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

  • Yongfang Liu and Chaosheng Zhu EMAIL logo
Published/Copyright: April 14, 2022
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Open Mathematics
From the journal Open Mathematics Volume 20 Issue 1

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.

  1. 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 ) .

  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 .

  1. 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 .

  2. 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 , q 2 , and q 3 .

Furthermore, according to Lemma 3.1, we know that when t → 0 , t − 1 − σ 2 e − A t P → 0 , and when t = 0 , t − 1 − σ 2 U 1 = 0 . Hence,

t − 1 − σ 2 e − A t P : L p 1 p 2 p 3 ( R 3 ) → P L q 1 q 2 q 3 ( R 3 )

is uniformly bounded. Similarly, when t → 0 , t − 1 2 D x e − A t P → 0 , and when t = 0 , t − 1 2 D x U 1 = 0 . Hence,

t − 1 − σ 2 D x e − A t P : L p 1 p 2 p 3 ( R 3 ) → P L p 1 p 2 p 3 ( R 3 )

is also uniformly bounded. Hence, we have U 0 ∈ χ p , q , ∞ and

(3.12) ‖ U 1 ‖ χ p , q , ∞ ≤ N 1 ∥ U 0 ∥ L p 1 p 2 p 3 ( R 3 ) .

Now let’s prove that bilinear G : χ p , q , ∞ × χ p , q , ∞ → χ p , q , ∞ is bounded.

Let β k = 1 , γ k = α k = p k q k ∈ ( 0 , 1 ] , we obtain p k γ k = q k ,

− α + β − γ 2 = − 1 2 ∑ k = 1 n = 3 α k p k + β k p k − γ k p k = − 1 2 ∑ k = 1 n = 3 1 q k + 1 p k − 1 q k = − 1 2 ∑ k = 1 n = 3 1 p k = − 1 2 .

By using 1 p 1 + 1 p 2 + 1 p 3 = 1 , 1 q 1 + 1 q 2 + 1 q 3 = δ ∈ ( 0 , 1 ) , (3.9) and the definition of X p , q , T , and applying (3.3), we can obtain

‖ G ( U , U ) ‖ L q 1 q 2 q 3 ( R 3 ) ≤ N ∫ 0 t ( t − s ) − 1 2 ( ‖ u ‖ L q 1 q 2 q 3 ( R 3 ) ∥ D x u ∥ L p 1 p 2 p 3 ( R 3 ) + 2 ‖ b ‖ L q 1 q 2 q 3 ( R 3 ) ∥ D x b ∥ L p 1 p 2 p 3 ( R 3 ) + ‖ u ‖ L q 1 q 2 q 3 ( R 3 ) ∥ D x b ∥ L p 1 p 2 p 3 ( R 3 ) + ‖ b ‖ L q 1 q 2 q 3 ( R 3 ) ∥ D x u ∥ L p 1 p 2 p 3 ( R 3 ) ) d s ≤ N ∫ 0 t ( t − s ) − 1 2 s 1 − δ 2 ‖ u ‖ L q 1 q 2 q 3 ( R 3 ) s 1 2 ∥ D x u ∥ L p 1 p 2 p 3 ( R 3 ) s − 1 − δ 2 s − 1 2 + 2 s 1 − δ 2 ‖ b ‖ L q 1 q 2 q 3 ( R 3 ) s 1 2 ∥ D x b ∥ L p 1 p 2 p 3 ( R 3 ) s − 1 − δ 2 s − 1 2 + s 1 − δ 2 ‖ u ‖ L q 1 q 2 q 3 ( R 3 ) s 1 2 × ∥ D x b ∥ L p 1 p 2 p 3 ( R 3 ) s − 1 − δ 2 s − 1 2 + s 1 − δ 2 ‖ b ‖ L q 1 q 2 q 3 ( R 3 ) s 1 2 ∥ D x u ∥ L p 1 p 2 p 3 ( R 3 ) s − 1 − δ 2 s − 1 2 d s ≤ N ( ∥ u ∥ χ p , q , ∞ 2 + ∥ b ∥ χ p , q , ∞ 2 + 2 ‖ u ‖ χ p , q , ∞ ‖ b ‖ χ p , q , ∞ ) ∫ 0 t ( t − s ) − 1 2 s − 1 + δ 2 d s ,

