Electronic Journal of Differential Equations, Vol. 2023 (2023), No. 78, pp. 1–12.

ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu

DOI: 10.58997/ejde.2023.78

SOLUTIONS FOR THE NAVIER-STOKES EQUATIONS WITH

CRITICAL AND SUBCRITICAL FRACTIONAL DISSIPATION

IN LEI-LIN AND LEI-LIN-GEVREY SPACES

WILBERCLAY G. MELO, NATÃ F. ROCHA, NATIELLE DOS SANTOS COSTA

Abstract. In this article, we prove the existence of a unique global solu-
tion for the critical case of the generalized Navier-Stokes equations in Lei-Lin

and Lei-Lin-Gevrey spaces, by assuming that the initial data is small enough.

Moreover, we obtain a unique local solution for the subcritical case of this
system, for any initial data, in these same spaces. It is important to point out

that our main result is obtained by discussing some properties of the solutions

for the heat equation with fractional dissipation.

1. Introduction

This work studies the existence of global and local in time solutions for the incom-
pressible Navier-Stokes equations in Lei-Lin-Gevrey and Lei-Lin spaces X sa,σ(R3),

ut +(−∆)αu+ u · ∇u+∇p = 0, x ∈ R3, t > 0,

div u = 0, x ∈ R3, t > 0,

u(x, 0) = u0(x), x ∈ R3,

(1.1)

where u(x, t) = (u1(x, t), u2(x, t), u3(x, t)) ∈ R3 denotes the incompressible velocity
field, and p(x, t) ∈ R the hydrostatic pressure; see [1, 2, 3, 4, 5, 6, 7, 8, 10, 11,
12, 13, 18, 20, 26, 28, 31] and their references. Here (−∆)α, with α ≥ 1/2, is the
fractional Laplacian, see (2.1). The initial data for the velocity field, given by u0

in (1.1), is assumed to be divergence free, i.e., div u0 = 0.
The fractional Laplacian (−∆)α has been studied in many works in the lit-

erature (see, for instance, [32, 34] and references therein). To cite some models
involving this kind of operator, we refer: Diffusion-reaction, Quasi-geostrophic,
Cahn-Hilliard, Porous medium, Schrödinger, Ultrasound, Magnetohydrodynamics
(MHD), Magnetohydrodynamics-α (MHD-α) and Navier-Stokes itself (see [1, 2, 3,
4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 21, 22, 23, 24, 25, 26, 27, 28, 29,
30, 31, 33, 35] and references therein). It is important to recall that, by applying
the Spectral Theorem, (−∆)α assumes the diagonal form in the Fourier variable,

2020 Mathematics Subject Classification. 35A01, 35Q35, 42B37.
Key words and phrases. Navier-Stokes equations; global and local solutions;

Lei-Lin-Gevrey spaces.
©2023. This work is licensed under a CC BY 4.0 license.

Submitted April 18, 2023. Published November 10, 2023.

1



2 W. G. MELO, N. F. ROCHA, N. S. COSTA EJDE-2023/78

i.e., this is a Fourier multiplier operator with symbol |ξ|2α (which extends Fourier
multiplier property of −∆).

It is also important to notice that system (1.1) becomes the usual Navier-Stokes
equations by replacing the fractional Laplacian operator by the usual one. More
precisely, these last equations are

ut + u · ∇u+∇p = ∆u, x ∈ R3, t > 0,

div u = 0, x ∈ R3, t > 0,

u(x, 0) = u0(x), x ∈ R3,

and play an important role in continuum mechanics. It is necessary to show that
singularities for the solutions of theses equations are not present in finite time (from
smooth initial data with finite energy) in order to make this system well-posed.
This is one of the most important open problems in Nonlinear Analysis. Thus, the
fractional Laplacian must be an interesting mathematical tool to understand better
this problem. In fact, Wu [35] showed that the generalized Navier-Stokes equations
(1.1) admit global classical solution provided that the initial data u0 is smooth and
α ≥ 5

4 . More precisely, [35] assumes that α ≥ 5/4 and u0 ∈ Hs(R3), with s > 2α,
to obtain a unique global classical solution for (1.1) (see also [1, 4, 5, 7, 8, 9, 15,
16, 17, 18, 27, 28, 29, 31, 35] and references therein).

Physically, (1.1) are the equations that describe the motion of a fluid with inter-
nal friction interaction and such motion is a chain of particles that are connected
by elastic springs (see, for example, [32] for more details).

