Home The asymptotic behavior for the Navier-Stokes-Voigt-Brinkman-Forchheimer equations with memory and Tresca friction in a thin domain
Article Open Access

The asymptotic behavior for the Navier-Stokes-Voigt-Brinkman-Forchheimer equations with memory and Tresca friction in a thin domain

  • Mohamed Dilmi , Youcef Djenaihi , Mourad Dilmi , Salah Boulaaras EMAIL logo and Hamid Benseridi
Published/Copyright: November 27, 2024
Download Article (PDF)
Become an author with De Gruyter Brill
Author Information Explore this Subject
Demonstratio Mathematica
From the journal Demonstratio Mathematica Volume 57 Issue 1

Abstract

In this article, we investigate the behavior of weak solutions for the three-dimensional Navier-Stokes-Voigt-Brinkman-Forchheimer fluid model with memory and Tresca friction law within a thin domain. We analyze the asymptotic behavior as one dimension of the fluid domain approaches zero. We derive the limit problem and obtain the specific Reynolds equation, while also establishing the uniqueness of the limit velocity and pressure distributions.

Keywords: asymptotic behavior; Navier-Stokes-Voigt-Brinkman-Forchheimer equations; Reynolds equation; mathematical model; nonlinear systems; Tresca friction law
MSC 2010: 35R35; 76F10; 78M35

1 Introduction

The Navier-Stokes equations are well known in the fields of applied mathematics and engineering, particularly for their effectiveness in modeling turbulence in fluid phenomena, which has been intensively studied for the past 90 years. In 1973, the mathematician Oskolkov [1] modified the Navier-Stokes equation by adding the pseudoparabolic regularization − ν Δ ∂ t u for velocity field u , and this model is called Navier-Stokes-Voigt. In recent years, this model has attracted significant attention from mathematicians, resulting in numerous research papers focused on studying the properties of its solutions. This surge in interest is partly due to the model yielding interesting results in various applications, including its recent use in image processing [2]. Let us briefly review the available literature on the mathematical analysis of Navier-Stokes-Voigt-type models. Anh and Thanh [3] demonstrated the existence and uniqueness of solutions for the Navier-Stokes-Voigt model, followed by an examination of the mean square exponential stability and the almost sure exponential stability of the stationary solutions. Zelati and Gal [4] investigated the long-term behavior of the three-dimensional Navier-Stokes-Voigt model. They established the existence of global and exponential attractors of optimal regularity. The existence and time decay rates of solutions for the Navier-Stokes-Voigt model have been studied in [5]. Layton and Rebholz [6] studied the analytically and numerically the relaxation time of flow evolution governed by the Navier-Stokes-Voigt model.

The Brinkman-Forchheimer equation [7,8] describes the motion of fluid flow in a saturated porous medium and has been studied by many researchers. Djoko and Razafimandimby [9] studied the Brinkman-Forchheimer equations driven under slip boundary conditions of the friction type. They proved the existence and uniqueness of weak solutions. Le [10] proved the existence of global weak solutions, the exponential stability of a stationary solution, and the existence of a global attractor for the three-dimensional convective Brinkman-Forchheimer equations with finite delay and fast growing nonlinearity in bounded domains with homogeneous Dirichlet boundary conditions. Cai and Jiu [11] incorporated the damping term ∣ u ∣ p − 2 u into the classical 3D Navier-Stokes equations and explored its impact on the well-posedness of the resulting equations. This damping term widely used in geophysical hydrodynamics arises from the resistance to the motion of the flow or by friction effects; for more details, see [12].

Regarding the memory term, its significance extends to various phenomena like non-Newtonian flows, soil mechanics, and heat conduction theory, as documented in [13]. Several researchers have delved into the behavior of solutions for the Navier-Stokes model incorporating memory effects. Notable studies include those referenced in [14,15].

