Home Existence of random attractors and the upper semicontinuity for small random perturbations of 2D Navier-Stokes equations with linear damping
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Existence of random attractors and the upper semicontinuity for small random perturbations of 2D Navier-Stokes equations with linear damping

  • Haiyan Li EMAIL logo and Bo Wang
Published/Copyright: December 31, 2021
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Open Mathematics
From the journal Open Mathematics Volume 19 Issue 1

Abstract

The incompressible 2D stochastic Navier-Stokes equations with linear damping are considered in this paper. Based on some new calculation estimates, we obtain the existence of random attractor and the upper semicontinuity of the random attractors as ε → 0 + on the two-dimensional space.

Keywords: random attractors; small random perturbations; upper semicontinuity; Navier-Stokes equation
MSC 2010: 35B41; 35B40; 76D05

1 Introduction

This paper considers the following stochastic Navier-Stokes equations with linear damping in a two-dimensional domain D ⊂ R 2 ,

(1.1) ∂ u ε ∂ t − ν Δ u ε + γ u ε + ( u ε ⋅ ∇ ) u ε + ∇ p = f ( x ) + ε ϕ d W ( t ) d t , x ∈ D , t > 0 , ∇ ⋅ u ε = 0 , x ∈ D , t > 0 , u ε ( x , t ) ∣ ∂ D = 0 , x ∈ ∂ D , t > 0 , u ε ( x , 0 ) = u 0 ( x ) , x ∈ D ,

where D is the abounded domain with boundary ∂ D , u ε = ( u 1 ε , u 2 ε ) T is the velocity, which depends on the disturbance parameter ε . p is the pressure, ν > 0 is the kinematic viscosity, γ u ε is the linear damping which parallels the velocity u ε , and the constant γ is positive. u 0 ( x ) is the initial velocity. The symbol W ( t ) is a real valued two-sided Wiener process.

Equations (1.1) describe the movement of incompressible fluids in geophysical dynamics. The constant γ is Rayleigh’s friction coefficient or Ekman suction/dissipation constant. The linear damping γ u is a simulation of the bottom friction in a two-dimensional ocean model or a line in a two-dimensional atmospheric model. Especially, γ = 0 , equations (1.1) are the classical stochastic 2D Navier-Stokes equation. For ε = 0 , equation (1.1) become non-stochastic system. In the past 20 years, many extensive and in-depth studies emerge. Ilyin et al. [1] studied the limit of small viscosity coefficient ν → 0 + and derived that the linear damping term γ u ε plays an important role in reducing the number of degrees of freedom in the two-dimensional model. The estimates for the number of determining modes and nodes are comparable to the sharp estimates for the fractal dimension of the global attractor. For details, we can refer to the literature [2]. Constantin and Ramos [3] derived that in R 2 space the rate of dissipation of enstrophy vanishes. The stationary statistical solutions of the damped and driven Navier-Stokes equations converge to renormalized stationary statistical solutions of the damped and driven Euler equations, and the solutions obey the enstrophy balance [3]. On arbitrary open sets, Rosa [4] deduced the existence of the global attractor. Under the condition f ( x ) ∈ ( L 2 ( R 2 ) ) , Zhao and Zheng [5] proved the existence of global attractor and studied the deformations of the Navier-Stokes equation by limit behavior. Li [6] established the existence of uniform random attractor for stochastic Navier-Stokes equations in the space H .

The theory on the stochastic dynamical system is investigated in [7,8,9, 10,11]. Our investigation of the Navier-Stokes equations with linear damping is inspired by [6,12,13]. We focus on the random attractor and its upper semicontinuity. By calculations, we derives the existence of random attractors and the upper semicontinuity for small random perturbations of Navier-Stokes equations with linear damping on the two-dimensional space, which enriches the theoretical results of the model.

This paper is arranged as follows. In Section 2, we recall some fundamental concepts and some lemmas which are used in the sequel. In Section 3, we conduce the existence of random attractors. In Section 4, we derive the upper semicontinuity for random attractors.

2 Preliminaries

This section introduces some basic related concepts for the random attractors, which were developed by Crauel and Flandoli [8,14].

Let ( Ω , ℱ , P ) be a probability space and { θ t : Ω → Ω , t ∈ R } a family of measures which preserves transformations. For all s , t ∈ R , the mapping ( t , ω ) → θ t ω is measurable, θ 0 = i d , θ t + s = θ t θ s . The θ t with the probability space ( Ω , ℱ , P , ( θ t ) t ∈ R ) is called a measurable dynamical system. For the integrity of knowledge, it introduces the following concepts.

Definition 2.1

For any ω ∈ Ω , if the function φ : R + × Ω × X → X satisfies the following conditions:

  1. φ ( 0 , ω , ⋅ ) is the identity of X ,

  2. φ ( t + s , ω , x ) = φ ( t , ω , x ) φ ( s , ω , x ) for all t , s ∈ R + , x ∈ X and P -almost every (a.e.) ω ∈ Ω ,

  3. φ ( t , ω , ⋅ ) : X → X is continuous for all t ∈ R + ,

the function φ : R + × Ω × X → X is called a continuous random dynamical system (RDS) on a metric dynamical system ( Ω , ℱ , P , ( θ t ) t ∈ R ) .

Next is the concept of a random absorption set [14].

Definition 2.2

The symbol D is a collection of families of random subsets { B ( ω ) } ω ∈ Ω of space X . If for every B = { B ( ω ) } ω ∈ Ω ∈ D , φ on RDS in D , and P -a.e. ω ∈ Ω , there exists t B ( ω ) ≥ 0 , for any t ≥ t B ( ω ) such that

(2.2) φ ( t , θ − t ω , B ( θ − t ω ) ) ⊂ K ( ω ) ,

we called the random set { K ( ω ) } ω ∈ Ω ∈ D a random absorbing set for φ on RDS in D .

The following three theorems were given and proved in [12].

Lemma 2.3

Let K ( ω ) be a random compact set which absorbs every bounded non-random set B ⊂ X , the set

(2.3) A ( ω ) = ⋃ B ⊂ X Λ ( B , ω )

is a random attractor for φ , where the union is taken over all B ⊂ X bounded, and Λ ( B , ω ) is the omega-limit set of B and is given by

(2.4) Λ ( B , ω ) = ⋂ T ≥ 0 ⋃ t ≥ T φ ( t , θ − t ω ) B ¯ .

When we add the random element which depends on a parameter ε ∈ ( 0 , 1 ] , the random term ε ϕ d W ( t ) d t perturbs the unperturbation system. According to the conventional theory, it derives an RDS which depends on the parameter ε ∈ ( 0 , 1 ]

φ ε : R + × Ω × X

such that for P -a.e. ω ∈ Ω and all t ∈ R + ,

φ ε ( t , θ − t ω , x ) → S ( t ) x , as ε → 0 + ; ( C 1 )

uniformly on bounded sets of X .