Recently, some authors have published works that study the usual Navier-Stokes
equations and their extensions in Lei-Lin and Lei-Lin-Gevrey spaces (see [1, 4, 7,
9, 18, 24, 28] and references therein). For example Melo, Souza and Santos [28], by
studying the MHD-α equations, proved the existence of a unique global solution
in Cb([0,∞);X sa,σ(R3)). In addition, [28] presents analyticity and decay rates for
global solutions in this same context (for more information, see [28] and references
therein).

Another motivating work was written by Melo and Rocha [24, 30]. This ar-
ticle proves the local existence, as well as blow-up criteria, for solutions of the
generalized Magnetohydrodynamics equations in [CT (X s(R3))∩L1

T (X s+2α(R3))]×
[CT (X s(R3))∩L1

T (X s+2β(R3))], where the fractional dissipations α and β belong to

the interval (1/2, 1], and s ∈ (max{1− 2α, 1− 2β, α(1−2β)
β , β(1−2α)

α }, 0] (see [24, 30]

for more information on the blow-up criteria proved in these works).
Motivated by these works, we present global and local solutions for the Navier-

Stokes equations (1.1), with fractional dissipation of order α ≥ 1/2, in Lei-Lin and
Lei-Lin-Gevrey spaces (we refer to [8, 28, 30, 33] and papers therein). Moreover, it
is worth to point out that we have adapted some of the ideas applied in the paper
[17].

Our main result proves the existence and uniqueness of solutions for the Navier-
Stokes equations (1.1) in Lei-Lin-Gevrey and Lei-Lin spaces and can be written as
follows.

Theorem 1.1. The following statements hold:
(i) Critical Case: global solution. Assume that α = 1/2, (a, s, σ) ∈

(
(0,+∞) ×

[−1, 0)× (1,+∞)
)
∪
(
[0,+∞)×{0}× [1,+∞)

)
and u0 ∈ X sa,σ(R3). Then there is a

constant Ca,σ,s > 0 such that if ‖u0‖X sa,σ < Ca,σ,s, then for all instant T > 0 there



EJDE-2023/78 CRITICAL AND SUBCRITICAL CASES FOR NAVIER-STOKES EQUATIONS 3

is a unique global solution

u ∈ CT (X sa,σ(R3)) ∩ L1
T (X s+1

a,σ (R3))

to the Navier-Stokes equations (1.1), it satisfies

‖u‖L∞T (X sa,σ) + ‖u‖L1
T (X s+1

a,σ ) ≤ 4‖u0‖X sa,σ .

Moreover, u ∈ LpT (X s+
1
p

a,σ (R3)) for all p ≥ 1.
(ii) Subcritical Case: local solution. Assume that α > 1/2, (a, s, σ) ∈

(
(0,+∞)×

[−1, 0) × (1,+∞)
)
∪
(
[0,+∞) × {0} × [1,+∞)

)
, and u0 ∈ X sa,σ(R3). Then there

exist an instant T > 0 and a unique local solution

u ∈ CT (X sa,σ(R3)) ∩ L1
T

(X s+2α
a,σ (R3))

to the Navier-Stokes equations (1.1), such that

‖u‖L∞
T

(X sa,σ) + ‖u‖L1
T

(X s+2α
a,σ ) ≤ 4‖u0‖X sa,σ .

Furthermore, u ∈ Lp
T

(X s+
2α
p

a,σ (R3)) for all p ≥ 1.

Let us recall that the Navier-Stokes equations (1.1) are invariant under the
change of time and space scaling. More precisely, if u and p solve (1.1); then,
for any λ > 0, the functions

uλ(x, t) = λ2α−1u(λx, λ2αt), pλ(x, t) = λ4α−2p(λx, λ2αt), uλ0 (x) = λ2α−1u0(λx)

also solve (1.1). By observing this same scaling, we say that (X, ‖ · ‖) is a critical
space for the Navier-Stokes equations (1.1) (see [19] and references therein for more
details) if ‖fλ‖ = ‖f‖, for all λ > 0, where fλ(x) = λ2α−1f(λx). Then, it is easy
to check that X 1−2α(R3) is a critical space for (1.1) [9]. In particular, for α = 1/2,
one has that X 0(R3) is also a critical space for (1.1).