On the other hand, the study of models placed in thin domains appears in many applications such as (thin elastic bodies, thin rods, plates, or shells) for solid mechanics, and (lubrication, meteorology, ocean dynamics) for fluid mechanics. During the past few decades, the lubrication theory was successfully applied to the modelling of dewetting processes in microscopic and nanoscopic liquid films on solid substrates. The previous works [16,17] addressed the study of three-dimensional Navier-Stokes equations in thin domains without friction forces. In studying the asymptotic behavior of Stokes equations in a thin domain with Tresca friction law, please refer to [18–20]. The reader can also explore related research studies concerning the asymptotic behavior elastic and viscosity materials in a thin domain with the Tresca friction law, as presented in [21–23].

Based on the principles of lubrication theory, which typically involves very thin flow domains, this article investigates the asymptotic behavior of solutions to the initial-boundary value problem governing the 3D Navier-Stokes-Voigt-Brinkman-Forchheimer equations. Incorporating a memory term and Tresca friction law, we aim to know the behavior of the solution as one dimension of the domain approaches zero and to establish the specific Reynolds equation.

This article is organized as follows. In Section 2, for convenience of the reader, we recall some auxiliary results on function spaces and some of the inequalities frequently used in this article. In Section 3, we give the mechanical problem governed by the Navier-Stokes-Voigt-Brinkman-Forchheimer model with a memory term and Tresca friction law. In Section 4, we derive the variational formulation of the problem and introduce the theorem of the existence and uniqueness of the weak solution. In Section 5, we use the change of variable z = x 3 ⁄ ε to transform the initial problem posed in the domain Ω ε , to a new problem posed on a fixed domain Ω independent of parameter ε . Then we establish some a priori estimates for the velocity and pressure fields, independently of ε . Finally, we derive and study the limit problem when ε tends to zero. Moreover, we show that the limit pressure is given as the unique solution of a Reynolds equation.

2 Mathematical setting

In this section, we describe the necessary function spaces needed to obtain the asymptotic behavior results of Navier-Stokes-Voigt-Brinkman-Forchheimer equations.

Let ω be a bounded domain in R 2 situated in the plane of equation x 3 = 0 . We suppose that ω represents the lower surface of the domain occupied by the substance. The upper surface ∂ Ω 1 ε is defined by

∂ Ω 1 ε = { ( x ′ , x 3 ) ∈ R 2 × R : x 3 = ε h ( x ′ ) and x ′ = ( x 1 , x 2 ) ∈ ω } ,

where h is a function defined and bounded on ω ¯ , such that:

0 < h ̲ ≤ h ( x ′ ) ≤ h ¯ , ∀ ( x ′ , 0 ) ∈ ω .

We study the velocity of a substance in

Ω ε = { ( x ′ , x 3 ) ∈ R 3 : x ′ = ( x 1 , x 2 ) ∈ ω , 0 < x 3 < ε h ( x ′ ) } ,

with Lipschitz boundary ∂ Ω ε = ∂ Ω 1 ε ∪ ∂ Ω L ε ∪ ω ¯ , where ∂ Ω L ε is the lateral surface of Ω .

Throughout this work, we denote by:

•   L p ( Ω ε ) is the Lebesgue space for the norm

∥ u ε ∥ L p ( Ω ε ) = ∫ Ω ε ∣ u ε ∣ p d x 1 p , p ∈ [ 1 , ∞ [ .

For p = 2 , we use ( ⋅ , ⋅ ) to denote the inner product, i.e.,

( u ε , v ε ) ≔ ∑ i = 1 3 ∫ Ω ε u i ε . v i ε d x , u ε = ( u 1 ε , u 2 ε , u 3 ε ) , v ε = ( v 1 ε , v 2 ε , v 3 ε ) ∈ L 2 ( Ω ε ) 3 .

•   H 1 ( Ω ε ) 3 is the Sobolev space that gives

H 1 ( Ω ε ) = { u ε ∈ L 2 ( Ω ε ) 3 : ∇ u ε ∈ L 2 ( Ω ε ) 3 × 3 } ,

with inner product and associated norm

⟨ u , v ⟩ H 1 ( Ω ε ) = ∫ Ω ε u . v d x + ∫ Ω ε ∇ u . ∇ v d x , ∥ u ∥ H 1 ( Ω ε ) = [ ∥ u ∥ L 2 ( Ω ε ) 3 + ∥ ∇ u ∥ L 2 ( Ω ε ) 3 × 3 ] 1 2 .