Lemma 2.4

For all ε ∈ ( 0 , 1 ] , let A ε ( ω ) be a random attractor of the system (1.1). Assume that there exists a compact set K such that, P-a.s.

lim ε → 0 + dist ( A ε ( ω ) , K ) = 0 . ( C 2 )

Then,

lim ε → 0 + dist ( A ε ( ω ) , A ) = 0 ,

with probability one.

Conditions ( C 1 ) and ( C 2 ) are necessary and sufficient for the upper semicontinuity property. Condition ( C 2 ) is a similar property for the random absorbing sets, which are used to derive the random attractors (see [12] for more details). Here, we just list the result as follows.

Lemma 2.5

Let K ε ( ω ) be a family of random compact absorbing sets which are uniformly in disturbance parameter ε , that is, for P-a.e. ω ∈ Ω and all B ⊂ X , there exists t B ( ω ) which is independent in disturbance parameter ε such that, for any t > t B ( ω ) and ε ∈ ( 0 , 1 ] ,

φ ε ( t , θ − t ω ) B ⊂ K ε ( ω ) ,

and there exists a compact set K such that P-a.s.

lim ε → 0 + dist ( K ε ( ω ) , K ) = 0 . ( C 2 ′ )

Then, for each ε ∈ ( 0 , 1 ] , there exists a random attractor A ε ( ω ) , and ( C 2 ) holds.

For convenience, it introduces the function spaces and inequalities. Denote by

( u , v ) = ∫ D ( u 1 v 1 + u 2 v 2 ) d x , u = ( u 1 , u 2 ) ∈ ( L 2 ( D ) ) , v = ( v 1 , v 2 ) ∈ ( L 2 ( D ) )

and

( ( u , v ) ) = ∫ D ( ∇ u 1 ∇ v 1 + ∇ u 2 ∇ v 2 ) d x , u , v ∈ ( H 0 1 ( D ) ) 2

and the associated norms ∣ u ∣ = ( u , u ) 1 2 , ‖ u ‖ = ( ( u , u ) ) 1 2 .

Let λ 1 be the first eigenvalue of the operator A = − P △ , where P is the orthogonal projection of ( L 2 ( D ) ) 2 . For all u , v , w , z ∈ V , according to Sobolev’s relevant knowledge, it has the following inequalities:

(2.5) λ 1 ∣ u ∣ 2 ≤ ‖ u ‖ 2 ,

(2.6) ∣ ( B ( u , v ) , w ) ∣ = ∣ b ( u , v , w ) ∣ ≤ c 0 ∣ u ∣ 1 2 ‖ u ‖ 1 2 ∣ v ∣ 1 2 ‖ v ‖ 1 2 ‖ w ‖ ,

and

(2.7) ( B ( u , z ) , z ) = 0 .

3 Existence of random attractors

We will derive computational inequalities and use Lemma 2.5 to study the random attractors of system (1.1) in this section. Assume f ∈ H , ϕ ∈ D ( A ) and set

v ε ( t , w ) = u ε ( t , w ) − ε ϕ z ( t , w ) ,

where

z ( t ) = ∫ − ∞ t exp ( − α ( t − τ ) ) d W ( τ ) .

The constant in the above inequality α > 0 is large enough and fixed.

The z = ( z 1 ( t , ω ) , z 2 ( t , ω ) , … , z n ( t , ω ) ) mentioned in the above formula is the Ornstein-Uhlenbeck process

z = ∑ j = 1 m ϕ j z j ,

with z j = ∫ − ∞ t e − α ( t − τ ) d w j ( τ ) and the real valued two-sided Wiener process W ( t ) = ( w 1 ( t , ω ) , w 2 ( t , ω ) , … , w n ( t , ω ) ) . It derives that

(3.8) d z d t = exp ( − α ( t − t ) ) d W ( t ) d t − α ∫ − ∞ t exp ( − α ( t − τ ) ) d W ( τ ) = d W ( t ) d t − α z .

As we know, z ( t ) is a stationary process and its trajectories are P -a.s. continuous. Introducing the projection operator P and the linear operator A , the first formula of equation (1.1) becomes the following form:

(3.9) d v ε d t + ν A v ε + γ v ε + B ( v ε + ε ϕ z , v ε + ε ϕ z ) = f + ( α − γ ) ε ϕ z − ε ν A ( ϕ z ) .

The existence and uniqueness of solutions for (3.9) are derived by using the method which is similar to [11,10]. We omit it here. Define an RDS ( φ ( t , τ ; ω ) ) t ≥ τ , ω ∈ Ω associated with (3.9).

φ ( t , τ ; ω ) u τ = u ε ( t , ω ) = v ε ( t , ω ) + ε ϕ z ( t , ω ) .

Taking the scalar product in (3.9), it derives

(3.10) 1 2 d d t ∣ v ε ∣ + ν ‖ v ε ‖ 2 + ( b ( v ε + ε ϕ z , v ε + ε ϕ z ) , v ε ) = ( f , v ε ) + ( α − γ ) ( ε ϕ z , v ε ) − ( ( ε ν ϕ z , v ε ) ) ,

(3.11) ∣ ( b ( v ε + ε ϕ z , v ε + ε ϕ z ) , v ε ) ∣ = ∣ ( b ( v ε + ε ϕ z , ε ϕ z ) , v ε + ε ϕ z ) ∣ = b v ε + ε ϕ z , ε ϕ ∑ j = 1 m ϕ j z j , v ε + ε ϕ z ≤ c 1 ε ∑ j = 1 m ∣ z j ∣ ∣ v ε + ε ϕ z ∣ 2 ≤ c 1 ε ∑ j = 1 m ∣ z j ∣ ( ∣ v ε ∣ 2 + ε 2 ∣ z ∣ 2 ) ≤ 2 c 1 ∑ j = 1 m ∣ z j ∣ ( ∣ v ε ∣ 2 + ∣ z ∣ 2 ) .

Using the Young inequality with ε , it follows

(3.12) ( f , v ε ) ≤ ν λ 1 8 ∣ v ε ∣ 2 + 2 ν λ 1 ∣ f ∣ 2 ,

(3.13) ( α − γ ) ( ε ϕ z , v ε ) ≤ ν λ 1 8 ∣ v ε ∣ 2 + 2 ( α − γ ) 2 ε 2 ν λ 1 ∣ z ∣ 2 ,

(3.14) ν ( ( ε ϕ z , v ε ) ) ≤ ν 2 ‖ v ε ‖ 2 + ν 2 ‖ z ‖ 2 .

