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Commun. Optim. Theory 2019 (2019), Article ID 12 https://doi.org/10.23952/cot.2019.12

THE UNIFORM GLOBAL WELL-POSEDNESS AND THE STABILITY OF THE 3DGENERALIZED MAGNETOHYDRODYNAMIC EQUATIONS WITH THE CORIOLIS

FORCE

AZZEDDINE EL BARAKA, MOHAMED TOUMLILIN∗

University Sidi Mohamed Ben Abdellah, FST Fes-Saiss, Laboratory AAFA, Department of Mathematics,B.P 2202 Route Immouzer, Fes 30000 Morocco

Abstract. This paper deals with the Cauchy problem of the 3D generalized magnetohydrodynamic equations with the Cori-olis force (GMHDC). By using the Fourier localization argument and the Littlewood-Paley theory, we obtain the uniform global

well-posedness results with small initial data (u0,b0) belonging to the critical Fourier-Besov-Morrey spaces F ˙N4−2α+ λ−3

p

p,λ ,q (R3).Moreover, the stability of global solutions is also discussed.Keywords. Magnetohydrodynamic equations; Fourier-Besov-Morrey space; Stability; Global solution.

2010 Mathematics Subject Classification. 35Q30, 76D05,76D03.

1. INTRODUCTION

We consider the generalized magnetohydrodynamic equations with the Coriolis force in the wholespace R3

ut +u ·∇u+µ(−∆)αu+Ωe3×u−b ·∇b+∇π = 0 in (0,+∞)×R3,

∇ ·u = 0, ∇ ·b = 0,bt +u ·∇b+ν(−∆)αb−b ·∇u = 0 in (0,+∞)×R3,

(u,b)|t=0 = (u0,b0),

(1.1)

where u denotes the velocity field of the fluid, π denotes the pressure function, b is the magnetic field,ν > 0 is the magnetic diffusivity, µ > 0 is the viscosity, ∇ · u = 0 and ∇ · b = 0 is the incompressibleconditions, Ω∈R denotes twice the speed of rotation around the vertical unit vector e3 = (0,0,1), and u0

and b0 denote the initial velocity and the initial magnetic field with ∇ ·u0 = 0 and ∇ ·b0 = 0, respectively.(−∆)α is the pseudo-differential operator with symbol |ξ |2α . Mathematically, the equation (1.1) is usedto explain why the earth has a non-zero large-scale magnetic field whose polarity turns out to invert overseveral hundred centuries when α = 1. For this concept, we refer readers to [1] and the references therein.

∗Corresponding author.E-mail addresses: [email protected] (A. El Baraka), [email protected] (M. Toumlilin).Received February 10, 2019; Accepted July 8, 2019.

c©2019 Communications in Optimization Theory

1

2 A. EL BARAKA, M. TOUMLILIN

If Ω = 0, then equation (1.1) becomes the generalized Magnetohydrodynamics equations, which consistto examine the magnetic properties of electrically conducting incompressible fluids. Let us take a timeto briefly cite some recent results. Duvaut and Lions [2] developed a global Leray-Hopf weak solutionto MHD system. Cao and Wu [3] showed global regularity of classical solutions for the MHD equationswith magnetic diffusion and mixed partial dissipation. Besides, numerous notable successes have beenachieved on the fundamental issues of the regularity criteria or blow-up criteria to system (1.1). For moreresults in this direction, see [4, 5, 6, 7, 8, 9, 10, 11, 12] and references therein.

When b = 0 and Ω 6= 0, Wang and Wu [13] investigated the global well-posedness and the Gevreyclass regularity of mild solutions to the 3D incompressible generalized Navier-Stokes equations with theCoriolis force in the Lei-Lin space X 1−2α defined by

X 1−2α = u ∈D ′(R3) :∫R3|ξ |1−2α |u(ξ )|dξ <+∞.

Recently, Wang and Wu [14] established the global well-posedness of the generalized rotating magneto-hydrodynamic equations in the Lei-Lin space X 1−2α .

The goal of this paper is to show the existence and the stability of an uniform global solution tothe 3D generalized rotating magnetohydrodynamic equations (1.1) in the critical Fourier-Besov-Morreyspaces F ˙N

sp,λ ,q(R3) with s = 4−2α + λ−3

p . In fact, this space contains many classical spaces, e.g., theFourier-Herz space Bs

q = F ˙Ns1,0,q, the Fourier-Besov-Lebesgue space FBs

p,q = F ˙Nsp,0,q and the Lei-

Lin’s space X s =F ˙Ns1,0,1. Motivated by the results [13, 14], we prove the uniform global existence in

the sense that the smallness condition on the initial data is independent of the size of the speed of rotationΩ. Specifically, this paper generalizes the results of existence given in [14] from Lei-Lin’s space X 1−2α

to Fourier Besov-Morrey spaces F ˙N4−2α+ λ−3

pp,λ ,q (R3).

2. PRELIMINARIES AND MAIN RESULTS

In this section, we briefly mention some notations and give the fundamental properties of the Fourier-Besov-Morrey spaces that will be used in the rest of the paper. In the following, we first introduce thehomogeneous decomposition of Littlewood-Paley decomposition.

Let χ, ϕ be two nonnegative smooth radial functions satisfying

suppϕ ⊂ ξ ∈ Rn :34≤ |ξ | ≤ 8

3, ∑

j∈Zϕ(2− j

ξ ) = 1, ξ ∈ Rn\0,

supp χ ⊂ ξ ∈ Rn : |ξ | ≤ 43, χ(ξ )+ ∑

j≥0ϕ(2− j

ξ ) = 1, ξ ∈ Rn .

We denote the space of tempered distributions by S′ and we designate ϕ j(ξ ) = ϕ(2− jξ ). P is the set ofall polynomials.