•   H 0 1 ( Ω ε ) 3 is the closure of D ( Ω ε ) 3 in H 1 ( Ω ε ) 3 , and H − 1 ( Ω ε ) 3 is the dual space of H 0 1 ( Ω ε ) 3 .

For any separable Banach space E equipped with the norm ∥ . ∥ E , we denote by L p ( 0 , T ; E ) the Banach space consisting of (classes of) functions t ⟶ f ( t ) measurable from [ 0 , T ] ⟼ E (for the measure d t ) such that

∥ u ε ∥ L p ( 0 , T ; E ) = ∫ 0 T ∥ u ε ( s ) ∥ E p d s 1 p , 1 ≤ p < ∞ .

Let us introduce the following spaces

V ε = { v ∈ H 1 ( Ω ε ) 3 : v = 0 on ∂ Ω 1 ε ∪ ∂ Ω L ε , v . n = 0 on ω } ,

where n = ( n 1 , n 2 , n 3 ) is the normal vector outward to ∂ Ω ε .

V div ε = { v ∈ V ε : div ( v ) = 0 }

and

L 0 2 ( Ω ε ) = φ ∈ L 2 ( Ω ε ) : ∫ Ω ε φ d x = 0 .

The following inequalities are frequently used in the article

• Cauchy-Schwarz inequality

∫ Ω ε ∣ u ε ∣ . ∣ v ε ∣ d x ≤ ∥ u ε ∥ L 2 ( Ω ε ) 3 . ∥ v ε ∥ L 2 ( Ω ε ) 3 .

• Poincaré’s inequality

(1) ∥ u ε ∥ L 2 ( Ω ε ) 3 ≤ ε h ¯ ∥ ∇ u ε ∥ L 2 ( Ω ε ) 3 × 3 .

• Young’s inequality

(2) a . b ≤ a 2 2 + b 2 2 , ∀ a , b > 0 .

• Sobolev’s inequality

(3) ∥ u ε ∥ L p ( Ω ε ) 3 ≤ c s ∥ ∇ u ε ∥ L 2 ( Ω ε ) 3 × 3 , for p ∈ [ 2 , 6 ] ⋅ .

3 The dynamical system

In the domain Ω ε , we study the asymptotic behavior of weak solutions to the following Navier-Stokes-Voight-Brinkman-Forchheimer equations with a memory:

Find the velocity field u ε : Ω ε × ] 0 , T [ → R 3 and the pressure π ε : Ω ε × ] 0 , T [ → R , such that

(4) ρ ∂ t u ε − div ( σ ε ( u ε , π ε ) ) + ε ( u ε . ∇ u ε ) + r ε u ε + α ε ∣ u ε ∣ p − 2 u ε = f ε in Ω ε × ] 0 , T [ ,

(5) σ ε ( u ε , π ε ) = μ ∇ u ε + ν ∇ ( ∂ t u ε ) + ϕ ⋆ ∇ u ε − π ε I 3 in Ω ε × ] 0 , T [ ,

(6) div ( u ε ) = 0 in Ω ε × ] 0 , T [ ,

(7) u ε = 0 on ∂ Ω 1 ε × ] 0 , T [ ,

(8) u ε = 0 on ∂ Ω L ε × ] 0 , T [ ,

(9) u ε ⋅ n = 0 on ω × ] 0 , T [ ,

(10) ∣ σ τ ε ∣ < k ε ⇒ u τ ε = 0 , ∣ σ τ ε ∣ = k ε ⇒ ∃ β > 0 , such that u τ ε = − β σ τ ε , on ω × ] 0 , T [ .