Substitute (3.11)–(3.14) into (3.10) and deduce

(3.15) 1 2 d d t ∣ v ε ∣ 2 + γ ∣ v ε ∣ 2 + ν ‖ v ε ‖ 2 ≤ 2 c 1 ∑ j = 1 m ∣ v ε ∣ 2 + ν λ 1 4 ∣ v ε ∣ 2 + ν 2 ‖ v ε ‖ 2 + g ,

where

(3.16) g = 2 c 1 ∑ j = 1 m ∣ z j ∣ ∣ z ∣ 2 + 2 ν λ 1 ∣ f ∣ 2 + 2 ( α − γ ) 2 ν λ 1 + ν 2 ‖ z ‖ 2

and λ 1 is the first eigenvalue of the operator A .

For ∀ v ε ∈ V , it satisfies

‖ v ε ‖ ≥ λ 1 ∣ v ε ∣ .

Noting

ν ‖ v ε ‖ 2 ≥ ν 4 ‖ v ε ‖ 2 + 3 ν 4 ‖ v ε ‖ 2 ≥ ν 4 ‖ v ε ‖ 2 + 3 λ 1 ν 4 ∣ v ε ∣ 2

and substituting the above inequality into (3.15), it derives

(3.17) d d t ∣ v ε ∣ 2 + ν 4 ‖ v ε ‖ 2 + 2 γ + ν λ 1 4 − 4 c 1 ∑ j = 1 m ∣ z j ∣ ∣ v ε ∣ 2 ≤ 2 g .

By Gronwall’s lemma, for s ≤ − 1 , and t ∈ [ − 1 , 0 ] ,

(3.18) ∣ v ε ( t ) ∣ 2 ≤ ∣ v ε ( s ) ∣ 2 exp − ∫ s t 2 γ + ν λ 1 4 − 4 c 1 ∑ j = 1 m ∣ z j ( σ ) ∣ d σ + 2 ∫ s t g ( σ ) exp − ∫ s t 2 γ + ν λ 1 4 − 4 c 1 ∑ j = 1 m ∣ z j ( τ ) ∣ d τ d σ ≤ c 2 ∣ v ε ( s ) ∣ 2 exp s 2 γ + ν λ 1 4 + 4 c 1 s ∫ s 0 ∑ j = 1 m ∣ z j ( σ ) ∣ d σ + 2 c 2 ∫ s 0 g ( σ ) exp − ∫ σ 0 2 γ + ν λ 1 4 − 4 c 1 ∑ j = 1 m ∣ z j ( τ ) ∣ d τ d σ ,

where

c 2 = exp 2 γ + ν λ 1 4 .

By the ergodic theorem, the stationary and ergodic process ∑ j = 1 m ∣ z j ∣ satisfies

(3.19) − 1 s ∫ s 0 ∑ j = 1 m ∣ z j ( σ ) ∣ d σ → E ∑ j = 1 m ∣ z j ( 0 ) ∣ ,

when s → − ∞ . Then, there exists s 0 ( ω ) such that for any s ≤ s 0 ( ω ) ,

(3.20) − 1 s ∫ s 0 ∑ j = 1 m ∣ z j ( σ ) ∣ d σ ≤ 2 E ∑ j = 1 m ∣ z j ( 0 ) ∣

and

(3.21) exp s 2 γ + ν λ 1 4 + 4 c 1 s ∫ s 0 ∑ j = 1 m ∣ z j ( σ ) ∣ d σ ≤ exp s 2 γ + ν λ 1 4 + 8 c 1 E ∑ j = 1 m ∣ z j ( 0 ) ∣ .

Note that

(3.22) E ∑ j = 1 m ∣ z j ( 0 ) ∣ ≤ ∑ j = 1 m E ( ∣ z j ( 0 ) ∣ 2 ) 1 2 = m ( 2 α ) 1 2 .

Taking the constant α large enough, it derives

(3.23) ∑ j = 1 m E ( ∣ z j ( 0 ) ∣ 2 ) 1 2 ≤ ν λ 1 64 c 1 .

Considering s → − ∞ , the last part of the index in the first term which comes from (3.18) decays to 0.

Now, we estimate the second term of (3.18).

z j = z j ( 0 ) − α ∫ t 0 z j ( s ) d s + w j ( t )

which shows that ∣ z j ( t ) ∣ t is bounded at − ∞ . The term g ( t ) grows at most polynomially. Since g ( t ) is multiplied by function which decays exponentially by (3.19) and (3.23) the integral converges. Thus for s < s 0 ( ω ) and t ∈ [ − 1 , 0 ] ,

(3.24) ∣ v ε ( t ) ∣ 2 ≤ c 2 ∣ v ε ( s ) ∣ 2 exp s 2 γ + ν λ 1 8 + 2 c 2 ∫ − ∞ 0 g ( σ ) exp σ 2 γ + ν λ 1 4 + 2 c 1 σ ∫ 0 σ ∑ j = 1 m ∣ z j ( τ ) ∣ d τ d σ ≤ 2 c 2 ∣ u ε ( s ) ∣ 2 exp s 2 γ + ν λ 1 8 + 2 c 2 ∣ z ( s ) ∣ 2 exp s 2 γ + ν λ 1 8 + 2 c 2 ∫ − ∞ 0 g ( σ ) exp σ 2 γ + ν λ 1 4 + 2 c 1 σ ∫ 0 σ ∑ j = 1 m ∣ z j ( τ ) ∣ d τ d σ ,

so, there exist s 1 ( ω , B ) which depends only on B and ω such that for s < s 1 ( ω , B ) , t ∈ [ − 1 , 0 ] and deduce

(3.25) ∣ v ε ( t ) ∣ 2 ≤ r 0 ( ω ) = 2 c 2 ∫ − ∞ 0 g ( σ ) exp σ 2 γ + ν λ 1 4 + 2 c 1 σ ∫ 0 σ ∑ j = 1 m ∣ z j ( τ ) ∣ d τ d σ + 2 c 2 sup s ∈ ( − ∞ , − 1 ] ∣ z ( s ) ∣ 2 exp s ν λ 1 8 + 1 .

Moreover, we can integrate (3.10) from − 1 to 0 and deduce

(3.26) ∫ − 1 0 ‖ v ε ( s ) ‖ 2 ≤ r 1 ( ω ) = 8 ν ∫ − 1 0 g ( σ ) d σ + 8 c 1 ν ∫ − 1 0 ∑ j = 1 m ∣ z j ( σ ) ∣ d σ r 0 ( ω ) .