We introduce some frequency localization operators, such as,

∆ ju = ϕ(2− jD)u = 2 jn∫

h(2 jy)u(x− y)dy,

S ju = ∑k≤ j−1

∆ku = χ(2− jD)u = 2 jn∫

h(2 jy)u(x− y)dy,

where h = F−1ϕ and h = F−1χ .

THE UNIFORM GLOBAL WELL-POSEDNESS AND THE STABILITY 3

First, we define an appropriate version of the classical Morrey spaces which are a complement to Lp

spaces.

Definition 2.1. For 1 ≤ p < ∞, 0 ≤ λ < n, the Morrey spaces Mλp = Mλ

p(Rn) is the set of functionsf ∈ Lp

loc(Rn) such that

‖ f‖Mλp= sup

x0∈Rnsupr>0

r−λ

p ‖ f‖Lp(B(x0,r)) < ∞, (2.1)

where B(x0,r) is the ball in Rn with center x0 and radius r.

The following lemma is devoted to some basic properties related to Morrey spaces, see [15].

Lemma 2.2. Let 1≤ p1, p2, p3 < ∞ and 0≤ λ1, λ2, λ3 < n.

(1) (Holder’s inequality) Let 1p3= 1

p1+ 1

p2and λ3

p3= λ1

p1+ λ2

p2. If fi ∈Mλi

pifor i = 1,2, then f1 f2 ∈Mλ3

p3

and

‖ f1 f2‖Mλ3p3≤ ‖ f1‖Mλ1

p1‖ f2‖Mλ2

p2.

(2) (Young’s inequality) Assume that ϕ ∈ L1 and g ∈Mλ1p1

, then

‖ϕ ∗g‖Mλ1

p1≤ ‖ϕ‖L1‖g‖

Mλ1p1, (2.2)

(3) (Bernstein-type inequality) Let 1≤ p2 ≤ p1 < ∞ such that

n−λ1

p1≤ n−λ2

p2.

If supp( f ) ⊂ ξ ∈ Rn : |ξ | ≤ A2 j then there is a constant C > 0 independent of f and j suchthat

‖(iξ )γ f‖Mλ2

p2≤C2 j|γ|+ j( n−λ2

p2− n−λ1

p1)‖ f‖

Mλ1p1, (2.3)

where γ is a multi-index and j ∈ Z.

Now, let us define the homogeneous Fourier-Besov-Morrey spaces.

Definition 2.3. Let s ∈R, 1≤ p <+∞, 1≤ q≤+∞, and 0≤ λ < n, the space F ˙Nsp,λ ,q(Rn) is defined

by

F ˙Nsp,λ ,q(Rn) =

u ∈S ′(Rn)/P; ‖u‖F ˙N

sp,λ ,q(Rn) < ∞

,

where

‖u‖F ˙Nsp,λ ,q(Rn) =

∑j∈Z

2 jqs‖∆ ju‖qMλ

p

1/qf or q < ∞,

supj∈Z

2 js‖∆ ju‖Mλp

f or q = ∞ ,

where P is the set of all polynomials on Rn.

We notice that the homogeneous Fourier-Besov-Morrey spaces are some refined functional spaceswhich cover many classical spaces, such as; Fourier-Herz space Bs

q, Fourier-Besov-Lebesgue spaceFBs

p,q and Lei-Lin’s space χ−1, thus such spaces are more suitable and more adapted for studying certainequations of fluid mechanics.

Now, we recall the definition of the mixed space-time spaces.


Definition 2.4. Let s∈R, 1≤ p<∞, 1≤ q,ρ ≤∞, 0≤ λ < n, and I = [0,T ), T ∈ (0,∞]. The space-timenorm is defined on u(t,x) by

‖u(t,x)‖L ρ (I,F ˙Nsp,λ ,q)

=

∑j∈Z

2 jqs‖∆ ju‖qLρ (I,Mλ

p )

1/q,

and denote by L ρ(I,F ˙Nsp,λ ,q) the set of distributions in S′(R×Rn)/P with finite ‖.‖L ρ (I,F ˙N

sp,λ ,q)

norm. Due to the Minkowski inequality, we have

‖u‖L ρ(I;F ˙Nsp,λ ,q)≤ ‖u‖Lρ(I;F ˙N

sp,λ ,q)

, if ρ ≤ q and ‖u‖Lρ(I;F ˙Nsp,λ ,q)≤ ‖u‖L ρ(I;F ˙N

sp,λ ,q)

, if ρ ≥ q ,

where ‖u(t,x)‖Lρ (I;F ˙Nsp,λ ,q)

:=(∫

I ‖u(τ, ·)‖ρ

F ˙Nsp,λ ,q

dτ

)1/ρ

.

Our first main result is on the global well-posedness result.

Theorem 2.5. Let 1≤ p < ∞, 1≤ q≤ 2, 0≤ λ < 3, and 12 < α ≤ 5

2 +λ−32p . Then there exists a constant

C0(α, p,q) such that, for any (u0,b0) ∈F ˙N4−2α+ λ−3

pp,λ ,q satisfying ∇ ·u0 = ∇ ·b0 = 0 and

‖(u0,b0)‖F ˙N

4−2α+ λ−3p

p,λ ,q

≤C0 minµ,ν,

and the Cauchy problem (1.1) admits a unique global solution (u,b),

(u,b) ∈ C([0,∞);F ˙N

4−2α+ λ−3p

p,λ ,q

)∩L 1([0,∞),F ˙N

4+ λ−3p

p,λ ,q

),

and it satisfies ∥∥∥(u,b)∥∥∥L ∞

([0,∞);F ˙N

4−2α+ λ−3p

p,λ ,q

)∩L 1

([0,∞),F ˙N

4+ λ−3p

p,λ ,q

)≤ 2(

1+(

169

)α )∥∥∥(u0,b0)∥∥∥

F ˙N4−2α+ λ−3

pp,λ ,q

.