and

∫ 0 t ( t − s ) − 1 2 s − 1 + δ 2 d s = ∫ 0 t 2 ( t − s ) − 1 2 s − 1 + δ 2 d s + ∫ t 2 t ( t − s ) − 1 2 s − 1 + δ 2 d s ≤ t 2 − 1 2 ∫ 0 t 2 s − 1 + δ 2 d s + t 2 − 1 + δ 2 ∫ t 2 t ( t − s ) − 1 2 d s ≤ N t 2 − 1 2 t 2 δ 2 + N t 2 − 1 + δ 2 t 2 1 2 ≤ N t − 1 − δ 2 .

So, we have

(3.13) ‖ G ( U , U ) ‖ L q 1 q 2 q 3 ( R 3 ) ≤ N t − 1 − δ 2 ( ∥ u ∥ χ p , q , ∞ 2 + 2 ∥ b ∥ χ p , q , ∞ 2 + 2 ‖ u ‖ χ p , q , ∞ ‖ b ‖ χ p , q , ∞ ) .

Similarly, let β k = γ k = 1 , α k = p k q k ∈ ( 0 , 1 ] , and p k γ k = q k ,

− 1 + α + β − γ 2 = − 1 2 1 + ∑ k = 1 n = 3 α k p k + β k p k − γ k p k = − 1 2 1 + ∑ k = 1 n = 3 1 q k + 1 p k − 1 p k = − 1 2 1 + ∑ k = 1 n = 3 1 q k = − 1 + δ 2 .

By using 1 p 1 + 1 p 2 + 1 p 3 = 1 , 1 q 1 + 1 q 2 + 1 q 3 = δ ∈ ( 0 , 1 ) , and (3.10), we also obtain

‖ D x G ( U , U ) ‖ L p 1 p 2 p 3 ( R 3 ) ≤ N ∫ 0 t ( t − s ) − 1 + δ 2 ( ‖ u ‖ L q 1 q 2 q 3 ( R 3 ) ∥ D x u ∥ L p 1 p 2 p 3 ( R 3 ) + 2 ‖ b ‖ L q 1 q 2 q 3 ( R 3 ) ∥ D x b ∥ L p 1 p 2 p 3 ( R 3 ) + ‖ u ‖ L q 1 q 2 q 3 ( R 3 ) ∥ D x b ∥ L p 1 p 2 p 3 ( R 3 ) + ‖ b ‖ L q 1 q 2 q 3 ( R 3 ) ∥ D x u ∥ L p 1 p 2 p 3 ( R 3 ) ) d s ≤ N ( ∥ u ∥ χ p , q , ∞ 2 + ∥ b ∥ χ p , q , ∞ 2 + 2 ‖ u ‖ χ p , q , ∞ ‖ b ‖ χ p , q , ∞ ) ∫ 0 t ( t − s ) − 1 + δ 2 s − 1 + δ 2 d s ,

and

∫ 0 t ( t − s ) − 1 + δ 2 s − 1 + δ 2 d s = ∫ 0 t 2 ( t − s ) − 1 + δ 2 s − 1 + δ 2 d s + ∫ t 2 t ( t − s ) − 1 + δ 2 s − 1 + δ 2 d s ≤ t 2 − 1 + δ 2 ∫ 0 t 2 s − 1 + δ 2 d s + t 2 − 1 + δ 2 ∫ t 2 t ( t − s ) − 1 + δ 2 d s ≤ N t 2 − 1 + δ 2 t 2 δ 2 + N t 2 − 1 + δ 2 t 2 1 + δ 2 ≤ N t − 1 2 .

