´en a mis compa˜neros doctorales, cuya amistad hace que
mi paso por la Universidad de Los Andes sea recordada siempre como una etapa
feliz y llena de camarader´ıa.
Por ´ultimo, gracias a todas las personas en el Departamento de Mat´ematicas de
La Universidad de los Andes. Todos ellos poseedores de grandes calidades personales
Contents
1. Introduction 1
2. Preliminaries on existence and regularity 6
2.1 Existence theorems . . . 6
2.2 Regularity . . . 16
3. Lower bounds for blow–up rates 18 3.1 Introduction . . . 18
3.2 Preliminaries . . . 20
3.3 The energy inequality . . . 22
3.4 The cases 1 2 < s < 5 2 and s > 5 2 . . . 26
3.4.1 A remark about the Euler equations . . . 27
3.5 The case s = 5 2 . . . 28
3.5.1 A proof of Theorem 5.2 . . . 31
3.6 The case s = 1 2 . . . 33
4. Self-similar solutions and backward uniqueness 36 4.1 Introduction . . . 36
4.2 Preliminaries . . . 37
4.3 The case of self-similar solutions revisited . . . 39
4.3.1 Some comments regarding the Euler equations . . . 42
4.4 Backward uniqueness . . . 43
5. Conclusions and future research 51
A. A Paley-Wiener type theorem 53
CHAPTER
1
Introduction
In this work we consider the equations which describe the state of a fluid under no
external forces. These are the incompressible Navier-Stokes equations which can be
formulated in three spatial dimensions as
ρut−ν∆u+ρ(u· ∇)u+∇p= 0 inR3 ×(0, T) or T3×(0, T),
∇ ·u= 0,
u(x,0) =u0(x),
whereT3 = [0,2π]3 denotes the three dimensional torus,u(x, t) = (u1, u2, u3) is the
velocity, p(x, t) is the pressure , ρ is the density of the fluid, and ν is the viscosity
coefficient; without loss of generality we will assume ν = 1 and ρ= 1. Notice that
the equation ∇ ·u= 0 represents the incompressibility condition of the fluid.
The modern history of the Navier-Stokes equations started with Lerays’s work
presented in his remarkable paper Sur le mouvement d’un liquide visqueux
emplis-sant l’espace[30] where he introduced the definition of a weak solution of the
Navier-Stokes equations, and showed its global existence. In this way, with Leray’s work
began a systematic study of the Navier-Stokes equations whose aim is to
lish when these weak solutions remain smooth or if there are weak solutions which
develop singularities at a finite time T.
There are two approaches to the study of the Navier-Stokes equations: one is
through weak solutions, represented by Leray’s work, where solutions with initial
data u0 ∈L2(R3) satisfying the energy inequality
||u(t1)||L2(
R3)+ Z t1
t0
||∇u(τ)||L2(
R3) ≤ ||u(t0)||L2(R3),
which holds fort0 = 0 and almost every t0 >0, are constructed. And the other one
is through mild solutions introduced by Fujita and Kato’s work (see [22] and [13]):
they construct solutions of the Navier-Stokes equation in the space C([0, T);X)
with initial datau0 ∈X, whereX is eitherLp(R3) (withp≥3) or the homogeneous
Sobolev spaces ˙Hs(R3) (withs≥ 1
2). One difference between these two approaches is that the existence of Leray’s weak solutions are global in time, whereas mild
solutions are local in time. Another difference is that mild solutions are unique in
their time of existence, in comparison with the fact that the Leray’s weak solutions
uniqueness problem remains open since the time they were introduced in [30].
From Fujita and Kato’s work several developments have been made in the mild
solutions of approach constructing solutions to the Navier-Stokes equations.
Exis-tence theorems have been carried over to more general Banach spaces X, spaces
which contain Lp(R3) or ˙Hs(R3), such as the Besov SpacesBp,qr,s (see Canone’s work in [3]) or most recently in the space BM O−1 (see Koch and Tataru’s work in [24]).
As mentioned before, Leray’s weak solutions are global in time, but there is the
possibility that the solutions develop a singularity at some finite time. Leray called
such turbulent solutions. Whether weak solutions develop finite-time singularities
is a problem which remains open to this day. To establish whether a solution has a
singularity at timeT there are several results and theorems, and among the plethora
of existing results, perhaps the most familiar is the Prodi-Serrin-Ladyzhenskaya
If we suppose that a solution u(x, t) of the Navier-Stokes equations becomes
singular at time T <∞, it is possible to find lower blow-up rates. Indeed, Leray in
[30] showed that
ku(t)kH1(
R3) ≥
C0
(T −t)14
for aC0 >0, and
ku(t)k ≥ p C1
(T −t) for aC1 >0.
Also, he stated without proof that
ku(t)kLp(
R3)≥
Cp
(T −t)12(1− 3
p)
,
for p > 3. Giga, in [16], gave a proof of these blow-up rates. One of the goals in
this dissertation is to give sharp blow-up rates for the homogeneous Sobolev spaces
˙
H32(T3) and ˙H 5
2(T3), problems which remained open until recently. Notice that
3 2 and
5
2 are critical Sobolev coefficients; therefore, u(x, t) and ∇u(x, t) are not continuously embedded into some Lp(T3), with p ≥ 2. More specifically we prove the following results.
Theorem. Let u(x, t) = (u1, u2, u3) be a solution Navier–Stokes equations whose
maximum interval of existence is (0, T), 0 < T < ∞, and such that u ∈
C((0, T),H˙s(T3)∩H˙s+1(T3)).
(a) For 1
2 < s < 5
2 the following estimate holds
Cs
t12(s− 1 2)
≤ ku(T −s)kH˙s(
T3), and
(b) For s > 5
2 the following estimate holds
||u(T −t)||s ≥Cs max
0<τ <T||u(τ)||
s−5/2
s
L2(
T3)t
−25s
Theorem. Let u be a smooth weak solution of (NS) whose maximal interval of existence is (0, T). Then, there is a constant c >0 and a t0 ∈(0, T) such that
||u(T −t)||5 2 ≥
c
t|log(t)| for all t∈(0, t0).
Another possibility thatu(x, t) can develop a singularity at timeT is whenu(x, t)
is a backwards self-similar solution, i.e.,
u(x, t) = p 1
2a(T −t)U
x
p
2a(T −t)
!
.
Notice that if u(x, t) is a backward self-similar solution we have that
ku(·, t)kLp(
R3) =
kUkLp(
R2)
(2a(T −t))12(1− 3
p)
,
which are the expected blow-up rates. In the case p= 3 the above equality states
that the L3-norm ofu(x, t) remains constant in time for t∈(0, T). Neˇcas, R˚uˇziˇcka, and ˇSver´ak studied the case p= 3 in [32], and showed that there are no backward
self-similar solutions such that U ∈ L3(R3). When 3< p ≤ ∞, Tsai in [39] proved a similar result. In this work, we have tried to give an alternative simpler proof of
this result. We did not succeed, but we were able to show the following theorem.
Theorem. Let u(x, t) = p 1
2a(T −t)U
x
p
2a(T −t)
!
be a self similar mild
solu-tion in L∞((0, T),H˙s(R3))∩L2((0, T),H˙s+1(R3)), with 1
2 ≤s < 3
2 such that
Z R3
(1 +|x|)U(x)dx <+∞. Then U is identically equal to zero.
From the proof of this theorem, we were able to deduce the following backward
uniqueness result.
Theorem. Let be u(x, t) be a mild solution of the Navier-Stokes equation in the Banach space L∞((0, T); ˙H12(R3)) such that
Z R3
(1 +|x|)|u(x, t)|dx < +∞ for all
Finally, we also proved the following result, which implies a case of backward
uniqueness, is the following.