Now, we estimate the ‖ v ε ‖ 2 . Taking the scalar product of (3.9) by v ε in V and using the following inequality:

∣ b ( u , u ) ∣ ≤ c 3 ∣ u ∣ 1 2 ∣ A u ∣ 1 2 ‖ u ‖ ,

it has

(3.27) 1 2 d d t ‖ v ε ‖ 2 + γ ‖ v ε ‖ 2 + ν ∣ A v ε ∣ 2 = ( ( f , v ε ) ) + ( α − γ ) ε ( ( ϕ z , v ε ) ) − ε ν ( A ϕ z , A v ε ) − ( b ( v ε + z , v ε + z ) , A v ε ) ≤ 4 ν ∣ f ∣ 2 + 4 ( α − γ ) 2 ε 2 ν ∣ z ∣ 2 + 4 ν ε ∣ A ϕ z ∣ 2 + 4 c 3 2 ν ε ∣ v ε + ε ϕ z ∣ ∣ A ϕ z ∣ ‖ v ε + ε ϕ z ‖ 2 + 32 c 3 4 ν ε ∣ v ε + ε ϕ z ∣ 2 ‖ ϕ z ‖ 4 + 32 c 3 4 ν 3 ε ∣ v ε + ε ϕ z ∣ 2 ‖ v ε ‖ 2 .

For any t ∈ [ − 1 , 0 ] , it deduces that

(3.28) ‖ v ε ( 0 ) ‖ 2 ≤ ‖ v ε ( s ) ‖ 2 e ∫ t 0 N ( σ ) d σ + ∫ t 0 M ( σ ) e ∫ σ 0 N ( τ ) d τ d σ ≤ ( ‖ v ε ( s ) ‖ 2 + ∫ − 1 0 M ( σ ) d σ ) e ∫ − 1 0 N ( σ ) d σ ,

with

M ( t ) = 4 ν ∣ f ∣ 2 + 4 ( α − γ ) 2 ε 2 ν ∣ z ∣ 2 + 4 ν ε ∣ A ϕ z ∣ 2 + 4 c 3 2 ν ε ∣ v ε + ε ϕ z ∣ ∣ A ϕ z ∣ ‖ v ε + ε ϕ z ‖ 2 + 32 c 3 4 ν ε ∣ v ε + ε ϕ z ∣ 2 ‖ ϕ z ‖ 4 , N ( t ) = 32 c 3 4 ν 3 ε ∣ v ε + ε ϕ z ∣ 2 ‖ v ε ‖ 2 .

Integration (3.28) with respect to t on [ − 1 , 0 ] ,

(3.29) ‖ v ε ( 0 ) ‖ 2 ≤ ∫ − 1 0 ‖ v ( t ) ‖ 2 d t + ∫ − 1 0 M ( σ ) d σ e ∫ − 1 0 N ( σ ) d σ .

Finally, it can deduce that there exists r ε ( ω ) such that, if t 0 < t ( ω ) ,

(3.30) ‖ v ε ( 0 ) ‖ 2 ≤ r ε 2 .

Taking K ε ( ω ) as the ball in space V of radius r ε ( ω ) + ε ‖ ϕ z ( 0 ) ‖ , it has a compact absorbing set which is uniformly in ε in H for φ . It is clear that

(3.31) lim ε → 0 + ( r ε ( ω ) + ε ‖ ϕ z ( 0 ) ‖ ) ≤ r d .

r d is independent of ω ∈ Ω . So, the assumption ( C 2 ′ ) of Lemma 2.5 holds. Then, for ε , there is a random attractor A ε ( ω ) . According to Lemma 2.5, the equation ( C 2 ) is also guaranteed.

From the discussion above, the following theorem on the existence of random attractors holds.

Theorem 3.1

Assume f ∈ H , ϕ ∈ D ( A ) , ε ∈ ( 0 , 1 ] the system (1.1) has random attractors.

4 Upper semicontinuity of attractors

Section 3 derives that ( C 2 ) which comes from Lemma 2.4 holds. In order to apply Lemma 2.4 to derive the upper semicontinuity of the random attractor, we just need to prove that C 1 is established. By calculations, it has the following Theorem 4.1.

Theorem 4.1

Let u ε ( 0 , ω ; − t 0 , u 0 ) be the solution of system (1.1) and u ( t 0 ; u 0 ) be the solution of the unperturbed problem ( ε = 0 ). When the perturbed parameter ε → 0 , u ε ( 0 , ω ; − t 0 , u 0 ) converges in space H P-a.s. to u ( t 0 ; u 0 ) , uniformly on bounded sets of initial conditions, that is, for P-a.e. ω ∈ Ω , t 0 ∈ R + and ∀ G ⊂ H bounded

(4.32) lim ε → 0 + ∣ u ε ( 0 , ω ; − t 0 , u 0 ) − u ( t 0 ; u 0 ) ∣ = 0 , u 0 ∈ G .

Proof

The proof of the theorem is similar to Section 3.3 in [12]. In order to the completeness of the description, we describe as follows.

Set v ε ( t , ω ) = u ε ( t , ω ) − u ( t ) as the difference between the solutions of the perturbed and the unperturbed equation with the same initial condition u 0 at − t 0 . It is clear that v ε satisfies

d d t v ε + γ v ε + A v ε + B ( v ε + u , v ε + u ) − B ( u , u ) = ε ϕ d W ( t ) d t , v ε ( − t 0 ) = 0 .

Considering the operator B is bilinear, it derives

d d t v ε + γ v ε + A v ε + B ( v ε , v ε ) + B ( v ε , u ) + B ( u , v ε ) = ε ϕ d W ( t ) d t , v ε ( − t 0 ) = 0 .

Let

z ε = v ε − ε ϕ W ( t ) .

It obtains

d z ε d t + γ z ε + ε γ ϕ W ( t ) + ε ν A ( ϕ W ( t ) ) + B ( z ε + ε ϕ W ( t ) , z ε + ε ϕ W ( t ) ) + B ( z ε + ε ϕ W ( t ) , u ) + B ( u , z ε + ε ϕ W ( t ) ) = 0 .

Multiplying (4.33) by z ε , and using the bilinearity of operater B , it derives

(4.33) 1 2 d d t ∣ z ε ∣ 2 + γ ∣ z ε ∣ 2 + γ ε ( ϕ W ( t ) , z ε ) + ν ( A z ε , z ε ) + ν ( ε W ( t ) A ϕ , z ε ) + ( B ( z ε , ε ϕ W ( t ) ) , z ε ) + ( B ( ε ϕ W ( t ) , ε ϕ W ( t ) ) , z ε ) + ( B ( z ε , u ) , z ε ) + ( B ( ε ϕ W ( t ) , u ) , z ε ) + ( B ( u , ε ϕ W ( t ) ) , z ε ) = 0 .