So, we have

(3.14) ‖ D x G ( U , U ) ‖ L p 1 p 2 p 3 ( R 3 ) ≤ N t − 1 2 ( ∥ u ∥ χ p , q , ∞ 2 + 2 ∥ b ∥ χ p , q , ∞ 2 + 2 ‖ u ‖ χ p , q , ∞ ‖ b ‖ χ p , q , ∞ ) .

From the estimates (3.13), (3.14), and Lemma 3.1, we can prove that t 1 − δ 2 G ( U , U ) : [ 0 , ∞ ) → P L q 1 q 2 q 3 ( R 3 ) is continuous and when t → 0 , t 1 − δ 2 G ( U , U ) → 0 . Similarly, we can also prove that t 1 2 D x G ( U , U ) : [ 0 , ∞ ) → P L p 1 p 2 p 3 ( R 3 ) is continuous and when t → 0 , t 1 2 D x G ( U , U ) → 0 . Therefore, we obtain G ( U , U ) ∈ χ p , q , ∞ and for all U ∈ χ p , q , ∞ :

(3.15) ‖ G ( U , U ) ‖ χ p , q , ∞ ≤ N ( ∥ u ∥ χ p , q , ∞ 2 + 2 ∥ b ∥ χ p , q , ∞ 2 + 2 ‖ u ‖ χ p , q , ∞ ‖ b ‖ χ p , q , ∞ ) ≤ N ‖ U ‖ χ p , q , ∞ ‖ U ‖ χ p , q , ∞ ≤ N 2 ∥ U ∥ χ p , q , ∞ 2 , ∀ U ∈ χ p , q , ∞ ,

where N 2 is a constant depending on p 1 , p 2 , p 3 , q 1 , q 2 , and q 3 . That is, the bilinear G : χ p , q , ∞ × χ p , q , ∞ → χ p , q , ∞ is bounded.

Next, let us choose a sufficiently small constant λ 0 so that

4 N 1 N 2 λ 0 < 1 ,

where N 1 is defined in (3.12) and N 2 is defined in (3.15). If ∥ U 0 ∥ L p 1 p 2 p 3 ( R 3 ) ≤ λ 0 , then it can be obtained by (3.12)

4 N 2 ‖ U 1 ‖ χ p , q , ∞ ≤ 4 N 1 N 2 ∥ U 0 ∥ L p 1 p 2 p 3 ( R 3 ) ≤ 4 N 1 N 2 λ 0 < 1 .

By this and Lemma 3.3, we can gain a unique time solution U ∈ χ p , q , ∞ of (3.2) such that

(3.16) ‖ U ‖ χ p , q , ∞ ≤ 2 ∥ U 1 ∥ χ p , q , ∞ ≤ 2 N 1 ∥ U 0 ∥ L p 1 p 2 p 3 ( R 3 ) .

Now, we need to prove that U ∈ Y p , ∞ . As U = U 1 + G ( U , U ) , we have

(3.17) ‖ U ‖ L p 1 p 2 p 3 ( R 3 ) ≤ ‖ U 1 ‖ L p 1 p 2 p 3 ( R 3 ) + ‖ G ( U , U ) ‖ L p 1 p 2 p 3 ( R 3 ) ,

(3.18) ‖ D x U ‖ L p 1 p 2 p 3 ( R 3 ) ≤ ‖ D x U 1 ‖ L p 1 p 2 p 3 ( R 3 ) + ‖ D x G ( U , U ) ‖ L p 1 p 2 p 3 ( R 3 ) .

Then, by Lemma 3.1, we see that

(3.19) ∥ U 1 ∥ L p 1 p 1 p 3 ( R 3 ) ≤ N ‖ U 0 ‖ L p 1 p 2 p 3 ( R 3 ) ,

(3.20) ∥ D x U 1 ∥ L p 1 p 2 p 3 ( R 3 ) ≤ N t − 1 2 ‖ U 0 ‖ L p 1 p 2 p 3 ( R 3 ) .