It is important to point out that Theorem 1.1 presents some information for
solutions of system (1.1) in the critical Lei-Lin space X 0(R3) and the specific Lei-
Lin-Gevrey space X 1−2α

a,σ (R3). More precisely, we have the statements below:
• Theorem 1.1 (i) (in Lei-Lin spaces) can be rewritten as follows: Assume that

α = 1/2, a = 0 and u0 ∈ X 0(R3). Thus, there is a constant C > 0 such that if
‖u0‖X 0 < C; then for all instant T > 0 there is a unique global solution

u ∈ CT (X 0(R3)) ∩ L1
T (X 1(R3))

to the Navier-Stokes equations (1.1), and it satisfies

‖u‖L∞T (X 0) + ‖u‖L1
T (X 1) ≤ 4‖u0‖X 0 .

(See [9] for the Quasi-geostrophic case). Under the hypotheses above, it follows
that u ∈ LpT (X 1/p(R3)), for all p ≥ 1.
• Theorem 1.1 (ii), with s = 1 − 2α, can be rewritten as follows: Assume that

α ∈ ( 1
2 , 1], a > 0, σ > 1, and u0 ∈ X 1−2α

a,σ (R3). Then there exist an instant T > 0
and a unique local solution

u ∈ CT (X 1−2α
a,σ (R3)) ∩ L1

T
(X 1

a,σ(R3))

to the Navier-Stokes equations (1.1) such that

‖u‖L∞
T

(X 1−2α
a,σ ) + ‖u‖L1

T
(X 1
a,σ) ≤ 4‖u0‖X 1−2α

a,σ
.



4 W. G. MELO, N. F. ROCHA, N. S. COSTA EJDE-2023/78

See [30] for a study of the generalized Magnetohydrodynamics equations. In this

case, one infers that u ∈ Lp
T

(X 1+
2(1−p)α

p
a,σ (R3)), for all p ≥ 1.

Remark 1.2. To obtain mild solutions for the Navier-Stokes equations (1.1), we
apply a standard fixed point theorem (see Lemma 3.2). To this end, we need to
prove the continuity of a bilinear operator (see (4.5)) related to the nonlinear term
of this same system (1.1) (see proof of Theorem 1.1). Since this statement is the key
point in the proof of our main result, we present some preliminary lemmas that are
useful to achieve this goal. More specifically, these results show us how to estimate
the solutions of the heat equation (see systems (3.2) and (4.6)) in Lei-Lin-Gevrey
and Lei-Lin spaces through its nonhomogeneous term and initial data (see Lemma
3.1), and help us to choose the values for a, σ and s such that X sa,σ(R3) ↪→ X 0

a
σ ,σ

(R3)

(see Lemma 3.3 and (4.7)). At last, it is also important to emphasize that this path
is taken because of the advantages of the use of Fourier analysis and some usual
techniques.

The outline of this article is as follows: Section 2 presents the most important
definitions and notations that are applied in this paper, Section 3 presents some
lemmas that play an important role in this work, and Section 4 presents the proof
of our main result (see Theorem 1.1).

2. Notation

In this section, we list the most important definitions and notation that are used
throughout this paper.
• S′(R3) is the space of tempered distributions.
• The Fourier transform and its inverse are defined by

F(f)(ξ) = f̂(ξ) :=

∫
R3

e−iξ·xf(x) dx,

F−1(g)(x) := (2π)−3

∫
R3

eiξ·xg(ξ) dξ,

• The fractional Laplacian (−∆)α (see [34]), for α ≥ 1/2, is defined by

F [(−∆)αf ](ξ) = |ξ|2αf̂(ξ), ∀ξ ∈ R3, (2.1)

where f ∈ S′(R3) and f̂ ∈ L1
loc(R3).

• The tensor product is

f ⊗ g := (g1f, g2f, g3f),

where f = (f1, f2, f3) and g = (g1, g2, g3) ∈ S′(R3).
• Let s ∈ R. The Lei-Lin spaces are

X s(R3) := {f ∈ S′(R3) : f̂ ∈ L1
loc(R3) and

∫
R3

|ξ|s|f̂(ξ)| dξ <∞}

and the X s(R3)-norm is

‖f‖X s =

∫
R3

|ξ|s|f̂(ξ)| dξ.