Now, we give some remarks about this result.

Remark 2.6. We not that for 12 ≤ α ≤ 1 and b = 0, the system (1.1) becomes the fractional Navier-

Stokes equations with the Coriolis force studied by Wang and Gang [13] in the Lei-Lin space X 1−2α .The proof of Theorem 2.5 is based on two methods, that is, the use of the dissipative equation and theclassical semigroup approach.

Theorem 2.5 extends the result of [14] from Lei-Lin spaces to Fourier-Besov-Morrey spaces, sincefor (λ = 0, q = 1, p = 1), F ˙N

s1,0,1 = X s. In addition, Theorem 2.5 also holds in the Fourier-Besov-

Lebesgue spaces since, for λ = 0, F ˙Nsp,0,q = FBs

p,q.We also remark that the fractional magnetohydrodynamic equations with the Coriolis force is uni-

formly globally well-posed in the sense that the smallness conditions are independent of Ω.

The second result of this paper is to give the stability of global solutions.

Theorem 2.7. Let T ∗ denote the maximal time of existence of a solution (u,b) in

L ∞

([0,T ∗);F ˙N

4−2α+ λ−3p

p,λ ,q

)∩L 1

([0,T ∗),F ˙N

4+ λ−3p

p,λ ,q

).


If T ∗ < ∞, then

‖(u,b)‖L 1([0,T ∗),F ˙N

4+ λ−3p

p,λ ,q )= ∞.

Besides, if (u,b) ∈C(R+,F ˙N4−2α+ λ−3

pp,λ ,q ) is a global solution of (1.1), and for all

(u0, b0) ∈F ˙N4−2α+ λ−3

pp,λ ,q

such that

‖u0−u0‖F ˙N

4−2α+ λ−3p

p,λ ,q

+‖b0−b0‖F ˙N

4−2α+ λ−3p

p,λ ,q

(2.4)

<C0minµ,ν

8exp∫ ∞

0− 1

C0(|Ω|+‖u‖

F ˙N4+ λ−3

pp,λ ,q

+‖b‖F ˙N

4+ λ−3p

p,λ ,q

),

for some constant C0 sufficiently small, then

‖u(t)−u(t)‖F ˙N

4−2α+ λ−3p

p,λ ,q

+µ

4‖u−u‖

L 1([0,t),F ˙N4+ λ−3

pp,λ ,q )

+‖b(t)−b(t)‖F ˙N

4−2α+ λ−3p

p,λ ,q

+ν

4‖b−b‖

L 1([0,t),F ˙N4+ λ−3

pp,λ ,q )

≤C(‖u0−u0‖F ˙N

4−2α+ λ−3p

p,λ ,q

+‖b0−b0‖F ˙N

4−2α+ λ−3p

p,λ ,q

)

× exp∫ ∞

0C(|Ω|+‖u‖

F ˙N4+ λ−3

pp,λ ,q

+‖b‖F ˙N

4+ λ−3p

p,λ ,q

),

where C is a positive constant.

Remark 2.8. When α = 1 and Ω= 0, Wang [16] obtained the same result in the space χ−1. Theorem 2.7

generalizes the stability of global solution of (1.1) to the Fourier-Besov-Morrey space F ˙N4−2α+ λ−3

pp,λ ,q .

In what follows, we consider the linear dissipative equation

ut +µ(−∆)αu = f (t,x) (t,x) ∈ R+×R3

u(0,x) = u0(x) x ∈ R3,(2.5)

for which we give the following result.

Lemma 2.9 ([17]). Let I = [0,T ), 0 < T ≤ ∞, s ∈ R, 0 ≤ λ < n, 1 ≤ p < ∞, and 1 ≤ q ≤ ∞. Ifu0 ∈F ˙N

sp,λ ,q and f ∈L 1(I,F ˙N

sp,λ ,q), then the solution u(t,x) to the Cauchy problem (2.5) satisfies

‖u‖L ∞(I,F ˙Nsp,λ ,q)

+µ‖u‖L 1(I,F ˙N

s+2α

p,λ ,q )

≤ (1+(169)α)(‖u0‖F ˙N

sp,λ ,q

+‖ f‖L 1(I,F ˙Nsp,λ ,q)

).(2.6)

If q is finite, then u belongs to C (I,F ˙Nsp,λ ,q).

Lemma 2.10 ([17]). Let 1≤ p < ∞, 1≤ ρ ≤∞ 1≤ q≤ 2, 0≤ λ < 3, I = [0,T ), 0 < T ≤+∞, 12 < α <

5+ λ−3p

4− 2ρ

and set

X = L ∞(I,F ˙N4−2α+ λ−3

pp,λ ,q )∩L ρ(I,F ˙N

4−2α+ 2α

ρ+ λ−3

p

p,λ ,q ),


with the norm

‖u‖X = ‖u‖L ∞(I,F ˙N

4−2α+ λ−3p

p,λ ,q )+minµ,ν‖u‖

L ρ (I,F ˙N4−2α+ 2α

ρ + λ−3p

p,λ ,q ).

There exists a constant C =C(α, p,q)> 0 depending on α, p,q such that

‖∇.(u⊗ v)‖L ρ (I,F ˙N

4−4α+ 2αρ + λ−3

pp,λ ,q )

≤C(minµ,ν)−1‖u‖X‖v‖X .

