Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 111, pp. 1–15.

ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, DOI: 10.58997/ejde.2025.111

WELL-POSEDNESS OF GENERALIZED MAGNETOHYDRODYNAMIC

EQUATIONS IN VARIABLE LEBESGUE SPACES

JINYI SUN, YUANWEI MAI, MINGHUA YANG

Abstract. This article concerns the well-posedness of the generalized magnetohydrodynamic
equations in variable Lebesgue spaces. By using some basic properties of variable Lebesgue

spaces and decay estimates of the fractional heat kernel, we prove the existence of local and

global solutions to the generalized magnetohydrodynamic equations in two different types of
variable Lebesgue spaces.

1. Introduction

In this article, we consider the Cauchy problem of the three-dimensional incompressible gener-
alized magnetohydrodynamic equations,

∂tu+ (−∆)αu+ (u · ∇)u− (B · ∇)B +∇P = 0 in R3 × R+,

∂tB + (−∆)βB + (u · ∇)B − (B · ∇)u = 0 in R3 × R+,

div u = 0, divB = 0 in R3 × R+,

u|t=0 = u0, B|t=0 = B0 in R3,

(1.1)

where u = (u1, u2, u3) denotes the velocity field of the fluid, B = (B1, B2, B3) denotes the magnetic
field, and P represents scalar pressure. The positive parameters α and β represent the fractional
dissipations corresponding to the velocity field and magnetic field, respectively. The fractional
Laplacian operator (−∆)γ is defined through the Fourier transform F [(−∆)γf ](ξ) = |ξ|2γF [f ](ξ).

Problem (1.1) describes the motion of electrically conducting incompressible fluids in a magnetic
field such as plasmas and liquid metals. Duvaut and Lions [15] established local well-posedness
results of problem (1.1) with α = β = 1 in Hs(R3) with s ≥ 3 and global well-posedness for
the small initial data. Zhai et al. [45] established global well-posedness of problem (1.1) with
α = β = 1 for initial data in critical Besov spaces and relaxed the smallness condition in the third
components of the initial velocity field and initial magnetic field. For more relevant studies on the
existence of solutions of the classical magnetohydrodynamic equations, we refer to [10, 21, 26, 44].
In [40], Wu established the global existence of weak solutions to problem (1.1) corresponding to
arbitrary L2 initial data. Furthermore if α, β ≥ 5

4 and initial data are sufficiently smooth, the
weak solutions are actually global classical solutions. Jin et al. [23] proved the unique existence
of global solutions to problem (1.1) for small initial data in Hs(R3) with 0 < α = β < 1 and
s ≥ 5

2 − α. In addition, if initial data (u0, B0) ∈ Lp for 1 ≤ p < 2, they established optimal decay
estimates for the solutions. See also [42, 43] for the global well-posedness of problem (1.1) with
1
2 ≤ α, β ≤ 1 for small initial data in Lei-Lin spaces and Fourier-Herz spaces. However, it remains

open whether there exists a global smooth solution for problem (1.1) with α+ β < 5
2 or not.

It is worth mentioning that variable Lebesgue spaces Lp(·)(RN ) were introduced by Orlicz [31],
then further studied by Musielak[28] and Nakano[29, 30]. Indeed, the modern development of
variable Lebesgue spaces began with Kováčik and Rákosńık[24], Cruz-Uribe[13], Diening[16], Fan

2020 Mathematics Subject Classification. 35Q35, 46E30, 76D03, 76W05.
Key words and phrases. Generalized magnetohydrodynamic equations; variable Lebesgue spaces;

well-posedness.
©2025. This work is licensed under a CC BY 4.0 license.
Submitted July 6, 2025. Published November 25, 2025.

1



2 J. SUN, Y. MAI, M. YANG EJDE-2025/111

and Zhao[20], et al. So far, there have been extensive literature and monographs on the research
of function spaces with variable exponents such as variable Lebesgue spaces [14], variable Sobolev
spaces[17], variable Triebel-Lizorkin spaces [18], variable Besov spaces [6, 41], variable Morrey
spaces [5] and among others. In addition to theoretical considerations, the study of these function
spaces can be motivated by applications to fluid dynamics[3, 34], image processing[12, 22], PDEs
and variational calculus [4, 19, 32, 46].

To the best of our knowledge, Ru and Abidin [33] first studied the Cauchy problem of the
fractional Navier-Stokes equations in function spaces with variable exponents. More precisely, by
introducing the definition of variable Fourier-Besov spaces and exploiting the good properties of
Stokes semigroup on the frequency space, they obtained the existence and uniqueness of global
solutions to incompressible fractional Navier-Stokes equations for small initial data in variable
Fourier-Besov spaces. Subsequently, Wang[39] obtained the global well-posedness and analyticity
of the generalized magnetohydrodynamics equations (1.1) in variable Fourier-Besov spaces. Abidin
and Chen[1] proved that the fractional Navier-Stokes equations are globally well-posed if the initial
data are small in variable Fourier-Besov-Morrey spaces. Abidin et al. [2] studied the global well-
posedness for the three-dimensional micropolar fluid equations with small initial data in variable
Fourier-Besov spaces.

Very recently, Chamorro and Vergara-Hermosilla [8, 35] established the well-posedness of in-
compressible (fractional) Navier-Stokes equations in variable Lebesgue spaces. In [9], they also
studied the Liouville-type theorems for the stationary Navier-Stokes equations in variable Lebesgue
spaces. Vergara-Hermosilla [36] showed the well-posedness results of the generalized nonlinear Heat
equations in variable Lebesgue spaces. Chen, Vergara-Hermosilla and Zhao [11] proved the local
existence and regularity of solutions to the two-dimensional dissipative surface quasi-geostrophic
equation in variable Lebesgue spaces. For the well-posedness results related to the fractional
Keller-Segel system, we refer to [37, 38].

Inspired by the above works, it is natural to consider the existence of solutions to the generalized
magnetohydrodynamic equations in the framework of the variable Lebesgue spaces. Now we state
our main results. The first result is the local existence of solutions for arbitrary initial data in
Lp(R3)× Lq(R3).

Theorem 1.1. Let α, β ∈ ( 12 ,
5
4 ], γ = min{α, β}, p > 3

2γ−1 , q ∈ [p, 2p] and p(·), q(·) ∈ P log[0,+∞)

satisfy p(·) ≤ q(·), p− > 2 and γ
p(·) +

3
2p < γ − 1

2 . Then for any (u0, B0) ∈ Lp(R3)× Lq(R3) with

div u0 = 0 and divB0 = 0, there exists a time T > 0 such that problem (1.1) possesses a unique
local mild solution (u,B) ∈ Lp(·)([0, T );Lp(R3))× Lq(·)([0, T );Lq(R3)).

Next, we obtain the existence of global solutions with small initial data in Lp(·)(R3).

Theorem 1.2. Let α = β ∈ ( 12 ,
5
4 ) and p(·) ∈ P log(R3). Then there exists a positive constant

ε such that if the initial value (u0, B0) ∈ Lp(·)(R3) ∩ L
3

2α−1 (R3) with div u0 = 0 and divB0 = 0
satisfying

max
{
∥(u0, B0)∥Lp(·) , ∥(u0, B0)∥

L
3

2α−1

}
≤ ε,

problem (1.1) possesses a unique global mild solution

(u,B) ∈ Lp(·)(R3;L∞[0,+∞)) ∩ L
3

2α−1 (R3;L∞[0,+∞)).

This article is organized as follows. In Section 2, we recall the definitions and properties of
the variable Lebesgue spaces. Moreover, we introduce some basic facts on the variable Lebesgue
spaces. In Section 3, by using the decay estimates of the fractional heat kernel, we derive some
linear and bilinear estimates of solutions. In Section 4, we complete the proofs of Theorems 1.1 and
1.2 by the contraction mapping principle. Throughout this paper the constant C in the estimates
may vary from line to line.