Let β k = γ k = 1 , α k = p k q k ∈ ( 0 , 1 ] , and p k γ k = p k ,

− α + β − γ 2 = − 1 2 ∑ k = 1 n = 3 α k p k + β k p k − γ k p k = − 1 2 ∑ k = 1 n = 3 1 q k + 1 p k − 1 p k = − 1 2 ∑ k = 1 n = 3 1 q k = − δ 2 .

By using (3.9), we can infer that

‖ G ( U , U ) ‖ L p 1 p 2 p 3 ( R 3 ) ≤ N ∫ 0 t ( t − s ) − δ 2 ( ‖ u ‖ L q 1 q 2 q 3 ( R 3 ) ∥ D x u ∥ L p 1 p 2 p 3 ( R 3 ) + 2 ‖ b ‖ L q 1 q 2 q 3 ( R 3 ) ∥ D x b ∥ L p 1 p 2 p 3 ( R 3 ) + ‖ u ‖ L q 1 q 2 q 3 ( R 3 ) ∥ D x b ∥ L p 1 p 2 p 3 ( R 3 ) + ‖ b ‖ L q 1 q 2 q 3 ( R 3 ) ∥ D x u ∥ L p 1 p 2 p 3 ( R 3 ) ) d s ≤ N ( ∥ u ∥ χ p , q , ∞ 2 + ∥ b ∥ χ p , q , ∞ 2 + 2 ‖ u ‖ χ p , q , ∞ ‖ b ‖ χ p , q , ∞ ) ∫ 0 t ( t − s ) − δ 2 s − 1 + δ 2 d s .

Since

∫ 0 t ( t − s ) − δ 2 s − 1 + δ 2 d s = ∫ 0 t 2 ( t − s ) − δ 2 s − 1 + δ 2 d s + ∫ t 2 t ( t − s ) − δ 2 s − 1 + δ 2 d s ≤ t 2 − δ 2 ∫ 0 t 2 s − 1 + δ 2 d s + t 2 − 1 + δ 2 ∫ t 2 t ( t − s ) − δ 2 d s ≤ N 1 t 2 − δ 2 t 2 δ 2 + N 2 t 2 − 1 + δ 2 t 2 − δ 2 + 1 ≤ N ,

by using (3.16), we obtain that

(3.21) ‖ G ( U , U ) ‖ L p 1 p 2 p 3 ( R 3 ) ≤ N ( ∥ u ∥ χ p , q , ∞ 2 + 2 ∥ b ∥ χ p , q , ∞ 2 + 2 ‖ u ‖ χ p , q , ∞ ‖ b ‖ χ p , q , ∞ ) ≤ N ∥ U ∥ χ p , q , ∞ 2 ≤ N ∥ U 0 ∥ L p 1 p 2 p 3 ( R 3 ) 2 .

Similarly, by using (3.14) and (3.16), we obtain that

(3.22) ‖ D x G ( U , U ) ‖ L p 1 p 2 p 3 ( R 3 ) ≤ N t − 1 2 ( ∥ u ∥ χ p , q , ∞ 2 + 2 ∥ b ∥ χ p , q , ∞ 2 + 2 ‖ u ‖ χ p , q , ∞ ‖ b ‖ χ p , q , ∞ ) ≤ N t − 1 2 ∥ U ∥ χ p , q , ∞ 2 ≤ N t − 1 2 ∥ U 0 ∥ L p 1 p 2 p 3 ( R 3 ) 2 .

From estimates (3.17), (3.19), and (3.21), we see that

‖ U ‖ L p 1 p 2 p 3 ( R 3 ) ≤ N ( ‖ U 0 ‖ L p 1 p 2 p 3 ( R 3 ) + ‖ U 0 ‖ L p 1 p 2 p 3 ( R 3 ) 2 ) .

From the estimates (3.18), (3.20), and (3.22), we see that

‖ D x U ‖ L p 1 p 2 p 3 ( R 3 ) ≤ N t − 1 2 ( ‖ U 0 ‖ L p 1 p 2 p 3 ( R 3 ) + ‖ U 0 ‖ L p 1 p 2 p 3 ( R 3 ) 2 ) .