• Let a > 0, σ ≥ 1, and s ∈ R. The Lei-Lin-Gevrey spaces are

X sa,σ(R3) := {f ∈ S′(R3) : f̂ ∈ L1
loc(R3) and

∫
R3

|ξ|sea|ξ|
1/σ

|f̂(ξ)| dξ <∞}



EJDE-2023/78 CRITICAL AND SUBCRITICAL CASES FOR NAVIER-STOKES EQUATIONS 5

and the X sa,σ(R3)-norm is

‖f‖X sa,σ =

∫
R3

|ξ|sea|ξ|
1/σ

|f̂(ξ)| dξ.

• Let s ∈ R. The homogeneous Sobolev space is

Ḣs(R3) = {f ∈ S′(R3) : f̂ ∈ L1
loc(R3) and

∫
R3

|ξ|2s|f̂(ξ)|2 dξ <∞}

and the Ḣs(R3)-norm is

‖f‖Ḣs :=
(∫

R3

|ξ|2s|f̂(ξ)|2 dξ
)1/2

.

• Let a > 0, σ ≥ 1 and s ∈ R. The Sobolev-Gevrey space is

Ḣs
a,σ(R3) = {f ∈ S′(R3) : f̂ ∈ L1

loc(R3) and

∫
R3

|ξ|2se2a|ξ|1/σ |f̂(ξ)|2 dξ <∞}

and the Ḣs
a,σ(R3)-norm is

‖f‖Ḣsa,σ :=
(∫

R3

|ξ|2se2a|ξ|1/σ |f̂(ξ)|2 dξ
)1/2

.

• Let T > 0, (X, ‖ · ‖X) a normed space and I ⊆ R an interval. We define

C(I;X) = {f : I → X continuous function},
and the C(I;X)-norm

‖f‖L∞(I;X) := sup
t∈I
{‖f(t)‖X}.

We denote CT (X) = C([0, T ];X) and ‖ · ‖L∞T (X) = ‖ · ‖L∞([0,T ];X).

• Let 1 ≤ p < ∞, T > 0, (X, ‖ · ‖X) a normed space and I ⊆ R an interval. We
define

Lp(I;X) = {f : I → X mensurable function :

∫
I

‖f(t)‖pX dt <∞},

and the Lp(I;X)-norm is given by

‖f‖Lp(I;X) :=
(∫

I

‖f(t)‖pX dt
)1/p

.

We denote LpT (X) = Lp([0, T ];X).
• The constants in this paper may change their values from line to line without
change of notation. For example, Cq denotes any constant that depends on q and
C is always a positive constant.

3. Preliminary lemmas

The most important result presented in this article, Theorem 1.1, is a con-
sequence of a study based on the solutions for the following heat equation with
fractional dissipation and initial data v0 ∈ X sa,σ(R3) (with a ≥ 0, σ ≥ 1 and s ∈ R):

vt + (−∆)αv = f, t ∈ (0, T ];

v(·, 0) = v0,
(3.1)

where f ∈ L1
T (X sa,σ(R3)) (provided that T > 0 is arbitrary).

It is worth to point out that the main ideas that will be presented below were
firstly established by Orf [31] and generalized a few years later by Guterres, Melo,



6 W. G. MELO, N. F. ROCHA, N. S. COSTA EJDE-2023/78

Rocha, and Santos [17], in the case of Sobolev-Gevrey spaces. See [36, Lemma 2.6]
for examples of particular cases.

Lemma 3.1. Assume that a ≥ 0, σ ≥ 1, T > 0, s ∈ R, α ∈ R, f ∈ L1
T (X sa,σ(R3))

and v0 ∈ X sa,σ(R3). Consider that v ∈ CT (S′(R3)) solves the system

vt + (−∆)αv = f, x ∈ R3, t ∈ (0, T ];

v(·, 0) = v0, x ∈ R3.
(3.2)

Then, v ∈ CT (X sa,σ(R3)) ∩ LpT (X s+
2α
p

a,σ (R3)) for all p ≥ 1. Furthermore,

(i) ‖v‖L∞T (X sa,σ) ≤ ‖v0‖X sa,σ + ‖f‖L1
T (X sa,σ),

(ii) ‖v‖
LpT (X

s+2α
p

a,σ )
≤ ‖v0‖X sa,σ + ‖f‖L1

T (X sa,σ).

