Well posedness of magnetohydrodynamic equations in 3D mixed-norm Lebesgue space

Yongfang Liu; Chaosheng Zhu

doi:10.1515/math-2022-0014

Open Access Published by De Gruyter Open Access April 14, 2022

Well posedness of magnetohydrodynamic equations in 3D mixed-norm Lebesgue space

Yongfang Liu and Chaosheng Zhu

From the journal Open Mathematics

https://doi.org/10.1515/math-2022-0014

Abstract

In this paper, we introduce a new metric space called the mixed-norm Lebesgue space, which allows its norm decay to zero with different rates as ∣ x ∣ → ∞ in different spatial directions. Then we study the well posedness for the system of magnetohydrodynamic equations in 3D mixed-norm Lebesgue spaces. By using some fundamental analysis theories in mixed-norm Lebesgue space such as Young’s inequality, time decaying of solutions for heat equations, and the boundedness of the Helmholtz-Leray projection, we prove local well posedness and global well posedness of the solutions.

Keywords: magnetohydrodynamic equations; local well posedness; global well posedness

MSC 2010: Primary 35D30; Secondary 35Q30; 76W05; 76D06

1 Introduction

Unlike a usual Lebesgue space [1,2,3], the mixed-norm Lebesgue space allows its norm decay to zero with different rates as ∣ x ∣ → ∞ in different spatial directions [4]. On the basis of the mixed-norm Lebesgue space feature, we investigate the following magnetohydrodynamic (MHD) equations in R 3 [1]:

(1.1) ∂ t u − 1 Re Δ u + u ⋅ ∇ u − S ( ∇ × b ) × b + ∇ p ˜ = 0 , in R 3 × ( 0 , + ∞ ) , ∂ t b − ∇ × ( u × b ) + 1 Rm ∇ × ( ∇ × b ) = 0 , in R 3 × ( 0 , + ∞ ) , u ( x , 0 ) = u 0 ( x ) , b ( x , 0 ) = b 0 ( x ) , in R 3 ,

where u , b , u 0 , and b 0 satisfy

(1.2) div u ( x , t ) = div b ( x , t ) = 0 , div u 0 = div b 0 = 0 ,

S = M 2 ReRm , Re > 0 is the Reynolds number, Rm > 0 is the magnetic Reynolds number, M is the Hartman number. u : R 3 × ( 0 , + ∞ ) → R 3 denote the velocity of the fluid, b : R 3 × ( 0 , + ∞ ) → R 3 denote the magnetic field, and p ˜ = p ˜ ( x , t ) ∈ R denote the pressure.

The main purpose of this paper is to study the well posedness of the solution for the equations (1.1) in 3D mixed-norm Lebesgue spaces. First, we review some of the relevant work of the MHD equations. If b = 0 , then the equations (1.1) can be reduced to the incompressible Navier-Stokes equations:

∂ t u − 1 Re Δ u + u ⋅ ∇ u + ∇ p ˜ = 0 , in R 3 × ( 0 , + ∞ ) , u ( x , 0 ) = u 0 ( x ) , in R 3 .

The mathematical theory of the Navier-Stokes equation has been much studied in recent years. For example, Leray first introduced the weak solution [5], and later, Hopf gained the existence of global weak solutions with u 0 ∈ H s ( R N ) [6], and Fujita and Kato demonstrated that the well posedness of the Cauchy problem when u 0 ∈ H s ( R N ) ( N ≥ 2 ) [7]. In addition, there are many monographs that study the Navier-Stokes equation, for example, Temam [8], Lions [9], and Constantin and Foias [10]. The mild and self-similar solutions in R 3 are obtained in references [11,12]. Especially, Jia and Šverák derived that the homogeneous classical Cauchy problem with initial value has a global scale invariant solution, and the solution is smooth in positive time [13].