In equation (4) representing the equilibrium of the fluid, f ε : Ω ε × ] 0 , T [ → R 3 represents an external body force, ρ is the mass density per unit of reference volum, μ > 0 is the Brinkman coefficient, ν > 0 is the length-scale parameter characterizing the elasticity of the fluid, r ε > 0 is the Darcy coefficient, α ε is the Forchheimer coefficient, and p ∈ [ 2 , 6 ] is a constant. Equation (5) represents the law of behavior of the fluid for the Navier-Stokes-Voigt model with a memory

( ϕ ⋆ ∇ u ) ( t ) ≔ ∫ 0 t ϕ ( t − s ) ∇ u ( s ) d s ,

where the function ϕ : [ 0 ; T ] → R , is usually called the memory kernel and satisfies certain conditions specified later. Equation (6) represents the incompressibility equation. The boundary conditions (7) and (8) are the nonslip boundary condition on ∂ Ω 1 ε × ] 0 , T [ and ∂ Ω L ε × ] 0 , T [ . Since there is a no-flux condition across ω , we have equation (9). Condition (10) represents a Tresca friction law on ω with a friction coefficient k ε , where ∣ . ∣ is the R 2 Euclidean norm and

u n ε = u ε . n , u τ ε = u ε − u n ε . n , σ n ε = ( σ n ε ) . n , σ τ ε = σ ε . n − ( σ n ε ) . n

are the normal and the tangential velocity u ε , and the components of the normal and the tangential stress tensor σ ε , respectively.

Equations (4)–(10) are supplemented with the initial condition

(11) u ε ( x , 0 ) = u 0 ε , ∀ x ∈ Ω ε .

Remark 1

On ω × ] 0 , T [ the third component of velocity equals zero. Indeed, according to condition (9), we have

u ε . n = u 1 ε . n 1 + u 2 ε . n 2 + u 3 ε . n 3 = 0 on ω × ] 0 , T [ ,

where n = ( n 1 , n 2 , n 3 ) = ( 0 , 0 , − 1 ) is the unit normal to ω . So u 3 ε = 0 , on ω × ] 0 , T [ .

Properties of the kernel ϕ ( ⋅ )

Let ϕ ( ⋅ ) satisfies the following conditions:

  1. ϕ ( t ) ∈ C 2 [ 0 , T ] ,

  2. ( − 1 ) k d k d t k ( ϕ ( t ) ) ≥ 0 , for t ∈ [ 0 , T ] , k = 0 ; 1 ; 2 .

  3. ϕ ( ⋅ ) is positive kernel, i.e.,

    ∫ 0 T ( ( ϕ ⋆ u ε ) ( t ) , u ε ( t ) ) d t = ∫ 0 T ∫ 0 t ϕ ( t − s ) ( u ε ( s ) , u ε ( t ) ) d s d t ≥ 0 ,

    for all u ε ∈ L 2 ( 0 , T ; L 2 ( Ω ε ) ) and every T > 0 .

By the Lemma 2.6, [24], we have

(12) ∫ 0 T ∥ ( ϕ ⋆ u ε ) ( t ) ∥ L 2 ( Ω ε ) 2 d t ≤ ∫ 0 T ∫ 0 t ϕ ( t − s ) ⋅ ∥ u ε ( s ) ∥ L 2 ( Ω ε ) d s 2 d t ≤ ∫ 0 T ϕ ( t ) d t 2 ∫ 0 T ∥ u ε ( t ) ∥ L 2 ( Ω ε ) 2 d t .

By using the Cauchy-Schwarz inequality and (12), we can obtain the following estimate:

∫ 0 t ( ( ϕ ⋆ u ε ) ( s ) , ψ ( s ) ) d s = ∫ 0 t ∥ ( ϕ ⋆ u ε ) ( s ) ∥ L 2 ( Ω ε ) 2 1 2 . ∫ 0 t ∥ ψ ( s ) ∥ L 2 ( Ω ε ) 2 d s 1 2 ≤ ∫ 0 t ϕ ( s ) d s ∫ 0 t ∥ u ε ( s ) ∥ L 2 ( Ω ε ) 2 d s 1 2 . ∫ 0 t ∥ ψ ( s ) ∥ L 2 ( Ω ε ) 2 d s 1 2 ,

for u ε , ψ ∈ L 2 ( 0 , T ; L 2 ( Ω ε ) ) .