Proof. At first, by applying the heat semigroup e−(t−τ)(−∆)α (where 0 ≤ τ ≤ t ≤ T )
to the first equation of system (3.2), using the Fourier transform and integrating
over [0, t] the result obtained, one concludes that

|v̂(t)| ≤ e−t|ξ|
2α

|v̂0|+
∫ t

0

e−(t−τ)|ξ|2α |f̂(τ)| dτ. (3.3)

Thereby, we can write

|v̂(t)| ≤ |v̂0|+
∫ T

0

|f̂(τ)| dτ.

Now, by multiplying the inequality above by |ξ|sea|ξ|1/σ , we obtain

|ξ|sea|ξ|
1/σ

|v̂(t)| ≤ |ξ|sea|ξ|
1/σ

|v̂0|+ |ξ|sea|ξ|
1/σ

∫ T

0

|f̂(τ)| dτ.

Applying the L1(R3)-norm, we have

‖v(t)‖X sa,σ ≤ ‖v0‖X sa,σ + ‖f‖L1
T (X sa,σ), ∀t ∈ [0, T ].

As a result, one concludes that

‖v‖L∞T (X sa,σ) ≤ ‖v0‖X sa,σ + ‖f‖L1
T (X sa,σ). (3.4)

This proves (i) and, furthermore, this shows that v ∈ CT (X sa,σ(R3)) (it is enough

to recall that the Fourier transform F is continuous, v ∈ CT (S′(R3)) and apply
Dominated Convergence Theorem) since v0 ∈ X sa,σ(R3) and f ∈ L1

T (X sa,σ(R3)).

To show (ii), we multiply (3.3) by |ξ|s+2αea|ξ|
1/σ

and obtain that

|ξ|s+2αea|ξ|
1/σ

|v̂(t)|

≤ |ξ|s+2αea|ξ|
1/σ

e−t|ξ|
2α

|v̂0|+ |ξ|s+2αea|ξ|
1/σ

∫ t

0

e−(t−τ)|ξ|2α |f̂(τ)| dτ.

By using the L1([0, T ])-norm, one has∫ T

0

|ξ|s+2αea|ξ|
1/σ

|v̂(t)| dt ≤ |ξ|sea|ξ|
1/σ

|v̂0|+ |ξ|s+2αea|ξ|
1/σ

∫ T

0

[e−t|ξ|
2α

] ∗ [|f̂(t)|] dt.

Apply Young’s inequality we obtain∫ T

0

|ξ|s+2αea|ξ|
1/σ

|v̂(t)| dt ≤ |ξ|sea|ξ|
1/σ

|v̂0|+ |ξ|sea|ξ|
1/σ

∫ T

0

|f̂(t)| dt.



EJDE-2023/78 CRITICAL AND SUBCRITICAL CASES FOR NAVIER-STOKES EQUATIONS 7

By taking the L1(R3)-norm, it follows that

‖v‖L1
T (X s+2α

a,σ ) ≤ ‖v0‖X sa,σ + ‖f‖L1
T (X sa,σ). (3.5)

On the other hand, Hölder’s inequality implies that

‖v‖
X
s+2α

p
a,σ

=

∫
R3

|ξ|s+
2α
p ea|ξ|

1/σ

|v̂(ξ)| dξ

≤
(∫

R3

|ξ|sea|ξ|
1/σ

|v̂(ξ)| dξ
)1− 1

p
(∫

R3

|ξ|s+2αea|ξ|
1/σ

|v̂(ξ)| dξ
)1/p

.

Hence, one has

‖v‖
X
s+2α

p
a,σ

≤ ‖v‖1−
1
p

X sa,σ
‖v‖1/pX s+2α

a,σ
.

As a result, we can write

‖v‖p
LpT (X

s+2α
p

a,σ )

=

∫ T

0

‖v‖p
X
s+2α

p
a,σ

dt ≤
∫ T

0

‖v‖p−1
X sa,σ
‖v‖X s+2α

a,σ
dt

≤ ‖v‖p−1
L∞T (X sa,σ)‖v‖L1

T (X s+2α
a,σ ).

By (3.4) and (3.5), one infers that

‖v‖
LpT (X

s+2α
p

a,σ )
≤ ‖v0‖X sa,σ + ‖f‖L1

T (X sa,σ),

for all p ≥ 1. This proves (ii). As a result, we can conclude that v ∈ LpT (X s+
2α
p

a,σ (R3))
since f ∈ L1

T (X sa,σ(R3)) and v0 ∈ X sa,σ(R3). �

In the proof of Theorem 1.1, we shall apply the following Fixed Point Theorem.

