International Journal of Analysis and Applications

Volume 18, Number 3 (2020), 421-438

URL: https://doi.org/10.28924/2291-8639

DOI: 10.28924/2291-8639-18-2020-421

STUDY OF THE BLOW UP OF THE MAXIMAL SOLUTION TO THE

THREE-DIMENSIONAL MAGNETOHYDRODYNAMIC SYSTEM IN

LEI-LIN-GEVREY SPACES

RIDHA SELMI1,2,3,∗, JAMEL BENAMEUR4

1Department of Mathematics, College of Sciences, Northern Border University, P.O. Box 1321, Arar,

73222, KSA

2Faculty of Sciences of Gabes, University of Gabes, 6072, Gabès, Tunisia

3Laboratory of partial differential equations and applications (LR03ES04), Faculty of sciences of Tunis,

University of Tunis El Manar, 1068 Tunis, Tunisia

4Department of Mathematics, ISSAT Gabès, University of Gabès, Tunisia

∗Corresponding author: Ridha.selmi@nbu.edu.sa

Abstract. In this paper, we investigate existence, uniqueness and blowup in finite time of the local solution

to the three dimensional magnetohydrodynamic system, in Gevrey-Lei-Lin spaces. To prove the blowup

results and give the blow profile as a function of time, two key points are used. The first is a frequency

decomposition of the spectrum of the initial data. This allows to use Leray theory. The second is a technical

lemma we proved to state that the Lei-Lin space is an interpolation space between the Gevrey-Lei-Lin and

the Lebesgue square integrable functions spaces. To prove uniqueness, we use a penalization procedure

and energy methods. About existence, we split the initial condition into low frequencies part and high

frequencies part. The former are considered as initial data to the linear part of the system. The latter will

be taken as small as needed, so that smallness theory applies and allows to run a fixed point argument.

Received February 8th, 2020; accepted February 28th, 2020; published May 1st, 2020.

2010 Mathematics Subject Classification. 35A01, 35A02, 35B40, 35B44, 35B45.

Key words and phrases. magnetohydrodynamic system; critical spaces; existence; uniqueness; blowup; Gevrey-Lein-Lin

space; frequency decomposition.

©2020 Authors retain the copyrights

of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

421

https://doi.org/10.28924/2291-8639
https://doi.org/10.28924/2291-8639-18-2020-421


Int. J. Anal. Appl. 18 (3) (2020) 422

Let us consider the following three dimensional incompressible magnetohydrodynamic system,

(MHD)




∂tu− ∆u + u ·∇u− b ·∇b + ∇(p + 12|b|
2) = 0, (t,x) ∈ R+ ×R3

∂tb− ∆b + u ·∇b− b ·∇u = 0, (t,x) ∈ R+ ×R3

div u = 0, (t,x) ∈ R+ ×R3

div b = 0, (t,x) ∈ R+ ×R3

(u,b)(0) = (u0,b0), x ∈ R3,

where u, b and p denote respectively the unknown velocity, the unknown magnetic field and the unknown

pressure at the point (t,x). If the initial data u0 and b0 are quite regular, the divergence free conditions

determine the pressure p. We aim to study the existence, uniqueness and blowup in finite time of local

solution to the MHD system, in the framework of Gevrey-Lei-Lin spaces. These spaces are defined for the

real numbers a > 0, σ > 1 and ρ, by

Xρa,σ(R
3) = {f ∈S′(R3); f̂ ∈ L1loc(R

3) ,

∫
R3
|ξ|ρea|ξ|

1/σ

|f̂(ξ)|dξ < ∞}

and endowed with its naturel norm

‖f‖Xρa,σ(R3) =
∫
R3
|ξ|ρea|ξ|

1/σ

|f̂(ξ)|dξ,

where f̂ denotes the Fourier transform of f.

In [15], the authors defined the Lei-Lin space by

X−1(R3) = {f ∈S′(R3), f̂ ∈ L1loc(R
3),

∫
R3

|f̂(ξ)|
|ξ|

dξ < ∞}

and endowed with its natural norm

‖f‖X−1(R3) =
∫
R3

|f̂(ξ)|
|ξ|

dξ

Here, L1loc(R
3) states for the set of locally R3-Lebesgue integrable distributions. In this critical space, the

distinguishable fact was that to obtain global well-posedness to the Navier-Stokes equations, the norm

of the initial data have to be exactly less than the viscosity of the fluid. However, in the wide fluid

mechanic literature, it was always assumed that the initial data must be very small, especially smaller

than the viscosity multiplied by a tiny positive constant. Such assumption is mandatory to run the

smallness argument and to obtain global well-posedness; see for example [11–14] and a complete survey

in [10]. For many fluid mechanics equations, well-posedness and asymptotic behavior, as time goes to

infinity or as small parameter goes to zero, were investigated by the authors, in various spaces; see for

example [6–9, 18–20]. About blowup, it is worthwhile to emphasize that several authors studied this

phenomena to the Navier-Stokes equations; see for example [1–3, 17] and references therein. The author

observed, in [4], that in the case of the Navier-Stokes equations, the blowup phenomena depends on the

chosen space not on the nonlinear part. To do so, he used Fourier analysis in Sobolev-Gevrey spaces, for



Int. J. Anal. Appl. 18 (3) (2020) 423

Sobolev index s > 3/2. His blowup result was improved later on, in [5], where the authors gave precisely an

exponential type of the blowup profile, in Sobolev-Gevrey spaces but with less regularity on the initial data

since they dealt with s = 1. In [16], authors studied the Cauchy problem for a two-components high-order

Camassa-Holm system. First, they proved the local well-posedness of the system in Besov spaces. Then,

using Littlewood-Paley theory, they derived a blowup criterion for the strong solution. Finally, they studied

Gevrey regularity and analyticity of the solutions to the Camassa-Holm system in the Gevrey-Sobolev spaces.

In this paper, we begin by addressing the problem of local well-posedness. Our result is summarized in

the following existence and uniqueness theorem.

Theorem 0.1. Let (u0,b0) ∈ (X−1a,σ(R3))2. Then, there exist a time T > 0 and a unique solution (u,b) ∈

C([0,T], (X−1a,σ(R3))2) of (MHD), such that (u,b) ∈ L1([0,T], (X1a,σ(R3))2).