It is known that ‖ U 0 ‖ L p 1 p 2 p 3 ( R 3 ) is sufficiently small, we can gain

‖ U ‖ Y p , ∞ ≤ N 0 ( ‖ U 0 ‖ L p 1 p 2 p 3 ( R 3 ) + ‖ U 0 ‖ L p 1 p 2 p 3 ( R 3 ) 2 ) .

Now, we have to prove (ii). Similar to the proof of (3.12), we found that U 1 ∈ χ p , q , ∞ , χ p , q , ∞ is continuous, and when t → 0 , t 1 − δ 2 U 1 → 0 , t 1 2 D x U 1 → 0 . There is a sufficiently small constant T 0 > 0 depending on p 1 , p 2 , p 3 , q 1 , q 2 , q 3 , and U 0 such that

‖ U 1 ‖ χ p , q , T 0 ≤ λ 0 .

Furthermore, similar to the proof of (3.15), we also found that the bilinear G : χ p , q , T 0 × χ p , q , T 0 → χ p , q , T 0 is bounded and

‖ G ( U , U ) ‖ χ p , q , T 0 ≤ N ‖ U ‖ χ p , q , T 0 ‖ U ‖ χ p , q , T 0 , ∀ U ∈ χ p , q , T 0 .

Then, by Lemma 3.3, we can obtain a unique local time solution U ∈ χ p , q , T 0 of (3.2) with

‖ U ‖ χ p , q , T 0 ≤ 2 N 1 ∥ U 0 ∥ L p 1 p 2 p 3 ( R 3 ) .

As in proof (i), we can gain U ∈ Y p , T 0 and

‖ 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 ) . □

Acknowledgements

The author wishes to thank the referees for their useful comments.

  1. Funding information: This work was supported by the National Natural Sciences Foundation of China (No. 11571283) and the Postdoctoral Research Foundation of Chongqing (No. yuXM201102006).

  2. Author contributions: The first author, Yongfang Liu, wrote the paper. Corresponding author Chaosheng Zhu proposed the ideas for this study and performed additional analysis to refine these ideas, and revised and proofread the paper.

  3. Conflict of interest: The author states no conflict of interest.

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Received: 2020-07-03
Revised: 2021-05-31
Accepted: 2021-06-29
Published Online: 2022-04-14