Lemma 3.2 ([13]). Let (X, ‖ · ‖) be a Banach space and B : X × X → X a
continuous bilinear operator, i.e., there exists a positive constant C such that

‖B(w, v)‖ ≤ C‖w‖‖v‖, ∀w, v ∈ X. (3.6)

Then, for each x0 ∈ X that satisfies 4C‖x0‖ < 1, the equation a = x0+B(a, a), with
a ∈ X, admits a solution u ∈ X. Moreover, u solves the inequality ‖u‖ ≤ 2‖x0‖
and it is the only one such that ‖u‖ ≤ 1

2C .

The next lemmas will be useful in the proof of our main result, Theorem 1.1.

Lemma 3.3 ([28]). Let a, σ and s be real numbers such that (a, s, σ) ∈
(
(0,+∞)×

(−∞, 0)×(1,+∞)
)
∪
(
[0,+∞)×{0}× [1,+∞)

)
. Assume that f ∈ X sa,σ(R3). Then,

f ∈ X 0
a
σ ,σ

(R3). Moreover, there exists a positive constant Ca,s,σ such that

‖f‖X 0
a
σ
,σ
≤ Ca,s,σ‖f‖X sa,σ .

Lemma 3.4 ([30]). Let a ≥ 0, σ ≥ 1 and s ≥ −1. Assume that f, g ∈ X s+1
a,σ (R3) ∩

X 0
a
σ ,σ

(R3). Then, fg ∈ X s+1
a,σ (R3). Moreover, there is a positive constant Cs such

that

‖fg‖X s+1
a,σ
≤ Cs[‖f‖X 0

a
σ
,σ
‖g‖X s+1

a,σ
+ ‖f‖X s+1

a,σ
‖g‖X 0

a
σ
,σ

].



8 W. G. MELO, N. F. ROCHA, N. S. COSTA EJDE-2023/78

4. Proof of main result

Proof of Theorem 1.1. First of all, it is necessary to apply the operator e−(t−τ)(−∆)α

(with τ ∈ [0, t]) to the first equation in (1.1) to obtain

e−(t−τ)(−∆)αuτ + e−(t−τ)(−∆)αP (u · ∇u) + e−(t−τ)(−∆)α(−∆)αu = 0, (4.1)

where P is the usual Helmholtz’s projector. It is known that this operator satisfies

|F [P (f)](ξ)| ≤ |f̂(ξ)|, ∀ξ ∈ R3. (4.2)

Integrate (4.1) over [0, t] to obtain

u(t) = e−t(−∆)αu0 −
∫ t

0

e−(t−τ)(−∆)αP (u · ∇u)(τ) dτ, (4.3)

On the other hand, (4.3) implies

u(t) = e−t(−∆)αu0 +B(u, u)(t), (4.4)

where

B(w, v)(t) = −
∫ t

0

e−(t−τ)(−∆)αP (v · ∇w)(τ) dτ, ∀w, v ∈ XT . (4.5)

Here XT := CT (X sa,σ(R3)) ∩ L1
T (X s+2α

a,σ (R3)) (for any arbitrary T > 0) denotes
Banach space endowed with the norm

‖g‖XT := ‖g‖L∞T (X sa,σ) + ‖g‖L1
T (X s+2α

a,σ ), ∀g ∈ XT .

Notice that, from (4.5) it is easy to check that B : XT × XT → XT is a bilinear
operator. Thus, our next goal is to show that the operator B is also continuous.
To prove this fact, we firstly observe that

∂tB(w, v)(t) = (−∆)α
∫ t

0

e−(t−τ)(−∆)αP (v · ∇w)(τ) dτ − P (v · ∇w)(t)

= −(−∆)αB(w, v)(t)− P (v · ∇w)(t).

As a consequence, we can write the following system related to the operator B:

∂tB(w, v)(t) + (−∆)αB(w, v)(t) = −P (v · ∇w)(t);

B(w, v)(0) = 0.
(4.6)

We shall apply Lemma 3.1 to (4.6) to prove that B is continuous. By observing
(4.2), we conclude that

‖P (v · ∇w)‖L1
T (X sa,σ) =

∫ T

0

∫
R3

|ξ|sea|ξ|
1/σ

|F [P (v · ∇w)(t)]| dξdt

≤
∫ T

0

∫
R3

|ξ|s+1ea|ξ|
1/σ

|F [(w ⊗ v)(t)]| dξdt.