To prove uniqueness of solution, we use a penalisation method. This allows to put the problem in a form

where Gronwall inequality can be applied. To establish existence of solution, we split the initial condition

into low and high frequencies. The former will be considered as initial data to the linear part of the MHD

system. The latter, taken as small as needed, will be the initial data to the remaining nonlinear part, for

which smallness theory applies and allows to run a fixed point argument, in (X−1a,σ(R3))2.

Then, we turn to the blowup result that we state in the following theorem.

Theorem 0.2. Let (u,b) ∈ C([0,T∗[, (X−1a,σ(R3))2) be the maximal solution of MHD system, where T∗ < ∞.

Then, there exists c0 > 0, such that

lim inf
t↗T∗

(T∗ − t)1/3e−c0(T
∗−t)−

1
3σ ‖(u,b)(t)‖X−1a,σ ≥

1

2
.

The proof is somewhat technical. Our idea is to use a suitable frequency decomposition and to impose

”the problematic” large frequencies part to be a square integrable function, so that we fully profit from

Leray theory. Some technical lemmas are specially derived to handle technical difficulties, mainly Lemma

1.1, where we proved that X0 is an interpolation space between the Lei-Lin spaces Xs and the space of

Lebesgue square integrable functions. The structure of our proof is as follows. Starting with the energy

estimates, we prove that the X−1a,σ norm of the solution blows up, in finite time. Gronwall type inequality

allows to infer that blowup holds also in X0a′,σ, for any a
′ ∈ (0,a). Using a particular choice of parameter a′,

we deduce that our solution blows up in the X0 norm, as a limit of X0a′,σ spaces. We split the initial data

into two parts, a large frequencies one that belongs to X−1a,σ ∩L2 and a small frequencies remainder that leis

in X−1a,σ. The smallness theory applies an leads to a global and continuous in time solution that belongs to

X−1a,σ. This continuity plays an important role. About the large frequencies part, the L2 theory applies and

allows to derive a Leray type energy estimates. We use the above two estimates to dominate the X0 norm



Int. J. Anal. Appl. 18 (3) (2020) 424

of the solution. Finally, Lemma 1.1 with a judicious choice of the index s, finish the proof of the blowup

result, in X−1a,σ and determine its profile as a function of time.

This paper is organized as follows. In section 2, we give some notations and useful preliminary results.

Section 3 is devoted to prove existence and uniqueness of local in time solution. In section 4, we establish

the blowup result.

1. Technical lemmas

In this section, we prove some technical lemmas that will be used later on.

Lemma 1.1. Let s ≥ 3/2, there exists C > 0 such that, ∀f ∈ L2(R3) ∩Xs(R3),

‖f‖X0 ≤ C‖f‖
s

s+ 3
2

L2
‖f‖

3
2

s+ 3
2

Xs .

Proof. For λ > 0, we consider the decomposition

‖f‖X0 =
∫
|ξ|<λ

|f̂(ξ)|dξ︸ ︷︷ ︸
I(λ)

+

∫
|ξ|>λ

|f̂(ξ)|dξ︸ ︷︷ ︸
J(λ)

.

By Cauchy-Schwarz inequality for I(λ) and direct computation for J(λ), we get

I(λ) ≤
√

4πλ3/2‖f‖L2,

J(λ) ≤ λ−s‖f‖Xs.

For A =
√

4π‖f‖L2 and B = ‖f‖Xs, let ϕ(λ) = Aλ3/2 + Bλ−s. Clearly the value λ0 =
(

2sB
3A

) 1
s+ 3

2 is a

minimum of ϕ and for all s ≥ 3/2, ϕ(λ0) ≤ CA
s

s+ 3
2 B

3/2

s+ 3
2 . �

Lemma 1.2. Let σ > 1, then there is a constant C > 0, such that for all f,g ∈X−1a,σ(R3), we have

‖fg‖X0a,σ ≤ C‖f‖X0a,σ‖g‖X0a
σ
,σ

+ C‖g‖X0a,σ‖f‖X0a
σ
,σ
.

Proof. In a first step, using the triangular inequality and the fact that

|ξ| ≤ max(|ξ −η|, |η|)(1 +
min(|ξ −η|, |η|)
max(|ξ −η|, |η|)

),

we obtain

‖fg‖X0a,σ ≤
∫
ξ

∫
η

e
a max(|ξ−η|,|η|)1/σ(1+ min(|ξ−η|,|η|)

max(|ξ−η|,|η|) )
1/σ

|f̂(ξ −η)|.|ĝ(η)|dηdξ.

In a second step, the inequality (1 + r)θ ≤ 1 + θrθ, ∀r ∈ [0, 1], ∀θ ∈]0, 1] leads to

a max(|ξ −η|, |η|)1/σ(1 + min(|ξ−η|,|η|)
max(|ξ−η|,|η|) )

1/σ ≤ a max(|ξ −η|, |η|)1/σ + a
σ

min(|ξ −η|, |η|)1/σ.

Finally, distinguishing the two cases |ξ −η| < |η| and |ξ −η| > |η|, we obtain the desired result. �



Int. J. Anal. Appl. 18 (3) (2020) 425

By Cauchy-Schwarz inequality, we prove the following lemma.

Lemma 1.3. For all f ∈Xs−1a,σ (R3) ∩Xs+1a,σ (R3), with σ > 1 and a > 0, we have

‖f‖Xsa,σ ≤‖f‖
1/2

Xs−1a,σ
‖f‖1/2

Xs+1a,σ
.

Using the fact that |ξ|1/σ ≤ (|ξ −η| + |η|)1/σ ≤ |ξ −η|1/σ + |η|1/σ, we have the lemma below.

Lemma 1.4. For all f ∈X0a,σ(R3), g ∈X1a,σ(R3), with σ > 1 and a > 0, we have

‖f.∇g‖X0a,σ ≤‖f‖X0a,σ‖g‖X1a,σ.

Lemma 1.5. Let u,v ∈ L∞([0,T],X−1a,σ(R3)) ∩L1([0,T],X1a,σ(R3)). Then

‖
∫ t

0

e(t−τ)∆div(uv)dτ‖X−1a,σ ≤‖u‖
1/2

L∞
T

(X−1a,σ)
‖u‖1/2

L1
T

(X1a,σ)
‖v‖1/2

L∞
T

(X−1a,σ)
‖v‖1/2

L1
T

(X1a,σ)
.