© 2022 Yongfang Liu and Chaosheng Zhu, published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/math-2022-0014
Keywords for this article
magnetohydrodynamic equations; local well posedness; global well posedness
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  31. On a more accurate half-discrete Hilbert-type inequality involving hyperbolic functions
  32. On split twisted inner derivation triple systems with no restrictions on their 0-root spaces
  33. Geometry of conformal η-Ricci solitons and conformal η-Ricci almost solitons on paracontact geometry
  34. Bifurcation and chaos in a discrete predator-prey system of Leslie type with Michaelis-Menten prey harvesting
  35. A posteriori error estimates of characteristic mixed finite elements for convection-diffusion control problems
  36. Dynamical analysis of a Lotka Volterra commensalism model with additive Allee effect
  37. An efficient finite element method based on dimension reduction scheme for a fourth-order Steklov eigenvalue problem
  38. Connectivity with respect to α-discrete closure operators
  39. Khasminskii-type theorem for a class of stochastic functional differential equations
  40. On some new Hermite-Hadamard and Ostrowski type inequalities for s-convex functions in (p, q)-calculus with applications
  41. New properties for the Ramanujan R-function
  42. Shooting method in the application of boundary value problems for differential equations with sign-changing weight function
  43. Ground state solution for some new Kirchhoff-type equations with Hartree-type nonlinearities and critical or supercritical growth
  44. Existence and uniqueness of solutions for the stochastic Volterra-Levin equation with variable delays
  45. Ambrosetti-Prodi-type results for a class of difference equations with nonlinearities indefinite in sign
  46. Research of cooperation strategy of government-enterprise digital transformation based on differential game
  47. Malmquist-type theorems on some complex differential-difference equations
  48. Disjoint diskcyclicity of weighted shifts
  49. Construction of special soliton solutions to the stochastic Riccati equation
  50. Remarks on the generalized interpolative contractions and some fixed-point theorems with application
  51. Analysis of a deteriorating system with delayed repair and unreliable repair equipment
  52. On the critical fractional Schrödinger-Kirchhoff-Poisson equations with electromagnetic fields
  53. The exact solutions of generalized Davey-Stewartson equations with arbitrary power nonlinearities using the dynamical system and the first integral methods
  54. Regularity of models associated with Markov jump processes
  55. Multiplicity solutions for a class of p-Laplacian fractional differential equations via variational methods
  56. Minimal period problem for second-order Hamiltonian systems with asymptotically linear nonlinearities
  57. Convergence rate of the modified Levenberg-Marquardt method under Hölderian local error bound
  58. Non-binary quantum codes from constacyclic codes over 𝔽q[u1, u2,…,uk]/⟨ui3 = ui, uiuj = ujui⟩
  59. On the general position number of two classes of graphs
  60. A posteriori regularization method for the two-dimensional inverse heat conduction problem
  61. Orbital stability and Zhukovskiǐ quasi-stability in impulsive dynamical systems
  62. Approximations related to the complete p-elliptic integrals
  63. A note on commutators of strongly singular Calderón-Zygmund operators
  64. Generalized Munn rings
  65. Double domination in maximal outerplanar graphs
  66. Existence and uniqueness of solutions to the norm minimum problem on digraphs
  67. On the p-integrable trajectories of the nonlinear control system described by the Urysohn-type integral equation
  68. Robust estimation for varying coefficient partially functional linear regression models based on exponential squared loss function
  69. Hessian equations of Krylov type on compact Hermitian manifolds
  70. Class fields generated by coordinates of elliptic curves
  71. The lattice of (2, 1)-congruences on a left restriction semigroup
  72. A numerical solution of problem for essentially loaded differential equations with an integro-multipoint condition
  73. On stochastic accelerated gradient with convergence rate
  74. Displacement structure of the DMP inverse
  75. Dependence of eigenvalues of Sturm-Liouville problems on time scales with eigenparameter-dependent boundary conditions
  76. Existence of positive solutions of discrete third-order three-point BVP with sign-changing Green's function
  77. Some new fixed point theorems for nonexpansive-type mappings in geodesic spaces
  78. Generalized 4-connectivity of hierarchical star networks
  79. Spectra and reticulation of semihoops
  80. Stein-Weiss inequality for local mixed radial-angular Morrey spaces
  81. Eigenvalues of transition weight matrix for a family of weighted networks
  82. A modified Tikhonov regularization for unknown source in space fractional diffusion equation
  83. Modular forms of half-integral weight on Γ0(4) with few nonvanishing coefficients modulo ℓ
  84. Some estimates for commutators of bilinear pseudo-differential operators
  85. Extension of isometries in real Hilbert spaces
  86. Existence of positive periodic solutions for first-order nonlinear differential equations with multiple time-varying delays
  87. B-Fredholm elements in primitive C*-algebras
