Một số tính chất định tính của phương trình navier stokes (tt)

VIETNAM ACADEMY OF SCIENCE AND TECHNOLOGY
INSTITUTE OF MATHEMATICS

DAO QUANG KHAI

SOME QUALITATIVE PROPERTIES OF SOLUTIONS
TO NAVIER-STOKES EQUATIONS

Speciality: Differential and Integral Equations
Speciality code: 62 46 01 03

SUMMARY
DOCTORAL DISSERTATION IN MATHEMATICS

HANOI 2017


Introduction
Navier-Stokes equations are useful because they describe the motion of fluids.
They may be used to model the weather, ocean currents, the design of aircraft and
cars, the study of blood flow, the analysis of pollution, and many other things.
The Navier-Stokes equations are also of great interest in a purely mathematical
sense. They have particular importance within the development of the modern
mathematical theory of partial differential equations. Although the theory of partial
differential equations has undergone a great development in the twentieth century,
some fundamental questions remain unresolved. They are essentially concerned
with the global existence and uniqueness of solutions, as well as their asymptotic
behavior. More precisely, given a smooth datum at time zero, will the solution
of the Navier-Stokes equations continue to be smooth and unique for all time?
This question was posed in 1934 by J. Leray and is still without answer, neither
in the positive nor in the negative. There is no uniqueness proof except for over

small time intervals and it has been questioned whether the Navier-Stokes equations
really describe general flows. But there is no proof for non-uniqueness either.
Uniqueness of the solutions of the equations of motion is the cornerstone of
classical determinism (J. Earman 1986). If more than one solution were associated
to the same initial data, the committed determinist will say that the space of the
solutions is too large, beyond the real physical possibility, and that uniqueness can
be restored if the unphysical solutions are excluded.
A question intimately related to the uniqueness problem is the regularity of the
solution. Do the solutions to the Navier-Stokes equations blow-up in finite time?
The solution is initially regular and unique, but at the instant T when it ceases to
be unique (if such an instant exists), the regularity could also be lost.
One may imagine that blow-up of initially regular solutions never happens, or that it
becomes more likely as the initial norm increases, or that there is
blow-up, but only on a very thin set of probability zero. The best result in this
direction concerning the possible loss of smoothness for the Navier-Stokes equations
was obtained by L. Caffarelli (1982), R. Kohn and L. Nirenberg (1998), who proved
that the one-dimensional Hausdorff measure of the singular set is zero.
We can say that if ”some quantity” turns out to ”be small”, then the NavierStokes equations are well-posed in the sense of Hadamard (existence, uniqueness and
stability of the corresponding solutions). For instance, the unique global solution
exists when the initial value and the exterior force are small enough, and the solution is smooth depending on smoothness of these data. Another quantity that
can be small is the dimension. If we are in dimension n = 2, the situation is easier
than in dimension n = 3 and completely understood (P. Lions (1966), R. Temam
1


2

(1979)). Finally, if the domain Ω ⊂ R3 is small, in the sense that Ω = ω × (0, ) is
thin in one direction, say, then the question is also settled by M. Wiegner (1999).
In this thesis, we study well-posedness for the Cauchy problem of incompressible

Navier-Stokes equations

 ∂t u = ∆u − ∇ · (u ⊗ u) − ∇p,
div(u) = 0,
(0.1)

u0 (0, x) = u0 ,
where t ∈ R+ , x ∈ Rd (d ≥ 2), u = (u1 , u2 , ..., ud ) denote the flow velocity vector
and p(t, x) describe the scalar pressure, ∇ = (∂1 , ∂2 , ..., ∂d ) is the gradient operator,
∆ = ∂12 + ∂22 + ... + ∂d2 is the Laplacian, u0 (x) = (u01 , u02 , ..., u0d ) is a given initial
datum with div(u0 ) = ∂1 u01 + ∂2 u02 + ... + ∂d u0d = 0. For a tensor F = (Fij ) we define
d
the vector ∇ · F by (∇ · F )i = j=1 ∂j Fij and for two vectors u and v, we define
their tensor product (u ⊗ v)ij = ui vj . It is to see that (0.1) can be rewritten in the
following equivalent form:
∂t u = ∆u − P∇ · (u ⊗ u),
u0 (0, x) = u0 ,

(0.2)

where the operator P is the Helmholtz-Leray projection onto the divergence-free
fields
(Pf )j = fj +
Rj Rk fk ,
1≤k≤d

here Rj are the Riesz transforms defined as
∂j
iξj
Rj = √

gˆ(ξ)
i.e. Rj g(ξ) =
|ξ|
−∆
with ˆ denoting the Fourier transform. It is known that (0.2) is essentially equivalent
to the following integral equation:
t
t∆

e(t−τ )∆ P∇ · (u ⊗ u)dτ,

u = e u0 −

(0.3)

0

where the heat kernel et∆ is defined as
2

et∆ u(x) = ((4πt)−d/2 e−|·|

/4t

∗ u)(x).

Note that (0.1) is scaling invariant in the following sense: if u solves (0.1), so does
uλ (t, x) = λu(λ2 t, λx) and pλ (t, x) = λ2 p(λ2 t, λx) with initial data λu0 (λx). A
function space X defined in Rd is said to be a critical space for (0.1) if its norm
is invariant under the action of the scaling f (x) → λf (λx) for any λ > 0, i.e.,

f (·) = λf (λx) . It is easy to see that the following spaces are critical spaces for
NSE:
d

d
−1,∞
−1,∞
(Rd )(q<∞) → BM O−1 (Rd ) → B˙ ∞
(Rd ). (0.4)
H˙ 2 −1 (Rd ) → Ld (Rd ) → B˙ qq

It is remarkable feature that the Navier-Stokes equations are well-posed in the sense
of Hadarmard (existence, uniqueness and continuous dependence on data) when the
initial data are divergence-free and belong to the critical function spaces (except


3
d

−1,∞
d
−1,∞
B˙ ∞
) listed in (0.4) (M. Cannone (1995) for H˙ 2 −1 (Rd ), Ld (Rd ), and B˙ qq
(Rd ),
see H. Koch (2001) for BM O−1 (Rd ), and the recent ill-posedness result J. Bourgain
−1,∞
(2008) for B˙ ∞
(Rd )). Very recently, ill-posedness of Navier-Stokes equations in
−1

critical Besov spaces B˙ ∞,q
was investigated B. Wang (2015) and finite time blowup
for an averaged three-dimensional Navier-Stokes equation was investigated T. Tao.
In the 1960s, mild solutions were first constructed by Kato and Fujita (1962) and
Kato and Fujita (1964) that are continuous in time and take values in the Sobolev
spaces H s (Rd ), (s ≥ d2 − 1), say u ∈ C([0, T ]; H s (Rd )). In 1992, a modern treatment
for mild solutions in H s (Rd ), (s ≥ d2 −1) was given by Chemin (1992). In 1995, using
the simplified version of the bilinear operator, Cannone proved the existence of mild
solutions in H˙ s (Rd ), (s ≥ d2 − 1), see M. Cannone (1995). Results on the existence
of mild solutions with value in Lq (Rd ), (q > d) were established in the papers of
Fabes, Jones and Rivi`ere (1972) and of Y. Giga (1986). Concerning the initial data
in the space L∞ , the existence of a mild solution was obtained by Cannone and
Meyer (1995). Moreover, in Cannone and Meyer (1995), they also obtained theorems on the existence of mild solutions with value in the Morrey-Campanato space
M2q (Rd ), (q > d) and the Sobolev space Hqs (Rd ), (q < d, 1q − ds < d1 ), and in general
in the so-called well-suited space W for the Navier-Stokes equations. The NavierStokes equations in the Morrey-Campanato spaces were also treated by T. Kato
(1992) and Taylor M. Taylor (1992). In 1981, F. Weissler (1981) gave the first existence result for mild solutions in the half space L3 (R3+ ). Then Giga and Miyakawa
(1985) generalized the result to L3 (Ω), where Ω is an open bounded domain in R3 .
Finally, in 1984, T. Kato (1984) obtained, by means of a purely analytical tool
(involving only H¨older and Young inequalities and without using any estimate of
fractional powers of the Stokes operator), an existence theorem in the whole space
L3 (R3 ). In (M. Cannone (1995), M. Cannone (1997), M. Cannone (1999)), Cannone
showed how to simplify Kato’s proof. The idea is to take advantage of the structure
of the bilinear operator in its scalar form. In particular, the divergence ∇ and heat
et∆ operators can be treated as a single convolution operator. In 1994, Kato and
Ponce (1994) showed that NSE are well-posed when the initial data belong to the
d
−1
˙
homogeneous Sobolev spaces Hqq (Rd ), (d ≤ q < ∞).
In this thesis, we use the progress achieved in the field of harmonic analysis for

the last fifteen years to study the Navier-Stokes equations. We mean the use of
the Fourier transform and its properties, better suited for the study of nonlinear
problems.
Chapter 1 is devoted to the recalling of some well-known results of harmonic
analysis.
In Chapter 2, we apply these tools to the study of the Cauchy problem for the
Navier-Stokes equations.
Section 2.1 presents the general shift-invariant space of distributions and some
Sobolev spaces over a shift-invariant Banach space of distributions.
From Sections 2.2 to Section 2.6, we construct mild solutions to (0.3), a natural
t
approach is to iterate the transform u → et∆ u0 − 0 e(t−τ )∆ P∇ · (u ⊗ u)dτ and to
find a fixed point u for this transform. This is the so-called Picard contraction
method already in use by C. Oseen (1927) to establish the local existence of a clas-