Theorem. There exists an 0 > 0 such that if a solution u of the Navier-Stokes
equations satisfies
sup
0≤t≤T
sup
ξ∈Z˙3
|ξ|2|u(ξ, t)| ≤b 0,
then there exist a K ≥ 1 and sequences {tn}∞n=1, {an}∞n=1 {ξn ∈ Z˙3}∞n=1 such that
tn → ∞, 1≤ |ξn| ≤K, an >0 and
|ξn|2|u(ξb n, tn)| ≥an.
Hereubk denote the Fourier coefficients ofu. We expect that the proof of this result
can be carried over to R3.
This thesis is organized as follows. In Chapter 2, we present a short survey about
some existence and regularity theorems for both weak solutions andmild solutions;
and we also give a proof of an existence result that implies the analyticity in space
of the solutions for the Navier-Stokes equations. In Chapter 3, we prove differential
inequalities for the ˙Hs(T3)-norm and use these to find blow-up rates in these spaces. Finally in Chapter 4, we use Fourier analysis to study the existence of backward
self-similar mild solutions in the homogeneous Sobolev space ˙Hs(R3) (with 1
2 ≤s < 3 2); in addition, we prove some result related to the backward uniqueness problem for the
Navier-Stokes equations; we also make some comments on the possible applications
of these methods to the existence problem of backward self-similar solutions for the
CHAPTER
2
Preliminaries on existence and regularity
The purpose of this chapter is to present a short survey on different existence and
regularity theorems for the Navier-Stokes equations. We want to remark that it
is not the purpose of this chapter to make an exhaustive presentation of all the
theory regarding existence and regularity about the Navier-Stokes equations, so we
have chosen a few theorems that we consider are meaningful to, and somewhat
representative of, the development of the history of the Navier-Stokes equations
regularity and uniqueness problem.
2.1
Existence theorems
The Navier-Stokes equation is the system of partial differential equations given by
ut−∆u+ (u· ∇)u+∇p= 0, in R3×(0, T) or T3×(0, T),
∇ ·u= 0,
u(x,0) =u0(x),
(2.1)
where T3 denotes the three-dimensional torus (and in this case periodic boundary conditions are imposed). For simplicity, we state the following in the R3 setting, but the same applies to domains Ω⊂R3 or T3 with adequate boundary conditions. A Leray-Hopf weak solution of the Navier-Stokes equations is a vector field u :
R3×(0, T)−→R3 such that
(a) u∈L∞((0, T);L2(R3))∩L2((0, T);H1(R3)), (b)
Z R3
u(x, t)· ∇φ(x)dx= 0,
with φ ∈ (C0∞(R3))3 and ∇ · φ = 0. The function t →
Z R3
u(x, t) · w(x)dx is
continuous on [0, T] for all w∈(L2(R3))3,
(c)
Z T
0
Z R3
(−u·∂tφ−u⊗u:∇φ+∇u:∇φ)dxdt= 0,
(d) 1 2
Z R3
|u(x, t0)|2dx+
Z t0
0
Z R3
|∇u|2dxdt≤ 1
2
Z R3
|u0(x)|2dx,
where the las energy inequality holds tot0 = 0 and almost everyt0 >0,
and (e) ||u(˙,t)−u0(·)||L2(
R3) →0 as t→0.
Above, we have used the following notation: if A, B are n×n matrices the double
dot product is defined by A:B = Tr(BTA). With this definition Leray’s existence
theorem can be stated as follows (see [30], [20]).
Theorem 1.1. Suppose that
u0 ∈L2(R3) and
Z R3
u0(x)· ∇φ(x)dx= 0,
where φ ∈ C0∞(R3). Then there exists at least one Leray-Hopf weak solution of the Cauchy problem (3.1) on R3×(0,∞).
Thirty years later, Kato and Fujita, in [22], used the theory of semigroups to show
the existence of mild solutions for the Navier-Stokes equations in the homogeneous
Sobolev spaces ˙Hs(R3), with s ≥ 1
Navier-Stokes equations in a Banach spaces X is a function u∈ C([0, T);X), the space of
continuous maps from [0, T) to X, such that
u(x, t) = eAtu0(x) +
Z t
0
eA(t−s)P∇(u(x, s)⊗u(x, s))ds,
divu(x, t) = 0,
(2.2)
where P is the Leray projector (i.e the projection into the subspace of functions in
X with ∇ ·u= 0 in the distributional sense), and A is the Stokes operator defined
as A=−P∆. Finally, they proved the following theorem (see [22],[13]).
Theorem 1.2. Let u0 ∈ H
1
2(R3). Then there is a T > 0 such that a solution
u(t)∈C([0, T); ˙H12(R3))of (2.2) exists for 0≤t < T. Moreover, there exits a >0 such that if ||u0||H˙1
2(R3) < , then there is a solution of (2.2) u(t) for 0< t <∞.
Afterwards, Kato in [21], extended his result to L3(R3), by proving the following theorem.
Theorem 1.3. Let u0 ∈ L3(R3) and ∇ ·u0 = 0 in the distributional sense. Then
there is a T > 0 and an unique solution u ∈ C([0, T);L3(R3))∩L5((0, T)×R3)
of (2.2). Moreover there is an > 0 such that if ||u0||L3(
R3) < there is a unique
solution in C([0,∞);L3(R3))∩L5((0,∞)×R3).
When we analyze the spaces in Theorems 1.2 and 1.3 where existence of mild
solutions is proved, we observe that the norm in these spaces is invariant under the
scaling λu0(λx); this is an important property because if u(x, t) is a solution of the
Navier-Stokes equation with initial data u0 the scaled solution λu(λx, λ2t) is also a
solution with initial data λu0(λx). Spaces for which this property holds are called
critical spaces.
Definition 1( see [15]). A Banach spacesX is called a critical space for the Navier-Stokes equation, if the following properties hold:
(a) The norm of space X is invariant under the scaling λu(λx): if u ∈ X and
(b) There exists an > 0 such that if an initial data u0 ∈ X satisfies ||u0||X < ,
then equation (2.2) has a solution in C([0,∞);X).
So, ˙H12(R3) and L3(R3) are critical spaces. There are some other interesting
critical spaces for which existence results for the Navier-Stokes equations can be
proved. But before we state the next existence results, we must give a couple of
definitions.
Definition 2 (see [14]). Let φ be a function in S(Rn) such that φb= 1 for |ξ| ≤ 1
andφb= 0for |ξ|>2, and define φj(x) = 2njφ(2jx). Then the frequency localization
operators are defined by
Sj =φj?·, ∆j =Sj+1−Sj.
Let f ∈ S0(Rn). We say f belongs to B˙s
p,q if and only if
• The partial sum
m
X
−m
∆j(f) converges to f as a tempered distribution if s <
n p
and after taking the quotient with polynomials if it does not.
• The sequence j = 2js||∆j(f)||Lp belongs to lq.
We also need a vector extension of these spaces.
Definition 3( see [14]). Letu(x, t)∈ S0(Rn+1)and let∆j be a frequency localization
operator with respect to the x variable. We will say that u ∈ Leρ((a, b),D˙p,qs ) if and
only if
2js||∆ju||Lρ((a,b),Lp
x =j ∈l
q,
• The partial sum
m
X
−m
∆j(f) converges tof as a tempered distribution if s <
n p,
and after taking the quotient with polynomials if it does not.
• The sequence j = 2js||∆j(f)||Lp belongs to lq.
We define
kukLeρ((a,b),B˙s
p,q) =||2
js||∆
For these spaces we have the following existence theorem (see [14]).
Theorem 1.4. Letuo ∈B˙
−1+p3
p,q (R3)be a divergence free vector field. There exists a
unique local in time solution to (2.2) such that
u∈C([0, T); ˙B−1+3p(R3))∩Ler((0, T); ˙Bsp+
2
r
p,q ) ∀r ∈[1,∞],
where sp =−1 +
3
p. Moreover, there exists a constantK0 such that if ||u0||B˙p,qsp ≤K0,
then we can choose T = +∞.