EJDE-2025/111 MAGNETOHYDRODYNAMIC EQUATIONS 3

2. Preliminaries

Firstly, we recall some standard notation in variable Lebesgue spaces. Let P(Rn) be the set of
all measurable functions p(·) : Rn → [1,+∞] such that

p− := ess infx∈Rn p(x) > 1, p+ := ess supx∈Rn p(x) < +∞.

Let p′(·) denote the conjugate exponent of the p(·) with 1
p(·) +

1
p′(·) = 1.

Definition 2.1. Let p(·) ∈ P(Rn). The Lebesgue space with variable exponent p(·) is defined by

Lp(·)(Rn) :=
{
f is measurable :

∫
Rn

|f(x)
λ

|p(x)dx <∞, for some λ > 0
}
.

Then Lp(·)(Rn) is a Banach space with the Luxemburg-Nakano norm

∥f∥Lp(·)(Rn) := inf
{
λ > 0 :

∫
Rn

|f(x)
λ

|p(x)dx ≤ 1
}
.

On the basis of classical Lebesgue spaces Lp(Rn) and variable Lebesgue spaces Lp(·)(Rn), let
us state the definition of the following mixed variable Lebesgue spaces introduced in [7].

Definition 2.2. Let p(·) ∈ P log(Rn) and 1 < q < +∞. Then the mixed Lebesgue space L
p(·)
q (Rn)

is defined by
Lp(·)q (Rn) = Lp(·)(Rn) ∩ Lq(Rn),

endowed with the norm
∥ · ∥

L
p(·)
q

= max{∥ · ∥Lp(·) , ∥ · ∥Lq}.

Next, we recall some properties of the variable Lebesgue spaces such as the Hölder inequality,
norm conjugate formula and embedding theorem(see [14, Corollaryies 2.28 and 2.48], [17, Lemma
3.2.20 and Corollary 3.2.14]).

Lemma 2.3. Let p(·), p1(·), p2(·) ∈ P(Rn) satisfy 1
p(x) =

1
p1(x)

+ 1
p2(x)

. Then there exists a constant

C such that for all f ∈ Lp1(·)(Rn) and g ∈ Lp2(·)(Rn), we have fg ∈ Lp(·)(Rn) and
∥fg∥Lp(·) ≤ C∥f∥Lp1(·)∥g∥Lp2(·) . (2.1)

Lemma 2.4. Let p(·) ∈ P(Rn). Then

∥f∥Lp(·) ≤ sup
∥g∥

Lp′(·)≤1

∫
Rn

|f(x)g(x)|dx, ∀f ∈ Lp(·), g ∈ Lp
′(·). (2.2)

Lemma 2.5. Given Ω ⊂ Rn and p1(·), p2(·) ∈ P(Ω) with 1 < p+1 , p
+
2 < +∞. Then Lp2(·)(Ω) ⊂

Lp1(·)(Ω) if and only if p1(x) ≤ p2(x) almost everywhere. Furthermore, in this case we have that

∥f∥Lp1(·) ≤ (1 + |Ω|)∥f∥Lp2(·) , ∀f ∈ Lp2(·)(Ω),

where |Ω| denotes Lebesgue measure of measurable set Ω.

Remark 2.6. Since Lp(·)(Rn) is not invariant to translation, not all properties of the Lebesgue
spaces Lp(Rn) can be generalized to the variable Lebesgue spaces. For example, the Young
inequality and the Plancherel formula are not valid anymore, see [17].

Note that the boundedness property of Hardy-Littlewood maximal operator on variable Lebesgue
spaces provides a rigorous foundation for numerous conclusions in classical harmonic analysis and
function spaces theory. Thus, to guarantee the boundedness of the singular integral operator in the
variable Lebesgue spaces, we need to impose the so-called log-Hölder continuity on the exponent
functions p(·).

Definition 2.7. Let p(·) ∈ P(Rn). Then p(·) ∈ P log(Rn) if the following conditions hold:

• (locally log-Hölder continuous) For all x, y ∈ Rn, x ̸= y,
∣∣ 1
p(x) −

1
p(y)

∣∣ ≤ C
log(e+ 1

|x−y| )
;

• (log-Hölder decay condition) For all x ∈ Rn,
∣∣ 1
p(x) − 1

p∞

∣∣ ≤ C
log(e+|x|) , where 1

p∞
=

lim
|x|→∞

1
p(x) .



4 J. SUN, Y. MAI, M. YANG EJDE-2025/111

Next, we introduce the following results on the Hardy-Littlewood maximal operator and Riesz
transforms (see [14, Theorems 3.16 and 5.42], [17, Theorem 4.3.8 and Corollary 6.3.10], [25, Lemma
7.4]).

Lemma 2.8. Let p(·) ∈ P log(Rn). Then for any f ∈ Lp(·)(Rn), there exists a positive constant C
such that

∥M(f)∥Lp(·) ≤ C∥f∥Lp(·) ,

where M is the Hardy-Littlewood maximal operator defined as

M(f)(x) := sup
x∈B

1

|B|

∫
B
|f(y)|dy,

and B ⊂ Rn is an open ball with center x. Furthermore, we have

∥Rj(f)∥Lp(·) ≤ C∥f∥Lp(·) , ∀1 ≤ j ≤ n,

where Rj := −∂xj
(−∆)−

1
2 is the Riesz transform.

Lemma 2.9. Let φ is a radially decreasing function on R3 and f is a locally integrable function,
then

|(φ ∗ f)(x)| ≤ ∥φ∥L1M(f)(x).

Next we review the definition of Riesz potential operator and its boundedness property on the
variable Lebesgue spaces (see [14, Theorem 5.46], [7, Theorem 4]).

Definition 2.10. Let 0 < δ < n, for any measurable function f , the Riesz potential operator Iδ
is defined as

Iδ(f)(x) :=
∫
Rn

|f(y)|
|x− y|n−δ

dy, x ∈ Rn.

Lemma 2.11. Let p(·) ∈ P log(Rn) and 0 < δ < n/p+. Then for any f ∈ Lp(·)(Rn), there exists
a positive constant C such that

∥Iδ(f)∥Lq(·) ≤ C∥f∥Lp(·) with
1

q(·)
=

1

p(·)
− δ

n
.

Lemma 2.12. Let 1 < p < +∞, p(·) ∈ P log(Rn) and 0 < δ < min{ n
p+ ,

n
p }. Given f ∈ L

p(·)
p (Rn)

and a function ρ(·) satisfying the condition

ρ(·) = np(·)
n− δp

,

then there exists a positive constant C such that

∥Iδ(f)∥Lρ(·) ≤ C∥f∥
L

p(·)
p
.

Remark 2.13. Note that the mixed spaces L
p(·)
p inherit the properties of the spaces Lp(·) and Lp.

In particular we have the Hölder inequality ∥f∥
L

p(·)
p

≤ ∥f∥
L

q(·)
q

∥f∥
L

r(·)
r

with 1
p(·) = 1

q(·) +
1
r(·) and

1
p = 1

q +
1
r and of course the Riesz transforms are also bounded in these spaces.

Finally, by Duhamel’s principle, the mild solutions (u,B) for the equations (1.1) can be repre-
sented as

u(t) = Gαt ∗ u0(x)−
∫ t

0

Gαt−τ ∗ P∇ · [u⊗ u−B ⊗B](x, τ)dτ,

B(t) = Gβt ∗B0(x)−
∫ t

0

Gβt−τ ∗ P∇ · [u⊗B −B ⊗ u](x, τ)dτ,

(2.3)

where P := Id − ∇(−∆)−1 div is the Leray-Hopf projection. In our proof we need the following
decay estimates for the fractional heat kernel Gαt (x), see [27].