3. THE WELL-POSEDNESS

First, we observe that the equation (1.1) reduces to the fractional Navier-Stokes equations with theCoriolis force when the magnetic field b = 0. We therefore introduce the corresponding generalizedStokes-Coriolis semigroup. More precisely, we investigate the following linear Stokes problem with theCoriolis force

ut +µ(−∆)αu+Ωe3×u+∇π = 0 (t,x) ∈ R+×R3,

∇.u = 0,u(0,x) = u0(x) x ∈ R3 .

(3.1)

The solution of equation (3.1) can be obtained from the generalized Stokes-Coriolis semigroup TΩ,α

[18, 19, 20], which is represented as

TΩ,α(t)u = F−1[cos(Ωξ3

|ξ |t)I + sin(Ω

ξ3

|ξ |t)R(ξ )]∗ (e−µ(−∆)α tu) ,

for divergence-free vector fields u and t ≥ 0. Here, I represents the identity matrix in R3 and R(ξ ) is theskew-symmetric matrix symbol for the Riesz transform, which is expressed by

R(ξ ) :=1|ξ |

0 ξ3 −ξ2

−ξ3 0 ξ1

ξ2 −ξ1 0

.

Thus, we are easy to get a semigroup

AΩ,α(t) =

(TΩ,α(t) 0

0 Sα(t)

),

where Sα(t) := e−µ(−∆)α t = F−1(e−µ|ξ |2α t).Now we can rewrite GMHDC equation (1.1) in the form of the integral(

ub

)= AΩ,α(t)

(u0

b0

)−∫ t

0AΩ,α(t− τ)P

(∇ · (u⊗u−b⊗b)∇ · (u⊗b−b⊗u)

)(·,τ)dτ,

where P= Id−∇∆−1div is the Leray-Hopf projector.The following estimate corresponds to the Stokes-Coriolis semigroup TΩ,α .

Lemma 3.1. Let 0 < T ≤ ∞, s ∈ R, 0≤ λ < 3,1≤ p < ∞, 1≤ q,ρ,r ≤ ∞ and

f ∈L r([0,T ),F ˙Nsp,λ ,q(R3)).

There exists a constant C > 0 such that

‖∫ t

0 TΩ,α(t− τ) f (τ)dτ‖L ρ ([0,T ),F ˙Nsp,λ ,q)≤C‖ f‖

L r([0,T ),F ˙Ns−2α− 2α

ρ + 2αr

p,λ ,q ).


Besides, if u0 ∈FN4−2α+ λ−3

pp,λ ,q (R3), then there exists a constant C > 0 such that

‖TΩ,α(t)u0‖L ∞([0,T ),F ˙N

4−2α+ λ−3p

p,λ ,q )≤C‖u0‖

F ˙N4−2α+ λ−3

pp,λ ,q

, (3.2)

‖TΩ,α(t)u0‖L 1([0,T ),F ˙N

4+ λ−3p

p,λ ,q )≤C‖u0‖

F ˙N4−2α+ λ−3

pp,λ ,q

, . (3.3)

Proof. The definition of the space-time norm of L ρ([0,T ),F ˙Nsp,λ ,q) and Young’s inequality give∥∥∥∫ t

0TΩ,α(t− τ) f (τ)dτ

∥∥∥L ρ ([0,T ),F ˙N

sp,λ ,q)

≤

∑j∈Z

2 jqs(∫ T

0‖ϕ j

∫ t

0e−µ|ξ |2α (t−τ) f (τ)dτ‖ρ

Mλpdt) q

ρ1/q

≤

∑j∈Z

2 jqs(∫ T

0‖ϕ j

∫ t

0e−µ22α j(t−τ) f (τ)dτ‖ρ

Mλpdt) q

ρ1/q

≤C

∑j∈Z

2 jq(s−2α− 2α

ρ+ 2α

r )‖ϕ j f (τ)‖qLr([0,T ),Mλ

p )

1/q

≤C‖ f‖L r([0,T ),F ˙N

s−2α− 2αρ + 2α

rp,λ ,q )

,

where we have used 1+ 1ρ= 1

ρ+ 1

r . To show inequality (3.2), it is sufficient to write that

‖TΩ,α(t)u0‖L ∞([0,T ),F ˙N

4−2α+ λ−3p

p,λ ,q )

≤ C(

∑j∈Z

2 j(4−2α+ λ−3p )q‖ϕ ju0‖q

Mλp

) 1q

≤ C‖u0‖F ˙N

4−2α+ λ−3p

p,λ ,q

.

Similarly, for proving (3.3), we write

‖TΩ,α(t)u0‖L 1([0,T ),F ˙N

4+ λ−3p

p,λ ,q )

≤(

∑j∈Z

2 j(4+ λ−3p )q(∫ T

0e−tµ22α j‖ϕ ju0‖Mλ

pdt)q) 1

q

≤ C‖u0‖F ˙N

4−2α+ λ−3p

p,λ ,q

.

Proof of Theorem 2.5. In usual practice, the mild solution (u,b) for GMHDC equation (1.1) can berewritten as follows

u = TΩ,α(t)u0−∫ t

0TΩ,α(t− τ)P∇ · (u⊗u−b⊗b)(·,τ)dτ := T1(u,b),

b = e−tν(−∆)α

b0−∫ t

0e−ν(t−τ)(−∆)α

P∇ · (u⊗b−b⊗u)(·,τ)dτ := T2(u,b),(3.4)

where P= Id−∇∆−1div is the Leray-Hopf projector.Let

X = L ∞

([0,∞);F ˙N

4−2α+ λ−3p

p,λ ,q

)∩L 1

([0,∞),F ˙N

4+ λ−3p

p,λ ,q

).


We define the norm of the vector (u,b) for u,b ∈ X , as follows

‖(u,b)‖X = ‖u‖X +‖b‖X .