By Young’s inequality, we have the following estimates:

(4.34) ε γ ( ϕ W ( t ) , z ε ) ≤ ε 2 4 ∣ ϕ ∣ 2 ∣ W ( t ) ∣ 2 + γ 2 ∣ z ε ∣ 2 ,

(4.35) ( ε ν W ( t ) A ϕ , z ε ) ≤ ε ∣ W ( t ) ∣ ∣ A ϕ ∣ ∣ z ε ∣ ≤ ε 2 ν 2 W ( t ) 2 ∣ A ϕ ∣ 2 2 + ∣ z ε ∣ 2 2 ,

(4.36) ( B ( z , ε ϕ W ( t ) ) , z ε ) ≤ ε ∣ W ( t ) ∣ ∣ ( B ( z ε , ϕ ) , z ε ) ∣ ≤ c 1 ε ∣ W ( t ) ∣ ∣ z ε ∣ 2 ,

(4.37) ( B ( ε ϕ W ( t ) , ε ϕ W ( t ) ) , z ε ) ≤ ε ∣ W ( t ) ∣ ∣ ( B ( z ε , ϕ ) , z ε ) ∣ ≤ c 3 ε 2 W ( t ) 2 ∣ ϕ ∣ 1 2 ‖ ϕ ‖ ∣ A ϕ ∣ 1 2 ∣ z ε ∣ ≤ c 3 2 2 ε 4 W ( t ) 4 ∣ ϕ ∣ ‖ ϕ ‖ 2 ∣ A ϕ ∣ + ∣ z ε ∣ 2 2 ,

(4.38) ( B ( z ε , u ) , z ε ) ≤ c 4 ∣ z ε ∣ ‖ u ‖ ‖ z ε ‖ ≤ ν ‖ z ε ‖ 2 + c 5 ‖ u ‖ 2 ∣ z ε ∣ 2 ,

(4.39) ( B ( ε ϕ W ( t ) , u ) , z ε ) ≤ ε ∣ W ( t ) ∣ ∣ ( B ( ϕ , u ) , z ε ) ∣ ≤ ε ∣ W ( t ) ∣ ∣ ϕ ∣ 1 2 ∣ A ϕ ∣ 1 2 ‖ u ‖ ∣ z ε ∣ ≤ 1 2 ε 2 W ( t ) 2 ∣ ϕ ∣ ∣ A ϕ ∣ ‖ u ‖ 2 + 1 2 ∣ z ε ∣ 2 ,

(4.40) ( B ( u , ε ϕ W ( t ) ) , z ε ) ≤ ε ∣ W ( t ) ∣ ∣ ( B ( u , ϕ ) , z ε ) ∣ ≤ ε ∣ W ( t ) ∣ ∣ u ∣ 1 2 ‖ u ‖ 1 2 ‖ ϕ ‖ 1 2 ∣ A ϕ ∣ 1 2 ∣ z ε ∣ ≤ ε ∣ W ( t ) ∣ 2 ∣ u ∣ ‖ u ‖ ‖ ϕ ‖ ∣ A ϕ ∣ + 1 2 ∣ z ε ∣ 2 .

Taking the estimates (4.34)–(4.40) into (4.33), it has

(4.41) d d t ∣ z ε ∣ 2 ≤ g ( t ) + h ( t ) ∣ z ε ∣ 2 ,

where

(4.42) g ( t ) = K ε ( W ( t ) 2 ) ∣ A ϕ ∣ 2 + W ( t ) 4 ∣ ϕ ∣ ‖ ϕ ‖ 2 ∣ A ϕ ∣ + W ( t ) 2 ∣ ϕ ∣ ∣ A ϕ ∣ ‖ u ‖ 2 + W ( t ) 2 ∣ u ∣ ‖ u ‖ ‖ ϕ ‖ ∣ A ϕ ∣ + W ( t ) 2 ∣ ϕ ∣ 2

and

(4.43) h ( t ) = 2 + γ 2 + c 1 ε ∣ W ( t ) ∣ + c 5 ‖ u ‖ 2 .

Using Gronwall’s lemma,

(4.44) ∣ z ε ∣ 2 ≤ g ( t ) + ∫ − t 0 t g ( s ) h ( s ) exp ∫ s t k ( τ ) d τ d s ,

and so ∣ z ε ∣ 2 → as ε → 0 + for all t ≥ − t 0 . Then

(4.45) lim ε → 0 + ∣ v ε ∣ 2 ≤ lim ε → 0 + 2 ( ∣ z ε ∣ 2 + ε 2 ∣ ϕ ∣ 2 W ( t ) 2 ) = 0 .

Taking t = 0 , it completes the proof.□

From the above analysis, we obtain the upper semicontinuity of the random attractors.

Acknowledgements

The authors would like to thank the referees for careful reading of this paper and for valuable suggestions to improve the paper.

  1. Funding information: Bo Wang was supported by the Outstanding Young Teacherąŕs Scientific Research Funds for the Ningxia Universities (NGY2018-155) and the Ministry of Education of Humanities and Social Science of China (grant number 17YJC630144). Haiyan Li was supported by the Natural Science Foundation of Ningxia (2020AAC03233), the Key Special Projects of North Minzu University (grant number ZDZX201901) and the Scientific Research Funds of North Minzu University (No. 2022XYZSX04).

  2. Author contributions: Both authors contributed equally and significantly in writing the paper. Both authors read and approved the final manuscript.

  3. Conflict of interest: Authors state no conflict of interest.

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Received: 2021-07-06
Accepted: 2021-08-26
Published Online: 2021-12-31