EJDE-2025/111 MAGNETOHYDRODYNAMIC EQUATIONS 5

Lemma 2.14. The fractional heat kernel Gαt (x) satisfies the point-wise estimate

|∇Gαt (x)| ≤ C
1

(t
1
2α + |x|)n+1

,

where

Gαt (x) = F−1(e−t|ξ|
2α

) = (2π)−
n
2

∫
Rn

eix·ξe−t|ξ|
2α

dξ, x ∈ Rn.

Lemma 2.15. For all α > 0, ν > 0, 1 ≤ p ≤ q ≤ ∞. Then for any f ∈ Lp(Rn), we have the
Lp-Lq estimates

∥Gαt ∗ f∥Lq ≤ Ct−
n
2α ( 1

p−
1
q )∥f∥Lp ,

∥(−∆)
ν
2Gαt ∗ f∥Lq ≤ Ct−

ν
2α− n

2α ( 1
p−

1
q )∥f∥Lp .

3. Key estimates

The aim of this section is to derive estimates used in the proof of Theorems 1.1 and 1.2.
Firstly, we establish the linear and bilinear estimates of the integral equations (2.3) for the local
well-posedness issue. For T > 0, we introduce the functional space Lp(·)

(
[0, T );Lp(R3)

)
with the

norm

∥f∥Lp(·)([0,T );Lp) = inf
{
λ > 0 :

∫ T

0

∣∣∥f(·, t)∥Lp

λ

∣∣p(t)dt ≤ 1
}
.

Lemma 3.1. Let α ∈ ( 12 ,
5
4 ], p(·) ∈ P log[0,+∞) and p ∈ [1,+∞]. Then for any f ∈ Lp(R3), there

exists a positive constant C such that

∥Gαt ∗ f∥Lp(·)([0,T );Lp) ≤ Cmax{T
1

p− , T
1

p+ }∥f∥Lp .

Proof. By applying the Young inequality, we obtain

∥Gαt ∗ f∥Lp
x
≤ ∥Gαt (x)∥L1

x
∥f∥Lp

x
= ∥f∥Lp

x
. (3.1)

Then taking the Lp(·)-norm to both sides of (3.1) with respect to the time variable t yields

∥Gαt ∗ f∥
L

p(·)
t (Lp

x)
≤ C∥1∥

L
p(·)
t

∥f∥Lp
x
.

Using the classical estimate

∥1∥Lp(·)[0,T ) ≤ Cmax{T
1

p− , T
1

p+ },

we deduce that

∥Gαt ∗ f∥
L

p(·)
t (Lp

x)
≤ Cmax{T

1

p− , T
1

p+ }∥f∥Lp
x
.

This completes the proof. □

Lemma 3.2. Let α ∈ ( 12 ,
5
4 ], p >

3
2α−1 and p(·) ∈ P log[0,+∞) satisfy p− > 2 and α

p(·)+
3
2p < α− 1

2 .

Then for any T > 0, there exists a positive constant C such that

∥
∫ t

0

Gαt−τ ∗ P∇ · (u⊗ u)(τ)dτ∥Lp(·)([0,T );Lp) ≤ C(1 + T )∥u∥2Lp(·)([0,T );Lp). (3.2)

Proof. For any 0 < t < T , by using Lemma 2.15 and the Hölder inequality, together with the
boundedness property of the Leray projector P, we obtain

∥
∫ t

0

Gαt−τ ∗ P∇ · (u⊗ u)(τ)dτ∥Lp
x
≤

∫ t

0

∥∇Gαt−τ ∗ (u⊗ u)(τ)∥Lp
x
dτ

≤
∫ t

0

1

(t− τ)
1
2α+ 3

2αp

∥(u⊗ u)(τ)∥
L

p
2
x

dτ

≤
∫ t

0

1

(t− τ)
1
2α+ 3

2αp

∥u(τ)∥2Lp
x
dτ.



6 J. SUN, Y. MAI, M. YANG EJDE-2025/111

Then taking the Lp(·)-norm with respect to the time variable t and using the norm conjugate
formula (2.2), we see that

∥
∫ t

0

Gαt−τ ∗ P∇ · (u⊗ u)(τ)dτ∥
L

p(·)
t (Lp

x)

≤ C∥
∫ t

0

1

(t− τ)
1
2α+ 3

2αp

∥u(τ)∥2Lp
x
dτ∥

L
p(·)
t

≤ C sup
∥ψ∥

L
p′(·)
t

≤1

∫ T

0

∫ t

0

|ψ(t)|
|t− τ |

1
2α+ 3

2αp

∥u(τ)∥2Lp
x
dτdt

= C sup
∥ψ∥

L
p′(·)
t

≤1

∫ T

0

∫ T

0

1{0<τ<t}|ψ(t)|
|t− τ |

1
2α+ 3

2αp

dt∥u(τ)∥2Lp
x
dτ.

(3.3)

For using lemma 2.11, we extend the function ψ(t) by zero on R\[0, T ), and see that the right-hand
side of (3.3) can be represented as

sup
∥ψ∥

L
p′(·)
t

≤1

∫ T

0

∫ T

0

1{0<τ<t}|ψ(t)|
|t− τ |

1
2α+ 3

2αp

dt∥u(τ)∥2Lp
x
dτ

= sup
∥ψ∥

L
p′(·)
t

≤1

∫ T

0

(∫ +∞

−∞

|ψ(t)|
|t− τ |

1
2α+ 3

2αp

dt
)
∥u(τ)∥2Lp

x
dτ

= sup
∥ψ∥

L
p′(·)
t

≤1

∫ T

0

Iδ1(|ψ|)∥u(τ)∥2Lp
x
dτ,

(3.4)

where δ1 = 1 − 1
2α − 3

2αp . Furthermore, for the right-hand side of (3.4), by using the Hölder

inequality (2.1) with 2
p(·) +

1
p̃1(·) = 1 yields

sup
∥ψ∥

L
p′(·)
t

≤1

∫ T

0

Iδ1(|ψ|)∥u(τ)∥2Lp
x
dτ ≤ C sup

∥ψ∥
L
p′(·)
t

≤1

∥Iδ1(|ψ|)∥Lp̃1(·)
t

∥u∥2
L

p(·)
t (Lp

x)

≤ C sup
∥ψ∥

L
p′(·)
t

≤1

∥ψ∥
L

r1(·)
t

∥u∥2
L

p(·)
t (Lp

x)
,

(3.5)

where 1
p̃1(·) =

1
r1(·) − (1− 1

2α − 3
2αp ).

Since 1
p̃1(·) = 1− 2

p(·) and 1
p′(·) = 1− 1

p(·) , we can deduce that r1(·) < p′(·) under the condition
α
p(·) +

3
2p < α− 1

2 . By using Lemma 2.5 with r1(·) < p′(·) and Ω = [0, T ), we obtain

sup
∥ψ∥

L
p′(·)
t

≤1

∥ψ∥
L

r1(·)
t

∥u∥2
L

p(·)
t (Lp

x)
≤ (1 + T ) sup

∥ψ∥
L
p′(·)
t

≤1

∥ψ∥
L

p′(·)
t

∥u∥2
L

p(·)
t (Lp

x)

≤ (1 + T )∥u∥2
L

p(·)
t (Lp

x)
.