This theorem includes the Besov spaces ˙B−1+
3
p
p,∞ (R3), 3< p < ∞, which are also
critical spaces. The largest critical Banach space where there is a local existence
theorem for the Navier-Stokes equations is the space BMO−1, whose definition is as follows.
Definition 4 ( see [24]). We say that the tempered distribution u is in BMO−1 if
sup
x,R
|B(x, R)|
−1
Z
B(x,R)
Z R2
0
(Φt? u)(x)dxdt
<+∞
with Φ(x) = π−n2e−|x| 2
, Φt(x) = t−nΦ
x
t
.
The norm in BMO−1 is defined as
kukBMO−1 = sup
x,R
|B(x, R)|
−1
Z
B(x,R)
Z R2
0
(Φt? u)(x)dxdt
1 2
.
This definition motivates looking for solutions of the Navier-Stokes equation in
the space X defined in Rn×R+ with norm
kukX = sup
t
t12 ku(t)k
∞+
sup
x,R>0
|B(x, R)|−1
Z R2
0
Z
B(x,R)
|u|2dydt
.
Theorem 1.5. The Navier-Stokes equation (2.2) have a unique global solution in
X for all initial data u0 with ∇ ·u0 = 0 which are small in BMO−1.
The following chain of continuous embeddings for critical spaces holds:
˙
H12(R3),→Lp(R3),→B˙−1+ 3
p(
R3),→BMO−1,
and all these spaces are continuously embedded in the Besov space ˙B−∞1,∞(R3) whose norm is also invariant under the scaling λu0(λx). It would be desirable to have
an existence result for the space ˙B∞−1,∞(R3) but Bourgain and Pavlovi´c proved in [2] that the Navier-Stokes equations are ill-posed in this space; they showed that
initial data in the Schwartz class S that are arbitrarily small in ˙B∞−1,∞ can produce solutions arbitrarily large in ˙B−∞1,∞ after an arbitrarily short time.
We shall give a proof of an existence theorem for the Navier-Stokes equations
in what follows, but we will need the definition of a direct family of Banach spaces
which we take from [17].
Definition 5. Let Z = Z(ω) be a Banach space of real scalar valued functions defined on a locally compact set ω with norm || · ||Z. For any x ∈ ω, let H(x) be
a Banach space with a norm || · ||H(x). Introduce the Banach space X of mappings
f :x∈ω →H(x), such that ||f(x)||H(x) ∈Z(ω), and X is equipped with the norm
||f||X =|| ||f(x)||H(x)||Z.
We will call X a direct family of spaces H(x) with the basic space Z(ω) and write
X(Z(ω);H(·)), and f = (f(x))x∈ω.
We will also need the following definition.
Definition 6. Define the space
e
Hts(R3) :=
u∈ S0(R3) :
Z R3
|ξ|2s|bu(ξ)|2e2|ξ|
√
t
dξ <∞
with the norm
kuk
e
Hs t =
Z R3
|ξ|2s|
b
u(ξ)|2e2|ξ|√tdξ
12
.
Next, as promised, we present a proof of an existence theorem in a direct family of
Banach spaces which we will define shortly and which is based on a similar presented
theorem in [23]. The merit of this proof is that in this space, the space analyticity
of solutions follows at once.
Theorem 1.6. Let u0 be a function in H˙
1
2(R3). Then there exits T ∈(0,+∞), and
u∈X(L∞(0, T);He
1 2
t (R
3))∩X(L2
(0;T);He
3 2
t (R
3
))such thatuis a mild solution of the
Navier-Stokes equation with initial data u0. Here the spaces X(L∞(0, T);He
1 2
t (R3))
andX(L2(0;T);He
3 2
t (R
3
))denote the direct family related toL∞((0, T); He
1 2
t (R
3
))and
L2((0, T);He
3 2
t (R3)) respectively.
Proof. First, we introduce some notation. For f ∈ S0, define
e
f(ξ, t) := f(ξ) exp(2π|ξ|b
√ t),
ET :=X(L∞(0, T);He
1 2
t (R3))∩X(L2(0, T);He
3 2
t (R3)), with norm
||f||ET =
||f(t)||2
X(L∞(0,T);
e
H
1 2
t )
+||f||2
X(L2(0,T);He 3 2
t ))
12
; and
Ft:=X(L4(0, T);Het1(R3)), with norm
||f||Ft =||f||X(L4(x,T);He1
t).
Step 1. If u0 ∈ H˙
1
2(R3), then exp(t∆)u0 ∈ X(L∞(0, T);He 1 2
t (R
3)) ∩
X(L2(0, T);He
3 2
t (R3)). Indeed, first we have that
Z R3
|ξ||ub0(ξ, t)|2exp(−8π2|ξ|2t) exp(2π|ξ|
√ t)dξ
≤exp(1 2)
Z R3
|ξ||bu0|2exp−2(2π|ξ| −
1 2)
2dξ
≤exp(1 2)||u0||
2 ˙
H12
which shows that exp(t∆)uo ∈X(L∞(0, T);He
1 2
t (R3)). On the other hand, we have
Z T
0
Z R3
|ξ|3|
b
u0|2exp(−8π2|ξ|2t) exp(4π|ξ|
√ t)dξdt
≤exp(1 2)
Z R3
|ξ|3|
b
u0|2
Z T
0
exp(−(2π|ξ|√t−1 2))dtdξ ≤ exp( 1 2) 2π2 Z R3
|ξ||bu0|2
Z 2π|ξ|
√
T
0
yexp(2(y−1 2))dydξ
≤ exp(1/2) 2π2 ||u0||
2 ˙
H12
Z ∞
0
yexp(2(y− 1 2)
2)dy.
Notice that as
Z 2π|ξ|
√
T
0
yexp(2(y− 1
2))dy →0 as T →0, then kukX(L2(0,T);He 1 2
t )
→0 as
T →0 as well. We will use this fact later to conclude that the map
eAtu0(x) +
Z t
0
eA(t−s)P∇(u(x, s)⊗u(x, s))ds
is a contraction for small T.
Step 2. If u ∈ X(L∞(0, T);He
1 2
t (R3)) ∩ X(L2(0, T);He
3 2
t (R3)) then u ∈
X(L4(0;T);Het1((R)3)). Indeed, since 1 =
1 2 1 2+ 1 2 3
2, by H¨older’s inequality
ku(t)kH˙1(
T3) ≤ ku(t)k
1 2
˙
H12(T3)
ku(t)k
1 2
˙
H32(T3)
.
Step 3. Recall the product rule for homogeneous Sobolev spaces. Let f ∈H˙s1(
R3)
and g ∈ H˙s2(
R3), with (s1, s2) ∈ (−3/2,3/2)2 and s1 +s2 ≥ 0 (see Corollary 2.55
p90 [1]), then
||f g||˙
Therefore
||f g||
L2((0,T),
e
H
1 2
t )
≤ ||f||FT||g||FT. (2.3)
Given ξ, q ∈R3,t > τ > 0 we have the following inequality (see [29], p249):
−4π2|ξ|2(t−τ)−√τ|ξ−q| −√τ|q| ≤2−2π√t|ξ| −2π2(t−τ)|ξ|2.
Now, we use all these inequalities to estimate the norms in X(L2(0, T);He
3 2
t (R)
3
)
and X(L∞(0, T);He
1 2
t (R)3) of the non-linear term. So, if we define
B(f, g) =
Z t
0
exp(−4π2|ξ|2
(t−τ))ξ
δml−
ξmξl
|ξ|2
(f ?b b
g)(ξ, τ)dqdτ,
for all h∈L2((0, T)×R3) we have that
Z T
0
Z R3
B(f, g)|ξ|32 exp(2π|ξ|
√
t)h(ξ, t)dξdt
≤
Z R3
Z T
0
Z t
0
exp(−2π2|ξ|2(t−τ))|ξ|2|ξ|12(
f
|f|?|g|)(ξ, τf )|h(ξ, t)|dτ dtdξ.