© 2021 Haiyan Li and Bo Wang, published by De Gruyter

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

Articles in the same Issue

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  2. Sharp conditions for the convergence of greedy expansions with prescribed coefficients
  3. Range-kernel weak orthogonality of some elementary operators
  4. Stability analysis for Selkov-Schnakenberg reaction-diffusion system
  5. On non-normal cyclic subgroups of prime order or order 4 of finite groups
  6. Some results on semigroups of transformations with restricted range
  7. Quasi-ideal Ehresmann transversals: The spined product structure
  8. On the regulator problem for linear systems over rings and algebras
  9. Solvability of the abstract evolution equations in Ls-spaces with critical temporal weights
  10. Resolving resolution dimensions in triangulated categories
  11. Entire functions that share two pairs of small functions
  12. On stochastic inverse problem of construction of stable program motion
  13. Pentagonal quasigroups, their translatability and parastrophes
  14. Counting certain quadratic partitions of zero modulo a prime number
  15. Global attractors for a class of semilinear degenerate parabolic equations
  16. A new implicit symmetric method of sixth algebraic order with vanished phase-lag and its first derivative for solving Schrödinger's equation
  17. On sub-class sizes of mutually permutable products
  18. Asymptotic solution of the Cauchy problem for the singularly perturbed partial integro-differential equation with rapidly oscillating coefficients and with rapidly oscillating heterogeneity
  19. Existence and asymptotical behavior of solutions for a quasilinear Choquard equation with singularity
  20. On kernels by rainbow paths in arc-coloured digraphs
  21. Fully degenerate Bell polynomials associated with degenerate Poisson random variables
  22. Multiple solutions and ground state solutions for a class of generalized Kadomtsev-Petviashvili equation
  23. A note on maximal operators related to Laplace-Bessel differential operators on variable exponent Lebesgue spaces
  24. Weak and strong estimates for linear and multilinear fractional Hausdorff operators on the Heisenberg group
  25. Partial sums and inclusion relations for analytic functions involving (p, q)-differential operator
  26. Hodge-Deligne polynomials of character varieties of free abelian groups
  27. Diophantine approximation with one prime, two squares of primes and one kth power of a prime
  28. The equivalent parameter conditions for constructing multiple integral half-discrete Hilbert-type inequalities with a class of nonhomogeneous kernels and their applications
  29. Boundedness of vector-valued sublinear operators on weighted Herz-Morrey spaces with variable exponents
  30. On some new quantum midpoint-type inequalities for twice quantum differentiable convex functions
  31. Quantum Ostrowski-type inequalities for twice quantum differentiable functions in quantum calculus
  32. Asymptotic measure-expansiveness for generic diffeomorphisms
  33. Infinitesimals via Cauchy sequences: Refining the classical equivalence
  34. The (1, 2)-step competition graph of a hypertournament
  35. Properties of multiplication operators on the space of functions of bounded φ-variation
  36. Disproving a conjecture of Thornton on Bohemian matrices
  37. Some estimates for the commutators of multilinear maximal function on Morrey-type space
  38. Inviscid, zero Froude number limit of the viscous shallow water system
  39. Inequalities between height and deviation of polynomials
  40. New criteria-based ℋ-tensors for identifying the positive definiteness of multivariate homogeneous forms
  41. Determinantal inequalities of Hua-Marcus-Zhang type for quaternion matrices
  42. On a new generalization of some Hilbert-type inequalities
  43. On split quaternion equivalents for Quaternaccis, shortly Split Quaternaccis
  44. On split regular BiHom-Poisson color algebras
  45. Asymptotic stability of the time-changed stochastic delay differential equations with Markovian switching
  46. The mixed metric dimension of flower snarks and wheels
  47. Oscillatory bifurcation problems for ODEs with logarithmic nonlinearity
  48. The B-topology on S∗-doubly quasicontinuous posets
  49. Hyers-Ulam stability of isometries on bounded domains
  50. Inhomogeneous conformable abstract Cauchy problem
  51. Path homology theory of edge-colored graphs
  52. Refinements of quantum Hermite-Hadamard-type inequalities
  53. Symmetric graphs of valency seven and their basic normal quotient graphs
  54. Mean oscillation and boundedness of multilinear operator related to multiplier operator
  55. Numerical methods for time-fractional convection-diffusion problems with high-order accuracy
  56. Several explicit formulas for (degenerate) Narumi and Cauchy polynomials and numbers
  57. Finite groups whose intersection power graphs are toroidal and projective-planar
  58. On primitive solutions of the Diophantine equation x2 + y2 = M
  59. A note on polyexponential and unipoly Bernoulli polynomials of the second kind
  60. On the type 2 poly-Bernoulli polynomials associated with umbral calculus
  61. Some estimates for commutators of Littlewood-Paley g-functions
  62. Construction of a family of non-stationary combined ternary subdivision schemes reproducing exponential polynomials
  63. On the evolutionary bifurcation curves for the one-dimensional prescribed mean curvature equation with logistic type
  64. On intersections of two non-incident subgroups of finite p-groups
  65. Global existence and boundedness in a two-species chemotaxis system with nonlinear diffusion
  66. Finite groups with 4p2q elements of maximal order
  67. Positive solutions of a discrete nonlinear third-order three-point eigenvalue problem with sign-changing Green's function
  68. Power moments of automorphic L-functions related to Maass forms for SL3(ℤ)
  69. Entire solutions for several general quadratic trinomial differential difference equations
  70. Strong consistency of regression function estimator with martingale difference errors
  71. Fractional Hermite-Hadamard-type inequalities for interval-valued co-ordinated convex functions
  72. Montgomery identity and Ostrowski-type inequalities via quantum calculus
  73. Universal inequalities of the poly-drifting Laplacian on smooth metric measure spaces
  74. On reducible non-Weierstrass semigroups
  75. so-metrizable spaces and images of metric spaces
  76. Some new parameterized inequalities for co-ordinated convex functions involving generalized fractional integrals
  77. The concept of cone b-Banach space and fixed point theorems
  78. Complete consistency for the estimator of nonparametric regression model based on m-END errors
  79. A posteriori error estimates based on superconvergence of FEM for fractional evolution equations
  80. Solution of integral equations via coupled fixed point theorems in 𝔉-complete metric spaces
  81. Symmetric pairs and pseudosymmetry of Θ-Yetter-Drinfeld categories for Hom-Hopf algebras
  82. A new characterization of the automorphism groups of Mathieu groups
  83. The role of w-tilting modules in relative Gorenstein (co)homology
  84. Primitive and decomposable elements in homology of ΩΣℂP∞
  85. The G-sequence shadowing property and G-equicontinuity of the inverse limit spaces under group action
  86. Classification of f-biharmonic submanifolds in Lorentz space forms
  87. Some new results on the weaving of K-g-frames in Hilbert spaces
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  89. Global optimization and applications to a variational inequality problem
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  92. On multi-step methods for singular fractional q-integro-differential equations
  93. Compact perturbations of operators with property (t)
  94. Entire solutions for several complex partial differential-difference equations of Fermat type in ℂ2
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  99. Structure of coincidence isometry groups
  100. Sharp function estimates and boundedness for Toeplitz-type operators associated with general fractional integral operators
  101. Oscillatory hyper-Hilbert transform on Wiener amalgam spaces
  102. Euler-type sums involving multiple harmonic sums and binomial coefficients
  103. Poly-falling factorial sequences and poly-rising factorial sequences
  104. Geometric approximations to transition densities of Jump-type Markov processes
  105. Multiple solutions for a quasilinear Choquard equation with critical nonlinearity
  106. Bifurcations and exact traveling wave solutions for the regularized Schamel equation
  107. Almost factorizable weakly type B semigroups
  108. The finite spectrum of Sturm-Liouville problems with n transmission conditions and quadratic eigenparameter-dependent boundary conditions
  109. Ground state sign-changing solutions for a class of quasilinear Schrödinger equations
  110. Epi-quasi normality
  111. Derivative and higher-order Cauchy integral formula of matrix functions
  112. Commutators of multilinear strongly singular integrals on nonhomogeneous metric measure spaces
  113. Solutions to a multi-phase model of sea ice growth
  114. Existence and simulation of positive solutions for m-point fractional differential equations with derivative terms
  115. Bernstein-Walsh type inequalities for derivatives of algebraic polynomials in quasidisks
  116. Review Article
  117. Semiprimeness of semigroup algebras
  118. Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part II)
  119. Third-order differential equations with three-point boundary conditions
  120. Fractional calculus, zeta functions and Shannon entropy
  121. Uniqueness of positive solutions for boundary value problems associated with indefinite ϕ-Laplacian-type equations
  122. Synchronization of Caputo fractional neural networks with bounded time variable delays
  123. On quasilinear elliptic problems with finite or infinite potential wells
  124. Deterministic and random approximation by the combination of algebraic polynomials and trigonometric polynomials
  125. On a fractional Schrödinger-Poisson system with strong singularity
  126. Parabolic inequalities in Orlicz spaces with data in L1
  127. Special Issue on Evolution Equations, Theory and Applications (Part II)
  128. Impulsive Caputo-Fabrizio fractional differential equations in b-metric spaces
  129. Existence of a solution of Hilfer fractional hybrid problems via new Krasnoselskii-type fixed point theorems
  130. On a nonlinear system of Riemann-Liouville fractional differential equations with semi-coupled integro-multipoint boundary conditions
  131. Blow-up results of the positive solution for a class of degenerate parabolic equations
  132. Long time decay for 3D Navier-Stokes equations in Fourier-Lei-Lin spaces
  133. On the extinction problem for a p-Laplacian equation with a nonlinear gradient source
  134. General decay rate for a viscoelastic wave equation with distributed delay and Balakrishnan-Taylor damping
  135. On hyponormality on a weighted annulus
  136. Exponential stability of Timoshenko system in thermoelasticity of second sound with a memory and distributed delay term
  137. Convergence results on Picard-Krasnoselskii hybrid iterative process in CAT(0) spaces
  138. Special Issue on Boundary Value Problems and their Applications on Biosciences and Engineering (Part I)
  139. Marangoni convection in layers of water-based nanofluids under the effect of rotation
  140. A transient analysis to the M(τ)/M(τ)/k queue with time-dependent parameters
  141. Existence of random attractors and the upper semicontinuity for small random perturbations of 2D Navier-Stokes equations with linear damping
  142. Degenerate binomial and Poisson random variables associated with degenerate Lah-Bell polynomials
  143. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
  144. On the mixed fractional quantum and Hadamard derivatives for impulsive boundary value problems
  145. The Lp dual Minkowski problem about 0 < p < 1 and q > 0
https://doi.org/10.1515/math-2021-0098
Keywords for this article
random attractors; small random perturbations; upper semicontinuity; Navier-Stokes equation
Creative Commons
BY 4.0