(3.6)

Finally, taking estimates (3.4)–(3.6) into (3.3), we obtain (3.2). This complete the proof. □

Lemma 3.3. Let α ∈ ( 12 ,
5
4 ], p >

3
2α−1 , q ∈ [p, 2p], p(·), q(·) ∈ P log[0,+∞) satisfy p(·) ≤ q(·),

q− > 2 and α
p(·) +

3
2p < α− 1

2 . Then for any T > 0, there exists a positive constant C such that

∥
∫ t

0

Gαt−τ ∗ P∇ · (B ⊗B)(τ)dτ∥Lp(·)([0,T );Lp) ≤ C(1 + T )∥B∥2Lq(·)([0,T );Lq). (3.7)

Proof. For any 0 < t < T , by using Lemma 2.15 and the Hölder inequality, along with the
boundedness property of the Leray projector P, it follows that

∥
∫ t

0

Gαt−τ ∗ P∇ · (B ⊗B)(τ)dτ∥Lp
x
dτ ≤ C

∫ t

0

∥∇Gαt−τ ∗ (B ⊗B)(τ)∥Lp
x
dτ



EJDE-2025/111 MAGNETOHYDRODYNAMIC EQUATIONS 7

≤ C

∫ t

0

1

(t− τ)
1
2α+ 3

2α ( 2
q−

1
p )
∥(B ⊗B)(τ)∥

L
q
2
x

dτ

≤ C

∫ t

0

1

(t− τ)
1
2α+ 3

2α ( 2
q−

1
p )
∥B(τ)∥2Lq

x
dτ.

Then taking Lp(·)-norm with respect to the time variable t and using the norm conjugate formula
(2.2), we have

∥
∫ t

0

Gαt−τ ∗ P∇ · (B ⊗B)(τ)dτ∥
L

p(·)
t (Lp

x)

≤ C∥
∫ t

0

1

(t− τ)
1
2α+ 3

2α ( 2
q−

1
p )
∥B(τ)∥2Lq

x
dτ∥

L
p(·)
t

≤ C sup
∥η∥

L
p′(·)
t

≤1

∫ T

0

∫ t

0

|η(t)|
|t− τ |

1
2α+ 3

2α ( 2
q−

1
p )
∥B(τ)∥2Lq

x
dτdt

= C sup
∥η∥

L
p′(·)
t

≤1

∫ T

0

∫ T

0

1{0<τ<t}|η(t)|
|t− τ |

1
2α+ 3

2α ( 2
q−

1
p )
dt∥B(τ)∥2Lq

x
dτ.

(3.8)

For using lemma 2.11, we extend the function η(t) by zero on R\[0, T ), and see that the right-hand
side of (3.8) can be represented as

sup
∥η∥

L
p′(·)
t

≤1

∫ T

0

∫ T

0

1{0<τ<t}|η(t)|
|t− τ |

1
2α+ 3

2α ( 2
q−

1
p )
dt∥B(τ)∥2Lq

x
dτ

= sup
∥η∥

L
p′(·)
t

≤1

∫ T

0

(∫ +∞

−∞

|η(t)|
|t− τ |

1
2α+ 3

2α ( 2
q−

1
p )
dt
)
∥B(τ)∥2Lq

x
dτ

= sup
∥η∥

L
p′(·)
t

≤1

∫ T

0

Iδ2(|η|)∥B(τ)∥2Lq
x
dτ,

(3.9)

where δ2 = 1− 1
2α − 3

2α (
2
q −

1
p ). Furthermore, for the right-hand side of (3.9), by using the Hölder

inequality (2.1) with 2
q(·) +

1
p̃2(·) = 1 yields

sup
∥η∥

L
p′(·)
t

≤1

∫ T

0

Iδ2(|η|)∥B(τ)∥2Lq
x
dτ ≤ C sup

∥η∥
L
p′(·)
t

≤1

∥Iδ2(|η|)∥Lp̃2(·)
t

∥B∥2
L

q(·)
t (Lq

x)

≤ C sup
∥η∥

L
p′(·)
t

≤1

∥η∥
L

r2(·)
t

∥B∥2
L

q(·)
t (Lq

x)
,

(3.10)

where 1
p̃2(·) =

1
r2(·) −

(
1− 1

2α − 3
2α (

2
q −

1
p )
)
.

Since 1
p̃2(·) = 1− 2

q(·) and 1
p′(·) = 1− 1

p(·) , we can deduce that r2(·) < p′(·) under the conditions

p(·) ≤ q(·) and p ≤ q. By using Lemma 2.5 with r2(·) < p′(·) and Ω = [0, T ), we obtain

sup
∥η∥

L
p′(·)
t

≤1

∥η∥
L

r2(·)
t

∥B∥2
L

q(·)
t (Lq

x)
≤ C(1 + T ) sup

∥η∥
L
p′(·)
t

≤1

∥η∥
L

p′(·)
t

∥B∥2
L

q(·)
t (Lq

x)

≤ C(1 + T )∥B∥2
L

q(·)
t (Lq

x)
.

(3.11)

Finally, taking estimates (3.9)–(3.11) into (3.8), we obtain (3.7). This completes the proof. □

Lemma 3.4. Let β ∈ ( 12 ,
5
4 ], q ∈ [1,+∞] and q(·) ∈ P log[0,+∞). Then for each f ∈ Lq(R3),

there exists a positive constant C such that

∥Gβt ∗ f∥Lq(·)([0,T );Lq) ≤ Cmax{T
1

q− , T
1

q+ }∥f∥Lq .

The proof of the above lemma is similar to Lemma 3.1, we omit it.



8 J. SUN, Y. MAI, M. YANG EJDE-2025/111

Lemma 3.5. Let β ∈ ( 12 ,
5
4 ], p >

3
2β−1 , q ≥

p
p−1 , p(·), q(·) ∈ P log[0,+∞) satisfy β

p(·) +
3
2p < β− 1

2 .

Then for any T > 0, there exists a positive constant C such that

∥
∫ t

0

Gβt−τ ∗ P∇ · (u⊗B)(τ)dτ∥Lq(·)([0,T );Lq) ≤ C(1 + T )∥u∥Lp(·)([0,T );Lp)∥B∥Lq(·)([0,T );Lq). (3.12)

Proof. For any 0 < t < T , by using Lemma 2.15 and the Hölder inequality, with the boundedness
property of the Leray projector P, we obtain

∥
∫ t

0

Gβt−τ ∗ P∇ · (u⊗B)(τ)dτ∥Lq
x
dτ ≤ C

∫ t

0

∥∇Gβt−τ ∗ (u⊗B)(τ)∥Lq
x
dτ

≤ C

∫ t

0

1

(t− τ)
1
2β+ 3

2βp

∥(u⊗B)(τ)∥
L

pq
p+q
x

dτ

≤ C

∫ t

0

1

(t− τ)
1
2β+ 3

2βp

∥u(τ)∥Lp
x
∥B(τ)∥Lq

x
dτ.

Then taking Lq(·)-norm with respect to the time variable t and using the norm conjugate formula
(2.2), we see that

∥
∫ t

0

Gβt−τ ∗ P∇ · (u⊗B)(τ)dτ∥
L

q(·)
t (Lq

x)

≤ C∥
∫ t

0

1

(t− τ)
1
2β+ 3

2βp

∥u(τ)∥Lp
x
∥B(τ)∥Lq

x
dτ∥

L
q(·)
t

≤ C sup
∥ϕ∥

L
q′(·)
t

≤1

∫ T

0

∫ t

0

|ϕ(t)|
|t− τ |

1
2β+ 3

2βp

∥u(τ)∥Lp
x
∥B(τ)∥Lq

x
dτdt

= C sup
∥ϕ∥

L
q′(·)
t

≤1

∫ T

0

∫ T

0

1{0<τ<t}|ϕ(t)|
|t− τ |

1
2β+ 3

2βp

dt∥u(τ)∥Lp
x
∥B(τ)∥Lq

x
dτ.