Let

Lµ(u,v) :=∫ t

0TΩ,α(t− τ)P∇ · (u⊗ v)(·,τ)dτ.

It is easy to rewrite the system (3.4) as follows

(u,b) = (T1(u,b),T2(u,b)) := T (u,b).

Lemma 3.1 and Lemma 2.10 lead to

‖Lµ(u,u)−Lµ(b,b)‖L 1([0,∞),F ˙N

4+ λ−3p

p,λ ,q )

=∥∥∥∫ t

0TΩ,α(t− τ)P∇.(u⊗u−b⊗b)(τ)dτ

∥∥∥L 1([0,∞),F ˙N

4+ λ−3p

p,λ ,q )

≤ C‖∇.(u⊗u−b⊗b)‖L 1([0,∞),F ˙N

4−2α+ λ−3p

p,λ ,q )

≤ C(minµ,ν)−1(‖u‖2X +‖b‖2

X) .

Similarly,

‖Lµ(u,u)−Lµ(b,b)‖L ∞([0,∞),F ˙N

4−2α+ λ−3p

p,λ ,q )

=∥∥∥∫ t

0TΩ,α(t− τ)P∇.(u⊗u−b⊗b)(τ)dτ

∥∥∥L ∞([0,∞),F ˙N

4−2α+ λ−3p

p,λ ,q )

≤ C‖∇.(u⊗u−b⊗b)‖L 1([0,∞),F ˙N

4−2α+ λ−3p

p,λ ,q )

≤ C(minµ,ν)−1(‖u‖2X +‖b‖2

X) .

Finally,

‖Lµ(u,u)−Lµ(b,b)‖X ≤C(minµ,ν)−1(‖u‖2X +‖b‖2

X) . (3.5)

Lemma 3.1 yields

‖TΩ,α(t)u0‖X ≤C‖u0‖F ˙N

4−2α+ λ−3p

p,λ ,q

. (3.6)

The estimates (3.5) and (3.6) allow us to obtain

‖T1(u,b)‖X ≤C‖u0‖F ˙N

4−2α+ λ−3p

p,λ ,q

+C(minµ,ν)−1(‖u‖2X +‖b‖2

X).

(3.7)

Similarly, let

Lν(u,v) :=∫ t

0e−ν(t−τ)(−∆)α

P∇.(u⊗ v)(τ,x)dτ,

For the second equation, we remark that Lν(u,b) can be considered as the solution to the heat equation(2.5) with u0 = 0 and force f = P∇.(u⊗v). According to Lemma 2.9 with s = 4−2α + λ−3

p and Lemma


2.10 with ρ = 1, we get

‖Lν(u,b)‖X ≤C‖P∇.(u⊗b)‖L 1(I,F ˙N

4−2α+ λ−3p

p,λ ,q )

≤C(minµ,ν)−1‖u‖X‖b‖X .(3.8)

We also notice that e−µt(−∆)α

b0 is the solution to the dissipative equation with f = 0 and b0 = b0. As aresult, Lemma 2.9 gives

‖e−µt(−∆)α

b0‖X ≤C‖b0‖F ˙N

4−2α+ λ−3p

p,λ ,q

. (3.9)

Through the estimates (3.8) and (3.9), we arrive at

‖T2(u,b)‖X ≤C‖b0‖F ˙N

4−2α+ λ−3p

p,λ ,q

+2C(minµ,ν)−1‖u‖X‖b‖X .(3.10)

Since ‖(u0,b0)‖F ˙N

4−2α+ λ−3p

p,λ ,q

≤C0 minµ,ν, we define

E =(u,b)|(u,b) ∈ X ,‖(u,b)‖X ≤ 2CC0 minµ,ν

,

where C0 is a constant that can be specified later. Combining (3.6), (3.7), and (3.10), it follows that

‖T (u,b)‖X

≤C‖(u0,b0)‖F ˙N

4−2α+ λ−3p

p,λ ,q

+C(minµ,ν)−1‖(u,b)‖2X

≤CC0 minµ,ν+4C3C20 minµ,ν,

which implies that T (u,b) ∈ E when we choose C0 small enough such that C0 <1

16C2 .On the other hand, for any (u1,b1), (u2,b2) ∈ E, we have

‖T1(u1,b1)−T1(u2,b2)‖X

≤ ‖Lµ(u1,u1)−Lµ(u2,u2)‖X +‖Lµ(b1,b1)−Lµ(b2,b2)‖X

≤ ‖Lµ(u1,u1−u2)+Lµ(u1−u2,u2)‖X +‖Bµ(b1−b2,b2)+Lµ(b1,b1−b2)‖X

≤C(minµ,ν)−1((‖u1‖X +‖u2‖X)‖u1−u2‖X

+(‖b1‖X +‖b2‖X)‖b1−b2‖X)

≤ 4C2C0(‖u1−u2‖X +‖b1−b2‖X)

≤ 14(‖u1−u2‖X +‖b1−b2‖X).

Similarly,

‖T2(u1,b1)−T2(u2,b2)‖X

≤ ‖Lν(u2,b2)−Lν(u1,b1)‖X +‖Lν(b2,u2)−Lν(b1,u1)‖X

≤ 4C2C0(‖u1−u2‖X +‖b1−b2‖X)

≤ 14(‖u1−u2‖X +‖b1−b2‖X).


Consequently,

‖T (u1,b1)−T (u2,b2)‖X ≤12(‖u1−u2‖X +‖b1−b2‖X).

Based on the above estimate, we obtain that T is a contraction mapping from E to E. From the Banachfixed point theorem, we deduce that T has a unique fixed point (u,b)∈ E which is the solution of system(1.1). The proof is finished.