Proof. First of all, let us prove that for f,g ∈X−1a,σ(R3) ∩X1a,σ(R3), we have

‖fg‖X0a,σ ≤‖f‖
1/2

X−1a,σ
‖f‖1/2X1a,σ‖g‖

1/2

X−1a,σ
‖g‖1/2X1a,σ. (1.1)

To do so, we recall that

‖fg‖X0a,σ ≤
∫
ξ

ea|ξ|
1/σ

(∫
η

|f̂(ξ −η)||ĝ(η)|dη
)
dξ.

Using the inequality ea|ξ|
1/σ

≤ ea|ξ−η|
1/σ

ea|η|
1/σ

, we obtain

‖fg‖X0a,σ ≤
∫
ξ

(

∫
η

ea|ξ−η|
1/σ

|f̂(ξ −η)|.ea|η|
1/σ

.|ĝ(η)|dη)dξ.

Put F(ξ) = ea|ξ|
1/σ

|f̂(ξ)| and G(ξ) = e
a|ξ|1/σ

|ξ| |f̂(ξ)|. It holds that,

‖fg‖X0a,σ ≤ ‖F ∗G‖L1

≤ ‖F‖L1‖G‖L1

≤ ‖f‖X0a,σ‖g‖X0a,σ.

By Cauchy-Schwarz inequality, we get (1.1). To continue proving lemma above, we have∫ t
0

‖e(t−τ)∆div(uv)dτ‖X−1a,σ ≤‖
∫ t

0

e(t−τ)∆div(uv)‖X−1a,σdτ

≤
∫ t

0

‖uv)‖X0a,σdτ

≤
∫ t

0

‖u‖X0a,σ‖v‖X0a,σdτ

≤
∫ t

0

‖u‖1/2
X−1a,σ
‖u‖1/2X1a,σ‖v‖

1/2

X−1a,σ
‖v‖1/2X1a,σdτ

≤‖u‖1/2
L∞
T

(X−1a,σ)
‖v‖1/2

L∞
T

(X−1a,σ)

∫ T
0

‖u‖1/2X1a,σ‖v‖
1/2
X1a,σ

dτ

≤‖u‖1/2
L∞
T

(X−1a,σ)
‖v‖1/2

L∞
T

(X−1a,σ)
‖u‖1/2

L1
T

(X1a,σ)
‖v‖1/2

L1
T

(X1a,σ)
.

�



Int. J. Anal. Appl. 18 (3) (2020) 426

Lemma 1.6. Let u,v ∈ L∞T (X
−1
a,σ(R

3)) ∩L1T (X
1
a,σ(R

3)). Then

‖
∫ t

0

e(t−τ)∆div(uv)dτ‖L1
T

(X1a,σ) ≤‖u‖
1/2

L∞
T

(X−1a,σ)
‖v‖1/2

L∞
T

(X−1a,σ)
‖u‖1/2

L1
T

(X1a,σ)
‖v‖1/2

L1
T

(X1a,σ)
.

Proof. Using the definition of X1a,σ norm and integrating the function e−(t−τ)|ξ|
2

twice with respect to

τ ∈ [0, t] and t ∈ [0,T], it follows that

∫ T
0

‖
∫ t

0

e(t−τ)∆div(uv)dτ‖X1a,σdt

≤
∫ T

0

∫ t
0

∫
R3
e−(t−τ)|ξ|

2

|ξ|2ea|ξ|
1/σ

|ûv(τ,ξ)|dτdtdξ

≤
∫
R3
|ξ|2ea|ξ|

1/σ

(∫ T
0

∫ t
0

e−(t−τ)|ξ|
2

|ûv(τ,ξ)|dτdt

)
dξ

≤
∫
R3
|ξ|2ea|ξ|

1/σ


∫ T

0

|ûv(τ,ξ)|


[−e−(t−τ)|ξ|2

|ξ|2

]T
τ


dτ


dξ

≤
∫
R3
|ξ|2ea|ξ|

1/σ

(∫ T
0

|ûv(τ,ξ)|(
1 −e−(T−τ)|ξ|

2

|ξ|2
)dτ

)
dξ

≤
∫ T

0

‖uv‖X0a,σdτ.

Using equation (1.1) finishes the proof. �

2. Well-posedness results

To prove uniqueness of solution to the (MHD), we consider two solutions (u1,b1) and (u2,b2) that belong

to (C([0,T],X−1a,σ(R3)))2 ∩ (L1([0,T],X1a,σ(R3)))2 and have the same initial data. Let δ = u1 − u2 and

η = b1 − b2, it follows that


∂tδ − ∆δ + δ ·∇u2 + u1 ·∇δ −η ·∇b2 − b1 ·∇η + ∇(p1 −p2 + 12 (|b1|
2 −|b2|2)) = 0

∂tη − ∆η + u2 ·∇η −η ·∇u1 + δ ·∇b1 − b2 ·∇δ = 0

div δ = 0

div η = 0

(δ,η) = (0, 0).

(2.1)

Taking the Fourier transform and using divergence free condition, one infers that

∂tδ̂ + |ξ|2δ̂ + ̂(δ ·∇u2) + ̂(u1 ·∇δ) − ̂(η.∇b2) − ̂(b1 ·∇η) = 0 (2.2)

∂tη̂ + |ξ|2η̂ + ̂(u2 ·∇η) − ̂(η ·∇u1) + ̂(δ ·∇b1) − ̂(b2 ·∇δ) = 0. (2.3)

Multiplying (2.2) (respectively (2.3)) by δ̂ (respectively η̂) and its conjugate by δ̂ (respectively η̂), summing

up the four resulting equations, and dominating the real part of any complex quantity by its modulus, we



Int. J. Anal. Appl. 18 (3) (2020) 427

obtain

1
2
∂t(|δ̂|2 + |η̂|2) + |ξ|2(|δ̂|2 + |η̂|2)

≤ | ̂(δ.∇u2)||δ̂| + | ̂(u1.∇δ)||δ̂| + | ̂(η.∇b2)||δ̂| + | ̂(b1.∇η)||δ̂|

+ | ̂(u2.∇η)||η̂| + | ̂(η.∇u1)||η̂| + 2| ̂(δ.∇b1)||η̂| + | ̂(b2.∇δ)||η̂|.