(3.13)

For using lemma 2.11, we extend the function ϕ(t) by zero on R\[0, T ), and see that the right-hand
side of (3.13) can be represented as

sup
∥ϕ∥

L
q′(·)
t

≤1

∫ T

0

∫ T

0

1{0<τ<t}|ϕ(t)|
|t− τ |

1
2β+ 3

2βp

dt∥u(τ)∥Lp
x
∥B(τ)∥Lq

x
dτ

= sup
∥ϕ∥

L
q′(·)
t

≤1

∫ T

0

(∫ +∞

−∞

|ϕ(t)|
|t− τ |

1
2β+ 3

2βp

dt
)
∥u(τ)∥Lp

x
∥B(τ)∥Lq

x
dτ

= sup
∥ϕ∥

L
q′(·)
t

≤1

∫ T

0

Iδ3(|ϕ|)∥u(τ)∥Lp
x
∥B(τ)∥Lq

x
dτ,

(3.14)

where δ3 = 1− 1
2β−

3
2βp . Furthermore, for the right-hand side of (3.14), using the Hölder inequality

(2.1) with 1
q(·) +

1
p(·) +

1
p̃3(·) = 1 yields

sup
∥ϕ∥

L
q′(·)
t

≤1

∫ T

0

Iδ3(|ϕ|)∥u(τ)∥Lp
x
∥B(τ)∥Lq

x
dτ

≤ C sup
∥ϕ∥

L
q′(·)
t

≤1

∥Iδ3(|ϕ|)∥Lp̃3(·)
t

∥u∥
L

p(·)
t (Lp

x)
∥B∥

L
q(·)
t (Lq

x)

≤ C sup
∥ϕ∥

L
q′(·)
t

≤1

∥ϕ∥
L

r3(·)
t

∥u∥
L

p(·)
t (Lp

x)
∥B∥

L
q(·)
t (Lq

x)
,

(3.15)

where 1
p̃3(·) =

1
r3(·) −

(
1− 1

2β − 3
2βp

)
.



EJDE-2025/111 MAGNETOHYDRODYNAMIC EQUATIONS 9

Since 1
p̃3(·) = 1 − 1

p(·) −
1
q(·) and 1

q′(·) = 1 − 1
q(·) , we can deduce that r3(·) < q′(·) under the

condition β
p(·) +

3
2p < β − 1

2 . By using Lemma 2.5 with r3(·) < q′(·) and Ω = [0, T ), we obtain

sup
∥ϕ∥

L
q′(·)
t

≤1

∥ϕ∥
L

r3(·)
t

∥u∥
L

p(·)
t (Lp

x)
∥B∥

L
q(·)
t (Lq

x)

≤ C(1 + T ) sup
∥ϕ∥

L
q′(·)
t

≤1

∥ϕ∥
L

q′(·)
t

∥u∥
L

p(·)
t (Lp

x)
∥B∥

L
q(·)
t (Lq

x)

≤ C(1 + T )∥u∥
L

p(·)
t (Lp

x)
∥B∥

L
q(·)
t (Lq

x)
.

(3.16)

Finally, taking estimates (3.14)–(3.16) into (3.13), we obtain (3.12). We complete the proof of
Lemma 3.5. □

Secondly, we shall establish the linear and bilinear estimates for global well-posedness issue. For
p(·) ∈ P log(R3), the function space Lp(·)(R3;L∞[0,+∞)) is defined to be the set of all measurable
functions f : [0,+∞)× R3 → R3 such that the following norm is finite

∥f∥Lp(·)(R3;L∞) = inf
{
λ > 0 :

∫
R3

∣∣∥f(·, x)∥L∞

λ

∣∣p(x)dx ≤ 1
}
.

Based on definition 2.2, for p ∈ (1,+∞), we introduce the mixed variable Lebesgue space

Lp(·)p (R3;L∞[0,+∞)) := Lp(·)(R3;L∞[0,+∞)) ∩ Lp(R3;L∞[0,+∞)),

with the norm

∥ · ∥
L

p(·)
p (R3;L∞)

= max
{
∥ · ∥

L
p(·)
x (L∞

t )
, ∥ · ∥Lp

x(L
∞
t )

}
.

Lemma 3.6. Let α = β ∈ ( 12 ,
5
4 ), p(·) ∈ P log(R3). Then for any f ∈ L

p(·)
3

2α−1

(R3), there exists a

positive constant C such that

∥Gαt ∗ f∥
L

p(·)
3

2α−1

(R3;L∞)
≤ C∥f∥

L
p(·)

3
2α−1

.

Proof. The proof is similar to that in [35, Proposition 3.4]. Since f ∈ L
3

2α−1 ⊂ L1
loc and the

fractional heat kernel Gαt (x) is a radially decreasing function, by Lemma 2.9 we see that

|(Gαt ∗ f)(t, x)| ≤ ∥Gαt ∥L1
x
M(f)(x) = M(f)(x),

which shows that

∥Gαt ∗ f∥L∞
t

≤ CM(f)(x). (3.17)

Then, taking the L
p(·)

3
2α−1

-norm to the both sides of (3.17) with respect to the space variable x

and using Lemma 2.8, together with the boundedness property of the maximal operator M in

L
3

2α−1 (R3), we obtain

∥Gαt ∗ f∥
L

p(·)
3

2α−1

(R3;L∞)
≤ C∥M(f)∥

L
p(·)

3
2α−1

≤ Cmax
{
∥M(f)∥Lp(·) , ∥M(f)∥

L
3

2α−1

}
≤ Cmax

{
∥f∥Lp(·) , ∥f∥

L
3

2α−1

}
≤ C∥f∥

L
p(·)

3
2α−1

.

This completes the proof. □

Lemma 3.7. Let α = β ∈ ( 12 ,
5
4 ) and p(·) ∈ P log(R3). Then there exists a positive constant C

such that

∥
∫ t

0

Gαt−τ ∗ P∇ · (v ⊗ w)(τ)dτ∥
L

p(·)
3

2α−1

(R3;L∞)
≤ C∥v∥

L
p(·)

3
2α−1

(R3;L∞)
∥w∥

L
p(·)

3
2α−1

(R3;L∞)
. (3.18)



10 J. SUN, Y. MAI, M. YANG EJDE-2025/111

Proof. Our proof is similar to that in [35, Proposition 3.6]. From the properties of the Leray
projector P, we have∫ t

0

Gαt−τ ∗P∇ · (v⊗w)(τ)dτ =

∫ t

0

P
(
Gαt−τ ∗ ∇ · (v ⊗ w)(τ)

)
dτ = P

(∫ t

0

Gαt−τ ∗∇ · (v⊗w)(τ)dτ
)
.

Then, using the Fubini theorem and Lemma 2.14, we obtain∣∣ ∫ t

0

Gαt−τ ∗ ∇ · (v ⊗ w)(τ)dτ
∣∣

≤ C

∫ t

0

∫
R3

|∇Gαt−τ (x− y)||v(τ, y)||w(τ, y)|dydτ

≤ C

∫
R3

∫ t

0

1

(|t− τ | 1
2α + |x− y|)4

|v(τ, y)||w(τ, y)|dτdy

≤ C

∫
R3

∫ t

0

1

(|t− τ | 1
2α + |x− y|)4

dτ∥v(·, y)∥L∞
t
∥w(·, y)∥L∞

t
dy.

(3.19)

Note that there exists a positive constant C such that∫ t

0

1

(|t− τ | 1
2α + |x− y|)4

dτ ≤
∫ +∞

0

1

(τ
1
2α + |x− y|)4

dτ

=

∫ +∞

0

|x− y|2α(
(|x− y|2αs) 1

2α + |x− y|
)4 ds

=
1

|x− y|4−2α

∫ +∞

0

1

(1 + s
1
2α )4

ds

≤ C

|x− y|4−2α
.