Articles in the same Issue

  1. Regular Articles
  2. Sharp conditions for the convergence of greedy expansions with prescribed coefficients
  3. Range-kernel weak orthogonality of some elementary operators
  4. Stability analysis for Selkov-Schnakenberg reaction-diffusion system
  5. On non-normal cyclic subgroups of prime order or order 4 of finite groups
  6. Some results on semigroups of transformations with restricted range
  7. Quasi-ideal Ehresmann transversals: The spined product structure
  8. On the regulator problem for linear systems over rings and algebras
  9. Solvability of the abstract evolution equations in Ls-spaces with critical temporal weights
  10. Resolving resolution dimensions in triangulated categories
  11. Entire functions that share two pairs of small functions
  12. On stochastic inverse problem of construction of stable program motion
  13. Pentagonal quasigroups, their translatability and parastrophes
  14. Counting certain quadratic partitions of zero modulo a prime number
  15. Global attractors for a class of semilinear degenerate parabolic equations
  16. A new implicit symmetric method of sixth algebraic order with vanished phase-lag and its first derivative for solving Schrödinger's equation
  17. On sub-class sizes of mutually permutable products
  18. Asymptotic solution of the Cauchy problem for the singularly perturbed partial integro-differential equation with rapidly oscillating coefficients and with rapidly oscillating heterogeneity
  19. Existence and asymptotical behavior of solutions for a quasilinear Choquard equation with singularity
  20. On kernels by rainbow paths in arc-coloured digraphs
  21. Fully degenerate Bell polynomials associated with degenerate Poisson random variables
  22. Multiple solutions and ground state solutions for a class of generalized Kadomtsev-Petviashvili equation
  23. A note on maximal operators related to Laplace-Bessel differential operators on variable exponent Lebesgue spaces
  24. Weak and strong estimates for linear and multilinear fractional Hausdorff operators on the Heisenberg group
  25. Partial sums and inclusion relations for analytic functions involving (p, q)-differential operator
  26. Hodge-Deligne polynomials of character varieties of free abelian groups
  27. Diophantine approximation with one prime, two squares of primes and one kth power of a prime
  28. The equivalent parameter conditions for constructing multiple integral half-discrete Hilbert-type inequalities with a class of nonhomogeneous kernels and their applications
  29. Boundedness of vector-valued sublinear operators on weighted Herz-Morrey spaces with variable exponents
  30. On some new quantum midpoint-type inequalities for twice quantum differentiable convex functions
  31. Quantum Ostrowski-type inequalities for twice quantum differentiable functions in quantum calculus
  32. Asymptotic measure-expansiveness for generic diffeomorphisms
  33. Infinitesimals via Cauchy sequences: Refining the classical equivalence
  34. The (1, 2)-step competition graph of a hypertournament
  35. Properties of multiplication operators on the space of functions of bounded φ-variation
  36. Disproving a conjecture of Thornton on Bohemian matrices
  37. Some estimates for the commutators of multilinear maximal function on Morrey-type space
  38. Inviscid, zero Froude number limit of the viscous shallow water system
  39. Inequalities between height and deviation of polynomials
  40. New criteria-based ℋ-tensors for identifying the positive definiteness of multivariate homogeneous forms
  41. Determinantal inequalities of Hua-Marcus-Zhang type for quaternion matrices
  42. On a new generalization of some Hilbert-type inequalities
  43. On split quaternion equivalents for Quaternaccis, shortly Split Quaternaccis
  44. On split regular BiHom-Poisson color algebras
  45. Asymptotic stability of the time-changed stochastic delay differential equations with Markovian switching
  46. The mixed metric dimension of flower snarks and wheels
  47. Oscillatory bifurcation problems for ODEs with logarithmic nonlinearity
  48. The B-topology on S∗-doubly quasicontinuous posets
  49. Hyers-Ulam stability of isometries on bounded domains
  50. Inhomogeneous conformable abstract Cauchy problem
  51. Path homology theory of edge-colored graphs
  52. Refinements of quantum Hermite-Hadamard-type inequalities
  53. Symmetric graphs of valency seven and their basic normal quotient graphs
  54. Mean oscillation and boundedness of multilinear operator related to multiplier operator
  55. Numerical methods for time-fractional convection-diffusion problems with high-order accuracy
  56. Several explicit formulas for (degenerate) Narumi and Cauchy polynomials and numbers
  57. Finite groups whose intersection power graphs are toroidal and projective-planar
  58. On primitive solutions of the Diophantine equation x2 + y2 = M
  59. A note on polyexponential and unipoly Bernoulli polynomials of the second kind
  60. On the type 2 poly-Bernoulli polynomials associated with umbral calculus
  61. Some estimates for commutators of Littlewood-Paley g-functions
  62. Construction of a family of non-stationary combined ternary subdivision schemes reproducing exponential polynomials
  63. On the evolutionary bifurcation curves for the one-dimensional prescribed mean curvature equation with logistic type
  64. On intersections of two non-incident subgroups of finite p-groups
  65. Global existence and boundedness in a two-species chemotaxis system with nonlinear diffusion
  66. Finite groups with 4p2q elements of maximal order
  67. Positive solutions of a discrete nonlinear third-order three-point eigenvalue problem with sign-changing Green's function
  68. Power moments of automorphic L-functions related to Maass forms for SL3(ℤ)
  69. Entire solutions for several general quadratic trinomial differential difference equations
  70. Strong consistency of regression function estimator with martingale difference errors
  71. Fractional Hermite-Hadamard-type inequalities for interval-valued co-ordinated convex functions
  72. Montgomery identity and Ostrowski-type inequalities via quantum calculus
  73. Universal inequalities of the poly-drifting Laplacian on smooth metric measure spaces