4

sical solution to the Navier-Stokes equations for a regular initial value. By Theorem
1.3.1 (see Section 1.5 of Chapter 1), to find a fixed point u for the equation (0.3),
we need to try to find a Banach space ET of functions defined on (0, T ) × Rd so that
the bilinear operator B defined by
t

e(t−s)∆ P∇ · (u ⊗ v)ds

B(u, v)(t) =

(0.5)

0

is bounded from ET × ET → ET . Section 2.2 to Section 2.6 are devoted to construct
examples of such spaces ET . The obtained results have a standard relation between
existence time and data size: large time with small data or large data with small
time.
In Section 2.2, we study local and global well-posedness for the Navier-Stokes equad
−1
tions with initial data in homogeneous Sobolev spaces H˙ qq (Rd ) for d ≥ 2, 1 < q ≤
d. The obtained result improves the known ones for q = 2 and q = d. These cases
were considered by many authors, see (M. Cannone (1995), J. Chemin (1992), H.
Fujita and T. Kato (1964),T. Kato (1984), P. G. Lemarie-Rieusset (2002)).
In Section 2.3, we study local well-posedness for the Navier-Stokes equations with arbitrary initial data in homogeneous Sobolev spaces H˙ ps (Rd ) for d ≥ 2, p > d2 , and dp −
d
1 ≤ s < 2p
. The obtained result improves the known ones for p > d and s = 0
(see M. Cannone (1995), M. Cannone and Y. Meyer (1995)). In the case of critical indexes s = dp − 1, we prove global well-posedness for Navier-Stokes equations
when the norm of the initial value is small enough. This result is a generalization of the ones in M. Cannone (1999) and P. G. Lemarie-Rieusset (2002) in which
(p = d, s = 0) and (p > d, s = dp − 1), respectively.
In Section 2.4, we introduce and study Sobolev-Fourier-Lorentz spaces H˙ Ls p,r (Rd ).
We then study local and global well-posedness for the Navier-Stokes equations with
d
−1
initial data in critical spaces H˙ Lp p,r (Rd ) with d ≥ 2, 1 ≤ p < ∞, and 1 ≤ r < ∞.
In Section 2.5, we study local well-posedness for the Navier-Stokes equations with
the arbitrary initial value in homogeneous Sobolev-Lorentz spaces H˙ Ls q,r (Rd ) :=
(−∆)−s/2 Lq,r for d ≥ 2, q > 1, s ≥ 0, 1 ≤ r ≤ ∞, and dq − 1 ≤ s < dq . This result
improves the known results for q > d, r = q, s = 0, see M. Cannone (1995) and for
q = r = 2, d2 − 1 < s < d2 , see M. Cannone (1995) and J. Chemin 1992.
In the case of critical indexes (s = dq − 1), we prove global well-posedness for NSE

when the norm of the initial value is small enough. The result is a generalization
of the result in M. Cannone (1997) for q = r = d, s = 0.
In Section 2.6, for 0 ≤ m < ∞ and index vectors q = (q1 , q2 , ..., qd ), r = (r1 , r2 , ..., rd ),
where 1 < qi < ∞, 1 ≤ ri ≤ ∞, and 1 ≤ i ≤ d, we introduce and study
mixed-norm Sobolev-Lorentz spaces H˙ Lmq,r . Then we investigate the existence and
uniqueness of solutions to the Navier-Stokes equations in the spaces Q := QT =
Lp ([0, T ]; H˙ Lmq,r ) where p > 2, T > 0, and initial data is taken in the class I = {u0 ∈
(S (Rd ))d , div(u0 ) = 0 : e·∆ u0 Q < ∞}. In the case when m = 0, q1 = q2 = ... =
qd = r1 = r2 = ... = rd , our results recover those of Faber, Jones and Riviere (1972).
In Chapter 3, using the method of Foias-Temam, we show the vanishing of
Hausdorff measure of the singular set in time of weak solutions to the Navier-Stokes
equations in the 3D torus.


Chapter 1

Preliminaries
1.1

This section is devoted to the recalling of some well-known results
of harmonic analysis.

This section is devoted to the recalling of some well-known results of harmonic
analysis
1.1.1.

The Littlewood-Paley decomposition

We take an arbitrary function ϕ in the Schwartz class S(Rd ) and whose Fourier
ˆ

= 0 if |ξ| ≥ 32 , and
transform ϕˆ is such that 0 ≤ ϕ(ξ)
ˆ
≤ 1, ϕ(ξ)
ˆ
= 1 if |ξ| ≤ 34 , ϕ(ξ)
let ψ(x) = 2d ϕ(2x) − ϕ(x), ϕj (x) = 2dj ϕ(2j x), j ∈ Z, ψj (x) = 2dj ψ(2j x), j ∈ Z. We
denote by Sj and ∆j , respectively, the convolution operators with ϕj and ψj . The
set {Sj , ∆j }j∈Z is the Littlewood-Paley decomposition.
1.1.2.

The Besov spaces

The Littlewood-Paley decomposition is very useful because we can define
(independently of the choice of the initial function ϕ) the following
(inhomogeneous) Besov spaces.
Definition 1.1.1. Let 0 < p, q ≤ ∞ and s ∈ R. Then a tempered distribution f
belongs to the (inhomogeneous) Besov space Bqs,p if and only if
S0 f

q

+

2

sj

∆j f

p
q

1
p

< ∞.

j≥0

For the sake of completeness, we also define the (inhomogeneous) Triebel-Lizorkin
spaces, even if we will not make a great use of them in the study of the Navier-Stokes
equations.
Definition 1.1.2. Let 0 < p ≤ ∞, 0 < q < ∞, and s ∈ R. Then a tempered
distribution f belongs to the (inhomogeneous) Triebel-Lizorkin space Fqs,p if and only
5


6

if
S0 f

q

sj

2 |∆j f |

+

p

1
p

< ∞.
q

j≥0

We are now ready to define the homogeneous version of the Besov and TriebelLizorkin spaces (G. Bourdaud (1993), G. Bourdaud (1988), M. Frazier (1991), J.
Peetre (1976)). If m ∈ Z, we denote by Pm the set of polynomials of degree ≤ m
with the convention that Pm = ∅ if m < 0. If p = 1 and s − d/q ∈ Z, we put
m = s − d/q − 1; if not, we put m = [s − d/q], with the brackets denoting the
integer part function.
Definition 1.1.3. Let 0 < p, q ≤ ∞ and s ∈ R. Then a tempered distribution f
belongs to the (homogeneous) Besov space B˙ qs,p if and only if
sj

2

1
p

p

∆j f

< ∞ and f =

q

j∈Z

∆j f in S /Pm .
j∈Z

Definition 1.1.4. Let 0 < p ≤ ∞, 0 < q < ∞, and s ∈ R. Then a tempered
distribution f belongs to the (homogeneous) Triebel-Lizorkin space F˙ qs,p if and only
if
sj

2 |∆j f |
j∈Z

p

1
p

< ∞ and f =
q

∆j f in S /Pm ,
j∈Z

with an analogous modification as in the inhomogeneous case when q = ∞.

1.2

The Navier-Stokes equations

This thesis studies the Cauchy problem of the incompressible Navier-tokes equations (NSE) in the whole space Rd for d ≥ 2,

 ∂t u = ∆u − ∇ · (u ⊗ u) − ∇p,
div(u) = 0,
(1.1)

u(0, x) = u0 ,
which is a condensed writing for

 1 ≤ k ≤ d, ∂t uk = ∆uk −
d
∂ u = 0,
 l=1 l l
1 ≤ k ≤ d, uk (0, x) = u0k .

d
l=1 ∂l (ul uk )

− ∂k p,

The unknown quantities are the velocity u(t, x) = (u1 (t, x), . . . , ud (t, x)) of the fluid
element at time t and position x and the pressure p(t, x). Taking the divergence
d
d
of (1.1), we obtain: ∆p = −∇ ⊗ ∇ · (u ⊗ u) = − k=1 l=1 ∂k ∂l (uk ul ). Thus, we
formally get the equations
∂t u = ∆u − P∇ · (u ⊗ u),

div(u) = 0,

(1.2)


7

where P is the Helmholtz Leray projection operator defined as Pf := f − ∇ ∆1 (∇ · f )
= (I − ∇⊗∇
∆ )f. We shall study the Cauchy problem for the equation (1.2) (looking
for a solution on (0, T ) × Rd with the initial value u0 ), and transform (1.2) into the
integral equation
t

e(t−s)∆ P∇ · (u ⊗ u)ds.

t∆

u = e u0 −

(1.3)

0

1.3

Outline of the dissertation

For T > 0, we say that u is a mild solution of NSE on [0, T ] corresponding to
a divergence-free initial datum u0 when u satisfies the integral equation (1.3). We

rewrite the equation (1.3) as following
u = U0 − B(u, u),
where

(1.4)

t

e(t−s)∆ P∇ · (u ⊗ v)ds and U0 = et∆ u0 .