For the MHD system, the coupling between u and b makes the situation more complicated. Because it describes abundant natural phenomena, as well as physical importance and mathematical challenges, the MHD system has become the subject of the study by physicists and mathematicians. Duraut and Lions derived the global weak solution and the local strong solution of the initial boundary value problem of equations (1.1) and derived the existence of the global strong solution in the case of the small initial value [14]. Nevertheless, it is still a challenging open problem whether a unique local solution of the exists globally when the initial value is large. Furthermore, Sermange and Temam derived the regularity of the weak solution ( u , b ) ∈ L ∞ ( [ 0 , T ] ; H 1 ( R 3 ) ) [3]. Kozono derived the existence of classical solutions in bounded domain Ω ⊂ R 3 for equations (1.1) [15]. For the appropriately weak solution, He and Xin gained different local regularity results [16]. Cao and Wu obtained the global well posedness of the MHD system for any initial value in H 2 ( R 2 ) , but it needs a condition of mixed partial dissipation and additional magnetic diffusion in R 2 [2].

Since the coefficients in the equations have no critical influence on the subsequent analysis, we can simply take Re = Rm = S = 1 . Futhermore, we can obtain the following equations [1]:

( ∇ × b ) × b = ( b ⋅ ∇ ) b − b ⋅ ( ∇ b ) , ∇ × ∇ × b = ∇ div b − Δ b , ∇ × ( u × b ) = ( b ⋅ ∇ ) u − ( u ⋅ ∇ ) b + u div b − b div u .

Then, (1.1) can be rewritten as follows:

(1.3) ∂ t u − Δ u + ( u ⋅ ∇ ) u − ( b ⋅ ∇ ) b + b ⋅ ( ∇ b ) + ∇ p ˜ , in R 3 × ( 0 , + ∞ ) , ∂ t b − Δ b + ( u ⋅ ∇ ) b − ( b ⋅ ∇ ) u = 0 , in R 3 × ( 0 , + ∞ ) , u ( x , 0 ) = u 0 ( x ) , b ( x , 0 ) = b 0 ( x ) , in R 3 .

In 2D and 3D cases, Ai et al. applied the semi-Galerkin approximation method to obtain the existence of weak solutions [1]. In 2D case, Ai et al. proved the global existence of strong solutions to (1.1), the continuous dependence of initial-boundary data, and the uniqueness of weak-strong solutions. On this basis, they also proved the existence of a uniform attractor for (1.1). Inspired by [1,4], the main purpose of this paper is to study the well posedness of the solution to (1.1) in 3D mixed-norm Lebesgue spaces. Specifically, in Section 2, we state some mixed-norm legesgue spaces and related properties. In Section 3, we prove the existence, uniqueness, and stability of the solution.

2 Preliminary

We define the 3D mixed-norm Lebesgue spaces as follows [4]:

L p 1 p 2 p 3 ( R 3 ) = f f : R 3 → R , ∥ f ∥ L p 1 p 2 p 3 ( R 3 ) = ∫ ∫ ∫ ∣ f ∣ p 1 d x 1 p 2 p 1 d x 2 p 3 p 2 d x 3 1 p 3 < + ∞ ,

where p 1 , p 2 , p 3 ∈ [ 1 , + ∞ ) .

Let PL p 1 p 2 p 3 ( R 3 ) as follows:

PL p 1 p 2 p 3 ( R 3 ) = { f ∈ L p 1 p 2 p 3 ( R 3 ) : div f = 0 } .

All the spaces that appear in this paper are invariant with respect to the scaling f ( ⋅ ) → λ f ( λ ⋅ ) , λ > 0 . The mixed-norm space L p 1 p 2 p 3 ( R 3 ) is invariant if and only if 1 p 1 + 1 p 2 + 1 p 3 = 1 [4]. For given T ∈ ( 0 , ∞ ] , 1 p 1 + 1 p 2 + 1 p 3 = 1 , 1 q 1 + 1 q 2 + 1 q 3 = δ ∈ ( 0 , 1 ) , p k ∈ ( 1 , + ∞ ) , q k ∈ ( p k , + ∞ ) , k = 1 , 2 , 3 , and f : R 3 × [ 0 , ∞ ) → R 3 denote the measurable vector field functions, we denote X p , q , T as follows [4]:

X p , q , T = f : g ( x , t ) ≔ t 1 − δ 2 f ( x , t ) , g ˜ ( x , t ) ≔ t 1 2 D x f ( x , t ) , ( x , t ) ∈ R 3 × ( 0 , T ) ,

then

g ∈ C ( [ 0 , T ] , PL q 1 q 2 q 3 ( R 3 ) ) , g ˜ ∈ C ( [ 0 , T ] , PL p 1 p 2 p 3 ( R 3 ) ) .