(2.4)

Let ε > 0 be a penalizing parameter. One has

∂t(|δ̂|2 + |η̂|2) = ∂t(|δ̂|2 + |η̂|2 + ε2) = 2
√
|δ̂|2 + |η̂|2 + ε2.∂t

√
|δ̂|2 + |η̂|2 + ε2. (2.5)

Substituting (2.5) in (2.4), dividing by

√
|δ̂|2 + |η̂|2 + ε2, integrating with respect to time, letting ε → 0 and

using that
|δ̂|+|η̂|√

2
≤
√
|δ̂|2 + |η̂|2, we infer that

|δ̂| + |η̂| +
∫ t

0

|ξ|2(|δ̂| + |η̂|)dτ

≤
√

2(

∫ t
0

(| ̂(δ.∇u2)||δ̂| + | ̂(u1.∇δ)||δ̂| + | ̂(η.∇b2)||δ̂| + | ̂(b1.∇η)||δ̂|)dτ

+

∫ t
0

(| ̂(u2.∇η)||η̂| + | ̂(η.∇u1)||η̂| + 2| ̂(δ.∇b1)||η̂| + | ̂(b2.∇δ)||η̂|dτ).

Multiplying by e
a|ξ|

1
σ

|ξ| and integrating with respect to ξ. By divergence free, we get

‖δ‖X−1a,σ + ‖η‖X−1a,σ +
∫ t

0

(‖∆δ‖X−1a,σ + ‖∆η‖X−1a,σ )dτ

≤
√

2(

∫ t
0

‖δu2‖X0a,σ + ‖u1δ‖X0a,σ + ‖ηb2‖X0a,σ + ‖b1η‖X0a,σdτ

+

∫ t
0

‖u2η‖X0a,σ + ‖ηu1‖X0a,σ + ‖δb1‖X0a,σ + ‖b2δ‖X0a,σdτ).

Using the product Young inequality, it follows that

‖δu2‖X0a,σ ≤ ‖δ‖X0a,σ‖u2‖X0a,σ

≤ ‖δ‖
1
2

X−1a,σ
‖∆δ‖

1
2

X−1a,σ
‖u2‖

1
2

X−1a,σ
‖∆u2‖

1
2

X−1a,σ

≤ 2‖δ‖X−1a,σ‖u2‖X−1a,σ‖∆u2‖X−1a,σ +
1

2
‖∆δ‖X−1a,σ,

and so on for the other terms. Thus,

‖δ‖X−1a,σ + ‖η‖X−1a,σ ≤ 2
√

2

∫ t
0

(‖δ‖X−1a,σ + ‖η‖X−1a,σ )
∑

1≤i≤2

‖ui‖X−1a,σ‖∆ui‖X−1a,σ + ‖bi‖X−1a,σ‖∆bi‖X−1a,σdτ.

Since the function t 7→
∑

1≤i≤2

‖ui‖X−1a,σ‖∆ui‖X−1a,σ + ‖bi‖X−1a,σ‖∆bi‖X−1a,σ belongs to L
1([0,T]), Gronwall

inequality implies that δ = 0 on [0,T]. Thus, uniqueness holds.

We turn to the existence result. To do so, let r ∈ (0, 1
10

), such that

(‖(u0,b0)‖X−1a,σ + r)
1/2r1/2 <

1

32
,



Int. J. Anal. Appl. 18 (3) (2020) 428

and N ∈ N, such that

∫
|ξ|>N

ea|ξ|
1/σ

|ξ|
|û0(ξ)|dξ +

∫
|ξ|>N

ea|ξ|
1/σ

|ξ|
|b̂0(ξ)|dξ <

r

5
.

Let v0 = F−1(1{|ξ|<N}û0(ξ)) (respectively c0 = F−1(1{|ξ|<N}b̂0(ξ))) the low frequencies part of u0 (re-

spectively b0) and w0 (respectively d0) its high frequencies part corresponding to (|ξ| > N). Clearly, one

has

‖(d0,w0)‖X−1a,σ <
r

5
. (2.6)

By a standard Fourier computation, one infers that (v,c) = eνt∆(v0,c0) is the unique solution to the following

linear system

(MHDL)




∂tv − ∆v = 0, (t,x) ∈ R+ ×R3

∂tc− ∆c = 0, (t,x) ∈ R+ ×R3

div v = 0, (t,x) ∈ R+ ×R3

div c = 0, (t,x) ∈ R+ ×R3

(v,c)(0) = (v0,c0), x ∈ R3

and that for all t ≥ 0,

‖(v,c)‖X−1a,σ ≤‖(u
0,b0)‖X−1a,σ. (2.7)

By definition of X1a,σ norm, expression of (v,c) and Tonelli’s theorem, we infers that

‖(v,c)‖L1
T

(X1a,σ) ≤
∫
R3

(1 −e−νT|ξ|
2

)|ξ|−1ea|ξ|
1/σ

(|û0(ξ)| + |b̂0(ξ)|)dξ.

Using the Dominated Convergence Theorem, we get

lim
T→0+

‖(v,c)‖L1
T

(X1a,σ) = 0. (2.8)

As it will be seen below, for instance, the stability condition of the fixed point argument requires a choice

of ε > 0 such that ε1/2‖u0‖1/2
X−1a,σ

< 1
18

. Moreover, for the operator Ψ to be a contraction mapping, ε has

to fulfill the supplementary condition (‖(u0,b0)‖X−1a,σ + r)
1/2ε1/2 < 1

32
. For this choice of ε, by (2.8), there

exists a time T = T(ε) > 0 such that

‖v‖L1
T

(X1a,σ) < ε. (2.9)



Int. J. Anal. Appl. 18 (3) (2020) 429

Put w = u−v and d = b−c, the 3+3 components vector (w,d) satisfies, for all (t,x), the following nonlinear

system denoted (MHDNL),




∂tw − ∆w + (w + v) ·∇(w + v) − (d + c) ·∇(d + c) + ∇(p + 12|d + c|
2) = 0

∂td− ∆d + (w + v) ·∇(d + c) − (d + c) ·∇(w + v) = 0

div w = 0

div d = 0

(u,b) = (u0,b0).