B(u, v)(t) =

(1.5)

0

Then we will find a fixed point u for the equation (1.4). This is the so-called Picard
contraction method already in use by Oseen at the beginning of the 20th century to
establish the (local) existence of a classical solution to the Navier-Stokes equations
for a regular initial value, see C. Oseen (1927).
Theorem 1.3.1. Let X be a Banach space, and let B : X × X → X be a continuous
bilinear form such that exists η so that B(x, y) ≤ η x y for any x and y ∈ X.
Then for any fixed y ∈ X such that y < 1/(4η), the equation
x = y − B(x, x) has
√
a unique solution x ∈ X satisfying x ≤ R, with R =

1−

1−4η y

2η

.

By above Theorem, we need to try to find a Banach space ET so that the bilinear
operator B which is defined by (1.5) is bounded from ET × ET → ET .
Chapter 2 is devoted to construct examples of such spaces ET . The solutions we
obtain through the Picard contraction principle are called mild solutions. We call
a space ET if we may apply the Picard contraction principle as an admissible path
space for the Navier-Stokes equations, and the associated space ET as an adapted
value space.
Let us review some results. We will indicate what are the admissible path space ET
and the associated adapted space ET .
• Classical admissible spaces are provided by the Lp theory of Kato (1984):
- For d < p < ∞, C([0; T ]; Lp ) is admissible with the associated adapted space
Lp (Rd ).
- For p = d, the space
√
√
{f ∈ C([0; T ]; Ld ) : sup t f L∞ (dx) < ∞ and lim t f L∞ (dx) = 0}
0
t→0

is admissible with the associated adapted space Ld (Rd ).


8

• Prodi (1959) gave the following admissible spaces, plus the corresponding the

associated adapted space
d

ET = Lq ([0, T ], Lp ), ET = B˙ pp

−1,q

with

d 2
+ = 1 and d < p < ∞.
p q

• Cannone (1997) studied the space
α

α

{f ∈ C([0; T ]; Ld ) : sup t 2 f

Lq (dx)

< ∞ and limt 2 f
t→0

0
Lq (dx)

= 0},

(1.6)

d
with q > d and α = 1 − ,
q

(1.7)

which is admissible with the associated adapted space Ld (Rd ).
• Gallagher and Planchon (2002) studied a Besov spaces scale
2 d
d
+ −1,q
−1,q
d 2
ET = Lq B˙ pq p
, ET = B˙ pp
with + > 1.
p q

In Chapter 2 of this thesis we study some other admissible spaces with other
associated adapted spaces.
In Section 2.2 of Chapter 2:
- For 2 < q ≤ d and p be such that q < p < min (d−2)q
d−q , d + 2 , we consider the
admissible space
2+d−p
p
˙

L [0, T ]; Hp p
∩ L∞ [0, T ]; H˙ d/q−1
q

d/q−1
which is admissible with the associated adapted space H˙ q
(Rd ).
- For 1 < q ≤ 2 we consider the admissible space

L2q [0, T ]; H˙

d+2−2q
q
dq
d+1−q

∩ L∞ [0, T ]; H˙ qd/q−1

d/q−1
which is admissible with the associated adapted space H˙ q
(Rd ).

In Section 2.3 of Chapter 2:
d
- For p > d2 , dp − 1 ≤ s < 2p
, 1q = p1 − ds , and r > max{p, q}, we consider the
r
admissible space Kq,T
∩ L∞ ([0, T ]; H˙ ps ) is admissible with the associated adapted
r

is made up of the functions u(t, x) such that
space H˙ ps (Rd ), where space Kq,T
α
α
sup t 2 u(t, x) Lr < ∞ and lim t 2 u(t, x) Lr = 0 with α = d( 1q − 1r ).

0
t→0

In Section 2.4 of Chapter 2: For s ∈ R and 1 ≤ p, r ≤ ∞ we introduce and
study the Sobolev-Fourier-Lorentz spaces H˙ Ls p,r (Rd ).
- For 1 < p ≤ d, 1 ≤ r < ∞, and
we consider the admissible space

[ dp ]−1
[ dp ] 1
[ dp ]−1
1
1
p˜ be such that 2p + 2d < p˜ < min d , 2 + 2d ,
d
−1
p˜
∞
˙
K d ,1,T ∩L [0, T ]; HLp p,r which is admissible with

[d
p]

d
−1
p
Lp,r
α
2

p˜
the associated adapted space H˙
(Rd ), where the space Kp,r,T
is made up by the
α
functions u(t, x) such that sup t u(t, x) dp −1 < ∞ and lim t 2 u(t, x) dp −1 = 0
0
H˙

˜
Lp,r

t→0

H˙

˜
Lp,r


9

with α = d

1
p

−

1
p˜

.
d

−1
q
- For p ≥ d, r ≥ 1, and q > p, we consider the admissible space Kd,1,T
∩L∞ ([0, T ]; H˙ Lp p,r )
d

−1
is admissible with the associated adapted space H˙ Lp p,r (Rd ).
- For d − 1 < s < d and r ≥ 1, we consider the admissible space Ks,r,T ∩
˙ d−1 d
L∞ ([0, T ]; H˙ Ld−1
1,r ) which is admissible with the associated adapted space HL1,r (R ),
α
where the space Ks,r,T is made up by the functions u(t, x) such that sup t 2 u(t, x) H˙ s

< ∞ and lim t
t→0

α
2

0
u(t, x)

H˙ Ls 1,r

= 0 with α = s + 1 − d.

In Section 2.5 of Chapter 2: For q > 1, 1 ≤ r ≤ ∞ and 0 ≤ s < dq , we
introduce and study the Sobolev-Lorentz spaces H˙ Ls q,r (Rd ), which are generalizations
of the classical Sobolev spaces H˙ qs (Rd ). For s ≥ 0, q > 1, r ≥ 1, ds < 1q ≤ s+1
d ,
s 1
+ ds < 1q˜ < min 12 + 2d
, q , we consider the admissible
s,˜
q
space Kq,1,T
∩ L∞ ([0, T ]; H˙ Ls q,r ) which is admissible with the associated adapted
s,˜
q
space H˙ Ls q,r (Rd ), where space Kq,r,T
is made up by the functions u(t, x) such that
α
α
sup t 2 u(t, x) H˙ s < ∞ and limt 2 u(t, x) H˙ s = 0 with α = d( 1q − 1q˜).

and q˜ be such that

Lq˜,r

0
1
2

1
q

t→0

Lq˜,r

In Section 2.6 of Chapter 2: For 0 ≤ m < ∞ and index vectors q =
(q1 , q2 , ..., qd ) and r = (r1 , r2 , ..., rd ), where 1 < qi < ∞, 1 ≤ ri ≤ ∞ for i = 1, 2, .., d,
we introduce and study mixed-norm Sobolev-Lorentz spaces H˙ Lmq,r . For q > 1, r ≥
d
d
1, 2 < p < ∞, and m ≥ 0 be such that m < 12 i=1 q1i , p2 − m + i=1 q1i ≤ 1, 2 <
qi
< ∞, i = 1, 2, .., d, we consider the admissible space Lp ([0, T ]; H˙ Lmq,r )
1−

m
d
1

i=1 qi

m− 2 ,p

which is admissible with the associated adapted space BLq,r p

(a Besov space).

In Chapter 3: Using the method of Foias-Temam, we investigate the Hausdorff dimension of the singular set in time of weak solutions to the Navier-Stokes
equations.

L1,r


Chapter 2

Mild solutions in some Sobolev spaces
over a shift-invariant Banach space
In this chapter we investigate mild solutions to the Navier-Stokes equations in
some Sobolev spaces over a shift-invariant Banach space of distributions.

2.1

The Sobolev spaces over a shift-invariant Banach space of
distributions

We shall often use Banach spaces of distributions whose norms are invariant
under translations f E = f (x − x0 ) E and on which dilations operate boundedly.
Definition 2.1.1. (Shift-invariant Banach spaces of distributions.)
A shift-invariant Banach space of test functions is a Banach space E such that we

have the continuous embeddings S(Rd ) → E → S (Rd ) and so that:
(a) for all x0 ∈ Rd and for all f ∈ E, f (x − x0 ) ∈ E and f E = f (x − x0 ) E ,
(b) for all λ > 0 there exists Cλ > 0 so that for all f ∈ E f (λx) ∈ E and
f (λx) E ≤ Cλ f E ,
(c) D(Rd ) is dense in E.
In the following definitions, we introduce the Sobolev spaces and their homogeneous spaces over a shift-invariant Banach space of distributions. Before proceeding
to the definition of the Sobolev spaces, let us introduce several necessary notations.
For a real number s, the operators Λ˙ s and (Id − ∆)s/2 are defined through the
Fourier transform by
Λ˙ s f

∧

(ξ) = |ξ|s fˆ(ξ) and (Id − ∆)s/2 f

∧

(ξ) = 1 + |ξ|2

s/2

fˆ(ξ).