Notice that the integral in the variable τ is the convolution between
W(ξ, t0) =
0 if t0 <0
|ξ|2exp(−2π2|ξ|2t0
) if t0 ≥0 and
V(ξ, t0) =
|ξ|12(f ?e
e
g)(ξ, t0) if 0≤t0 < t
0 otherwise.
Moreover, the Fourier transform of W over the t0 variable is
F {W(ξ,·)}(τ) = |ξ|
2
Then, by the formula for the Fourier transform for convolutions, and Parseval’s
identity applied to the integral in the variable t, we get
Z T 0 Z R3
B(f, g)|ξ|32 exp(2π|ξ|
√ t)h(ξ, t)dξdt ≤ Z R3 Z ∞ −∞
F {W(ξ,·)?|V(ξ,·)|}(τ)F {|h(ξ, .)|}(τ)dτ dξ
≤ Z R3 Z ∞ −∞ |ξ|2
|ξ|2+i2πτF {|V(ξ,·)|}(τ)F {|h(ξ,·)|}(τ)dτ
dξ ≤ 1 2π2 Z R3 Z ∞ −∞
|F {|V(ξ,·)|}(τ)F {|h(ξ,·)|}(τ)|dτ dξ
≤ 1 2π2 Z R3 Z ∞ −∞
|F {|V(ξ,·)}(τ)||2dτ dξ
12 Z
R3 Z ∞
−∞
|F {|h(ξ,·)|}(τ)|2dτ dξ
12 ≤ 1 2π2 Z T 0 Z R3
|ξ||(f ?e e
g)(ξ, s)|2dξds
12
||h||L2((0,T)×
R3).
So, by inequality (2.3) we have shown that
||B(u, u)||
X(L2(0,T);He 3 2
t )
≤ kukF
T kukFT.
Additionally, let φ be a function in L2(R3). We can estimate
Z R3
B(u, u)|ξ|12 exp(2π|ξ|
√ t)φ(ξ)dξ ≤ Z t 0 Z R3
exp(2π2|ξ|2(t−s))|ξ||φ(ξ)||ξ|12|(f ?e
e g)(ξ, s)|dξds ≤ Z t 0 Z R3
|ξ||(f ?e e
g)(ξ, s)|2dξds
12 Z
R3
|π|2|ξ|2
Z t
0
exp(2π2|ξ|2(t−s))dsdξ
12
.
From the calculations above, we finally get
||B(u, u)||ET ≤CkukFTkukFT
This implies, by the remark made at the end of Step 1, that forT > 0 small enough
the mapping u 7−→ exp(∆t)u0 +B(u, u) is a contraction in FT, which proves the
This proof mixes ideas from [29] (chapters 7, 14 and 24) and the standard proof
of existence of mild solutions in ˙H12(R3) (see [23]). As announced before this proof
gives the analyticity of the solutions constructed as a consequence of theorem A (see
the appendix).
2.2
Regularity
Now, the question is whether Leray-Hopf weak solutions or mild solutions develop
or not a singularity at time T. The proofs for weak and mild solutions include
that the solutions are also solutions in the classical sense until a singularity occurs.
A sufficient condition for a Leray-Hopf weak solution to not develop a singularity
at time T is the Prodi-Serrin-Ladyzhenskaya criteria which is formulated in the
following theorem.
Theorem 2.1. Let v a weak solution in Rn ×(0, T) to the homogeneous Navier-Stokes equation, and the initial equation vo ∈ L2(Rn) which is divergence free in
the distributional sense. Assume that v satisfies at least one of the following two
conditions:
(i) v ∈Lr((0, T);Ls(Rn)), for some r, s such that n
s + 2
r ≤1, s ∈(n,∞];
(ii) v ∈C((0, T];Ln(Rn)).
Then, we have
v ∈C∞(R3×(0, T]).
Preliminary versions of this theorem were given by Serrin in [38], Prodi in [33]
and Ladyzhenskaya in [27]; the version as stated above was proved by Giga in [16].
Note that the limit cases=nis not included in the first condition, and the theorem
gives the weaker conditionC((0, T];Ln(R3)) which is a subset ofL∞((0, T);Ln(Rn)). This limit case was finally proved in 2003 by Escauriaza, Seregin and ˇSver´ak in [20].
Theorem 2.2. Suppose that v is a weak Leray-Hopf Solution of the Cauchy problem
vt(x, t)−∆v(x, t) + (v(x, t)· ∇)v(x, t)−p(x, t) = 0 in R3×(0, T)
∇ ·v(x, t) = 0
v(x,0) =v0(x),
with v0 ∈L2(R3) and ∇v0 = 0; and v ∈L∞((0, T);L3(R3)). Then
v ∈L5(R3×(0, T)),
and hence it is smooth and unique on R3×(0, T).
We want to point out that our blow-up results imply a regularity criterion for
the Navier-Stokes equations similar to the Prodi-Serrin-Ladyzhenskaya result in the
CHAPTER
3
Lower bounds for blow–up rates
3.1
Introduction
The problem that concerns us in this chapter is to find lower bounds on putative
blow–up solutions of homogeneous Navier–Stokes equation
ut−∆u+ (u· ∇)u+∇p= 0 in X×(0, T)
∇ ·u= 0 in X×(0, T),
(3.1)
where X = R3 or X = T3 = [0,2π]3 the three dimensional torus, i.e., we impose periodic boundary conditions. In his beautiful paper Sur le mouvement d’un
liq-uide visqueux emplissant l’espace Leray ([30] p227) established, without proof , the
following lower bounds for Lp(R3) solutions to the Navier-Stokes equations which become singular at time T:
||u(T −t)||Lp(
R3)≥cpt
−(p−3)/2p
, 3< p <∞.
A proof for this bound was given by Giga [16] in 1986. Recently, Robinson, Sadowski
and Silva in [35] gave a lower bound for ˙Hs(T3) blow–up solutions that lose regularity 18
at timeT, where ˙Hs(T3) refers to the homogeneous Sobolev spaces. Their estimates are
||u(T −t)||H˙s(
T3)≥ct
−(s+1 2),
for 1/2< s <3/2 and 3/2< s <5/2, and
||u(T −t)||˙
H32(T3) ≥ct
−(12−), for any >0.
In [11] the bound for ˙H32(T3) solutions was improved to
ku(T −t)k˙
H32(T3)≥
c
(t|log(t)|)12
,
and the following bound was given for the case of s= 5 2
ku(T −t)k˙
H52(T3) ≥
c t|log(t)|,
which is one of the results of this work.
These blowup rates can be interpreted as a regularity criterion in the following
sense: if the norm ||u(t−T)||H˙s(
T3) grows slower than ct
−(s+12) then the solution to
the Navier–Stokes equation u is regular at T. A difficult question arises when we
consider the limit case s= 1/2: a conjectured possible lower bound for the blow up
rate is given by
||u(T −t)||˙
H12(T3) ≥C|log(t)|.
The best result known so far for the s = 1
2 case was established by Escauriaza, Seregin and ˇSver´ak in [20], where they proved that if the L3(R3) norm of a Navier– Stokes solutions is bounded in (0, T) then the solution is regular at time T. Since
˙
H12(R3) is continuously embedded into L3(R3) this result is also true for solutions
with ˙H12(R3) norm uniformly bounded in (0, T).
Our aim in this chapter is to find optimal lower bounds for the blow-up rate
Sobolev spaces ˙Hs(T3) when s = 3 2 and
5
2, and also for s > 5
2. Our methods also apply to give optimal blow-up rates for solutions to Euler equation when s > 5
2.
3.2
Preliminaries
As is customary, we denote by ˙Hs(T3) the space of periodic distributions such that
˙
Hs(T3) :=
(
u∈ S0(T3) : bu0 = 0 and
X
k∈Z3
|k|2s|
b
uk|2 <∞
)
,
with norm
kuks= X
k∈Z3
|k|2s|
b
uk|2
!12
.