Hence,

‖P (v · ∇w)‖L1
T (X sa,σ) ≤

∫ T

0

‖(w ⊗ v)(t)‖X s+1
a,σ

dt.

By using Lemma 3.4, it follows that

‖P (v · ∇w)‖L1
T (X sa,σ) ≤ Cs

∫ T

0

[‖v‖X 0
a
σ
,σ
‖w‖X s+1

a,σ
+ ‖v‖X s+1

a,σ
‖w‖X 0

a
σ
,σ

] dt,



EJDE-2023/78 CRITICAL AND SUBCRITICAL CASES FOR NAVIER-STOKES EQUATIONS 9

since a ≥ 0, σ ≥ 1 and s ≥ −1. Apply Lemma 3.3 and Hölder’s inequality to obtain

‖P (v · ∇w)‖L1
T (X sa,σ) ≤ Ca,σ,s‖v‖L∞T (X sa,σ)‖w‖

1− 1
2α

L∞T (X sa,σ)

∫ T

0

‖w‖
1
2α

X s+2α
a,σ

dt

+ Ca,σ,s‖w‖L∞T (X sa,σ)‖v‖
1− 1

2α

L∞T (X sa,σ)

∫ T

0

‖v‖
1
2α

X s+2α
a,σ

dt,

(4.7)

where (a, s, σ) ∈
(
(0,+∞)× (−∞, 0)× (1,+∞)

)
∪
(
[0,+∞)× {0} × [1,+∞)

)
and

α ≥ 1
2 . By using Hölder’s inequality once again, one infers that

‖P (v · ∇w)‖L1
T (X sa,σ) ≤ Ca,σ,sT 1− 1

2α ‖v‖L∞T (X sa,σ)‖w‖
1− 1

2α

L∞T (X sa,σ)‖w‖
1
2α

L1
T (X s+2α

a,σ )

+ Ca,σ,sT
1− 1

2α ‖w‖L∞T (X sa,σ)‖v‖
1− 1

2α

L∞T (X sa,σ)‖v‖
1
2α

L1
T (X s+2α

a,σ )
.

As a result, one has

‖P (v · ∇w)‖L1
T (X sa,σ) ≤ Ca,σ,sT 1− 1

2α ‖w‖XT ‖v‖XT , ∀w, v ∈ XT . (4.8)

By applying Lemma 3.1 (ii) (with p = 1) to system (4.6) and, by using (4.8), we
obtain

‖B(w, v)‖L1
T (X s+2α

a,σ ) ≤ Ca,σ,sT
1− 1

2α ‖w‖XT ‖v‖XT , ∀w, v ∈ XT . (4.9)

By using Lemma 3.1 (i),(4.6) and (4.8), one infers that

‖B(w, v)‖L∞T (X sa,σ) ≤ Ca,σ,sT 1− 1
2α ‖w‖XT ‖v‖XT , ∀w, v ∈ XT . (4.10)

From (4.9) and (4.10), one obtains

‖B(w, v)‖XT ≤ Ca,σ,sT 1− 1
2α ‖w‖XT ‖v‖XT , ∀w, v ∈ XT . (4.11)

This inequality shows that the operator B is continuous.
Therefore, we only need to estimate the term e−t(−∆)αu0 given in (4.4), by

considering the space XT , to apply Lemma 3.2. Thereby,

‖e−t(−∆)αu0‖X sa,σ =

∫
R3

|ξ|sea|ξ|
1/σ

e−t|ξ|
2α

|û0(ξ)| dξ ≤
∫
R3

|ξ|sea|ξ|
1/σ

|û0(ξ)| dξ,

for all t ∈ [0, T ]. Then, we conclude that

‖e−t(−∆)αu0‖L∞T (X sa,σ) ≤ ‖u0‖X sa,σ . (4.12)

On the other hand,

‖e−t(−∆)αu0‖L1
T (X s+2α

a,σ ) =

∫
R3

|ξ|s+2αea|ξ|
1/σ

|û0(ξ)|
(∫ T

0

e−t|ξ|
2α

dt
)
dξ

≤
∫
R3

|ξ|sea|ξ|
1/σ

|û0(ξ)| dξ = ‖u0‖X sa,σ .