Definition 2.1.2. (Sobolev spaces.)
Let E be a shift-invariant Banach space of distributions. Then, for s ∈ R, the
space HEs is defined as the space (Id − ∆)−s/2 E, equipped with the norm f H s =
E

s/2

(Id − ∆)

f

E

.

Definition 2.1.3. ( Homogeneous Sobolev spaces.) Let E be a shift-invariant Banach space of distributions. Then, for s ∈ R, the space H˙ Es is defined as the closure
10


11

of the space S0 = f ∈ S : 0 ∈
/ Suppfˆ in the norm f

s
H˙ E

= Λ˙ s f

E

.

In the rest of this chapter, we investigate mild solutions to NSE in the following
Sobolev spaces over a shift-invariant Banach space of distributions:
- Sobolev spaces and homogeneous Sobolev spaces over the Lebesgue spaces,
(Sections 2.2 and 2.3).

- Homogeneous Sobolev spaces over the Fourier-Lorentz spaces, (Section 2.4).
- Homogeneous Sobolev spaces over the Lorentz spaces, (Section 2.5).
- Homogeneous Sobolev over the mixed-norm Lorentz spaces, (Section 2.6).
d

2.2

d

−1
−1
Mild solutions in H˙ qq and Hqq (1 < q ≤ d)

∞

d

−1
[0, T ]; H˙ qq

In this section, we investigate mild solutions to NSE in the spaces L
d
−1
when the initial data belong to the homogeneous Sobolev spaces H˙ qq (Rd ), (d ≥
2, 1 < q ≤ d). We obtain the existence of local mild solutions with arbitrary initial
value and existence of global mild solutions when the norm of the initial value is
small enough. The main results of this section are Theorems 2.2.3, 2.2.4, 2.2.5,
and 2.2.6, the lemmas we need in order to prove these theorems are Lemmas 2.2.1
and 2.2.2 devoted to the study of the bilinear operator B(u, v)(t) defined by (1.5).
We prove these theorems by combining Lemmas 2.2.1 and 2.2.2 with fixed point

algorithm Theorem 1.3.1.
Lemma 2.2.1. Let d ≥ 3, s ≥ 0, p > 1, r > 2, and T > 0 be such that ds <
1
s
2
d
1
p < 2 + 2d and r + p − s ≤ 1. Then the bilinear operator B(u, v)(t) is continuous
from Lr ([0, T ]; Hps ) × Lr ([0, T ]; Hps ) into Lr ([0, T ]; Hps ), and the following inequality
1

2

d

holds B(u, v) Lr ([0,T ];Hps ) ≤ CT 2 (1+s− r − p ) u
positive constant independent of T.

Lr ([0,T ];Hps )

v

Lr ([0,T ];Hps )

, where C is a

Lemma 2.2.2. Let d ≥ 3, 0 ≤ s < d, p > 1, r > 2, and T > 0 be such that p1 <
1
s 2
s+1

2 d
2 + 2d , p ≥ d , and r + p −s = 1. Then the bilinear operator B(u, v)(t) is continuous
d

from Lr [0, T ]; H˙ ps ×Lr [0, T ]; H˙ ps into L∞ [0, T ]; B˙ p˜p˜
have the inequality B(u, v)

d −1, r
2

L∞ [0,T ];B˙ p˜p˜

≤C u

−1, 2r

, where

Lr ([0,T ];H˙ ps )

v

1
p˜

= p2 − ds , and we

Lr ([0,T ];H˙ ps )

, where

C is a positive constant independent of T.
Theorem 2.2.3. Let 3 ≤ d ≤ 4 and 2 ≤ q ≤ d. There exists a positive constant
d/q−1
δq,d such that for all T > 0 and for all u0 ∈ H˙ q
(Rd ) with div(u0 ) = 0 satisfying
e·∆ u0

d/q−1
L4 [0,T ];H˙ 2dq/(2d−q)

≤ δq,d ,

(2.1)

d/q−1
d/q−1
NSE has a unique mild solution u ∈ L4 [0, T ]; H˙ 2dq/(2d−q) ∩L∞ [0, T ]; H˙ q
. Ded/q−1
d/q−1,2
noting w = u−e·∆ u0 , then we have w ∈ L4 [0, T ]; H˙ 2dq/(2d−q) ∩L∞ [0, T ]; B˙ q
.
Finally, we have e·∆ u0 4
u0 B˙ d/q−3/2,4
u0 H˙ qd/q−1 , in partic˙ d/q−1

ular, for arbitrary u0 ∈

L [0,T ];H2dq/(2d−q)
d/q−1

H˙ q
(Rd ) the inequality

2dq/(2d−q)

(2.1) holds when T (u0 ) is small


12

enough;
u0

and

d/q−3/2,4
B˙ 2dq/(2d−q)

there

exists

a

positive

constant

σq,d

such

that

for

all

≤ σq,d we can take T = ∞.

Theorem 2.2.4. Let 3 ≤ d ≤ 4 and 2 ≤ q ≤ d. There exists a positive constant
d
q −1

δq,d such that for all T > 0 and for all u0 ∈ Hq
e·∆ u0

(Rd ) with div(u0 ) = 0 satisfying
≤ δq,d ,

d/q−1

L4 [0,T ];H2dq/(2d−q)

(2.2)

d/q−1

d/q−1

NSE has a unique mild solution u ∈ L4 [0, T ]; H2dq/(2d−q) ∩ L∞ [0, T ]; Hq
.
Finally, we have e·∆ u0 4
≤ e·∆ u0 4
u0 Hqd/q−1 ,
d/q−1
d/q−1
L

[0,T ];H2dq/(2d−q)
d
q −1

in particular, for arbitrary u0 ∈ Hq
enough.

L

[0,∞);H2dq/(2d−q)

the inequality (2.2) holds when T (u0 ) is small

Theorem 2.2.5. Let d ≥ 3 and 2 < q ≤ d. Then for any p be such that q < p <
min (d−2)q
d−q , d + 2 , there exists a constant δq,p,d > 0 such that for all T > 0 and
d/q−1
for all u0 ∈ H˙ q
(Rd ) with div(u0 ) = 0 satisfying

e·∆ u0

≤ δq,p,d ,

2+d−p
p

Lp [0,T ];H˙ p

(2.3)

2+d−p
p

d/q−1
∩ L∞ [0, T ]; H˙ q
.

NSE has a unique mild solution u ∈ Lp [0, T ]; H˙ p

2+d−p
p

Denoting w = u−e·∆ u0 , then we have w ∈ Lp [0, T ]; H˙ p
Finally, we have e·∆ u0
u0

d −1
H˙ qq

Lp

2+d−p
p

[0,T ];H˙ p

≤ e·∆ u0

∩L∞ [0, T ]; B˙

d+p−2
−1, p2
p
dp
d+p−2

u0

2+d−p
p

Lp [0,∞);H˙ p

.

d

−1,p
B˙ pp

d/q−1
, in particular, for arbitrary u0 ∈ H˙ q
the inequality (2.3) holds when

T (u0 ) is small enough; and there exists a positive constant σq,p,d such that for all
u0 dp −1,p ≤ σq,p,d we can take T = ∞.
B˙ p

Remark 2.2.1. The case q = d was treated by several authors, see for example
Hongjie Dong (2007), T. Kato (1984), and M. Cannone (1995).
Theorem 2.2.6. Let d ≥ 3 and 1 < q ≤ 2. There exists a positive constant δq,d
d/q−1
such that for all T > 0 and for all u0 ∈ H˙ q
(Rd ) with div(u0 ) = 0 satisfying
e·∆ u0

L2q [0,T ];H˙

NSE has a unique mild solution u ∈ L
·∆

d+2−2q
q
dq
d+1−q

2q

Denoting w = u − e u0 , then we have w ∈ L
Finally, we have e·∆ u0

L2q [0,T ];H˙

d+2−2q
q
dq
d+1−q

≤ δq,d ,

[0, T ]; H˙

2q

d+2−2q
q
dq
d+1−q
d+2−2q
q
dq
d+1−q

[0, T ]; H˙

≤ e·∆ u0

L2q [0,∞);H˙

(2.4)

d/q−1
∩ L∞ [0, T ]; H˙ q
.
∞

∩L
d+2−2q
q
dq
d+1−q

d

−1,q
[0, T ]; B˙ qq
.

u0

(d+1)/q−2,2q
B˙ dq/(d+1−q)

d/q−1
u0 H˙ qd/q−1 , in particular, for arbitrary u0 ∈ H˙ q
(Rd ) the inequality (2.4) holds
when T (u0 ) is small enough; and there exists a positive constant σq,d such that for all
u0 B˙ (d+1)/q−2,2q ≤ σq,d we can take T = ∞.
dq/(d+1−q)

13

Remark 2.2.2. The case q = 2 was treated by several authors, see for example P.
G. Lemarie-Rieusset (2002), H. Fujita (1964), and M. Cannone (1995).