Moreover, g ( x , 0 ) = 0 , g ˜ ( x , 0 ) = 0 , and the norm

‖ f ‖ χ p , q , T = sup t ∈ ( 0 , T ) [ ‖ g ( ⋅ , t ) ‖ L q 1 q 2 q 3 ( R 3 ) + ‖ g ˜ ‖ L p 1 p 2 p 3 ( R 3 ) ] < + ∞ .

We denote Y p , T as follows [4]:

Y p , T = f : f ∈ C ( [ 0 , T ] , PL p 1 p 2 p 3 ( R 3 ) ) , t 1 2 D x f ∈ C ( [ 0 , T ] , PL p 1 p 2 p 3 ( R 3 ) ) ,

and the norm

‖ f ‖ Y p , T = sup t ∈ ( 0 , T ) [ ‖ f ( t ) ‖ L p 1 p 2 p 3 ( R 3 ) + t 1 / 2 ∥ D x f ( t ) ∥ L p 1 p 2 p 3 ( R 3 ) ] < + ∞ .

We state the following result on Young’s inequality in mixed-norm Lebesgue spaces [4].

Lemma 2.1

[4] Let p k , r k , and q k be given numbers in [ 1 , + ∞ ] that satisfy

1 p k + 1 = 1 q k + 1 r k , k = 1 , 2 , 3 .

Then,

(2.1) ∥ f ∗ g ∥ L p 1 p 2 p 3 ( R 3 ) = ∥ f ∥ L q 1 q 2 q 3 ( R 3 ) ∥ g ∥ L r 1 r 2 r 3 ( R 3 ) ,

for all f ∈ L q 1 q 2 q 3 ( R 3 ) and g ∈ L r 1 r 2 r 3 ( R 3 ) .

We state the following results on heat equations in mixed-norm Lebesgue spaces. First, we see the Cauchy problem for the heat equations:

(2.2) u t − Δ u = 0 , in R 3 × ( 0 , + ∞ ) , b t − Δ b = 0 , in R 3 × ( 0 , + ∞ ) , u ( x , 0 ) = u 0 ( x ) , b ( x , 0 ) = b 0 ( x ) , in R 3 .

We see that (2.2) can be written as follows:

(2.3) U t − Δ U = 0 , in R 3 × ( 0 , + ∞ ) , U ( x , 0 ) = U 0 ( x ) , in R 3 ,

where U = u b , U 0 = u 0 b 0 . It is well known that a solution of (2.3) is

(2.4) U ( x , t ) = e Δ t U 0 ( x ) = ( G t * U 0 ) ( x , t ) , ( x , t ) ∈ R 3 × ( 0 , + ∞ ) ,

where

G t ( x ) = 1 ( 4 π t ) n 2 e − ∣ x ∣ 2 4 t , ( x , t ) ∈ R 3 × ( 0 , + ∞ ) .

Next, we state the following fundamental results of the solution of the heat equation in the mixed-norm Lebesgue space.

Lemma 2.2

[4] Let 1 ≤ p k ≤ q k ≤ + ∞ . There exists a positive constant N depending only on p 1 , p 2 , p 3 , q 1 , q 2 , and q 3 such that for every solution U ( x , t ) = e Δ t U 0 ( x ) defined in (2.4) of the Cauchy problem (2.3) with U 0 ∈ L q 1 q 2 q 3 ( R 3 ) , then for t > 0

(2.5) ∥ U ( x , t ) ∥ L p 1 p 2 p 3 ( R 3 ) ≤ N t − 1 2 ∑ k = 1 n = 3 1 q k − 1 p k ∥ U 0 ∥ L q 1 q 2 q 3 ( R 3 ) ,

(2.6) ∥ D x U ( x , t ) ∥ L p 1 p 2 p 3 ( R 3 ) ≤ N t − 1 2 − 1 2 ∑ k = 1 n = 3 1 q k − 1 p k ∥ U 0 ∥ L q 1 q 2 q 3 ( R 3 ) .