To run a fixed point argument, we introduce the following operator Ψ defined for all (w,d)T by the right

hand side of the following integral equation
 w

d


 = eνt∆


 w0

d0


−∫ t

0

eν(t−τ)∆


 (w + v) ·∇(w + v) − (d + c) ·∇(d + c)

(w + v) ·∇(d + c) − (d + c) ·∇(w + v)


dτ,

and we consider the space XT := C([0,T], (X−1a,σ(R3))2) ∩ L1([0,T], (X1a,σ(R3))2), endowed with its naturel

norm ‖f‖XT := ‖f‖L∞
T

((X−1a,σ)2) + ‖f‖L1T ((X1a,σ)2).

In a first step, let us prove that XT is stable under the operator Ψ. To do so, we denote by Br the subset

of XT defined by

Br = {(u,b) ∈XT ;‖(u,b)‖L∞
T

(X−1a,σ) ≤ r;‖(u,b)‖L1T (X1a,σ) ≤ r}.

For (w,d) ∈ Br, we have ψ((w,d)) ∈ Br. In fact, it holds that

‖Ψ(w,d)(t)‖X−1a,σ ≤ Iww + Iwv + Ivw + Ivv + Idd + Idc + Icd + Icc

+ Iwd + Iwc + Ivd + Ivc + Idw + Idv + Icw + Icv,
(2.10)

where we denoted, for any divergence free vector field υ and ω,

Iυω =

∫ t
0

‖e(t−τ)∆υ ·∇ω‖X−1a,σdτ.

To estimate ‖Ψ(w,d)(t)‖X−1a,σ , we recall that according to the choice of N, we have

‖w0‖X−1a,σ + ‖d
0‖X−1a,σ <

r

9
.

Using divergence free condition and lemma 1.5, we obtain that

Ivv ≤‖v‖L∞
T

(X−1a,σ)‖v‖L1T (X1a,σ) ≤ ε‖u
0‖X−1a,σ <

r

18
.

The same holds for Icc, Ivc and Icv. Moreover,

Iww ≤‖w‖L∞
T

(X−1a,σ)‖w‖L1T (X1a,σ) ≤ r
2 <

r

18
,

and the same holds for Idd, Iwd and Idw. Furthermore,

Ivw ≤‖v‖
1/2

L∞
T

(X−1a,σ)
‖v‖1/2

L1
T

(X1a,σ)
‖w‖1/2

L∞
T

(X−1a,σ)
‖w‖1/2

L1
T

(X1a,σ)
≤ rε1/2‖u0‖1/2

X−1a,σ
<

r

18
,



Int. J. Anal. Appl. 18 (3) (2020) 430

and the same holds for the seven remaining integrals. Finally, we obtain

‖Ψ(w,d)(t)‖X−1a,σ ≤ r. (2.11)

Let us estimate ‖Ψ(w)(t)‖L1(X1a,σ). As above, we have

‖Ψ(w)(t)‖L1(X1a,σ) ≤ Jww + Jwv + Jvw + Jvv + Jdd + Jdc + Jcd + Jcc

+ Jwd + Jwc + Jvd + Jvc + Jdw + Jdv + Jcw + Jcv,
(2.12)

where we denoted, for any divergence free vector field υ and ω,

Jυω =

∫ T
0

‖
∫ t

0

e(t−τ)∆υ ·∇ωdτ‖X1a,σdt.

By the facts that ‖v‖L1
T

(X1a,σ),‖c‖L1T (X1a,σ) < ε, we can take

‖v‖L1
T

(X1a,σ) + ‖c‖L1T (X1a,σ) <
r

9
.

Using lemma 1.6 and the fact that w ∈ Br, we get

Jvv ≤‖v‖L∞
T

(X−1a,σ)‖v‖L1T (X1a,σ) ≤ ε‖u
0‖X−1a,σ <

r

18

and so on for Jcc, Jvc and Jcv. Also, we get

Jww ≤‖w‖L∞
T

(X−1a,σ)‖w‖L1T (X1a,σ) ≤ r
2 <

r

18

and so on for Jdd, Jwd and Jdw. Moreover,

Jvw ≤‖v‖
1/2

L∞
T

(X−1a,σ)
‖v‖1/2

L1
T

(X1a,σ)
‖w‖1/2

L∞
T

(X−1a,σ)
‖w‖1/2

L1
T

(X1a,σ)
≤ rε1/2‖u0‖1/2

X−1a,σ
<

r

18

and so on for the seven remaining integrals. Thus,

‖Ψ(w,d)(t)‖L1(X1a,σ) ≤ r. (2.13)

Combining (2.11) and (2.13), we deduce that Ψ(Br) ⊂ Br.

In a second step, to prove that Ψ is a contraction mapping on Br. One has

Ψ(w2,d2) − Ψ(w1,d1) = −
∫ t

0

e(t−τ)∆


 αww + αdd

αwd + αdw


dτ,

where

αww = (w2 + v) ·∇(w2 −w1) − (w2 −w1) ·∇(w1 + v)

αdd = −(d2 + c) ·∇(d2 + c) + (d1 + c) ·∇(d1 + c)

αwd = (w2 + v) ·∇(d2 + c) − (w1 + v) ·∇(d1 + c)

αdw = −(d2 + c) ·∇(w2 + v) + (d1 + c) ·∇(w1 + v).



Int. J. Anal. Appl. 18 (3) (2020) 431

Or equivalently, in an adequate form to be estimated, one has

αww = (w2 + v) ·∇(w2 −w1) − (w2 −w1) ·∇(w1 + v)

αdd = −(d2 + c) ·∇(d2 −d1) + (d2 −d1) ·∇(d1 + c)

αwd = (w2 −w1) ·∇(d2 + c) + (w1 −v) ·∇(d2 −d1)

αdw = −(d2 −d1) ·∇(w2 + v) − (d1 + c) ·∇(w2 −w1).

It follows that

‖Ψ((w2,d2)) − Ψ((w1,d1))‖X−1a,σ ≤
2∑
i=1

K(i)ww + K
(i)
dd + K

(i)
wd + K

(i)
dd ,

where

K(1)ww = ‖
∫ t

0

e(t−τ)∆(w2 + v)∇(w2 −w1)dτ‖X−1a,σ,

K(2)ww = ‖
∫ t

0

e(t−τ)∆(w2 −w1)∇(w1 + v)dτ‖X−1a,σ,

and so on for the other integrals. Using lemma 1.5, triangle inequality, the fact that wi belongs to Br,

inequalities (2.7) and (2.9), we infer that

K
(1)
ww, K

(2)
ww

≤‖v + w2‖
1/2

L∞
T

(X−1a,σ)
‖v + w2‖

1/2

L1
T

(X1a,σ)
‖w2 −w1‖

1/2

L∞
T

(X−1a,σ)
‖w2 −w1‖

1/2

L1
T

(X1a,σ)

≤ (‖u0‖X−1a,σ + r)
1/2(ε + r)1/2‖w2 −w1‖XT

≤ (‖(u0,b0)‖X−1a,σ + r)
1/2(ε + r)1/2‖w2 −w1‖XT .