2.3

Mild solutions in Sobolev spaces of negative order

In this section, we present an different algorithm for constructing mild solutions
in the spaces L∞ ([0, T ]; H˙ ps (Rd )) to the Cauchy problem for NSE when the initial
d
.
datum belongs to the Sobolev spaces H˙ ps (Rd ), with d ≥ 2, p > d2 , and dp −1 ≤ s < 2p
This result improves the known results of Cannone (1997).
2.3.1.

Solutions to the Navier-Stokes equations with the initial value in the Sobolev
d
spaces H˙ ps (Rd ) for d ≥ 2, p > d2 , and dp − 1 ≤ s < 2p

The main result of this subsection is Theorem 2.3.4, The lemmas we need in
order to prove Theorem 2.3.4 are Lemmas 2.3.1, 2.3.2 and 2.3.3 devoted to the
study of the bilinear operator B(u, v)(t) defined by (1.5). We prove Theorem 2.3.4
by combining these lemmas with fixed point algorithm Theorem 1.3.1.
s
We define the space Np,T
which is made up of the functions u(t, x) such that
u N s := sup u(t, x) H˙ ps < ∞, and lim u(t, x) H˙ ps = 0, with p > 1 and
p,T

t→0

0
q˜
which is made up of the functions
s ≥ dp − 1. We now define the auxiliary space Kq,T
α
α
u(t, x) such that u Kq˜ := sup t 2 u(t, x) Lq˜ < ∞, and lim t 2 u(t, x) Lq˜ = 0,
q,T

with q˜ ≥ q ≥ d and α =

d( 1q

t→0

0− 1q˜).

Lemma 2.3.1. Suppose that u0 ∈ H˙ ps (Rd ) with p > 1 and dp − 1 ≤ s < dp . Then for
q˜
all q˜ satisfying q˜ > max{p, q}, where 1q = p1 − ds , we have e·∆ u0 ∈ Kq,∞
.
Lemma 2.3.2. Let p >
is continuous from

q˜

Kq,T

d
2

and

q˜
× Kq,T

d
p

−1 ≤ s <

d
2p .

Then the bilinear operator B(u, v)(t)
1
1
q = p
1
d
2 (1+s− p )

s
into Np,T
, where

following inequality holds B(u, v) N s ≤ CT
p,T
positive constant and independent of T.

− ds and q < q˜ ≤ 2p, and the
u

q˜
Kq,T

v

q˜
Kq,T

, where C is a

Lemma 2.3.3. Let d ≤ q ≤ q˜2 < ∞ and q < q˜1 < ∞ be such that one of the
d˜
q1
˜1 ≤ 2q, q ≤ q˜2 < ∞,
following conditions q < q˜1 < 2d, q ≤ q˜2 < 2d−˜
q1 , or 2d ≤ q
q˜1
or 2q < q˜1 < ∞, 2 < q˜2 < ∞, is satisfied. Then the bilinear operator B(u, v)(t) is
q˜1
q˜1
q˜2
continuous from Kq,T
× Kq,T

into Kq,T
, and we have the inequality B(u, v) Kq˜2 ≤
q,T

CT

1
d
2 (1− q )

u

q˜

1
Kq,T

v

q˜

1
Kq,T

, where C is a positive constant and independent of T.

d
Theorem 2.3.4. Let p > d2 and dp − 1 ≤ s < 2p
. Set 1q = p1 − ds .
(a) For all q˜ > max{p, q}, there exists a positive constant δq,˜q,d such that for all

T > 0 and for all u0 ∈ H˙ ps (Rd ) with div(u0 ) = 0 satisfying
1

d

d 1

s

1

T 2 (1+s− p ) sup t 2 ( p − d − q˜ ) et∆ u0
0
Lq˜

≤ δq,˜q,d ,

(2.5)


14
r
Kq,T
∩ L∞ ([0, T ]; H˙ ps ). In particular,

NSE has a unique mild solution u ∈

r>max{p,q}
arbitrary u0 ∈ H˙ ps (Rd )

the inequality (2.5) holds for
when T (u0 ) is small enough.
d
(b) If s = p − 1 then for all q˜ > max{p, d} there exists a constant σq˜,d > 0 such that
if u0 dq˜ −1,∞ ≤ σq˜,d and T = +∞ then the inequality (2.5) holds.
B˙ q˜

2.4

Mild solutions in the Sobolev-Fourier-Lorentz spaces

In this section, for s ∈ R and 1 ≤ p, r ≤ ∞, we introduce and study SobolevFourier-Lorentz spaces H˙ Ls p,r (Rd ). After that we show that the Navier-Stokes equations are well-posed when the initial datum belongs to the critical Sobolev-Fourierd
−1
Lorentz spaces H˙ Lp p,r (Rd ) with d ≥ 2, 1 ≤ p < ∞, and 1 ≤ r < ∞. The result that
is a generalization of the known results for the cases p = 1 and p = ∞ studied by
Le Jan (1997) and Zhen Lei Fang (2011), respectively.
2.4.1.

The Sobolev-Fourier-Lozentz Space

Definition 2.4.1. (Fourier-Lebesgue spaces). (See Lars Hormander (1976).)
For 1 ≤ p ≤ ∞, the Fourier-Lebesgue spaces Lp (Rd ) are defined as the space
F −1 (Lp (Rd )), ( p1 + p1 = 1), equipped with the norm f Lp (Rd ) := F(f ) Lp (Rd ) ,
where F and F −1 denote the Fourier transform and its inverse.
Definition 2.4.2. (Sobolev-Fourier-Lebesgue spaces).
For s ∈ R, and 1 ≤ p ≤ ∞, the Sobolev-Fourier-Lebesgue spaces H˙ Ls p (Rd ) are
defined as the space Λ˙ −s Lp (Rd ), equipped with the norm u H˙ s p := Λ˙ s u Lp .
L

Definition 2.4.3. (Fourier-Lorentz spaces). For 1 ≤ p, r ≤ ∞, the Fourier-Lorentz
spaces Lp,r (Rd ) are defined as the space F −1 (Lp ,r (Rd )), ( p1 + p1 = 1), equipped with
the norm f Lp,r (Rd ) := F(f ) Lp ,r (Rd ) .
Definition 2.4.4. (Sobolev-Fourier-Lorentz spaces).
For s ∈ R and 1 ≤ r, p ≤ ∞, the Sobolev-Fourier-Lorentz spaces H˙ Ls p,r (Rd ) are
defined as the space Λ˙ −s Lp,r (Rd ), equipped with the norm u H˙ s p,r := Λ˙ s u Lp,r .
L

2.4.2.

Solutions to the Navier-Stokes equations with the initial value in the critical
d
−1
spaces H˙ pp,r (Rd ) with 1 < p ≤ d and 1 ≤ r < ∞
L

The main result of this subsection is Theorem 2.4.3, The lemmas we need in
order to prove Theorem 2.4.3 are Lemmas 2.4.1 and 2.4.2 devoted to the study
of the bilinear operator B(u, v)(t) defined by (1.5). We prove Theorem 2.4.3 by
combining these lemmas with fixed point algorithm Theorem 1.3.1.
p˜
We define an auxiliary space Kp,r,T
which is made up by the functions u(t, x) such
that

u

α

p˜

Kp,r,T

:= sup t 2 u(t, x)
0
α

d −1
H˙ pp,r
L˜

< ∞, and lim t 2 u(t, x)
t→0

d

−1
H˙ pp,r
˜
L

= 0, with


15

1 < p ≤ p˜ < ∞, p1 − d1 < p1˜, 1 ≤ r ≤ ∞, T > 0, and α = α(p, p˜) = d p1 − p1˜ .
In the following lemmas, denote by [x] the integer part of x and by {x} the fraction
part of x.
1

2p

Lemma 2.4.1. Let 1 < p ≤ d. Then for all p˜ be such that
[ dp ] 1
d ,2

min
Kp˜d
[d
p]

×

,∞,T

C u

[ dp ]−1
2d
p˜
K d ,∞,T
d

+

[d
p]

v
,∞,T

[d
p]

p
into Kp,1,T
and the following inequality holds B(u, v)

Kp˜d

<

p
Kp,1,T

[d
p]

≤

, where C is a positive constant and independent of T.
,∞,T

[ dp ]−1
d

, the bilinear operator B(u, v)(t) is continuous from Kp˜d
[d
p]

,1,T

1
p˜

<

, the bilinear operator B(u, v)(t) is continuous from

Lemma 2.4.2. Let 1 < p ≤ d. Then for all p˜ be such that

Kp˜d

[ dp ]−1
2d

[p]

Kp˜d

[ dp ]−1
2d

+

and the following inequality holds B(u, v)

Kp˜d
[d
p]

<

< min
× Kp˜d

,∞,T

≤C u
,1,T

1
p˜

[d
p]

[d
p]

,∞,T

+

into

,∞,T

v

Kp˜d

[ dp ] 1
d ,2

Kp˜d
[d
p]

,
,∞,T

where C is a positive constant and independent of T.
Theorem 2.4.3. Let 1 < p ≤ d and 1 ≤ r < ∞. Then for all p˜ be such that
1
2p

+

[ dp ]−1
2d

<

1
p˜

< min

[ dp ] 1

d ,2

for all T > 0 and for all

[ dp ]−1
+ 2d , there exists a positive constant
d
−1
˙
u0 ∈ HLp p,r (Rd ) with div(u0 ) = 0 satisfying
1

d

d

sup t 2 ([ p ]− p˜ ) et∆ u0
0
d

[ ]−1
H˙ pp,∞
˜

≤ δp,˜p,d ,

δp,˜p,d such that

(2.6)

L

NSE has a unique mild solution u ∈ Kp˜d
[d
p]

d

,1,T

−1

∩ L∞ [0, T ]; H˙ Lp p,r . In particular, the
d

−1
inequality (2.6) holds for arbitrary u0 ∈ H˙ Lp p,r (Rd ) when T (u0 ) is small enough,
and there exists a positive constant σp,˜p,d such that we can take T = ∞ whenever

u0

d −1,∞

B˙ p˜p,∞
˜

≤ σp,˜p,d .