Lemma 2.3

[4] Let p 1 , p 2 , p 3 ∈ ( 1 , ∞ ) , and U 0 ∈ L p 1 p 2 p 3 ( R 3 ) . Let U ( x , t ) = e Δ t U 0 ( x ) be the solution of the heat equation (2.3) defined in (2.4). Then, U ∈ C ( [ 0 , ∞ ) , L p 1 p 2 p 3 ( R 3 ) ) and

(2.7) lim t → 0 + ∥ U ( x , t ) − U 0 ∥ L p 1 p 2 p 3 ( R 3 ) = 0 .

The following consequence illustrates the boundedness of Helmholtz-Leray projection P in mixed-norm Lebesgue spaces.

Lemma 2.4

[4] Let P = Id − ∇ Δ − 1 ∇ ⋅ be the Helmholtz-Leray projection onto the divergence-free vector fields. Let p 1 , p 2 , p 3 ∈ ( 1 , ∞ ) . Then, one has

(2.8) ∥ P ( f ) ∥ L p 1 p 2 p 3 ( R 3 ) ≤ N ∥ f ∥ L p 1 p 2 p 3 ( R 3 ) ,

for all f ∈ ( L p 1 p 2 p 3 ( R 3 ) ) 3 , where N = N ( p 1 , p 2 , p 3 ) is a positive constant.

3 MHD equations in 3D mixed-norm Lebesgue space

We apply P on the system (1.3), and then (1.3) can be expressed as follows:

(3.1) U t + A U + F ( U , U ) = 0 , in R 3 × ( 0 , + ∞ ) , U ( x , 0 ) = U 0 ( x ) , in R 3 ,

as P ∇ p ˜ = 0 , where

U = u b , U 0 = u 0 b 0 , A = − P Δ 0 0 − P Δ , F ( U , U ) = P ( u ⋅ ∇ ) u − P ( b ⋅ ∇ ) b + P b ⋅ ( ∇ b ) P ( u ⋅ ∇ ) b − P ( b ⋅ ∇ ) u .

By the Duhamel’s principle, the system (3.1) can be transformed into the following integral equations:

(3.2) U = U 1 + G ( U , U ) ,

where

U 1 = e − A t U 0 ( x ) = e Δ t u 0 ( x ) e Δ t b 0 ( x )

and

(3.3) G ( U , U ) = − ∫ 0 t e − ( t − s ) A F ( U , U ) d s

= − ∫ 0 t e ( t − s ) Δ ( P ( u ⋅ ∇ ) u − P ( b ⋅ ∇ ) b + P b ⋅ ( ∇ b ) ) d s − ∫ 0 t e ( t − s ) Δ ( P ( u ⋅ ∇ ) b − P ( b ⋅ ∇ ) u ) d s .

The main results in the paper are as follows.

Theorem 3.1

Let p k ∈ ( 1 , + ∞ ) , q k ∈ [ p k , + ∞ ) , k = 1 , 2 , 3 , and

1 p 1 + 1 p 2 + 1 p 3 = 1 , 1 q 1 + 1 q 2 + 1 q 3 = δ ∈ ( 0 , 1 ) .

Then, there are a sufficiently small constant λ 0 > 0 and a number N > 0 , depending on p 1 , p 2 , p 3 , q 1 , q 2 , and q 3 , such that the following results hold.