Similarly,

K
(1)
dd , K

(2)
dd ≤ (‖(u

0,b0)‖X−1a,σ + r)
1/2(ε + r)1/2‖d2 −d1‖XT

K
(1)
wd, K

(2)
dw ≤ (‖(u

0,b0)‖X−1a,σ + r)
1/2(ε + r)1/2‖w2 −w1‖XT

K
(2)
wd, K

(1)
dw ≤ (‖(u

0,b0)‖X−1a,σ + r)
1/2(ε + r)1/2‖d2 −d1‖XT .

Thus,

‖Ψ(w2,d2) − Ψ(w1,d1)‖L∞
T

(X−1a,σ) ≤

4(‖(u0,b0)‖X−1a,σ + r)
1/2(ε + r)1/2(‖w2 −w1‖XT + ‖d2 −d1‖XT ).

(2.14)

To estimate the L1T (X
1
a,σ) norm, we proceed as above;

‖Ψ((w2,d2)) − Ψ((w1,d1))‖L1(X1a,σ) ≤
2∑
i=1

L(i)ww + L
(i)
dd + L

(i)
wd + L

(i)
dd,

where

L(1)ww =

∫ T
0

‖
∫ t

0

e(t−τ)∆(w2 + v)∇(w2 −w1)dτ‖X1a,σdt

L(2)ww =

∫ T
0

‖
∫ t

0

e(t−τ)∆(w2 −w1)∇(w1 + v)dτ‖X1a,σdt,



Int. J. Anal. Appl. 18 (3) (2020) 432

and so on for the other integrals. Using lemma 1.6, triangle inequality, the fact that wi belongs to Br,

inequalities (2.7) and (2.9), we infer that

L(1)ww, L
(2)
ww ≤ (‖(u

0,b0)‖X−1a,σ + r)
1/2(ε + r)1/2‖w2 −w1‖|XT .

The same holds for L
(i)
dd,L

(i)
wd and L

(i)
dw, and one obtains

‖Ψ(w2,d2) − Ψ(w1,d1)‖L1
T

(X1a,σ) ≤

4(‖(u0,b0)‖X−1a,σ + r)
1/2(ε + r)1/2(‖w2 −w1‖XT + ‖d2 −d1‖XT ).

(2.15)

By (2.14) and (2.15), we infer that

‖Ψ(w2,d2) − Ψ(w1,d1)‖XT ≤

8(‖(u0,b0)‖X−1a,σ + r)
1/2(ε + r)1/2(‖w2 −w1‖XT + ‖d2 −d1‖XT ).

This implies that

‖Ψ(w2,d2) − Ψ(w1,d1)‖XT ≤
1

2
(‖w2 −w1‖XT + ‖d2 −d1‖XT ) (2.16)

and Ψ is a contraction mapping. The fixed point theorem implies that there is a unique (w,d) ∈ Br, such

that (u,b) = (v + w,c + d) is the solution of (MHD) with (u,b) ∈XT .

3. Blowup results

In this section we prove theorem 0.2. First of all, the following energy estimates holds in X−1a,σ,

‖(u,b)(t)‖X−1a,σ +
∫ t

0

‖(u,b)(t)‖X1a,σ ≤ ‖(u0,b0)‖X−1a,σ

+

∫ t
0

Luu(τ) + Lbb(τ) + Lub(τ) + Lbu(τ)dτ,

where Lυω = ‖div (υω)‖X−1a,σ . By lemma 1.2, we have

Lυω ≤ ‖υ‖X0a,σ‖ω‖X0a
σ
,σ

+ ‖ω‖X0a,σ‖υ‖X0a
σ
,σ

≤ 2‖(υ,ω)‖X0a,σ‖(u,b)‖X0a
σ
,σ
.

It follows that

‖(u,b)(t)‖X−1a,σ +
∫ t

0

‖(u,b)(t)‖X1a,σ ≤‖(u0,b0)‖X−1a,σ + 8
∫ t

0

‖(u,b)‖X0a
σ
,σ
‖(u,b)‖X0a,σdτ.

Using Lemma 1.3 and product Young inequality, we obtain

‖(u,b)(t)‖X−1a,σ +
1
2

∫ t
0

‖(u,b)(z)‖X1a,σdz ≤‖(u0,b0)‖X−1a,σ

+ 32

∫ t
0

‖(u,b)‖2X0a
σ
,σ
‖(u,b)‖X−1a,σdτ.

(3.1)



Int. J. Anal. Appl. 18 (3) (2020) 433

However, a direct computation implies that ‖(u,b)‖X0a
σ
,σ
≤ M0‖(u,b)‖X−1a,σ, where the constant M =

M0(a,σ) = supr≥0 re
−a( 1

σ
−1)r1/σ . Then, estimation (3.1) becomes

‖(u,b)(t)‖X−1a,σ +
1

2

∫ t
0

‖(u,b)(z)‖X1a,σdz ≤‖(u0,b0)‖X−1a,σ + 32M
2
0

∫ t
0

‖(u,b)‖3X−1a,σdz.

Let T = sup{t ∈ [0,T∗[; supz∈[0,t] ‖(u,b)(z)‖X−1a,σ ≤ 2‖(u0,b0)‖X−1a,σ}. By continuity of (u,b), we have T ∈

]0,T∗] and

2‖(u0,b0)‖X−1a,σ ≤‖(u0,b0)‖X−1a,σ + 128M
2
0 T‖(u0,b0)‖

3
X−1a,σ

.

We infer that

(128M20 )
−1

‖(u0,b0)‖2X−1a,σ
≤ T ≤ T∗.