L

2.4.3.

Solutions to the Navier-Stokes equations with the initial value in the critical
d
−1
spaces H˙ pp,r (Rd ) with d ≤ p < ∞ and 1 ≤ r < ∞
L

The main result of this subsection is Theorem 2.4.7, The lemmas we need in
order to prove Theorem 2.4.7 are Lemmas 2.4.4, 2.4.5, and 2.4.6 devoted to the
study of the bilinear operator B(u, v)(t) defined by (1.5). We prove Theorem 2.4.7
by combining these lemmas with fixed point algorithm Theorem 1.3.1.
d

−1
Lemma 2.4.4. Suppose that u0 ∈ H˙ Lp p,r with d ≤ p < ∞ and 1 ≤ r < ∞. Then
p˜
e·∆ u0 ∈ Kd,1,∞
for all p˜ > p.


16

Lemma 2.4.5. Let p ≥ d and d < p˜ < 2p. Then the bilinear operator B(u, v)(t) is
p˜
p˜
p
continuous from Kd,∞,T
×Kd,∞,T

into Kp,1,T
, and we have the inequality B(u, v) Kp

p,1,T

≤C u

p˜
Kd,∞,T

v

p˜
Kd,∞,T

, where C is a positive constant and independent of T.

Lemma 2.4.6. Let d < p˜1 < ∞ and d ≤ p˜2 < ∞ be such that one of the followd˜
p1
˜1 = 2d, d ≤ p˜2 < ∞, or 2d <
ing conditions d < p˜1 < 2d, d ≤ p˜2 < 2d−˜
p1 , or p
p˜1 < ∞, p˜21 < p˜2 < ∞ is satisfied. Then the bilinear operator B(u, v)(t) is continup˜1
p˜1
p˜2
ous from Kd,∞,T
× Kd,∞,T
into Kd,1,T
, and we have the inequality B(u, v) Kp˜2 ≤
d,1,T

C u

p˜

1
Kd,∞,T

v

p˜

1
Kd,∞,T

, where C is a positive constant and independent of T.

Theorem 2.4.7. Let p ≥ d and 1 ≤ r < ∞. Then for any p˜ such that p˜ > p, there
d
−1
˙
exists a positive constant δp˜,d such that for all T > 0 and for all u0 ∈ HLp p,r (Rd ),
with div(u0 ) = 0 satisfying
1

d

sup t 2 (1− p˜ ) et∆ u0
0

˜
Lp,∞

≤ δp˜,d ,

(2.7)
d

−1
q
NSE has a unique mild solution u ∈ ∩ Kd,1,T
∩ L∞ ([0, T ]; H˙ Lp p,r ). In particular,
q>p

d

−1
the inequality (2.7) holds for arbitrary u0 ∈ H˙ Lp p,r (Rd ) with T (u0 ) is small enough,
and there exists a positive constant σp˜,d such that we can take T = ∞ whenever

u0

d −1,∞

B˙ p˜p,∞
˜

≤ σp˜,d .

L

2.4.4.

Solutions to the Navier-Stokes equations with initial value in the critical
d
spaces H˙ Ld−1
1,r (R ) with 1 ≤ r < ∞

The main result of this subsection is Theorem 2.4.11, The lemmas we need in
order to prove Theorem 2.4.11 are Lemmas 2.4.8, 2.4.9, and 2.4.10 devoted to the
study of the bilinear operator B(u, v)(t) defined by (1.5). We prove Theorem 2.4.11
by combining these lemmas with fixed point algorithm Theorem 1.3.1.
We define an auxiliary space Ks,r,T which is made up by the functions u(t, x) such
that

u

α

α

Ks,r,T

:= sup t 2 u(t, x)
0
H˙ Ls 1,r

< ∞, and lim t 2 u(t, x)
t→0

d − 1 ≤ s < d, 1 ≤ r ≤ ∞, T > 0, and α = α(s) = s + 1 − d.

H˙ Ls 1,r

= 0, with

d
·∆
Lemma 2.4.8. Suppose that u0 ∈ H˙ Ld−1
u0 ∈ Ks,r,∞
1,r (R ) with 1 ≤ r < ∞. Then e
with d − 1 < s < d.

Lemma 2.4.9. Let d − 1 < s < d. Then the bilinear operator B(u, v)(t) is continuous from Ks,∞,T × Ks,∞,T into Ks,1,T and we have the inequality B(u, v) Ks,1,T ≤
C u

Ks,∞,T

v

Ks,∞,T

, where C is a positive constant and independent of T.

Lemma 2.4.10. Let d−1 < s < d. Then the bilinear operator B(u, v)(t) is continuous from Ks,∞,T ×Ks,∞,T into Kd−1,1,T and we have the inequality B(u, v) Kd−1,1,T ≤
C u

Ks,∞,T

v

Ks,∞,T

, where C is a positive constant and independent of T.


17

Theorem 2.4.11. Let d − 1 < s < d and 1 ≤ r < ∞. Then there exists a positive
d
constant δs,d such that for all T > 0 and for all u0 ∈ H˙ Ld−1
1,r (R ), with div(u0 ) = 0
satisfying
1
sup t 2 (s+1−d) et∆ u0 H˙ s ≤ δs,d ,
(2.8)
L1

0
NSE has a unique mild solution u ∈ Ks,r,T ∩ L∞ ([0, T ]; H˙ Ld−1
1,r ).
d
In particular, the inequality (2.8) holds for arbitrary u0 ∈ H˙ Ld−1
1,r (R ) when T (u0 ) is
small enough, and there exists a positive constant σs,d such that we can take T = ∞
whenever u0

2.5

H˙ Ld−1
1

≤ σs,d .

Mild solutions in Sobolev-Lorentz spaces

In this section, we study local well-posedness for the Navier-Stokes equations
(NSE) with the arbitrary initial value in homogeneous Sobolev-Lorentz spaces
H˙ Ls q,r (Rd ) := Λ˙ −s Lq,r for d ≥ 2, q > 1, s ≥ 0, 1 ≤ r ≤ ∞, and dq − 1 ≤ s < dq ,
this result improves the known results for q > d, r = q, s = 0 (see M. Cannone
(1995)) and for q = r = 2, d2 − 1 < s < d2 (see Chemin (1992)).
In the case of critical indexes (s = dq − 1), we prove global well-posedness for NSE
provided the norm of the initial value is small enough. The result that is a generalization of the result for q = r = d, s = 0 (see M. Cannone (1997)).
The main result of this section is Theorem 2.5.4, The lemmas we need in order to
prove Theorem 2.5.4 are Lemmas 2.5.1, 2.5.2, and 2.5.3 devoted to the study of the
bilinear operator B(u, v)(t) defined by (1.5). We prove Theorem 2.5.4 by combining
these lemmas with fixed point algorithm Theorem 1.3.1.

2.5.1.

The Sobolev-Lorentz spaces

Definition 2.5.1. (Sobolev-Lorentz spaces).
For q > 1, r ≥ 1, and 0 ≤ s < dq , the Sobolev-Lorentz space
the space Is (Lq,r (Rd )), equipped with the norm f H˙ s q,r :=
L

H˙ Ls q,r (Rd ) is defined as

Λ˙ s f Lq,r .

s,˜
q
We define the auxiliary space Kq,r,T
which is made up by the functions u(t, x)
α
α
such that u Ks,˜q := sup t 2 u(t, .) H˙ s < ∞, and lim t 2 u(t, .) H˙ s = 0, where
q,r,T

Lq˜,r

0
t→0

Lq˜,r

r, q, q˜, s being fixed constants satisfying q, q˜ ∈ (1, +∞), r ≥ 1, s ≥ 0, ds <
s+1
d ,

and α = α(q, q˜) = d

1
q

−

1
q˜

1
q˜

≤

1
q

≤

.