4. STABILITY OF GLOBAL SOLUTIONS

In this section, we prove the result of the stability for the global solutions and the blow up criteriawhen the maximal time of existence is finite.

Let T ∗ be the maximal existence time of solutions of (1.1) in

L ∞

([0,T ∗);F ˙N

4−2α+ λ−3p

p,λ ,q

)∩L 1

([0,T ∗),F ˙N

4+ λ−3p

p,λ ,q

).

To prove Theorem 2.7, we assume that T ∗ < ∞ and

‖(u,b)‖L 1([0,T ∗),F ˙N

4+ λ−3p

p,λ ,q )< ∞.

Then, we can find 0 < T0 < T ∗ satisfying

‖(u,b)‖L 1([T0,T ∗),F ˙N

4+ λ−3p

p,λ ,q )<

14.

For t ∈ [T0,T ∗), we expressly consider the integral equationu = TΩ,α(t)u(T0)−

∫ tT0

TΩ,α(t− τ)P∇ · (u⊗u−b⊗b)(τ, ·)dτ,

b = e−tν(−∆)α

b(T0)−∫ t

T0e−ν(t−τ)(−∆)αP∇ · (u⊗b−b⊗u)(τ, ·)dτ .

(4.1)

The same method as for Lemma 2.10 gives

‖u‖L ∞([T0,t),F ˙N

4−2α+ λ−3p

p,λ ,q ). ‖u(T0)‖

F ˙N4−2α+ λ−3

pp,λ ,q

+‖u‖L ∞([T0,t),F ˙N

4−2α+ λ−3p

p,λ ,q )‖u‖

L 1([T0,t),F ˙N4+ λ−3

pp,λ ,q )

+‖b‖L ∞([T0,t),F ˙N

4−2α+ λ−3p

p,λ ,q )‖b‖

L 1([T0,t),F ˙N4+ λ−3

pp,λ ,q )

,

and

‖b‖L ∞([T0,t),F ˙N

4−2α+ λ−3p

p,λ ,q ). ‖b(T0)‖

F ˙N4−2α+ λ−3

pp,λ ,q

+‖u‖L ∞([T0,t),F ˙N

4−2α+ λ−3p

p,λ ,q )‖b‖

L 1([T0,t),F ˙N4+ λ−3

pp,λ ,q )

+‖b‖L ∞([T0,t),F ˙N

4−2α+ λ−3p

p,λ ,q )‖u‖

L 1([T0,t),F ˙N4+ λ−3

pp,λ ,q )

,

As a result, we have

‖(u,b)‖L ∞([T0,t),F ˙N

4−2α+ λ−3p

p,λ ,q ). ‖(u(T0),b(T0))‖

F ˙N4−2α+ λ−3

pp,λ ,q

+14‖(u,b)‖

L ∞([T0,t),F ˙N4−2α+ λ−3

pp,λ ,q )

.


We can deduce from this that

supT0≤s≤t

‖(u,b)(s)‖F ˙N

4−2α+ λ−3p

p,λ ,q

. 2‖(u(T0),b(T0))‖F ˙N

4−2α+ λ−3p

p,λ ,q

,∀t ∈ [T0,T ∗) .

Putting

M = max(2‖(u(T0),b(T0))‖F ˙N

4−2α+ λ−3p

p,λ ,q

, maxt∈[0,T0]

‖(u,b)‖F ˙N

4−2α+ λ−3p

p,λ ,q

),

we have

‖(u(t),b(t))‖F ˙N

4−2α+ λ−3p

p,λ ,q

.M, ∀t ∈ [0,T ∗).

From (1.1), we obtainu(t ′)−u(t) =−µ

∫ t ′t (−∆)αu(τ)dτ−

∫ t ′t P∇ · (u⊗u−b⊗b)dτ−Ω

∫ t ′t P(e3×u)dτ,

b(t ′)−b(t) =−ν∫ t ′

t (−∆)αb(τ)dτ−∫ t ′

t ∇ · (u⊗b−b⊗u)dτ.(4.2)

From (4.2), we obtain that

‖u(t ′)−u(t)‖F ˙N

4−2α+ λ−3p

p,λ ,q

+‖b(t ′)−b(t)‖F ˙N

4−2α+ λ−3p

p,λ ,q

≤ µ‖u‖L 1([t,t ′),F ˙N

4+ λ−3p

p,λ ,q )+‖P∇ · (u⊗u−b⊗b)‖

L 1([t,t ′),F ˙N4−2α+ λ−3

pp,λ ,q )

+ν‖b‖L 1([t,t ′),F ˙N

4+ λ−3p

p,λ ,q )+‖∇ · (u⊗b−b⊗u)‖

L 1([t,t ′),F ˙N4−2α+ λ−3

pp,λ ,q )

+ |Ω|‖Pe3×u‖L 1([t,t ′),F ˙N

4−2α+ λ−3p

p,λ ,q )

. µ‖u‖L 1([t,t ′),F ˙N

4+ λ−3p

p,λ ,q )+ν‖b‖

L 1([t,t ′),F ˙N4+ λ−3

pp,λ ,q )

+ |Ω|‖e3×u‖L 1([t,t ′),F ˙N

4−2α+ λ−3p

p,λ ,q )

+‖(u,b)‖L ∞([t,t ′),F ˙N

4−2α+ λ−3p

p,λ ,q )‖(u,b)‖

L 1([t,t ′),F ˙N4+ λ−3

pp,λ ,q )

. (µ +ν +M+ |Ω|)‖(u,b)‖L 1([t,t ′),F ˙N

4+ λ−3p

p,λ ,q ).

The dominated convergence theorem yields

limsupt,t ′T ∗,t≤t ′

(‖u(t)−u(t ′)‖F ˙N

4−2α+ λ−3p

p,λ ,q

+‖b(t)−b(t ′)‖F ˙N

4−2α+ λ−3p

p,λ ,q

) = 0 .