Hence, we are able to write the inequality

‖e−t(−∆)αu0‖L1
T (X s+2α

a,σ ) ≤ ‖u0‖X sa,σ . (4.13)

Therefore, by (4.12) and (4.13), one concludes that

‖e−t(−∆)αu0‖XT ≤ 2‖u0‖X sa,σ . (4.14)



10 W. G. MELO, N. F. ROCHA, N. S. COSTA EJDE-2023/78

Now, let us prove Theorem 1.1 (i) (α = 1/2 in this case). Thus, assume that
‖u0‖X sa,σ < [8Ca,σ,s]

−1 (where Ca,σ,s is given in (4.11)) to apply Lemma 3.2 and

obtain a unique global solution u ∈ XT for the equation (4.4) that satisfies

‖u‖XT ≤ 2‖e−t(−∆)1/2u0‖XT ,

and, consequently, by (4.14), we have

‖u‖L∞T (X sa,σ) + ‖u‖L1
T (X s+1

a,σ ) ≤ 4‖u0‖X sa,σ .

Analogously to the proof above, we can prove that the solution u also belongs

to LpT (X s+
1
p

a,σ (R3)), for all p ≥ 1. In fact, observe that the Navier-Stokes equations
(1.1) (with α = 1/2) can be rewritten as

ut + (−∆)1/2u = −P (u · ∇u);

u(·, 0) = u0.

Hence, similarly to (4.8), we have

‖P (u · ∇u)‖L1
T (X sa,σ) ≤ Ca,σ,s[‖u‖L∞T (X sa,σ(R3)) + ‖u‖L1

T (X s+1
a,σ (R3))]

2.

Therefore, P (u · ∇u) ∈ L1
T (X sa,σ(R3)) since u ∈ CT (X sa,σ(R3))∩L1

T (X s+1
a,σ (R3)). By

using that u0 ∈ X sa,σ(R3), and applying Lemma 3.1 (ii), the proof of Theorem 1.1
i) is complete.

We are ready to show Theorem 1.1 (ii) (α > 1/2 in this case). Thereby, by

taking 0 < T < [8Ca,σ,s‖u0‖X sa,σ ]
2α

1−2α (where Ca,σ,s is given in (4.11)), Lemma 3.2

provides a unique local solution u ∈ XT for equation (4.4) such that

‖u‖XT ≤ 2‖e−t(−∆)αu0‖XT .

Therefore, by (4.14), one obtains

‖u‖L∞
T

(X sa,σ) + ‖u‖L1
T

(X s+2α
a,σ ) ≤ 4‖u0‖X sa,σ .

This solution u belongs to Lp
T

(X s+
2α
p

a,σ (R3)), for all p ≥ 1. In fact, at first, rewrite

the Navier-Stokes equations (1.1) as

ut + (−∆)αu = −P (u · ∇u);

u(·, 0) = u0.

Secondly, applying similar arguments as in (4.8) we obtain

‖P (u · ∇u)‖L1
T

(X sa,σ) ≤ Ca,σ,sT
1− 1

2α [‖u‖L∞
T

(X sa,σ(R3)) + ‖u‖L1
T

(X s+2α
a,σ (R3))]

2.

Thereby, P (u · ∇u) ∈ L1
T

(X sa,σ(R3)) since u ∈ CT (X sa,σ(R3)) ∩ L1
T

(X s+2α
a,σ (R3)).

By using that u0 ∈ X sa,σ(R3) and Lemma 3.1 ii), the proof of Theorem 1.1 ii) is
complete. �

Acknowledgments. W. G. Melo was partially supported by CNPq grant 309880/
2021-1. N. dos Santos Costa was partially supported by CAPES grant 88887.620886/
2021-00.



EJDE-2023/78 CRITICAL AND SUBCRITICAL CASES FOR NAVIER-STOKES EQUATIONS 11

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Wilberclay G. Melo

Departamento de Matemática, Universidade Federal de Sergipe, São Cristóvão, SE

49100-000, Brazil
Email address: wilberclay@academico.ufs.br

Natã F. Rocha
Campus Clóvis Moura, Universidade Estadual do Piaúı, Teresina, PI 64078-213, Brazil

Email address: natafirmino@ccm.uespi.br

Natielle dos Santos Costa

Departamento de Matemática, Universidade Federal de Sergipe, São Cristóvão, SE

49100-000, Brazil
Email address: natielle.scosta@academico.ufs.br


	1. Introduction
	2. Notation
	3. Preliminary lemmas
	4. Proof of main result
	Acknowledgments

	References