Lemma 2.5.1. If u0 ∈ H˙ Ls q,r (Rd ) with q > 1, r ≥ 1, s ≥ 0, and
for all q˜ satisfying ds < 1q˜ < 1q , we have
s−( d − d ),∞
map H˙ Ls q,r (Rd ) → B˙ q˜ q q˜ (Rd ).

e·∆ u0 ∈

s,˜
q
Kq,1,∞
,

s
d

<

1
q

≤

s+1
d

then

and the following imbedding


18

Lemma 2.5.2. Let s, q ∈ R be such that s ≥ 0, q > 1, and
q˜ satisfying

s
d

s,˜
q
from Kq,˜
q ,T ×
1

1

1
s 1
q˜ < min 2 + 2d , q
s,˜
q
s,˜
q
Kq,˜
q ,T into Kq,1,T and

<

d

C.T 2 (1+s− q ) u

s,˜
q
Kq,˜
q ,T

v

s,˜
q
Kq,˜
q ,T

s
d

<

s+1
d .

Then for all

the following inequality holds B(u, v)

s,˜
q
Kq,1,T

≤

, where C is a positive constant independent of T.
s
d

<

1
q

≤

s+1
d .

Then

1
1
s
1
s 1
q + d < q˜ < min 2 + 2d , q , the bilinear operator B(u, v)(t)
s,˜
q
s,˜
q
s,q
from Kq,˜
q ,T × Kq,˜
q ,T into Kq,1,T and the following inequality holds
d
1
≤ C.T 2 (1+s− q ) u Ks,˜q v Ks,˜q , where C is a positive constant indeq,˜
q ,T
q,˜
q ,T

for all q˜ satisfying
B(u, v) Ks,q
q,1,T
pendent of T.

≤

, the bilinear operator B(u, v)(t) is continuous

Lemma 2.5.3. Let s, q ∈ R be such that s ≥ 0, q > 1, and
is continuous

1
q

1
2

s
1
s+1
d < q ≤ d .
1
s 1
2 + 2d , q , there exists a positive
all u0 ∈ H˙ Ls q,r (Rd ) with div(u0 ) = 0

Theorem 2.5.4. Let s ≥ 0, q > 1, r ≥ 1 be such that
(a) For all q˜ satisfying

1
2

1
q

+

s
d

<

1
q˜

< min

constant δs,q,˜q,d such that for all T > 0 and for
satisfying
1
d
d 1 1
T 2 (1+s− q ) sup t 2 ( q − q˜ ) et∆ u0
0
H˙ q˜s

≤ δs,q,˜q,d ,

(2.9)

s,˜
q
NSE has a unique mild solution u ∈ Kq,1,T
∩ L∞ [0, T ]; H˙ Ls q,r . In particular, for
arbitrary u0 ∈ H˙ Ls q,r with div(u0 ) = 0, there exists T (u0 ) small enough such that the

inequality (2.9) holds.
1
< 1q˜ < min 12 +
(b) If 1 < q ≤ d, and s = dq − 1 then for any q˜ be such that 1q − 2d
1
2q

−

1 1
2d , q

, there exists a positive constant σq,˜q,d such that if u0

d −1,∞

B˙ q˜q˜

≤ σq,˜q,d

and T = ∞ then the inequality (2.9) holds.

2.6

Mild solutions in mixed-norm Sobolev-Lorentz spaces

In this section, for 0 ≤ m < ∞ and index vectors q = (q1 , q2 , ..., qd ), r =
(r1 , r2 , ..., rd ), where 1 < qi < ∞, 1 ≤ ri ≤ ∞, and 1 ≤ i ≤ d, we introduce
and study mixed-norm Sobolev-Lorentz spaces H˙ Lmq,r . Then we investigate the
existence and uniqueness of solutions to the NSE in the spaces Lp ([0, T ]; H˙ Lmq,r ). In

the case when m = 0, q1 = q2 = ... = qd = r1 = r2 = ... = rd , our results recover
those of Faber, Jones and Riviere (1972).
2.6.1.

Mixed-norm Lorentz spaces

Given a measurable function u on Rd and index vectors q = (q1 , q2 , ..., qd ), r =
(r1 , r2 , ..., rd ), where 1 < qi < ∞, 1 ≤ ri ≤ ∞, 1 ≤ i ≤ d, we can define the
norm u Lq,r by calculating first the Lq1 ,r1 - Lorentz norm of u(x1 , x2 , ..., xd ) with
respect to the variable x1 , and then Lq2 ,r2 - Lorentz norm of the resulting quantity
with respect to the variable x2 , and so on, finishing with the Lqd ,rd - Lorentz norm


19

with respect to the variable xd :

u

Lq,r

= ...

u

q ,r1

Lx11

notation, for a vector q = (q1 , q2 , ..., qd ) we will write

2.6.2.

1
q

q ,r2

Lx22

...

q ,rd

Lxdd

for the vector

. For a short
1
1
q1 , . . . , qd

.

Lp Lq,r solutions of the Navier-Stokes equations

Definition 2.6.1. For m ∈ R and q, r ∈ Rd , 1 < q < ∞, 1 ≤ r ≤ ∞, the space
H˙ Lmq,r is defined as the space Λ˙ −m Lq,r , equipped with the norm u H˙ mq,r = Λ˙ m u Lq,r .
L

Lemma 2.6.1. Let q = (q1 , q2 , ..., qd ), r = (r1 , r2 , ..., rd ), 2 < p < ∞, m ≥
d
d
0, and 0 < T < ∞, be such that m < 21 i=1 q1i , p2 − m + i=1 q1i ≤ 1,
qi
1 ≤ ri ≤ ∞, 2 <
< ∞, i = 1, 2, .., d. Then the bilinear operator B is
1−

m
d
1
i=1 qi

continuous from Lp ([0, T ]; H˙ Lmq,r ) × Lp ([0, T ]; H˙ Lmq,r ) to Lp ([0, T ]; H˙ Lmq,r ) and we have
the inequality
B(u, v)

1

Lp ([0,T ];H˙ Lmq,r )

T2

(1+m− p2 −

d
1
i=1 qi )

u

Lp ([0,T ];H˙ Lmq,r )

. v

Lp ([0,T ];H˙ Lmq,r )

. (2.10)

Combining Lemma 2.6.1 with Theorem 1.3.1 we obtain the following existence
result
Theorem 2.6.2. Let q = (q1 , q2 , ..., qd ), r = (r1 , r2 , ..., rd ), 2 < p < ∞, and m ≥ 0
d
d
qi
<
be such that m < 12 i=1 q1i , p2 − m + i=1 q1i ≤ 1, 1 ≤ ri ≤ ∞, 2 <
1−

m
d
1
i=1 qi

∞, i = 1, 2, .., d.
(a) There exists a positive constant δ(m,q,r,p) > 0 such that for all T > 0 and for all
u0 ∈ S (Rd ) with div(u0 ) = 0, satisfying
1

T2

(1+m− p2 −

d
1
i=1 qi )

e·∆ u0

Lp ([0,T ];H˙ Lmq,r )

≤ δ(m,q,r,p) ,

(2.11)

there is a unique mild solution u ∈ Lp ([0, T ]; H˙ Lmq,r ) for NSE.
If e·∆ u0 ∈ Lp ([0, 1]; H˙ Lmq,r ), then the inequality (2.11) holds when T (u0 ) is small
enough.
d
(b) If p2 + i=1 q1i − m = 1 then there exists a positive δ(m,q,r,p) > 0 such that we
can take T = ∞ whenever e·∆ u0 Lp ([0,∞];H˙ mq,r ) ≤ δ(m,q,r,p) .
L

2.6.3.

Uniqueness theorems

In this subsection, we give a theorem on the uniqueness of solutions.
Theorem 2.6.3. If u ∈ Lp (0, T ), (Lq,∞ (Rd ))d is a Leray weak solution associated

d
with u0 , where p ∈ R, q ∈ Rd , 2 < q < ∞, 2 < p < ∞ and p2 + i=1 q1i = 1,
then the condition iii) of Theorem in the book of P. G. Lemarie-Rieusset (2002) is
d
satisfied, and u ∈ Lp (0, T ), (Xr )d where r = i=1 q1i ∈ (0, 1), p2 + r = 1, and u is
the unique Leray solution associated with u0 on (0, T ).


Chapter 3

Hausdorff dimension of the set of
singularities for a weak solutions
In this chapter we investigate the Hausdorff dimension of the possible
singular set in time of weak solutions to the Navier-Stokes equation on the three
dimensional torus under some regularity conditions of Serrin’s type. The results
in the paper relate the regularity conditions of Serrin’s type to the Hausdorff
dimension of the singular set set in time. The result that is a generalization of
the results of V. Scheffer (1977) and J. Leray (1934).