These spaces are know as homogeneous Sobolev spaces. By Fr we will denote the
Banach space of periodic distribution such that
Fr:=
(
u∈ S0(T3) : bu0 = 0 and
X
k∈Z3
|k|r|
b
uk| ≤ ∞
)
with norm
kukFr =
X
k∈Z3
|k|r|
b
uk|.
These spaces are related by the following interpolation inequalities.
Proposition 2.1. Let u∈H˙s(T3)∩H˙s+1, s > 1
2. Then:
(a) For 1
2 < s < 3 2
kukF0 ≤2πcsc 1 2
π
s−1 2
kuks−12
s kuk
3 2−s
s+1 . (3.2)
(b) For 1
2 < s < 5 2
kuk
F
1
2(s−12) ≤2πcsc 1 2
π 2(s−
1 2)
kuk
1 2(s−
1 2) s kuk 1 2( 5 2−s)
(c) For s > 5 2
kukF1 ≤2π
kukL2(
T3) s−s5/2
(kuks)25s
1 (s−1)csc
π(2s−5) 2s−2
12
. (3.4)
Proof. First we prove (b). To this end, we use the interpolation technique employed
by Hardy in his proof of Carlson’s inequality (see [28]) to obtain:
X
k
|buk| |k|
1 2(s−
1
2) =X
k
|buk| |k|
1 2(s−
1 2)
q
a|k|s+1
2 +b|k|s+ 5 2
q
a|k|s+1
2 +b|k|s+ 5 2
≤ akuk2s+bkuk2s+112 X
k
1
a|k|s+12 +b|k|s+ 5 2
!12
≤ akuk2s+bkuk2s+112 √4π
ab
√
a √
b
32−sZ ∞
0
y32−s
1 +y2dy
!12
.
Now we choosea =kuk2s+1 and b =kuk2s, and note that after the change of variable y2 →y, we get
Z ∞
0
y32−s
1 +y2dy=
1 2
Z ∞
0
y12( 1 2−s)
1 +y dy=B
1− 1 2
s−1 2
,1 2
s− 1 2
,
where by B we denote the beta function. By well know properties of the Beta and
Gamma functions, we obtain
Z ∞
0
y32−s
1 +y2dy=
π 2 sin π2(s− 1
2)
,
For (a), we use the same technique. In this case, choosing the factor
p
a|k|2s+b|k|2s+2 gives:
X
k
|ubk| ≤ akuk2s+bkuk2s+1
12 X
k
1
a|k|2s+b|k|2s+2
!12
,
≤ akuk2s+bkuk2s+112
2π2 √ ab a b
1−s
B
1−
s− 1 2
, s− 1 2
12
.
Again, we choose a=kuk2s+1 and b =kuk2s and inequality (3.2) follows.
Finally, to prove (c), we apply Hardy’s technique with the factor pa+b|k|2s−2:
X
k
|ubk||k| ≤(akuk
2
1+bkuk 2 s) 1 2 4π2
(2s−2)a
a
b
2s3−2
B
1− 2s−5 2s−2,
2s−5 2s−2
12
.
So, if we picka =kuk2s and b=kuk21 and recall the interpolation inequality between homogeneous Sobolev spaces given by
kuk1 ≤ kuk
s−1
s
L2(
T3)kuk
1
s
s ,
we get inequality (3.4).
3.3
The energy inequality
In what follows, we shall denote by P the Leray projector and by A the Stokes
operator. Recall thatP is the projection to the subspace of ˙Hs(T3) of free divergence functions and A=−P∆ = −∆P.
Consider the Galerkin approximation of the periodic Navier–Stokes system
d
dtum(x, t) +Aum(x, t) +
X
|k|∞≤m k∈Z˙3
exp(2πik·x){P(um(·, t)· ∇)um(·, t)}
∧
k = 0
divum = 0
where ˙Z3 = Z3 \ {0} and |k|∞ = max{k1, k2, k3}. Taking the scalar product in
˙
Hs(T3) between the first identity in (3.5) andbum,k, yields
1 2
dkum(t)k
2
s
dt +kum(t)k
2
s+1 ≤2π
X
|k|∞≤m k∈Z3
X
|q|∞≤m q∈Z3
|k|2s(k·
b
um,(k−q)(t))(ubm,q(t)·bum,k(t))
.
(3.6)
To obtain a bound on the right hand side of the previous inequality we use the
following result from [35].
Proposition 3.1. For any s >1 and 0≤r≤1, we have
X
k∈Z˙3
X
q∈Z˙3
|k|2s(k·buk−q)(ubq·buk)
.≤cs
X
k∈Z˙3
|k|r|ubk|
kukskuks+1−r, (3.7)
for all u∈H˙s+1−r(T3)∩Fr.
As we said before, a proof of this proposition is given in [35], but for the sake of
completeness, we present these arguments here.
Proof. Since P is self-adjoint and um(x, t) is divergence free , we have that (see [9],
chapter 6 p. 53)
X
k∈Z˙3
X
q∈Z˙3
|q|s|k|s(
b
uk−q·q)(buq·buk) = 0.
Using the inequality ( see [18], p.39)
||x|s− |y|s| ≤s(2s)|x−y| |x−y|s−1+|y|s−1
, s >1;
and the well know inequality
we obtain: X
k∈Z˙3
X
q∈Z˙3
|k|2s(k·
b
uk−q)(buq·ubk) = X
k∈Z˙3 X
q∈Z˙3
|k|2s(k·ubk−q)(ubq·ubk)− X
k∈Z˙3
X
q∈Z˙3
|q|s|k|s(ubk−q·q)(buq·buk) ≤ X
k∈Z˙3 X
q∈Z˙3
|k|s|(q·
b
uk−q)| |ubq| |buk| ||k|
s− |q|s|
≤s2s−1 X
k∈Z˙3 X
q∈Z˙3
|k|s|(q·
b
uk−q)||ubq||buk||k−q|(|k−q|
s−1+|q|s−1)
≤s2sX
q∈Z˙3
X
k∈Z˙3
|k|s|k−q|s|q||
b
uk−q||buq||buk|
≤s2sX
q∈Z˙3 X
k∈Z˙3
|k|s|k−q|s|q|r|q|1−r|buk−q||ubq||buk|
≤s2sX
q∈Z˙3 X
k∈Z˙3
|k|s|k−q|s|q|r(|k−q|1−r+|k|1−r)|
b
uk−q||buq||buk|
≤s2s+1X
q∈Z˙3
|q|r|
b
uq|
X
k∈Z˙3
|k−q|s|
b
uk−q||k|s+1−r|buk|
≤s2s+1
X
k∈Z˙3
|k|r|
b
uk|
kukskuks+1−r,
which is what we wanted to prove.
By combining (3.6) and (3.7), we get the following useful differential inequalities
1 2
dkum(t)k2s
dt +kum(t)k
2
s+1 ≤cskum(t)kFrkum(t)kskum(t)ks+1−r.
Now, we consider the following three cases:
(a) 1
2 < s≤1. We choose r= 0 and use part (a) of Proposition 2.1 to get
1 2
d
dt(kum(t)k
2
s) + 4π
2ku
m(t)k2s+1 ≤Cskum(t)ks+
1 2
s kum(t)k
5 2−s
(b) 1 ≤ s < 5
2. We pick r = 1 2
s− 1 2
and use Proposition 2.1 part (b), so the
energy inequality for um becomes
1 2
d
dt(kum(t)k
2
s) + 4π
2ku
m(t)k2s+1 ≤cskum(t)k
s
2+ 3 4
s kum(t)k
5 4−
s
2
s+1 ku(t)ks
2+ 5 4 .
Observe that s 2+ 5 4 = s 2− 1 4 s+ 5 4− s 2
(s+1), so by interpolation between
homogeneous Sobolev spaces, we get
kumks
2+ 5
4 ≤ kumk
s
2− 1 4
s kumk
5 4−
s
2
s+1 .