(3.20)

Substituting (3.20) into (3.19) and by Definition 2.10 of the Riesz potential operator with n = 3,
we have

P
(∫ t

0

Gαt−τ ∗ ∇ · (v ⊗ w)(τ)dτ
)
≤ CP

(∫
R3

1

|x− y|4−2α
∥v(·, y)∥L∞

t
∥w(·, y)∥L∞

t
dy

)
= CP

(
I2α−1

(
∥v∥L∞

t
∥w∥L∞

t

)
(x)

)
.

(3.21)

Furthermore, applying Lemma 2.12 and the boundedness property of the Leray projector P, we
obtain∥∥P(∫ t

0

Gαt−τ ∗ ∇ · (v ⊗ w)(τ)dτ
)∥∥

L
p(·)
x (L∞

t )
≤ C∥I2α−1(∥v∥L∞

t
∥w∥L∞

t
)∥
L

p(·)
x

≤ C∥∥v∥L∞
t
∥w∥L∞

t
∥
L

p(·)
2

3
2(2α−1)

,x

≤ C∥v∥
L

p(·)
3

2α−1
,x
(L∞

t )
∥w∥

L
p(·)

3
2α−1

,x
(L∞

t )
,

(3.22)

since 1
2 < α < 5

4 and 2(2α−1)
3 = 2α−1

3 + 2α−1
3 , the Hardy-Littlewood-Sobolev inequality and the

Hölder inequality imply that∥∥P(∫ t

0

Gαt−τ ∗ ∇ · (v ⊗ w)(τ)dτ
)∥∥

L
3

2α−1
x (L∞

t )
≤ C∥I2α−1(∥v∥L∞

t
∥w∥L∞

t
)∥
L

3
2α−1
x

≤ C∥∥v∥L∞
t
∥w∥L∞

t
∥
L

3
2(2α−1)
x

≤ C∥v∥
L

3
2α−1
x (L∞

t )
∥w∥

L
3

2α−1
x (L∞

t )
.

(3.23)

Combining the above estimates (3.22) and (3.23), we obtain (3.18). This completes the proof. □



EJDE-2025/111 MAGNETOHYDRODYNAMIC EQUATIONS 11

4. Proof of main results

Proof of Theorem 1.1. For (u0, B0) ∈ Lp(R3) × Lq(R3), it follows from Lemmas 3.1 and 3.4 that
there exist positive constants C0 and C1 such that

∥Gαt ∗ u0∥Lp(·)([0,T );Lp) ≤ C0 max{T
1

p− , T
1

p+ }∥u0∥Lp , (4.1)

∥Gβt ∗B0∥Lq(·)([0,T );Lq) ≤ C1 max{T
1

q− , T
1

q+ }∥B0∥Lq . (4.2)

We define the mapping B and the solutions space Z by

B(u,B)(t) :=
(
B1(u,B)(t),B2(u,B)(t)

)
,

Z :=
{
(u,B) ∈ X × Y := Lp(·)

(
[0, T );Lp(R3)

)
× Lq(·)

(
[0, T );Lq(R3)

)
:

∥u∥X ≤ 2C0 max{T
1

p− , T
1

p+ }∥u0∥Lp , ∥B∥Y ≤ 2C1 max{T
1

q− , T
1

q+ }∥B0∥Lq

}
,

endowed with the norm ∥(u,B)∥Z := ∥u∥X + ∥B∥Y , where T will be determined later,

B1(u,B)(t) := Gαt ∗ u0(x)−
∫ t

0

Gαt−τ ∗ P∇ · [u⊗ u−B ⊗B](x, τ)dτ,

B2(u,B)(t) := Gβt ∗B0(x)−
∫ t

0

Gβt−τ ∗ P∇ · [u⊗B −B ⊗ u](x, τ)dτ.

Taking γ = min{α, β} and let p > 3
2γ−1 satisfying γ

p(·) +
3
2p < γ − 1

2 , it holds that p ≥ p
p−1 ,

applying lemmas 3.2, 3.3, and 3.5 yields that, for any 0 < T < +∞, there exist positive constants
C2, C3 and C4 such that

∥B1(u,B)∥X ≤ C0 max{T
1

p− , T
1

p+ }∥u0∥Lp + C2(1 + T )∥u∥2X + C3(1 + T )∥B∥2Y

≤ C0 max{T
1

p− , T
1

p+ }∥u0∥Lp

{
1 + 4C0C2(1 + T )max{T

1

p− , T
1

p+ }∥u0∥Lp

+ 4C−1
0 C2

1C3(1 + T )
max{T

2

q− , T
2

q+ }

max{T
1

p− , T
1

p+ }
∥u0∥−1

Lp ∥B0∥2Lq

} (4.3)

and

∥B2(u,B)∥Y ≤ C1 max{T
1

q− , T
1

q+ }∥B0∥Lq + C4(1 + T )∥u∥X∥B∥Y

≤ C1 max{T
1

q− , T
1

q+ }∥B0∥Lq

{
1 + 4C0C4(1 + T )max{T

1

p− , T
1

p+ }∥u0∥Lp

} (4.4)

for all (u,B) ∈ Z.
On the other hand, under the similar argument, for all (u1, B1), (u2, B2) ∈ Z, it is easy to see

that

∥B(u1, B1)− B(u2, B2)∥Z

≤
∥∥∫ t

0

Gαt−τ ∗ P∇ · [u1(τ)⊗ (u1(τ)− u2(τ)) + (u1(τ)− u2(τ))⊗ u2(τ)]dτ
∥∥
X

+
∥∥∫ t

0

Gαt−τ ∗ P∇ · [B1(τ)⊗ (B1(τ)−B2(τ)) + (B1(τ)−B2(τ))⊗B2(τ)]dτ
∥∥
X

+
∥∥∫ t

0

Gβt−τ ∗ P∇ · [u1(τ)⊗ (B1(τ)−B2(τ)) + (u1(τ)− u2(τ))⊗B2(τ)]dτ
∥∥
Y

+
∥∥∫ t

0

Gβt−τ ∗ P∇ · [B1(τ)⊗ (u1(τ)− u2(τ)) + (B1(τ)−B2(τ))⊗ u2(τ)]dτ
∥∥
Y

≤
[
C2(1 + T )(∥u1∥X + ∥u2∥X) + C4(1 + T )(∥B1∥Y + ∥B2∥Y )

]
∥u1 − u2∥X

+
[
C4(1 + T )(∥u1∥X + ∥u2∥X) + C3(1 + T )(∥B1∥Y + ∥B2∥Y )

]
∥B1 −B2∥Y

≤
{
4C0C2C

p
T ∥u0∥Lp + 4C1C4C

q
T ∥B0∥Lq

}
∥u1 − u2∥X



12 J. SUN, Y. MAI, M. YANG EJDE-2025/111

+
{
4C0C4C

p
T ∥u0∥Lp + 4C1C3C

q
T ∥B0∥Lq

}
∥B1 −B2∥Y , (4.5)

where CpT := (1 + T )max{T
1

p− , T
1

p+ }, CqT := (1 + T )max{T
1

q− , T
1

q+ }.
If we choose T > 0 small enough such that

∥u0∥Lp ≤ 1

CpT
min

{ 1

16C0C2
,

1

16C0C4

}
and

∥B0∥Lq ≤ 1

CqT
min

{ 1

16C1C4
,

1

16C1C3
,
C

1/2
0 (1 + T )1/2 max{T

1

2p− , T
1

2p+ }∥u0∥1/2Lp

2
√
2C1C

1/2
3

}
,

then (4.3)–(4.5) imply

∥B1(u,B)∥X ≤ 2C0 max{T
1

p− , T
1

p+ }∥u0∥Lp ,

∥B2(u,B)∥Y ≤ 2C1 max{T
1

q− , T
1

q+ }∥B0∥Lq ,

∥B(u1, B1)− B(u2, B2)∥Z <
1

2
∥(u1, B1)− (u2, B2)∥Z

for all (u1, B1) and (u2, B2) ∈ Z. Therefore, by the contraction mapping principle, there exists a
unique solution (u,B) ∈ Z satisfying (2.3). This completes the proof. □