This implies that u(t) and b(t) satisfies the Cauchy criterion at T ∗. Since F ˙N4−2α+ λ−3

pp,λ ,q is a Banach

space, there exists an element u∗,b∗ in F ˙N4−2α+ λ−3

pp,λ ,q such that u(t)→ u∗,b(t)→ b∗ in F ˙N

4−2α+ λ−3p

p,λ ,q ast→ T ∗. Now set u(T ∗) = u∗,b(T ∗) = b∗ and consider the generalized magnetohydrodynamic equationswith the Coriolis force starting by u∗,b∗. Using the well-posedness, we obtain a solution existing on alarger time interval than [0,T ∗), which is a contradiction.

Now, let u, b∈C([0,T ∗);F ˙N

4−2α+ λ−3p

p,λ ,q

)∩L 1

([0,T ∗),F ˙N

4+ λ−3p

p,λ ,q

)be the maximal solution of (1.1)

corresponding to the initial condition u0, b0. We need to prove T ∗ = ∞. Set w = u− u and d = b− b,which satisfies

∂tw+µ(−∆)αw+Ωe3×w+(u ·∇)w+(w ·∇)u+∇π = (b ·∇)d +(d ·∇)b,∂td +ν(−∆)αd +(u ·∇)d +(w ·∇)b = (b ·∇)w+(d ·∇)u,∇ ·w = 0, ∇ ·d = 0 .


We apply P to the above system. Then

∂tw+µ(−∆)αw =−ΩPe3×w−P∇.(u⊗w)−P∇.(w⊗u)+P∇.(b⊗d)+P∇.(d⊗b),∂td +ν(−∆)αd =−P∇.(u⊗d)−P∇.(w⊗b)+P∇.(b⊗w)+P∇.(d⊗u) .

For t ∈ [0,T ∗), we get

‖w(t)‖F ˙N

4−2α+ λ−3p

p,λ ,q

+µ‖w‖L 1([0,t),F ˙N

4+ λ−3p

p,λ ,q )

+‖d(t)‖F ˙N

4−2α+ λ−3p

p,λ ,q

+ν‖d‖L 1([0,t),F ˙N

4+ λ−3p

p,λ ,q )

≤C‖∇.(b⊗w)‖

L 1([0,t),F ˙N4−2α+ λ−3

pp,λ ,q )

+‖∇.(w⊗b)‖L 1([0,t),F ˙N

4−2α+ λ−3p

p,λ ,q )

+‖∇.(b⊗d)‖L 1([0,t),F ˙N

4−2α+ λ−3p

p,λ ,q )+‖∇.(d⊗b)‖

L 1([0,t),F ˙N4−2α+ λ−3

pp,λ ,q )

+‖∇.(u⊗w)‖L 1([0,t),F ˙N

4−2α+ λ−3p

p,λ ,q )+‖∇.(w⊗u)‖

L 1([0,t),F ˙N4−2α+ λ−3

pp,λ ,q )

+‖∇.(u⊗d)‖L 1([0,t),F ˙N

4−2α+ λ−3p

p,λ ,q )+‖∇.(d⊗u)‖

L 1([0,t),F ˙N4−2α+ λ−3

pp,λ ,q )

+‖∇.(w⊗w)‖L 1([0,t),F ˙N

4−2α+ λ−3p

p,λ ,q )+‖∇.(d⊗w)‖

L 1([0,t),F ˙N4−2α+ λ−3

pp,λ ,q )

+‖∇.(d⊗d)‖L 1([0,t),F ˙N

4−2α+ λ−3p

p,λ ,q )+‖∇.(w⊗d)‖

L 1([0,t),F ˙N4−2α+ λ−3

pp,λ ,q )

+ |Ω|‖e3×w‖L 1([0,t),F ˙N

4−2α+ λ−3p

p,λ ,q )+‖(w0,d0)‖

F ˙N4−2α+ λ−3

pp,λ ,q

≤CL1 +L2 +L3 +L4 +L5 +L6 + |Ω|‖e3×w‖

L 1([0,t),F ˙N4−2α+ λ−3

pp,λ ,q )

+‖(w0,d0)‖F ˙N

4−2α+ λ−3p

p,λ ,q

,

where

L1 = ‖∇.(b⊗w)‖L 1([0,t),F ˙N

4−2α+ λ−3p

p,λ ,q )+‖∇.(w⊗b)‖

L 1([0,t),F ˙N4−2α+ λ−3

pp,λ ,q )

,

L2 = ‖∇.(b⊗d)‖L 1([0,t),F ˙N

4−2α+ λ−3p

p,λ ,q )+‖∇.(d⊗b)‖

L 1([0,t),F ˙N4−2α+ λ−3

pp,λ ,q )

,

L3 = ‖∇.(u⊗w)‖L 1([0,t),F ˙N

4−2α+ λ−3p

p,λ ,q )+‖∇.(w⊗u)‖

L 1([0,t),F ˙N4−2α+ λ−3

pp,λ ,q )

,

L4 = ‖∇.(u⊗d)‖L 1([0,t),F ˙N

4−2α+ λ−3p

p,λ ,q )+‖∇.(d⊗u)‖

L 1([0,t),F ˙N4−2α+ λ−3

pp,λ ,q )

,

L5 = ‖∇.(w⊗w)‖L 1([0,t),F ˙N

4−2α+ λ−3p

p,λ ,q )+‖∇.(d⊗w)‖

L 1([0,t),F ˙N4−2α+ λ−3

pp,λ ,q )

,

L6 = ‖∇.(d⊗d)‖L 1([0,t),F ˙N

4−2α+ λ−3p

p,λ ,q )+‖∇.(w⊗d)‖

L 1([0,t),F ˙N4−2α+ λ−3

pp,λ ,q )

.