Therefore, we finally obtain
1 2
d
dt(kum(t)k
2
s) + 4π
2ku
m(t)k2s+1 ≤Cskum(t)ks+
1 2
s kum(t)k
5 2−s
s+1 .
(c) s > 5
2. For this case we taker= 1 and employ inequality (3.4), so we have
d
dtkum(t)k
2
s ≤cs kum(0)k
2 0
2s4−s5 ku
m(t)k
2
s
1+45s
.
In this way, using standard arguments and the Galerkin method, we can conclude
that there exists a solution of Navier–Stokes system which satisfies
1 2
d
dt(ku(t)k
2
s) + 4π
2ku(t)k2
s+1 ≤Cskum(t)k
s+12
s ku(t)k
5 2−s
s+1 (3.8)
for all 1
2 < s < 5
2, and if s > 5 2
d
dt ku(t)k
2
s ≤cs ||u(0)||20
2s4−s5
ku(t)k2s1+45s
3.4
The cases
1
2
< s <
5
2
and
s >
5
2
In this section we shall give a proof of the following theorem.
Theorem 4.1. Let u(x, t) = (u1, u2, u3) be a solution of the Navier–Stokes (3.1)
equations whose maximum interval of existence is (0, T), 0< T <∞, and such that
u∈C((0, T); ˙Hs(T3)∩H˙s+1(T3)).
(a) For 1
2 < s < 5
2 the estimate
Cs
t12(s− 1 2)
≤ ku(T −s)kH˙s(
T3) (3.10)
holds.
(b) For s > 5
2 the estimate
||u(T −t)||s≥Cs max
0<τ <T||u(τ)||
s−5/2
s
L2(
T3)t
−2s
5 (3.11)
holds.
A different proof of this theorem for the case s = 3
2 was given by McCormick, Olson, Robinson, Rodrigo, Vidal-Lopez, and Zhou in [31], and by Cheskidov and
Zaya in [8].
Proof of Theorem 4.1. For part (a) we have, using Young’s inequality,
ab≤ a
p
p + bq
q
1 p+
1 q = 1
with the choice p= 2 s+
1 2
s− 1 2
and q= 5 2
2 −s
, and inequality (3.8) yields
1 2
d
dt(ku(t)k
2
s)≤Cs ku(t)k
2
s
1+ 1 s−1
2
.
To prove (b) we use the energy inequality (3.9) instead of (3.8) to obtain
1 2
d
dt(ku(t)k
2
s)≤Csku(t)k
s−5/2
s
L2(
R3)(kuk
2
s)
1+45s,
and the result follows by integration.
3.4.1
A remark about the Euler equations
Recall the Euler equation, which describes the inviscid, incompressible fluid
dynam-ics, are given by
ut(x, t) + (u(x, t)· ∇)u(x, t) +∇p(x, t) = 0, in X×(0, T)
∇ ·u(x, t) = 0 inX×(0, T),
(3.12)
whereX =R3orX =T3. Since inequality (3.9) is also valid for the Euler equations, we have:
Corollary 4.1. Let u(x, t) be a solution of the periodic the Euler equations (3.12) in the class
C1((0, T); ˙H52+δ(T3))∩C((0, T); ˙H 7
2+δ(T3))
and let T be the minimum time of blow–up. Then there exists a finite, positive
constant C(δ,||u(0)||0)>0 such that
||u(·, t)||5
2+δ ≥C(δ,||u(0)||0)
1 T −t
2+25δ
.
A proof of the above result was given by Chen and Pavlovic in [7]. Let us give an
(see [7] Theorem 1.2)
ku(t)ks ≤ ||u0||sexp
Cs||u0||L2
Z t
0
(Ls−5 2(τ))
−5 2dτ
where (Ls−5
2(l)) = min
1, ||ω(t)||Cs−52
||u0||L2
!−1s
,
||ω||
Cs−52 = sup
|x−y|<1
|ω(x)−ω(y)| |x−y|s−5
2
,
and ω(x, t) = ∇ ×u(x, t) is the vorticity of u and δ = s− 5
2. Next, they use the above inequality and the local in time existence theorem for the Euler equations in a
recursive argument to create a monotonically increasing sequence {tn}∞n=0, tn→T,
such that
T −t0 ≥
N
X
n=0
tn+1−tn,
≥ 1
Cδ||u(t0)||s N
X
n=1
1 ρn,
where ρ = C(δ,||u(t0)||L2)||u(t0)|| 5 2s−1
s > 1. Our methods give then an alternative
and simpler proof of Chen and Pavlovic’s result.
3.5
The case
s
=
5
2
If we had that the space F1 were embedded in ˙H52(T3), the energy inequality (3.8)
would produce the following lower blowup–up rate for a singular solution of the
Navier-Stokes equation:
||u(T −t)||5
2 ≥K(T −t)
−1.
But, since this is not true, the above relation cannot be verified. Our goal in this
section is to prove a similar estimate with a logarithmic correction of this inequality.
Theorem 5.1. Let u be a smooth weak solution of (NS) whose maximal interval of existence is (0, T). Then, there is a constant c >0 and a t0 ∈(0, T) such that
||u(T −t)||5 2 ≥
c
t|log(t)| for all t∈(0, t0).
A comment must be made before we give a proof of this result. Recently
Cheski-dov and Zaya in [8] give another proof of this blow-up rate. Furthermore,
Mc-Cormick, Olson, Robinson, Rodrigo Vidal-L´opez, Zhou proved in [31] that ifu is a
blow-up solution to the Navier-Stokes equation, then there is a c >0 such that
lim sup
t→T
(T −t)ku(t)k5 2
≥c,
where T is the time of blow-up. With the purpose of giving the reader a broader
point of view we present the proof of this blow-up result, which is contained in [31].
It is an argument by contradiction, so we suppose that lim
t→0(T −t)ku(t)k52 = 0.
Then given >0, there exists a τ >0 such that
ku(t)k5 2 ≤
T −t.
Moreover, we have the energy inequality for the case s= 3
2 which is
1 2
dku(t)k23 2
dt ≤c32 ku(t)k
2
3
2 ku(t)k 5
2 − ku(t)k
2
5 2 .
Next, we choose >0 such that 2c3
2 <1. By assumption we have that
ku(t)k5 2 ≤
(T −t) ≤
1 2c3
2(T −t)
.
Using the blow-up rate for the norm in ˙H32(T3) we get
ku(t)k5 2 ≤
c3
2 ku(t)k
2
3 2
Now, we use the fact that ax−x2 is an increasing function for 0≤ x≤ a
2. In this way, by the above calculations we obtain
c3
2 ku(t)k
2
3
2 ku(t)k 5
2 − ku(t)k
2
≤c3
2 ku(t)k 3 2
T −t −
2
(T −t)2.
Therefore, we have that
dku(t)k23 2
dt −2c32 ku(t)k 3 2
T −t ≤ −2 2
(T −t)2 ≤0,
so using the integrating factor (T −t)2c32 and integrating between τ and t we get
ku(t)k3 2 ≤
||u(τ)||2
3 2
(T −τ)2c32
(T −t)2c32
.
Since 2c3
2 < 1 we can choose τ such that ||u(τ)||
2
3 2
(T −τ)2c32 > c3
2, so the above
inequality contradicts the blow-up rate for ˙H32(T3).
Proof of the Theorem 5.1. This result is a consequence of the following.
Theorem 5.2. Let u(x, t) be a smooth solution to the Navier-Stokes equation with maximum interval of existence(0, T),T < +∞. Then there exists a timet0 ∈(0, T)
and a constant C > 0 such that
t > 1
C||u(T −t)||5
2 log(C||u(T −t)|| 5 2)
for all t ∈(0, t0) (3.13)
Indeed, assume Theorem 5.2, and recall that the Lambert W function is the
inverse of the function xlog(x) (x >0). Then we have that
exp
W
1 t
≤expWc||u(T −t)||5
2 log(c||u(T −t)|| 5 2)
=c||u(T −t)||5 2.