  88. Unique solvability for an inverse problem of a nonlinear parabolic PDE with nonlocal integral overdetermination condition
  89. An algebraic semigroup method for discovering maximal frequent itemsets
  90. Class-preserving Coleman automorphisms of some classes of finite groups
  91. Exponential stability of traveling waves for a nonlocal dispersal SIR model with delay
  92. Existence and multiplicity of solutions for second-order Dirichlet problems with nonlinear impulses
  93. The transitivity of primary conjugacy in regular ω-semigroups
  94. Stability estimation of some Markov controlled processes
  95. On nonnil-coherent modules and nonnil-Noetherian modules
  96. N-Tuples of weighted noncommutative Orlicz space and some geometrical properties
  97. The dimension-free estimate for the truncated maximal operator
  98. A human error risk priority number calculation methodology using fuzzy and TOPSIS grey
  99. Compact mappings and s-mappings at subsets
  100. The structural properties of the Gompertz-two-parameter-Lindley distribution and associated inference
  101. A monotone iteration for a nonlinear Euler-Bernoulli beam equation with indefinite weight and Neumann boundary conditions
  102. Delta waves of the isentropic relativistic Euler system coupled with an advection equation for Chaplygin gas
  103. Multiplicity and minimality of periodic solutions to fourth-order super-quadratic difference systems
  104. On the reciprocal sum of the fourth power of Fibonacci numbers
  105. Averaging principle for two-time-scale stochastic differential equations with correlated noise
  106. Phragmén-Lindelöf alternative results and structural stability for Brinkman fluid in porous media in a semi-infinite cylinder
  107. Study on r-truncated degenerate Stirling numbers of the second kind
  108. On 7-valent symmetric graphs of order 2pq and 11-valent symmetric graphs of order 4pq
  109. Some new characterizations of finite p-nilpotent groups
  110. A Billingsley type theorem for Bowen topological entropy of nonautonomous dynamical systems
  111. F4 and PSp (8, ℂ)-Higgs pairs understood as fixed points of the moduli space of E6-Higgs bundles over a compact Riemann surface
  112. On modules related to McCoy modules
  113. On generalized extragradient implicit method for systems of variational inequalities with constraints of variational inclusion and fixed point problems
  114. Solvability for a nonlocal dispersal model governed by time and space integrals
  115. Finite groups whose maximal subgroups of even order are MSN-groups
  116. Symmetric results of a Hénon-type elliptic system with coupled linear part
  117. On the connection between Sp-almost periodic functions defined on time scales and ℝ
  118. On a class of Harada rings
  119. On regular subgroup functors of finite groups
  120. Fast iterative solutions of Riccati and Lyapunov equations
  121. Weak measure expansivity of C2 dynamics
  122. Admissible congruences on type B semigroups
  123. Generalized fractional Hermite-Hadamard type inclusions for co-ordinated convex interval-valued functions
  124. Inverse eigenvalue problems for rank one perturbations of the Sturm-Liouville operator
  125. Data transmission mechanism of vehicle networking based on fuzzy comprehensive evaluation
  126. Dual uniformities in function spaces over uniform continuity
  127. Review Article
  128. On Hahn-Banach theorem and some of its applications
  129. Rapid Communication
  130. Discussion of foundation of mathematics and quantum theory
  131. Special Issue on Boundary Value Problems and their Applications on Biosciences and Engineering (Part II)
  132. A study of minimax shrinkage estimators dominating the James-Stein estimator under the balanced loss function
  133. Representations by degenerate Daehee polynomials
  134. Multilevel MC method for weak approximation of stochastic differential equation with the exact coupling scheme
  135. Multiple periodic solutions for discrete boundary value problem involving the mean curvature operator
  136. Special Issue on Evolution Equations, Theory and Applications (Part II)
  137. Coupled measure of noncompactness and functional integral equations
  138. Existence results for neutral evolution equations with nonlocal conditions and delay via fractional operator
  139. Global weak solution of 3D-NSE with exponential damping
  140. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
  141. Ground state solutions of nonlinear Schrödinger equations involving the fractional p-Laplacian and potential wells
  142. A class of p1(x, ⋅) & p2(x, ⋅)-fractional Kirchhoff-type problem with variable s(x, ⋅)-order and without the Ambrosetti-Rabinowitz condition in ℝN
  143. Jensen-type inequalities for m-convex functions
  144. Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part III)
  145. The influence of the noise on the exact solutions of a Kuramoto-Sivashinsky equation
  146. Basic inequalities for statistical submanifolds in Golden-like statistical manifolds
  147. Global existence and blow up of the solution for nonlinear Klein-Gordon equation with variable coefficient nonlinear source term
  148. Hopf bifurcation and Turing instability in a diffusive predator-prey model with hunting cooperation
  149. Efficient fixed-point iteration for generalized nonexpansive mappings and its stability in Banach spaces
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