The same calculus of the proof of Lemma 2.10 lead to

L1 +L2 . 2∫ t

0(‖w‖

F ˙N4−2α+ λ−3

pp,λ ,q

+‖d‖F ˙N

4−2α+ λ−3p

p,λ ,q

)‖b‖F ˙N

4+ λ−3p

p,λ ,q

,

L3 +L4 ≤ 2∫ t

0(‖w‖

F ˙N4−2α+ λ−3

pp,λ ,q

+‖d‖F ˙N

4−2α+ λ−3p

p,λ ,q

)‖u‖F ˙N

4+ λ−3p

p,λ ,q

,

L5 . ‖w‖L ∞([0,t),F ˙N

4−2α+ λ−3p

p,λ ,q )‖w‖

L 1([0,t),F ˙N4+ λ−3

pp,λ ,q )

+‖d‖L ∞([0,t),F ˙N

4−2α+ λ−3p

p,λ ,q )‖w‖

L 1([0,t),F ˙N4+ λ−3

pp,λ ,q )

,

L6 . ‖d‖L ∞([0,t),F ˙N

4−2α+ λ−3p

p,λ ,q )‖d‖

L 1([0,t),F ˙N4+ λ−3

pp,λ ,q )

+‖w‖L ∞([0,t),F ˙N

4−2α+ λ−3p

p,λ ,q )‖d‖

L 1([0,t),F ˙N4+ λ−3

pp,λ ,q )

.

Then

‖w(t)‖F ˙N

4−2α+ λ−3p

p,λ ,q

+µ‖w‖L 1([0,t),F ˙N

4+ λ−3p

p,λ ,q )

+‖d(t)‖F ˙N

4−2α+ λ−3p

p,λ ,q

+ν‖d‖L 1([0,t),F ˙N

4+ λ−3p

p,λ ,q )

≤C‖(w0,d0)‖

F ˙N4−2α+ λ−3

pp,λ ,q

+(‖w‖L 1([0,t),F ˙N

4+ λ−3p

p,λ ,q )+‖d‖

L 1([0,t),F ˙N4+ λ−3

pp,λ ,q )

)

× (‖w‖L ∞([0,t),F ˙N

4−2α+ λ−3p

p,λ ,q )+‖d‖

L ∞([0,t),F ˙N4−2α+ λ−3

pp,λ ,q )

)

+∫ t

0(‖w‖

F ˙N4−2α+ λ−3

pp,λ ,q

+‖d‖F ˙N

4−2α+ λ−3p

p,λ ,q

)(‖b‖F ˙N

4+ λ−3p

p,λ ,q

+‖u‖F ˙N

4+ λ−3p

p,λ ,q

)

+ |Ω|‖w‖L 1([0,t),F ˙N

4−2α+ λ−3p

p,λ ,q )

.

Let

T = sup

t ∈ [0,T ∗),‖(w,d)‖L ∞([0,t),F ˙N

4−2α+ λ−3p

p,λ ,q )<

minµ,ν4C

. (4.3)

For t ∈ [0,T ), we have

‖w(t)‖F ˙N

4−2α+ λ−3p

p,λ ,q

+µ

4‖w‖

L 1([0,t),F ˙N4+ λ−3

pp,λ ,q )

+‖d(t)‖F ˙N

4−2α+ λ−3p

p,λ ,q

+ν

4‖d‖

L 1([0,t),F ˙N4+ λ−3

pp,λ ,q )

≤C‖(w0,d0)‖F ˙N

4−2α+ λ−3p

p,λ ,q

+∫ t

0C‖(w,d)‖

F ˙N4−2α+ λ−3

pp,λ ,q

(|Ω|+‖b‖F ˙N

4+ λ−3p

p,λ ,q

+‖u‖F ˙N

4+ λ−3p

p,λ ,q

) .


Gronwall’s Lemma gives

‖w(t)‖F ˙N

4−2α+ λ−3p

p,λ ,q

+µ

4‖w‖

L 1([0,t),F ˙N4+ λ−3

pp,λ ,q )

+‖d(t)‖F ˙N

4−2α+ λ−3p

p,λ ,q

+ν

4‖d‖

L 1([0,t),F ˙N4+ λ−3

pp,λ ,q )

≤C‖(w0,d0)‖F ˙N

4−2α+ λ−3p

p,λ ,q

exp∫ t

0C(|Ω|+‖u‖

F ˙N4+ λ−3

pp,λ ,q

+‖b‖F ˙N

4+ λ−3p

p,λ ,q

)

≤C‖(w0,d0)‖F ˙N

4−2α+ λ−3p

p,λ ,q

exp∫ ∞

0C(|Ω|+‖u‖

F ˙N4+ λ−3

pp,λ ,q

+‖b‖F ˙N

4+ λ−3p

p,λ ,q

).

Taking C0 sufficiently small in (2.4), we have

‖w(t)‖F ˙N

4−2α+ λ−3p

p,λ ,q

+µ

4‖w‖

L 1([0,t),F ˙N4+ λ−3

pp,λ ,q )

+‖d(t)‖F ˙N

4−2α+ λ−3p

p,λ ,q

+ν

4‖d‖

L 1([0,t),F ˙N4+ λ−3

pp,λ ,q )

<minµ,ν

8C,

which contradicts the definition (4.3). Then T = T ∗ and

‖(u,b)‖L 1([0,T ∗),F ˙N

4+ λ−3p

p,λ ,q )< ∞.

Therefore T ∗ = ∞. This completes the proof of Theorem 2.7.

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