The Lambert W function satisfies the following inequality (see [19])
So exp log 1 t −log log 1 t ≤exp W 1 t
≤c||u(T −t)||5 2,
and Theorem 5.1 follows.
3.5.1
A proof of Theorem 5.2
To finish the proof of the Theorem 5.1 we will prove Theorem 5.2. We start with
the energy inequality
1 2 d dt ku(t)k 2 5
2 +ku(t)k
2
7
2 ≤Cku(t)kF1kuk
2
5 2 .
Using the same technique employed to proof the Proposition 2.1 to estimate the F1 norm of u, we get
X
k∈Z˙3
|k||ubk| ≤(akuk25 2
+bkuk27 2
)12
X
k∈Z˙3
1 a|k|3+b|k|5
1 2
≤(akuk25
2 +bkuk
2 7 2) 1 2 8 Z
|x|>12
1
a|x|3+b|x|5dx
!12
≤(akuk25
2 +bkuk
2 7 2) 1 2 32π Z ∞ 1 2 1
r(a+br2)dr
!12
≤(akuk25
2 +bkuk
2 7 2) 1 2 16π a log 4a b + 1
12
.
As consequence of the elementary inequalities (|x|+|y|)12 ≤ |x| 1 2 +|y|
1
2 and |2ab| ≤
|a|2+|b|2 the energy inequality can be rewritten as
1 2 d dtku(t)k 2 5 2 ≤C
ku(t)k35 2 log
1 2
4a b + 1
+ b aku(t)k 4 5 2 log 4a b + 1.
Now, we choose a =ku(t)k5
2 and b = 4. Since limt→Tku(t)k
5
2 = +∞, there exists
a t0 such that ||u(T −t)||5
2 > e−1 for all t ∈ (0, t0), so log (ku(t)k+ 1) > 1 if
t∈(T −t0, T). Then we have
d
dtku(t)k
2
5 2 ≤C
ku(t)k35 2 log
ku(t)k5 2 + 1
for all t ∈(T −t0, T).
Next, we integrate between (T −t, T) witht ∈(0, t0):
Ct≥
Z T
T−t
1 ||u(s)||3
5 2
log(||u(s)||5 2 + 1)
d dt||u(s)|| 2 5 2 dt, ≥ Z T
T−t
1 (||u(s)||5
2 + 1)
2log(||u(s)||5
2 + 1)||u(s)|| 5 2 d dt||u(s)|| 2 5 2 dt.
We want to estimate the last integral in the above inequality, so we make the change
of variable ||u(s)||5
2 + 1 =e
v, which produces
Ct >
Z ∞
log(||u(T−t)||5 2+1)
e−v
v dv.
Since
Z ∞
log(||u(t−t)||5 2
e−vdv <+∞and the function f(y) = 1
y is convex on R
+, we can
use Jensen’s inequality, so the inequality becomes
ct > 1
(||u(T −t)||5 2 + 1)
2R∞
log(||u(T−t)||5 2
+1)ve
−vdv,
> 1
(||u(T −t)||5
2 + 1)(log(||u(T −t)|| 5
2 + 1) + 1)
,
as long as||u(T−t)||5
2 > e−1>1, since this implies that log(||u(T −t)|| 5
2 + 1)>1.
Finally, the previous inequality yields:
(c+1)t > 1
4||u(T −t)||5
2 log(2||u(T −t)|| 5 2)
> 1
4||u(T −t)||5
2 log(4(c+ 1)||u(T −t)|| 5 2)
.
3.6
The case
s
=
1
2
In the final section of this chapter we want to show an inequality which suggests a
possible lower blow up rate for the cases= 1
2. In passing, we shall also give another proof a Prodi-Serrin-Ladyzhenskaya type result for the Navier-Stokes equations (the
one implied by our blow-up result for the case s= 3 2 ).
We recall from [18] (p. 39) that we have:
rxr−1(x−y)< xr−yr < ryr−1(x−y);
therefore,
||k|r− |q|r| ≤r|k−q| |k|r−1+|q|r−1
.
Using the last inequality to estimate the nonlinear term in the energy inequality for
0< s <1 we have,
X
q
X
k
|k|2shq.buk−qihubkbuqi
≤X
q
X
k
|k|shq.
b
uk−qihbukbuqi ||k|
s− |q|s|
≤sX
q
X
k
|k|shq.
b
uk−qihubkbuqi|k−q|
1 |k|1−s +
1 |q|1−s
≤s
( X
q
|buq|
|q|1−s
X
k
|k|s+1|
b
uk||k−q||ubk−q|
+X
k
|ubk|
|k|1−s
X
q
(|k−q|s+|q|s)|q||
b
uq||k−q||ubk−q| )
.
≤3sX
k
|ubk|
On the other hand,
X
k
|ubk|
|k|1−s =
X
k
b
uk
p
a|k|2s+b|k|2s+2
|k|1−spa|k|2s+b|k|2s+2
≤C akuk
2
s+b|uk
2
s+1
√ ab
!12
.
If a=kuk2s+1 and b =kuk2s, we obtain
X
k
|buk|
|k|1−s ≤Ckuk
1 2
s kuk
1 2
s+1.
As a consequence, the energy inequality yields
dkuk2s
dt +c1kuk
2
s+1 ≤c2skuk
1 2
s kuk
3 2
s+1kuk1. (3.14)
Notice that the last inequality implies that ku(t)ks is bounded as long as
Z t
0
ku(τ)k41dτ <∞. Sincekuk1 ≤ kuk
1 2 1 2 kuk 1 2 3 2
, fors = 1
2 from (3.14) we obtain
dkuk21 2
dt
1 c2kuk1
2 −c1
!
≤ kuk23 2 .
Moreover, we have that
Z dy
c2y
1 2 −c1
= 2 c1
y12 + c1
c2
lnc2y
1 2 −c1
.
Therefore, we can conclude that
2 c2 ku(t)k1 2 + c1 c2
ln(c2ku(t)k1
2 −c1)−
ku(t0)k1 2 +
c1
c2
ln(c2ku(t0)k1 2 −c1)
≤
Z t
t0
ku(τ)k23 2 dτ.
(3.15)
The last inequality suggests the possibility of
Z t
t0
ku(τ)k23 2
dτ ≤ 2 c2
ku(t)k1 2
+ c1 c2
ln(c2ku(t)k1 2
−c1)
Thus, from the blow-up rate for the case s = 3
2 it would be possible to show that
ln
1 T −t
−ln
1 T −t0
≤ 2 c2
ku(t)k1 2 +
c1
c2
ln(c2ku(t)k1 2 −c1)
;
which gives a possible lower blow-up rate for the case s= 1 2.
As a final comment, notice that inequality (3.15) via the
Escauriaza-Seregin-ˇ
Sver´ak theorem, we obtain another proof the following result for the Navier-Stokes
equations (which is a consequence of the blow-up rates proved in this chapter).
Theorem. Letu(x, t)be a solution to the Navier-Stokes equations defined on(0, T). If u∈L2(0, T); ˙H32(R3)
CHAPTER
4
Self-similar solutions and backward
uniqueness
4.1
Introduction
In a remarkable paper [32], Neˇcas, R˚uˇziˇcka and ˇSver´ak proved the non existence of
Leray’s backward self-similar solutions to the Navier-Stokes equations which belongs
to L3(R3) space. Later Tsai generalized their results to the Lp(R3) spaces, 3< p ≤ ∞, in [39]. Recently, results in [20], [23], [36] and [37] generalized the result in [32]
showing that solutions which remain bounded in the space L3(R3) are regular. In addition, Chemin and Planchon in [6] proved an extension of the regularity theorem
in [36] and [37] for solutions to the Navier-Stokes equations which remain uniformly
bounded in the homogeneous Besov spaces ˙B
3
p−1
p,q (R3).