Proof of Theorem 1.2. For (u0, B0) ∈ Lp(·)(R3)∩L
3

2α−1 (R3), it follows from Lemma 3.6 that there
exists a positive constant C0 such that

∥Gαt ∗ u0∥Lp(·)
3

2α−1

(R3;L∞)
≤ C0∥u0∥Lp(·)

3
2α−1

, (4.6)

∥Gαt ∗B0∥Lp(·)
3

2α−1

(R3;L∞)
≤ C0∥B0∥Lp(·)

3
2α−1

. (4.7)

We define the mapping B and the solutions space X by

B(u,B)(t) :=
(
B3(u,B)(t),B4(u,B)(t)

)
,

X :=
{
(u,B) ∈ L

p(·)
3

2α−1

(R3;L∞[0,+∞)) : ∥u∥X ≤ 2C0∥u0∥Lp(·)
3

2α−1

, ∥B∥X ≤ 2C0∥B0∥Lp(·)
3

2α−1

}
,

endowed with the norm

∥(u,B)∥X := ∥u∥
L

p(·)
3

2α−1

(R3;L∞)
+ ∥B∥

L
p(·)

3
2α−1

(R3;L∞)
,

where

B3(u,B)(t) := Gαt ∗ u0(x)−
∫ t

0

Gαt−τ ∗ P∇ · [u⊗ u−B ⊗B](τ, x)dτ,

B4(u,B)(t) := Gαt ∗B0(x)−
∫ t

0

Gαt−τ ∗ P∇ · [u⊗B −B ⊗ u](τ, x)dτ.

Now applying lemma 3.7, (4.6) and (4.7) yields that, there exists a positive constant C1 such that

∥B3(u,B)∥X ≤ C0∥u0∥Lp(·)
3

2α−1

+ C1∥u∥2X + C1∥B∥2X

≤ C0∥u0∥Lp(·)
3

2α−1

{
1 + 4C0C1∥u0∥Lp(·)

3
2α−1

+ 4C0C1∥u0∥−1

L
p(·)

3
2α−1

∥B0∥2Lp(·)
3

2α−1

} (4.8)

and

∥B4(u,B)∥X ≤ C0∥B0∥Lp(·)
3

2α−1

+ 2C1∥u∥X ∥B∥X ≤ C0∥B0∥Lp(·)
3

2α−1

{
1 + 8C0C1∥u0∥Lp(·)

3
2α−1

}
(4.9)

for all (u,B) ∈ X .



EJDE-2025/111 MAGNETOHYDRODYNAMIC EQUATIONS 13

On the other hand, under the similar argument, for all (u1, B1), (u2, B2) ∈ X , it is easy to see
that

∥B(u1, B1)− B(u2, B2)∥X

≤
∥∥ ∫ t

0

Gαt−τ ∗ P∇ · [u1(τ)⊗ (u1(τ)− u2(τ)) + (u1(τ)− u2(τ))⊗ u2(τ)]dτ
∥∥
X

+
∥∥∫ t

0

Gαt−τ ∗ P∇ · [B1(τ)⊗ (B1(τ)−B2(τ)) + (B1(τ)−B2(τ))⊗B2(τ)]dτ
∥∥
X

+ ∥
∫ t

0

Gαt−τ ∗ P∇ · [u1(τ)⊗ (B1(τ)−B2(τ)) + (u1(τ)− u2(τ))⊗B2(τ)]dτ
∥∥
X

+ ∥
∫ t

0

Gαt−τ ∗ P∇ · [B1(τ)⊗ (u1(τ)− u2(τ)) + (B1(τ)−B2(τ))⊗ u2(τ)]dτ
∥∥
X

≤ C1(∥u1∥X + ∥u2∥X + ∥B1∥X + ∥B2∥X )(∥u1 − u2∥X + ∥B1 −B2∥X )

≤
{
4C0C1∥u0∥Lp(·)

3
2α−1

+ 4C0C1∥B0∥Lp(·)
3

2α−1

}
∥(u1, B1)− (u2, B2)∥X .

(4.10)

If (u0, B0) ∈ L
p(·)

3
2α−1

(R3) satisfies

∥u0∥Lp(·)
3

2α−1

≤ 1

16C0C1
,

and

∥B0∥Lp(·)
3

2α−1

≤ min

{
1

16C0C1
,

1

2
√
2C

1/2
0 C

1/2
1

∥u0∥1/2
L

p(·)
3

2α−1

}
,

then (4.8)–(4.10) imply that

∥B3(u,B)∥X ≤ 2C0∥u0∥Lp(·)
3

2α−1

,

∥B4(u,B)∥X ≤ 2C0∥B0∥Lp(·)
3

2α−1

,

∥B(u1, B1)− B(u2, B2)∥X <
1

2
∥(u1, B1)− (u2, B2)∥X

for all (u1, B1) and (u2, B2) ∈ X . Therefore, by the contraction mapping principle, there exists a
unique solution (u,B) ∈ X satisfying (2.3) with α = β. This completes the proof. □

Acknowledgments. Jinyi Sun was supported by the National Natural Science Foundation of
China (Grant No. 12361050), by the Outstanding Youth Fund Project of Gansu Province (Grant
No. 24JRRA121), and by the Funds for Innovative Fundamental Research Group Project of Gansu
Province (Grant No. 24JRRA778), by the Gansu Province University Teachers Innovation Fund
Project (Grant No. 2023A-002). Yuanwei Mai was supported by the Postgraduate Research Fund-
ing Program of Northwest Normal University (Grant No. 2022KYZZ-S120). Minghua Yang was
supported by the National Natural Science Foundation of China (Grant No. 12161041) and by the
Jiangxi Province Natural Science Foundation (Grant No. 20252BAC250004).

Author contributions. All authors designed and performed this research. Mai wrote the text,
then Sun and Yang reviewed the manuscript.

References

[1] M. Z. Abidin, J. Chen; Global well-posedness for fractional Navier-Stokes equations in variable exponent
Fourier-Besov-Morrey spaces, Acta Math. Sci. Ser. B (Engl. Ed.), 41(2021), no. 1, 164-176.

[2] M. Z. Abidin, M. Marwan, N. Ullah. A. M. Zidan; Well-posedness in variable-exponent function spaces for

the three-dimensional micropolar fluid equations, J. Math., 2023, Paper No. 4083997, 11 pp.
[3] E. Acerbi, G. Mingione; Regularity results for a class of functionals with non-standard growth, Arch. Ration.

Mech. Anal., 156(2001), no. 2, 121-140.

[4] E. Acerbi, G. Mingione; Gradient estimates for the p(x)-Laplacean system, J. Reine Angew. Math., 584(2005),
117-148.



14 J. SUN, Y. MAI, M. YANG EJDE-2025/111

[5] J. A. R. Almeida, J. J. Hasanov, S. G. Samko; Maximal and potential operators in variable exponent Morrey

spaces, Georgian Math. J., 15(2008), no. 2, 195-208.

[6] J. A. R. Almeida, P. A. Hästö; Besov spaces with variable smoothness and integrability, J. Funct. Anal.,
258(2010), no. 5, 1628-1655.

[7] D. Chamorro; Mixed Sobolev-like inequalities in Lebesgue spaces of variable exponents and in Orlicz spaces,

Positivity, 26(2022), 5.
[8] D. Chamorro, G. Vergara-Hermosilla; Lebesgue spaces with variable exponent: some applications to the

Navier-Stokes equations, Positivity, 28(2024), no. 2, 24.