For all U 0 ∈ L p 1 p 2 p 3 ( R 3 ) with div U 0 = 0 , if ∥ U 0 ∥ L p 1 p 2 p 3 ( R 3 ) ≤ λ 0 , then (3.1) has an unique global time solution U ∈ X p , q , ∞ ∩ Y p , ∞ with
‖ U ‖ χ p , q , ∞ ≤ N ∥ U 0 ∥ L p 1 p 2 p 3 ( R 3 ) , ‖ U ‖ Y p , ∞ ≤ N ( ∥ U 0 ∥ L p 1 p 2 p 3 ( R 3 ) + ∥ U 0 ∥ L p 1 p 2 p 3 ( R 3 ) 2 ) .
For all U 0 ∈ L p 1 p 2 p 3 ( R 3 ) with div U 0 = 0 , there is a sufficiently small T 0 > 0 depending on p 1 , p 2 , p 3 , q 1 , q 2 , and q 3 such that (3.1) has an unique local time solution U ∈ X p , q , T 0 ∩ Y p , T 0 with
‖ U ‖ χ p , q , T 0 ≤ N ∥ U 0 ∥ L p 1 p 2 p 3 ( R 3 ) , ‖ U ‖ Y p , T 0 ≤ N ( ∥ U 0 ∥ L p 1 p 2 p 3 ( R 3 ) + ∥ U 0 ∥ L p 1 p 2 p 3 ( R 3 ) 2 ) .

To prove Theorem 3.1, we need the following lemmas.

Lemma 3.1

[4] Let p 1 , p 2 , p 3 , q 1 , q 2 , and q 3 be given numbers and 1 < p k ≤ q k < ∞ . Also, let σ ≥ 0 be defined by σ = ∑ k = 1 n = 3 1 p k − 1 q k .

There exists a number N depending only on p 1 , p 2 , p 3 , q 1 , q 2 , and q 3 such that
(3.4) ∥ e − A t P f ∥ L q 1 q 1 q 3 ( R 3 ) ≤ N t − σ 2 ‖ f ‖ L p 1 P 2 p 3 ( R 3 ) ,

(3.5) ∥ D x e − A t P f ∥ L q 1 q 1 q 3 ( R 3 ) ≤ N t − 1 2 ( 1 + σ ) ‖ f ‖ L p 1 p 2 p 3 ( R 3 ) ,
for all f ∈ L p 1 p 2 p 3 ( R 3 ) 3 .
For all f ∈ L p 1 p 2 p 3 ( R 3 ) 3 , the following assertions hold: if σ > 0 , then
(3.6) lim t → 0 + t σ 2 ∥ e − A t P f ∥ L q 1 q 1 q 3 ( R 3 ) = 0 ,

(3.7) lim t → 0 + ∥ [ e − A t P f ] − P f ∥ L p 1 P 1 p 3 ( R 3 ) = 0 ,

(3.8) lim t → 0 + t − 1 2 ( 1 + σ ) ∥ D x e − A t P f ∥ L q 1 q 1 q 3 ( R 3 ) = 0 .

Lemma 3.2

Let p k ∈ ( 1 , ∞ ) , α k , β k , γ k ∈ ( 0 , 1 ] be given numbers satisfying γ k ≤ α k + β k < p k , k = 1 , 2 , 3 . Let

α = α 1 p 1 + α 2 p 2 + α 2 p 2 , β = β 1 p 1 + β 2 p 2 + β 2 p 2 , γ = γ 1 p 1 + γ 2 p 2 + γ 2 p 2 .

Then,

(3.9) ‖ G ( U , U ) ‖ L p 1 γ 1 p 2 γ 2 p 3 γ 3 ( R 3 ) ≤ N ∫ 0 t ( t − s ) − α + β − γ 2 ‖ u ‖ L p 1 α 1 p 2 α 2 p 3 α 3 ( R 3 ) ∥ D x u ∥ L p 1 β 1 p 2 β 2 p 3 β 3 ( R 3 ) + 2 ‖ b ‖ L p 1 α 1 p 2 α 2 p 3 α 3 ( R 3 ) ∥ D x b ∥ L p 1 β 1 p 2 β 2 p 3 β 3 ( R 3 ) + ‖ u ‖ L p 1 α 1 p 2 α 2 p 3 α 3 ( R 3 ) ∥ D x b ∥ L p 1 β 1 p 2 β 2 p 3 β 3 ( R 3 ) + ‖ b ‖ L p 1 α 1 p 2 α 2 p 3 α 3 ( R 3 ) ∥ D x u ∥ L p 1 β 1 p 2 β 2 p 3 β 3 ( R 3 ) d s ,