3.1

Functional setting of the equations

In this chapter, we consider the initial value problem for the non stationary
Navier-Stokes equations on the torus T3 = R3 /Z3 , or in other words in R3 with
periodic boundary conditions
∂ui
+
∂t

3

uj
j=1

∂ui
∂p
− ∆ui +
− fi = 0 on T3T := T3 × (0, T ), i = 1, 3
∂xj
∂xi
3

div(u) =
i=0

∂ui
= 0 on T3T ,
∂xi

u(x, 0) = u0 (x) in T3 × {0},

(3.1)

(3.2)
(3.3)

where fi (x, t) = (f1 (x, t), f2 (x, t), f3 (x, t)), u0 (x) are given functions with u0 (x) sat˙ 3 ) the space of all infinitely
isfying the condition div(u0 ) = 0. Denote by V(T
˙ 3 ) the space

differentiable solenoidal vector fields with zero averaging on T3 ; by V(T
T
of all compactly supported in T3T infinitely differentiable solenoidal vector fields
˙ 3)
with zero averaging on T3 for each t ∈ [0, T ]; H, V are the closures of the set V(T
in the spaces L2 (T3 ), H 1 (T3 ), respectively. Assume that f ∈ L∞ (0, T ; V ), u0 ∈ H,
where V is the dual space of V . A weak solution of the problem (3.1) - (3.3) in T3T

20


21

is a vector field such that
u ∈ L2 ((0, T ); V ) ∩ L∞ ((0, T ); H) ∩ C([0, T ]; L2w );
3

−
T3T

1
2

i=1

3

∂vi
∂ui ∂vi
∂vi

ui
+
+ ui uj
dxdt =< f, v >, ∀v ∈ C˙ ∞ (T3T );
∂t i,j=1 ∂xj ∂xj
∂xj

3

3

∂ui 2
1
dxdt ≤
∂xj
2

2

|ui (x, t1 )| dx +
T3 i=1

T3 ×(t0 ,t1 ) i,j=1

3

|ui (x, t0 )|2 dx;
T3 i=1

∀t0 ∈ [0, T ]\Σ, t1 ∈ [t0 , T ], where Σ has Lebesgue measure zero and 0 ∈

/ Σ;
u(x, t) − u0 (x) L2 (T3 ) → 0 as t → 0,
where < ., . > is the pairing between V and V . It was proved by Leray that there
exists at least one weak solution of the problem (3.1) - (3.3).

3.2

Weak solutions in Lr H α

Lemma 3.2.1. Assume that f ∈ L∞ (0, T ; Vα−1 ), u0 ∈ Vα , 21 < α < 32 . Then there
exists a unique strong solution to the Navier-Stokes equations, satisfying
u ∈ L2 (0, T ∗∗ ; Vα+1 ) ∩ C(0, T ∗∗ ; Vα )
where T ∗∗ = min(T, T ∗ ), with T ∗ =

C(α,N )
2

(|u(0)|2α +1) 2α−1

.

Let α ∈ ( 12 , 32 ). We say that a weak solution u is H α (T3 )-regular on (t1 , t2 ) if
u ∈ C((t1 , t2 ), H α (T3 )). We obtaine the following theorem by using Lemma 3.2.1.
Theorem 3.2.2. Assume that α ∈ ( 12 , 23 ), u0 ∈ H, f ∈ L∞ (0, T ; Vα−1 ) and u is
a weak solution of the Navier-Stokes equations and satisfies the condition u ∈
Lr (0, T ; Vα ), with r > 0 and r(2α−1) < 4. Then there exists a closed set Sα ⊂ [0, T ]
such that u ∈ C([0, T ] \ Sα ; Vα ) and µ1− r(2α−1) (Sα ) = 0.
4

3.3

Weak solutions in Lr W 1,q

Lemma 3.3.1. Assume that f ∈ L∞ (0, T ; Lq (T3 )), u0 ∈ W 1,q (T3 ), q ∈ [2, 3). Then
q
there exists a constant T ∗∗ depending only on ∇u(0) Lq (T3 ) , q, sup0≤t≤T f (t) Lq (T3 )
and a unique strong solution to the Navier-Stokes equations, satisfying
˜ 2,q ) ∩ C(0, T ∗∗ ; W 1,q ).
u ∈ L2 (0, T ∗∗ ; W
Let q ∈ [2, 3). We say that a weak solution u is W 1,q (T3 ) - regular on (t1 , t2 ) if
u ∈ C((t1 , t2 ), W 1,q (T3 )). We obtaine the following theorem by using Lemma 3.3.1
Theorem 3.3.2. Assume that q ∈ [2, 3), u0 ∈ H, f ∈ L∞ (0, T ; Lp (T3 )) and u is
a weak solution of the Navier-Stokes equations and satisfies the following condition
u ∈ Lr (0, T ; W 1,q ), r(2q−3)
< 1. Then there exists a closed set Sq ⊂ [0, T ] such that
2q
u ∈ C([0, T ] \ Sq ; W 1,q ) and µ1− r(2q−3) (Sq ) = 0.
2q


Conclusions
In this thesis, we construct mild solutions to the Navier-Stokes equations by applying the Picard contraction principle. For the Sobolev spaces H˙ qs (q > 1, dq − 1 ≤
s < dq ), we obtain the local existence of mild solutions in the spaces L∞ [0, T ]; H˙ qs (Rd )
with arbitrary initial value in H˙ qs (Rd ), in the case of critical indexes (q > 1, s = dq −1)
d

−1
we get the existence of global mild solutions in the spaces L ([0, ∞); H˙ qq (Rd ))
when the norm of the initial value is small enough. The same argument is applied
to following spaces:

∞

d

−1
- Critical Sobolev-Fourier-Lorentz spaces H˙ Lp p,r (Rd ), (r ≥ 1, 1 ≤ p < ∞).
- Sobolev-Lorentz spaces H˙ Ls q,r (Rd ), (s ≥ 0, q > 1, r ≥ 1, dq − 1 ≤ s < dq ) with
critical indexes s = dq − 1.

- For 0 ≤ m < ∞ and index vectors q = (q1 , q2 , ..., qd ), r = (r1 , r2 , ..., rd ),
where 1 < qi < ∞, 1 ≤ ri ≤ ∞, and 1 ≤ i ≤ d, we introduce and study
mixed-norm Sobolev-Lorentz spaces H˙ Lmq,r . Then we investigate the existence and
uniqueness of solutions to the Navier-Stokes equations in the spaces Q := QT =
Lp ([0, T ]; H˙ Lmq,r ) where p > 2, T > 0, and initial data is taken in the class I = {u0 ∈
(S (Rd ))d , div(u0 ) = 0 : e·∆ u0 Q < ∞}. The results have a standard relation between existence time and data size: large time with small data or large data with
d
small time. In the case with T = ∞ and critical indexes p2 + i=1 q1i − m = 1, the
2

m− ,p
space I coincides with the homogeneous Besov space B˙ Lq,r p .

Finally, we investigate the Hausdorff dimension of the possible singular set in
time of weak solutions to the Navier-Stokes equations on the three dimensional
torus under some regularity conditions of Serrin’s type. The results in the chapter
relate the regularity conditions of Serrin’s type to the Hausdorff dimension of the
singular set in time.

22

List of the author’s publications related
to the dissertation
[1] D. Q. Khai and N. M. Tri, Well-posedness for the Navier-Stokes equations
with data in Sobolev-Lorentz spaces, Nonlinear Analysis, 149 (2017), 130-145.
[2] D. Q. Khai, Well-posedness for the Navier-Stokes equations with datum in
the Sobolev spaces, Acta Math Vietnam (2016). doi:10.1007/s40306-016-0192-x.
[3] D. Q. Khai and N. M. Tri, Well-posedness for the Navier-Stokes equations
with datum in Sobolev-Fourier-Lorentz spaces, Journal of Mathematical Analysis
and Applications, 437 (2016), 854-781.
[4] D. Q. Khai and N. M. Tri, On the initial value problem for the Navier-Stokes
equations with the initial datum in critical Sobolev and Besov spaces, Journal of
Mathematical Sciences the University of Tokyo, 23 (2016), 499-528.
[5] D. Q. Khai and N. M. Tri, On the Hausdorff dimension of the singular
set in time for weak solutions to the nonstationary Navier-Stokes equations on
torus,Vietnam Journal of Mathematics, 43 (2015), 283-295.
[6] D. Q. Khai and N. M. Tri, Solutions in mixed-norm Sobolev-Lorentz spaces to
the initial value problem for the Navier-Stokes equations, Journal of Mathematical
Analysis and Applications, 417 (2014), 819-833.

Author’s other relevant papers
[7] D. Q.Khai, N.M. Tri, On general axisymmetric explicit solutions for the
Navier-Stokes equations, International Journal of Evolution Equations, 6 (2013),
325-336..
[8] D. Q. Khai and V. T. T. Duong,On the initial value problem for the NavierStokes equations with the initial datum in the Sobolev spaces,
preprint arXiv:1603.04219.
[9] D. Q. Khai and N. M. Tri, The existence and decay rates of strong solutions
for Navier-Stokes Equations in Bessel-potential spaces, preprint, arXiv:1603.01896.
[10] D. Q. Khai and N. M. Tri The existence and space-time decay rates of strong
solutions to Navier-Stokes Equations in weighed L∞ (|x|γ dx) ∩ L∞ (|x|β dx) spaces,

preprint, arXiv:1601.01441.

23


24

The results of the dissertation have been presented at
1) PhD. Students Conference, Hanoi Institute of Mathematics, Nov 07, 2012.
2) PhD. Students Conference, Hanoi Institute of Mathematics, Oct 25, 2013.
3) PhD. Students Conference, Hanoi Institute of Mathematics, Oct 30, 2014.
4) Seminar on Differential equations and its application, Hanoi Institute of
Mathematics.