Our aim in this chapter is to show a possible approach to the non existence
of backward self-similar solutions. In our approach, we use the mild formulation
of the Navier-Stokes equation in the Fourier space with initial condition in the
homogeneous Sobolev space ˙H12(R3). The idea, then, is to study the behavior of
the solutions for frequencies close to 0. The weakness of this formulation is that
our assumptions and hypothesis are less general than those that were necessary to
prove the non existence theorem in [32]. Also, we were led, via this approach, to a
backward uniqueness result for the Navier-Stokes equation. In section 4.4.1 of this
chapter, we also prove a backward uniqueness result using Fourier analysis and some
elementary arguments.
4.2
Preliminaries
Recall that a mild solution to the Navier-Stokes equations in a Banach spaces X is
a functionu∈C([0, T);X) such that
u(x, t) =e∆tu0(x) +
Z t
0
e∆(t−τ)P∇(u(x, τ)⊗u(x, τ))dτ,
∇ ·u(x, t) = 0,
(4.1)
where P is the Leray projector. The corresponding formulation in the Fourier
do-main is
b
ul(ξ, t) =ubl(ξ,0)e−4π2|ξ|2t+i2πξj
δml−
ξmξl
|ξ|2
Z t
0
e−4π2|ξ|2(t−τ)(ubj ?bum)(ξ, τ)dτ,
b
u1(ξ, t)ξ1+ub
2(ξ, t)ξ 2 +ub
3(ξ, t)ξ 3 = 0,
(4.2)
where bu(ξ, t) is the Fourier transform of u with respect to the spatial variable. In addition, a backward self-similar solution is a solution which can be written as
u(x, t) = p 1
2a(T −t)U
x
p
2a(T −t)
!
, U ∈H˙ 12(R3),
where T, a >0. The Fourier transform of this type of solution is
b
u(ξ, t) = 2a(T −t)Ub
In this way
(ubj ?bum)(ξ) =
Z R3
b
Uj(ξ−q)p2a(T −t)Ubm
qp2a(T −t)dq.
So, after the change of variableqp2a(T −τ)→win the integral which defines the convolution Ubj ?Ubm, we have that (4.2) becomes
2a(T −t)Ubl
ξp2a(T −t)e4π2|ξ|2t−2aTUbl
ξ√2aT
=i2πξj
δml−
ξlξm
|ξ|2
Z t
0
e4π2|ξ|2τ(2a(T −τ))12(Ubj ?Ubm)
ξp2a(T −τ)dτ, (4.3)
where ? denotes the convolution. First, we do some calculations to give us an idea
about the behavior of the Fourier transform Ub of U near zero. Recall the product
rule for homogeneous Sobolev spaces, i.e. if f, g∈ H˙s(R3) with 0< s < 3
2 we have that the product f g ∈H˙2s−32 (see Corollary 2.55 p. 90 [1]). Therefore, we obtain
| · |
2s−32
b
U ?Ub
L2(
R3)
≤C(s)kUk2Hs(
R3). (4.4)
Now, let h(ξ) ∈ L2(R3) and 1
2 ≤ s ≤ 5
4. Taking the product with e
−4π2|ξ|2T
h(ξ),
integrating, and changing variables (ξp2a(T −τ)→y), from (4.3) we deduce
Z R3
2a(T −t)Ub(ξ p
2a(T −t))e−4π2|ξ|2(T−t)−2aTU(ξb
√
2aT)e−4π2|ξ|2Th(ξ)dξ
≤ Z t 0 Z R3
|y|e−2π2|y|2/a
|y|32−2s
(2a(T −τ))32
|y|2s−3 2
b
U ?Ub
(y)h p y 2a(T −τ)
! dydτ ≤ Z t 0 M(s)
(2a(T −τ))32
| · |
2s−3 2
b
U ?Ub
L2(
R3)
h p ·
2a(T −τ)
!
L2(
R3)
dτ,
where M(s) = sup
ξ∈R3
e−2π2|ξ|2/a|ξ|52−2s
. From the identity
h p ·
2s(T −τ)
!
L2(
R3)
= (2a(T −τ))34 khk
L2(
and inequality (4.4), the last estimate yields
Z
R3
2a(T −t)Ub(ξ p
2a(T −t))e−4π2|ξ|2(T−t)−2aTUb(ξ
√
2aT)e−4π2|ξ|2T
h(ξ)dξ
≤C(T14 −(T −t) 1
4)kUk2˙
Hs||h||L2(
R3),
(4.5)
where C is a constant which depends ons and a. In this way, we can conclude
Z R3
2a(T −t)Ub(ξ p
2a(T −t))e−4π2|ξ|2(T−t)−2aTU(ξb
√
2aT)e−4π2|ξ|2T
2
dξ
≤CT12 kUk4˙
Hs.
(4.6)
If we suppose that U(ξ) can be expressed asb Ub(ξ) =
g(ξ)
|ξ|β with g continuous and
g(0) 6= 0, from (4.6) we have that.
Z R3
1 |ξ|2β
2a(T −t)
1−β
2g(ξp2a(T −t))e−4π
2|ξ|2(T−t)
−2aT1−β2g(ξ
√
2aT)e−4π2|ξ|2T
2
dξ
≤CT12 kUk4˙
Hs.
From the procedure to get inequality (4.6) we notice that the left-hand side of
equation (4.3) is an L2(R3) function, therefore we can apply to the last inequality the Dominated Convergence Theorem. In this way, if β ≥2 we get a contradiction.
Thus, if Ub(ξ) =
g(ξ)
|ξ|β (with g(ξ) 6= 0 and continuous), we have to impose the
conditionβ < 3
2. This fact seems to indicate that ifu(x, t) is a backward self-similar solutions of (4.2),U should be aL2(R3) function, hence the following conjecture: As a consequence we would obtain, from the Escauriaza-Seregin-ˇSver´ak Theorem (see
Theorem 1.2 in [20]), that a bacwkard self-similar solution which belongs to ˙Hs(R3), 1
2 ≤s < 3
2, must be identically equal to zero.
4.3
The case of self-similar solutions revisited
Theorem 3.1. Let u(x, t) = p 1
2a(T −t)U
x
p
2a(T −t)
!
be a backward
self-similar mild solution in L∞((0, T),H˙s(R3))∩L2((0, T),H˙s+1(R3)), with 1
2 ≤s < 3 2
such that
Z
R3U(x)dx= 0 and
Z R3
(1 +|x|)|U(x)|dx < +∞. Then U is identically
equal to zero.
Proof of Theorem 3.1. As a result of the condition
Z R3
(1 +|x|)|U(x)|dx < ∞ the
function Ub(ξ) is uniformly continuous on R3, which implies that it is square
inte-grable on the ball with center zero and radius one. Since U ∈H˙s(R3) we have that
Z
|ξ|≥1
|Ub(ξ)|2dξ < ∞; therefore, Ub ∈ L2(R3). Then U(x)U(x) is a L1(R3) function.
It follows that the convolution
Z R3
b
U(ξ−q)Ub(q)dq
is continuous in ξ. So, if we fix ξ=|ξ|v, wherev belongs to the unit sphere, divide
equation (4.3) by |ξ|,and then let|ξ| →0, by the Dominated Convergence Theorem
we get
(2a(T −t1))
3
2DvUbl(0)−(2aT) 3
2DvUbl(0)
=i2πvj(δlm−vmvl)
Z t1
0
(2a(T −s))12
Z R3
b
Uj(−q)Ubm(q)dqds,
where Dv denotes the directional derivative. Therefore, if t1 =T,
DvUbl(0) =−i
2π
3avj(δml−vmvl)
Z R3
b
U
j
(q)Ubm(q)dq. (4.7)
Moreover, since theU is divergence free and as the hipothesis
Z R3
U(x)dx = 0 implies
that Ub(0) = 0. If Ub is differentiable atξ = 0, we have that
b
U1(ξ)
|ξ| ξ1
|ξ| +
b
U2(ξ)
|ξ| ξ2
|ξ| +
b
U3(ξ)
|ξ| ξ3
|ξ| = 0,
taking |ξ| →0, we get