(3.10) ‖ D x G ( U , U ) ‖ L p 1 γ 1 p 2 γ 2 p 3 γ 3 ( R 3 ) ≤ N ∫ 0 t ( t − s ) − 1 + α + β − γ 2 ‖ u ‖ L p 1 α 1 p 2 α 2 p 3 α 3 ( R 3 ) ∥ D x u ∥ L p 1 β 1 p 2 β 2 p 3 β 3 ( R 3 ) + 2 ‖ b ‖ L p 1 α 1 p 2 α 2 p 3 α 3 ( R 3 ) ∥ D x b ∥ L p 1 β 1 p 2 β 2 p 3 β 3 ( R 3 ) + ‖ u ‖ L p 1 α 1 p 2 α 2 p 3 α 3 ( R 3 ) ∥ D x b ∥ L p 1 β 1 p 2 β 2 p 3 β 3 ( R 3 ) + ‖ b ‖ L p 1 α 1 p 2 α 2 p 3 α 3 ( R 3 ) ∥ D x u ∥ L p 1 β 1 p 2 β 2 p 3 β 3 ( R 3 ) d s ,

where N > 0 is a constant depending on p k , α k , β k , γ k , k = 1 , 2 , 3 .

Proof

We first prove (3.9) in Lemma 3.2. For γ k ≤ α k + β k < p k , we can gain

p k γ k ≥ p k α k + β k , ∑ k = 1 n = 3 α k + β k p k − γ k p k = α + β − γ .

By (3.4), we can obtain

(3.11) ‖ G ( U , U ) ‖ L p 1 γ 1 p 2 γ 2 p 3 γ 3 ( R 3 ) ≤ N ∫ 0 t ( t − s ) − α + β − γ 2 ‖ F 1 ( U , U ) ‖ L p 1 α 1 + β 1 p 2 α 2 + β 2 p 3 α 3 + β 13 ( R 3 ) d s ,

where F 1 ( U , U ) = ( u ⋅ ∇ ) u − ( b ⋅ ∇ ) b + b ⋅ ( ∇ b ) ( u ⋅ ∇ ) b − ( b ⋅ ∇ ) u .

By using Hölder’s inequality repeatedly, we can find that

‖ ( u ⋅ ∇ ) b ‖ L p 1 α 1 + β 1 p 2 α 2 + β 2 p 3 α 3 + β 3 ( R 3 ) = ∫ ∫ ∫ ∣ u ⋅ ( ∇ b ) ∣ p 1 α 1 + β 1 d x 1 p 2 ( α 1 + β 1 ) p 1 ( α 2 + β 2 ) d x 2 p 3 ( α 2 + β 2 ) p 2 ( α 3 + β 3 ) d x 3 ( α 3 + β 3 ) p 3 ≤ ∫ ∫ ∫ ∣ u ∣ p 1 α 1 d x 1 α 1 p 1 ∫ ∣ D x b ∣ p 1 β 1 d x 1 β 1 p 1 p 2 ( α 2 + β 2 ) d x 2 p 3 ( α 2 + β 2 ) p 2 ( α 3 + β 3 ) d x 3 ( α 3 + β 3 ) p 3 ≤ ∫ ∫ ∫ ∣ u ∣ p 1 α 1 d x 1 p 2 α 1 p 1 α 2 d x 2 α 2 p 2 ∫ ∫ ∣ D x b ∣ p 1 β 1 d x 1 p 2 β 1 p 1 β 2 d x 2 β 2 p 2 p 3 ( α 3 + β 3 ) d x 3 ( α 3 + β 3 ) p 3

≤ ∫ ∫ ∫ ∣ u ∣ p 1 α 1 d x 1 p 2 α 1 p 1 α 2 d x 2 p 3 α 2 p 2 α 3 d x 3 α 3 p 3 ∫ ∫ ∫ ∣ D x b ∣ p 1 β 1 d x 1 p 2 β 1 p 1 β 2 d x 2 p 3 β 2 p 2 β 3 d x 3 β 3 p 3 = ‖ u ‖ L p 1 α 1 p 2 α 2 p 3 α 3 ( R 3 ) ‖ D x b ‖ L p 1 β 1 p 2 β 2 p 3 β 3 ( R 3 ) .