If we consider the magnetohydrodynamic system starting at initial time t, with the data (u,b)(t) , we get

(128M20 )
−1

‖(u,b)(t)‖2
X−1a,σ

≤ T∗ − t,

or equivalently

(128M20 )
−1

T∗ − t
≤‖(u,b)(t)‖2X−1a,σ.

Therefore, we infers that

lim
t→T∗

‖(u,b)(t)‖X−1a,σ = +∞. (3.2)

Applying Gronwall inequality to inequality (3.1), we get

‖(u,b)(t)‖X−1a,σ +
1

2

∫ t
0

‖(u,b)(t)‖X1a,σ ≤‖(u0,b0)‖X−1a,σ exp(32
∫ t

0

‖(u,b)‖2X0a
σ
,σ
dz).

By equation (3.2), we obtain ∫ T∗
0

‖(u,b)‖2X0a
σ
,σ
dz = ∞. (3.3)

This implies that

lim
t→T∗

‖(u,b)(t)‖X0a
σ
,σ

= +∞. (3.4)

At this point, we proved that X0a
σ
,σ norm of the solution blows up, in finite time.

Let a′ = a
σ
∈ (0,a), using the same method, we obtain

‖(u,b)(t)‖X0
a′,σ

+

∫ t
0

‖(u,b)(t)‖X2
a′,σ

≤ ‖(u0,b0)‖X0
a′,σ

+

∫ t
0

Ruu(z) + Rbb(z) + Rub(z) + Rbu(z)dz,

where Rxy(t) = ‖x.∇y‖X0
a′,σ

. Using Lemma 1.4, we get

Rxy(t) ≤ ‖x‖X0
a′,σ
‖y‖X1

a′,σ

≤ ‖(x,y)‖X0
a′,σ
‖(x,y)‖X1

a′,σ
.



Int. J. Anal. Appl. 18 (3) (2020) 434

It follows that

‖(u,b)(t)‖X0
a′,σ

+

∫ t
0

‖(u,b)(t)‖X2
a′,σ
≤‖(u0,b0)‖X0

a′,σ
+ 4

∫ t
0

‖(u,b)‖X0
a′,σ
‖(u,b)‖X1

a′,σ
dz.

By lemma 1.3, we obtain

‖(u,b)(t)‖X0
a′,σ

+

∫ t
0

‖(u,b)‖X2
a′,σ
≤‖(u0,b0)‖X0

a′,σ
+ 4

∫ t
0

‖(u,b)‖3/2X0
a′,σ
‖(u,b)‖1/2X2

a′,σ
.

Using product Young inequality, we get

‖(u,b)(t)‖X0
a′,σ

+
1

2

∫ t
0

‖(u,b)‖X2
a′,σ
≤‖(u0,b0)‖X0

a′,σ
+ 4

∫ t
0

‖(u,b)‖3X0
a′,σ

.

Gronwall lemma gives

‖(u,b)(t)‖X0
a′,σ
≤‖(u0,b0)‖X0

a′,σ
exp

(
4

∫ t
0

‖(u,b)‖2X0
a′,σ

dz
)
.

Or equivalently,

8‖(u,b)(t)‖2X0
a′,σ

exp
(
− 8

∫ t
0

‖(u,b)‖2X0
a′,σ

dz
)
≤ 8‖(u0,b0)‖2X0

a′,σ
.

Integrating over [0,T∗) and using (3.4), we infer that

1 ≤ 8‖(u0,b0)‖2X0
a′,σ

T∗.

Since a
σn

< ... < a
σ
< a, by the same method we used for a′ ∈ (0,a), we prove that

1 ≤ 8‖(u0,b0)‖2X0a
σn

,σ
T∗, ∀n ∈ N∗.

By Dominate Convergence Theorem, we obtain

1

2
√

2
√
T∗
≤‖(u0,b0)‖X0.

Consider the (MHD) system starting at t ∈ [0,T∗), by time translation, we have

1

2
√

2
√
T∗ − t

≤‖(u,b)(t)‖X0. (3.5)

At this point, we proved that the X0 norm of the solution blows up, in finite time.

Let k ∈ N∗, we consider the subset Ak defined by

Ak = {ξ ∈ R3; |ξ| ≤ k and |û0(ξ)| ≤ k}

and v0 and c0 in L2(R3)∩X−1a,σ(R3), such that (v0k,c
0
k) = F

−1
(
1Ak(ξ)((û

0, b̂0)
)
. Let (w0k,d

0
k) = (u

0 −v0k,b
0 −

c0k), one has limk→∞‖(w
0
k,d

0
k)‖X−1a,σ = 0. So, there exists k ∈ N, such that ‖(w

0
k,d

0
k)‖X−1a,σ <

1
16

. Using



Int. J. Anal. Appl. 18 (3) (2020) 435

smallness theory, we prove that a unique and global in time solution (wk,dk) to the system

(MHD1)




∂tw − ∆w + w ·∇w −d ·∇d = −∇π1, in R+ ×R3

∂td− ∆d + w ·∇d−d ·∇w = 0, in R+ ×R3

div w = div d = 0, in R+ ×R3

(w,d)(0,x) = (w0k,d
0
k)(x), in R

3,

exists in Cb(R+,X−1a,σ(R3)) ∩L1(R+,X1a,σ(R3)) and satisfies for t ≥ 0,

‖(wk,dk)(t)‖X−1a,σ +
1

2

∫ t
0

‖(wk,dk)(z)‖X1a,σdz ≤‖(w
0
k,d

0
k)‖X−1a,σ. (3.6)

Consider (vk,ck) = (u−wk,b− ck), it belongs to C([0,T∗),X−1a,σ(R3)) and satisfies, for all (t,x) ∈ R × R3,

the following (MHD2) system,


∂tvk − ∆vk + vk ·∇vk + vk ·∇wk + wk ·∇vk − ck ·∇ck − ck ·∇dk −dk ·∇ck = −∇π2

∂tck − ∆ck + vk ·∇ck + vk ·∇dk + wk ·∇ck − ck ·∇vk − ck ·∇wk −dk ·∇vk = 0

div w = div d = 0

(vk,ck)(0,x) = (v
0
k,c

0
k)(x).