  74. On reducible non-Weierstrass semigroups
  75. so-metrizable spaces and images of metric spaces
  76. Some new parameterized inequalities for co-ordinated convex functions involving generalized fractional integrals
  77. The concept of cone b-Banach space and fixed point theorems
  78. Complete consistency for the estimator of nonparametric regression model based on m-END errors
  79. A posteriori error estimates based on superconvergence of FEM for fractional evolution equations
  80. Solution of integral equations via coupled fixed point theorems in 𝔉-complete metric spaces
  81. Symmetric pairs and pseudosymmetry of Θ-Yetter-Drinfeld categories for Hom-Hopf algebras
  82. A new characterization of the automorphism groups of Mathieu groups
  83. The role of w-tilting modules in relative Gorenstein (co)homology
  84. Primitive and decomposable elements in homology of ΩΣℂP∞
  85. The G-sequence shadowing property and G-equicontinuity of the inverse limit spaces under group action
  86. Classification of f-biharmonic submanifolds in Lorentz space forms
  87. Some new results on the weaving of K-g-frames in Hilbert spaces
  88. Matrix representation of a cross product and related curl-based differential operators in all space dimensions
  89. Global optimization and applications to a variational inequality problem
  90. Functional equations related to higher derivations in semiprime rings
  91. A partial order on transformation semigroups with restricted range that preserve double direction equivalence
  92. On multi-step methods for singular fractional q-integro-differential equations
  93. Compact perturbations of operators with property (t)
  94. Entire solutions for several complex partial differential-difference equations of Fermat type in ℂ2
  95. Random attractors for stochastic plate equations with memory in unbounded domains
  96. On the convergence of two-step modulus-based matrix splitting iteration method
  97. On the separation method in stochastic reconstruction problem
  98. Robust estimation for partial functional linear regression models based on FPCA and weighted composite quantile regression
  99. Structure of coincidence isometry groups
  100. Sharp function estimates and boundedness for Toeplitz-type operators associated with general fractional integral operators
  101. Oscillatory hyper-Hilbert transform on Wiener amalgam spaces
  102. Euler-type sums involving multiple harmonic sums and binomial coefficients
  103. Poly-falling factorial sequences and poly-rising factorial sequences
  104. Geometric approximations to transition densities of Jump-type Markov processes
  105. Multiple solutions for a quasilinear Choquard equation with critical nonlinearity
  106. Bifurcations and exact traveling wave solutions for the regularized Schamel equation
  107. Almost factorizable weakly type B semigroups
  108. The finite spectrum of Sturm-Liouville problems with n transmission conditions and quadratic eigenparameter-dependent boundary conditions
  109. Ground state sign-changing solutions for a class of quasilinear Schrödinger equations
  110. Epi-quasi normality
  111. Derivative and higher-order Cauchy integral formula of matrix functions
  112. Commutators of multilinear strongly singular integrals on nonhomogeneous metric measure spaces
  113. Solutions to a multi-phase model of sea ice growth
  114. Existence and simulation of positive solutions for m-point fractional differential equations with derivative terms
  115. Bernstein-Walsh type inequalities for derivatives of algebraic polynomials in quasidisks
  116. Review Article
  117. Semiprimeness of semigroup algebras
  118. Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part II)
  119. Third-order differential equations with three-point boundary conditions
  120. Fractional calculus, zeta functions and Shannon entropy
  121. Uniqueness of positive solutions for boundary value problems associated with indefinite ϕ-Laplacian-type equations
  122. Synchronization of Caputo fractional neural networks with bounded time variable delays
  123. On quasilinear elliptic problems with finite or infinite potential wells
  124. Deterministic and random approximation by the combination of algebraic polynomials and trigonometric polynomials
  125. On a fractional Schrödinger-Poisson system with strong singularity
  126. Parabolic inequalities in Orlicz spaces with data in L1
  127. Special Issue on Evolution Equations, Theory and Applications (Part II)
  128. Impulsive Caputo-Fabrizio fractional differential equations in b-metric spaces
  129. Existence of a solution of Hilfer fractional hybrid problems via new Krasnoselskii-type fixed point theorems
  130. On a nonlinear system of Riemann-Liouville fractional differential equations with semi-coupled integro-multipoint boundary conditions
  131. Blow-up results of the positive solution for a class of degenerate parabolic equations
  132. Long time decay for 3D Navier-Stokes equations in Fourier-Lei-Lin spaces
  133. On the extinction problem for a p-Laplacian equation with a nonlinear gradient source
  134. General decay rate for a viscoelastic wave equation with distributed delay and Balakrishnan-Taylor damping
  135. On hyponormality on a weighted annulus
  136. Exponential stability of Timoshenko system in thermoelasticity of second sound with a memory and distributed delay term
  137. Convergence results on Picard-Krasnoselskii hybrid iterative process in CAT(0) spaces
  138. Special Issue on Boundary Value Problems and their Applications on Biosciences and Engineering (Part I)
  139. Marangoni convection in layers of water-based nanofluids under the effect of rotation
  140. A transient analysis to the M(τ)/M(τ)/k queue with time-dependent parameters
  141. Existence of random attractors and the upper semicontinuity for small random perturbations of 2D Navier-Stokes equations with linear damping
  142. Degenerate binomial and Poisson random variables associated with degenerate Lah-Bell polynomials
  143. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
  144. On the mixed fractional quantum and Hadamard derivatives for impulsive boundary value problems
  145. The Lp dual Minkowski problem about 0 < p < 1 and q > 0
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