Then,

‖ F 1 ( U , U ) ‖ L p 1 α 1 + β 1 p 2 α 2 + β 2 p 3 α 3 + β 13 ( R 3 ) ≤ ∥ ∣ ( u ⋅ ∇ ) u ∣ + ∣ ( b ⋅ ∇ ) b ∣ + ∣ b ⋅ ( ∇ b ) ∣ + ∣ ( u ⋅ ∇ ) b ∣ + ∣ ( b ⋅ ∇ ) u ∣ ∥ L p 1 α 1 + β 1 p 2 α 2 + β 2 p 3 α 3 + β 13 ( R 3 ) ≤ ∥ ( u ⋅ ∇ ) u ∥ L p 1 α 1 + β 1 p 2 α 2 + β 2 p 3 α 3 + β 13 ( R 3 ) + ∥ ( b ⋅ ∇ ) b ∥ L p 1 α 1 + β 1 p 2 α 2 + β 2 p 3 α 3 + β 13 ( R 3 ) + ∥ b ⋅ ( ∇ b ) ∥ L p 1 α 1 + β 1 p 2 α 2 + β 2 p 3 α 3 + β 13 ( R 3 ) + ∥ ( u ⋅ ∇ ) b ∥ L p 1 α 1 + β 1 p 2 α 2 + β 2 p 3 α 3 + β 13 ( R 3 ) + ∥ ( b ⋅ ∇ ) u ∥ L p 1 α 1 + β 1 p 2 α 2 + β 2 p 3 α 3 + β 13 ( R 3 ) ≤ ‖ u ‖ L p 1 α 1 p 2 α 2 p 3 α 3 ( R 3 ) ‖ D x u ‖ L p 1 β 1 p 2 β 2 p 3 β 3 ( R 3 ) + 2 ‖ b ‖ L p 1 α 1 p 2 α 2 p 3 α 3 ( R 3 ) ‖ D x b ‖ L p 1 β 1 p 2 β 2 p 3 β 3 ( R 3 ) + ‖ u ‖ L p 1 α 1 p 2 α 2 p 3 α 3 ( R 3 ) ‖ D x b ‖ L p 1 β 1 p 2 β 2 p 3 β 3 ( R 3 ) + ‖ b ‖ L p 1 α 1 p 2 α 2 p 3 α 3 ( R 3 ) ‖ D x u ‖ L p 1 β 1 p 2 β 2 p 3 β 3 ( R 3 ) .

By substituting the aforementioned formula into (3.11), we can obtain (3.9). Similarly, (3.10) can be proved by (3.5).□

Lemma 3.3

[4] Let X be a Banach space with norm ‖ ⋅ ‖ X . Let G : X × X → X be a bilinear map such that there is N 0 > 0 so that

‖ G ( U , V ) ‖ X ≤ N 0 ‖ U ‖ X ‖ V ‖ X , ∀ U , V ∈ X .

Then, for all U 1 ∈ X with 4 N 0 ‖ U 1 ‖ X < 1 , the equation

U = U 1 + G ( U , U )

has an unique solution U with

‖ U ‖ X ≤ 2 ‖ U 1 ‖ X .

Proof of Theorem 3.1

We now prove (i). First, we start from the proof that U ∈ χ p , q , ∞ . From Lemma 2.2 and σ = ∑ k = 1 n = 3 1 p k − 1 q k = 1 − δ , we have

∥ U 1 ∥ L q 1 q 1 q 3 ( R 3 ) ≤ N 1 t − 1 − δ 2 ‖ U 0 ‖ L p 1 P 2 p 3 ( R 3 ) , ∥ D x U 1 ∥ L q 1 q 2 q 3 ( R 3 ) ≤ N 1 t − 1 2 ‖ U 0 ‖ L p 1 p 2 p 3 ( R 3 ) ,

where N 1 > 0 is a constant depending on p 1 , p 2 , p 3 , q 1