[9] D. Chamorro, G. Vergara-Hermosilla; Liouville-type theorems for stationary Navier-Stokes equations with
Lebesgue spaces of variable exponent, Doc. Math., 2025, DOI: 10.4171/DM/1018.

[10] J. Y. Chemin, D. S. McCormick, J. C. Robinson, J. L. Rodrigo; Local existence for the non-resistive MHD

equations in Besov spaces, Adv. Math., 286(2016), 1-31.
[11] H. Chen, G. Vergara-Hermosilla, J. Zhao; Well-posedness of the 2D surface quasi-geostrophic equation in

variable Lebesgue spaces, Rend. Circ. Mat. Palermo (2), 74(2025), no. 4, 134, 12 pp.
[12] Y. Chen, S. E. Levine, M. Rao; Variable exponent, linear growth functionals in image restoration, SIAM J.

Appl. Math., 66(2006), no. 4, 1383-1406.

[13] D. V. Cruz-Uribe; The Hardy-Littlewood maximal operator on variable-Lp spaces, in Seminar of Mathemat-
ical Analysis (Malaga/Seville, 2002/2003), 147-156, Colecc. Abierta, 64, Univ. Sevilla Secr. Publ., Seville,

2003.

[14] D. V. Cruz-Uribe, A. Fiorenza; Variable Lebesgue Spaces: Foundations and Harmonic Analysis, Birkhäuser-
Springer, Heidelberg, 2013.

[15] G. Duvaut, J. L. Lions; Inéquations en thermoélasticité et magnétohydrodynamique, Arch. Rational Mech.

Anal., 46(1972), 241-279.

[16] L. Diening; Maximal function on generalized Lebesgue spaces Lp(·), Math. Inequal. Appl., 7(2004), no. 2,

245-253.

[17] L. Diening, P. Harjulehto, P. Hästö, M. Ruzicka; Lebesgue and Sobolev Spaces with Variable Exponents,
Lecture Notes in Mathematics, 2017, Springer, Heidelberg, 2011.

[18] L. Diening, P. A. Hästö, S. A. Roudenko; Function spaces of variable smoothness and integrability, J. Funct.

Anal., 256(2009), no. 6, 1731-1768.
[19] X. L. Fan; Global C1,α regularity for variable exponent elliptic equations in divergence form, J. Differential

Equations, 235(2007), no. 2, 397-417.

[20] X. L. Fan, D. Zhao; On the spaces Lp(x)(Ω) and Wm,p(x)(Ω), J. Math. Anal. Appl., 263(2001), no. 2, 424-446.
[21] C. L. Fefferman, D. S. McCormick, J. C. Robinson, J. L. Rodrigo; Higher order commutator estimates and

local existence for the non-resistive MHD equations and related models, J. Funct. Anal., 267(2014), no. 4,

1035-1056.
[22] P. Harjulehto, P. Hästö, V. Latvala, O. Toivanen; Critical variable exponent functionals in image restoration,

Appl. Math. Lett., 26(2013), no. 1, 56-60.

[23] M. Jin, Q. Jiu, Y. Xie; Global well-posedness and optimal decay for incompressible MHD equations with
fractional dissipation and magnetic diffusion, Z. Angew. Math. Phys., 75(2024), no. 2, 73.

[24] O. Kováčik, J. Rákosńık; On spaces Lp(x) and Wk,p(x), Czechoslovak Math. J., 41(116)(1991), no. 4, 592-618.

[25] P. G. Lemarié-Rieusset; The Navier-Stokes problem in the 21st century, CRC Press, Boca Raton, FL, 2016.
[26] Q. Liu, J. Zhao; Global well-posedness for the generalized magneto-hydrodynamic equations in the critical

Fourier-Herz spaces, J. Math. Anal. Appl., 420(2014), no. 2, 1301-1315.

[27] C. Miao, B. Yuan, B. Zhang; Well-posedness of the Cauchy problem for the fractional power dissipative
equations, Nonlinear Anal., 68(2008), no. 3, 461-484.

[28] J. Musielak; Orlicz spaces and modular spaces, Lecture Notes in Mathematics, 1034, Springer, Berlin, 1983.
[29] H. Nakano; Modulared Semi-Ordered Linear Spaces, Maruzen, Tokyo, 1950.

[30] H. Nakano; Topology of linear topological spaces, Maruzen, Tokyo, 1951.

[31] W. Orlicz, Über konjugierte exponentenfolgen; Studia Mathematica, 3(1931), 200-212.
[32] V. D. Rădulescu, D. D. Repovš; Partial differential equations with variable exponents, Monographs and

Research Notes in Mathematics, CRC Press, Boca Raton, FL, 2015.

[33] S. Ru, M. Z. Abidin; Global well-posedness of the incompressible fractional Navier-Stokes equations in Fourier-
Besov spaces with variable exponents, Comput. Math. Appl., 77(2019), no. 4, 1082-1090.

[34] M. Růžička; Electrorheological fluids: modeling and mathematical theory, Lecture Notes in Mathematics,

1748, Springer, Berlin, 2000.
[35] G. Vergara-Hermosilla; Remarks on variable Lebesgue spaces and fractional Navier-Stokes equations, to

appear in ESAIM: Proceedings and Surveys.
[36] G. Vergara-Hermosilla; On Variable Lebesgue Spaces and Generalized Nonlinear Heat Equations. J. Dynam.

Differential Equations, 2025, DOI: 10.1007/s10884-025-10426-6.

[37] G. Vergara-Hermosilla, J. Zhao; On the Cauchy problem for the fractional Keller-Segel system in variable
Lebesgue spaces, arXiv:2405.01209.

[38] G. Vergara-Hermosilla, J. Zhao; Global well-posedness of the fractional dissipative system in the framework

of variable Fourier-Besov spaces, Mediterr. J. Math., 22(2025), no. 6, Paper No. 160, 25 pp.



EJDE-2025/111 MAGNETOHYDRODYNAMIC EQUATIONS 15

[39] W. H. Wang; Global well-posedness and analyticity for the 3D fractional magnetohydrodynamics equations

in variable Fourier-Besov spaces, Z. Angew. Math. Phys., 70(2019), no. 6, 163, 16 pp.

[40] J. Wu; Generalized MHD equations, J. Differential Equations, 195(2003), 284-312.
[41] J. Xu; Variable Besov and Triebel-Lizorkin spaces, Ann. Acad. Sci. Fenn. Math., 33(2008), no. 2, 511-522.

[42] Z. Ye; Global well-posedness and decay results to 3D generalized viscous magnetohydrodynamic equations,

Ann. Mat. Pura Appl., 195(2016), no. 4, 1111-1121.
[43] Z. Ye, X. Zhao; Global well-posedness of the generalized magnetohydrodynamic equations, Z. Angew. Math.

Phys., 69(2018), no. 5, 126, 26 pp.

[44] X. Zhai; Stability for the 2D incompressible MHD equations with only magnetic diffusion, J. Differential
Equations, 374(2023), 267-278.

[45] X. Zhai, Y. Li, W. Yan; Global well-posedness for the 3-D incompressible MHD equations in the critical Besov

spaces, Commun. Pure Appl. Anal., 14(2015), no. 5, 1865-1884.
[46] V. V. Zhikov; Averaging of functionals of the calculus of variations and elasticity theory, Izv. Akad. Nauk

SSSR Ser. Mat., 50(1986), no. 4, 675-710, 877.

Jinyi Sun (corresponding author)
College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China

Email address: sunjinyi333@163.com

Yuanwei Mai

College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China

Email address: maiyuanwei0425@163.com

Minghua Yang

Department of Mathematics, Jiangxi University of Finance and Economics, Nanchang 330032, China
Email address: ymh20062007@163.com


	1. Introduction
	2. Preliminaries
	3. Key estimates
	4. Proof of main results
	Acknowledgments
	Author contributions

	References