Having that (v0k,c
0
k) ∈ L

2(R3), we take the scalar product and use L2 theory. Under divergence free condition,

we infers that

1

2

d

dt
‖(vk,ck)(t)‖2L2 + ‖(∇vk,∇ck)(t)‖

2
L2 ≤ C‖(dk,wk)‖

2
L∞‖(vk,ck)(t)‖

2
L2.

The Gronwall lemma implies that

‖(vk,ck)(t)‖2L2 +
∫ t

0

‖(∇vk,∇ck)(z)‖2L2dz ≤‖(v
0
k,c

0
k)‖

2
L2 exp(C

∫ t
0

‖(wk,dk)‖2L∞dz).

Using that ‖f‖L∞ ≤‖f̂‖L1 = ‖f‖X0 ≤‖f‖
1/2
X−1‖f‖

1/2
X1 , we obtain

‖(vk,ck)(t)‖2L2 +
∫ t

0

‖(∇vk,∇ck)(z)‖2L2dz

≤ ‖(v0k,c
0
k)‖

2
L2
e
C

∫ t
0

‖(wk,dk)‖X−1‖(wk,dk)‖X1dz

≤ ‖(v0k,c
0
k)‖

2
L2
e
C

∫ t
0

‖(wk,dk)‖X−1a,σ‖(wk,dk)‖X1a,σdz
.

By the energy estimates (3.6), we obtain the following L2 energy estimates

‖(vk,ck)(t)‖2L2 +
∫ t

0

‖(∇vk,∇ck)(z)‖2L2dz ≤ α0, (3.7)

where α0 = ‖(v0k,c
0
k)‖

2
L2

exp(2C‖(w0k,d
0
k)‖

2
X−1a,σ

). At this point, thanks to the properties of small frequencies

part in X−1a,σ, we closed the L2 energy estimate of the large frequencies part of the solution. Using X−1a,σ



Int. J. Anal. Appl. 18 (3) (2020) 436

energy estimates (3.6), we obtain

‖(u,b)‖X0 ≤ ‖(vk,ck)‖X0 + ‖(wk,dk)‖X0

≤ ‖(vk,ck)‖X0 + M1‖(wk,dk)‖X−1a,σ
≤ ‖(vk,ck)‖X0 + M1‖(w0k,d

0
k)‖X−1a,σ,

where M1 = M1(a,σ) = supr≥0 re
−ar1/σ.

Inequality (3.5) implies that the function t → ‖(u,b)(t)‖X0 is continuous on [0,T∗) and tends to infinity

when t approaches T∗. Thus, there is T0 ∈ [0,T∗), such that

1/4

(T∗ − t)1/2
≤‖(vk,ck)(t)‖X0, ∀t ∈ [T0,T∗). (3.8)

At this point, we proved that the high frequencies part considered above blows up in the X0 norm. Thus,

according to equation (3.5), one can infers that only high frequencies are responsible for this phenomena.

Now, using Lemma 1.1 and (3.8), we can involve Xs. Mainly, for s ≥ 3/2 and t ∈ [T0,T∗),

1/4

(T∗ − t)1/2
≤‖(vk,ck)(t)‖

s

s+ 3
2

L2
‖(vk,ck)(t)‖

3
2

s+ 3
2

Xs .

Using inequality (3.7), we can omit the L2 norm and obtain

1/4

(T∗ − t)1/2
≤ α

s

s+ 3
2

0 ‖(vk,ck)(t)‖
3
2

s+ 3
2

Xs .

This implies for n ∈ N, such that n
σ
− 1 ≥ 3/2 or n ≥ n0 = [ 52σ] + 1,

4−1/3

(T∗ − t)1/3
an

n!

( 4− 23σ α− 23σ0
(T∗ − t)

1
3σ

)n
≤
an

n!
‖(vk,ck)(t)‖X nσ−1.

Summing up for n ≥ n0, we get

4−1/3

(T∗−t)1/3
∑
n≥n0

an

n!

( 4− 23σ α− 23σ0
(T∗ − t)

1
3σ

)n
≤

∑
n≥n0

an

n!
‖(vk,ck)(t)‖X nσ−1

≤ ‖(vk,ck)(t)‖X−1a,σ.

Using triangular inequality and X−1a,σ energy estimates (3.6), we have

‖(vk,ck)(t)‖X−1a,σ ≤‖(u,b)(t)‖X−1a,σ + ‖(w
0
k,d

0
k)‖X−1a,σ.

Dividing both sides of the resulting inequality by exp
(
a

4
− 2

3σ α
− 2

3σ
0

(T∗−t)
1
3σ

)
, we infer that

lim inf
t→T∗

(T∗ − t)1/3 exp
(
−

4−
2
3σ α
− 2

3σ
0

(T∗ − t)
1
3σ

)
‖(u,b)(t)‖X−1a,σ ≥ 4

−1/3.

This gives the blowup profile and finishes the proof.



Int. J. Anal. Appl. 18 (3) (2020) 437

Remark 3.1. Let U0 = (u0,b0) ∈X−1a,σ(R3), where a > 0, σ ≥ 1. Let U = (u,b) be the maximal solution of

(MHD) system. Using the fact

X−1a,σ(R
3) ↪→X−1a′,σ(R

3) ↪→X−1(R3), ∀0 < a′ < a,

one infers that U ∈ C([0,T∗a,σ),X−1a,σ(R3)), U ∈ C([0,T∗a′,σ),X
−1
a′,σ(R

3)) and U ∈ C([0,T∗),X−1(R3)), where

the maximal times of existence T∗a,σ, T
∗
a′,σ and T

∗ belong all of them to (0, +∞]. These times satisfy T∗a,σ ≤

T∗a′,σ ≤ T
∗. By the method we used in the proof of the blowup result (a′ = a

σn
), we proved that T∗a,σ = T

∗,

if σ > 1. However, we note that if σ = 1 our technics failes. So, this critical case seems to need an other

approach different from us.

Remark 3.2. Using the same technics we can prove existence, uniqueness and blowup results in finite

time of solution to the periodic MHD, the three dimensional Navier-Stokes system, the two dimensional

quasi-geostrophic equations with subcritical dissipation, in the Lei-Lin-Gevrey space.

Acknowledgement: The authors gratefully acknowledge the approval and the support of this research

study by the grant number SCI-2017-1-7-F-7081 from the Deanship of Scientific Research at Northern Border

University, Arar, K. S. A.

Conflicts of Interest: The author(s) declare that there are no conflicts of interest regarding the publication

of this paper.

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	1. Technical lemmas
	2. Well-posedness results
	3. Blowup results
	References