We can take ϕ ( t ) = c exp ( − δ t ) as an example of a kernel, using Lemma 4.1, [25], it is easy to see that this function is a positive kernel.

4 Weak formulation

To derive the weak formulation of the problem, we will need the following lemma.

Lemma 1

[26] The condition of Tresca (10) is equivalent to

σ τ ε . u τ ε + k ε ∣ u τ ε ∣ = 0 on ω × ] 0 , T [ .

By multiplying equation (4) by test-functions ( φ − u ε ) and integrating over Ω ε , then applying Green’s formula and the aforementioned lemma, we obtain the following weak formulation of problems (4)–(11):

Find ( u ε ( t ) , π ε ( t ) ) ∈ L 2 ( 0 , T ; V div ε ) × L 2 ( 0 , T ; L 0 2 ( Ω ε ) ) , such that for almost all t

(13) ( ρ ∂ t u ε ( t ) , φ − u ε ( t ) ) + μ A ( u ε ( t ) , φ − u ε ( t ) ) + ν A ( ∂ t u ε ( t ) , φ − u ε ( t ) ) + A ( ϕ ⋆ u ε ( t ) , φ − u ε ( t ) ) + ε ℬ ( u ε ( t ) , u ε ( t ) , φ − u ε ( t ) ) + ( ( r ε + α ε ∣ u ε ( t ) ∣ p − 2 ) u ε ( t ) , φ − u ε ( t ) ) − ( π ε ( t ) , div ( φ ) ) + j ε ( φ ) − j ε ( u ε ( t ) ) ≥ ( f ε ( t ) , φ − u ε ( t ) ) , ∀ φ ∈ V ε , u ε ( x , 0 ) = u 0 ε , ∀ x ∈ Ω ε .

With

A ( u , v ) = ∫ Ω ε ∇ u . ∇ v d x , j ε ( v ) = ∫ ω k ε ∣ v ∣ d x ′ , ( f , v ) = ∫ Ω ε f . v d x , ℬ ( u , v , w ) = ∫ Ω ε u . ∇ v . w d x = ∑ i , j = 1 3 ∫ Ω ε u i . ∂ x j v i . w j d x .

The bilinear form A ( ⋅ , ⋅ ) is continuous and coercive on V div ε × V div ε and j ε ( ⋅ ) is a functional continuous convex, but nondifferentiable on V div ε . The trilinear form ℬ ( ⋅ , ⋅ , ⋅ ) satisfies

ℬ ( u , v , v ) = 0 , ∀ u , v ∈ V div ε , ∣ ℬ ( u , v , w ) ∣ ≤ ∥ u ∥ L 3 ( Ω ε ) 3 ∥ ∇ v ∥ L 2 ( Ω ε ) 3 ∥ w ∥ L 6 ( Ω ε ) 3 , ∀ u , v , w ∈ V div ε .

Theorem 1

Under the assumptions

(14) f ε ( t ) , ∂ t f ε ( t ) ∈ L 2 ( 0 , T ; L 2 ( Ω ε ) 3 ) , u 0 ∈ H 1 ( Ω ε ) 3 , ( u 0 ) τ = 0 , k ε ∈ L ∞ ( ω ) , k ε > 0 .

Problem (13) admits a unique solution u ε ( t ) ∈ L 2 ( 0 , T ; V div ε ) , such that

∂ t u ε ( t ) ∈ L 2 ( 0 , T ; V div ε ) .

Proof

The proof is based on the regularization method, which is based on an approximation of nondifferentaible term j ε ( ⋅ ) by a family of differentaible once j ξ ε ( ⋅ ) , where

j ξ ε ( v ) = ∫ ω k ε ( x ′ ) ϕ ξ ( ∣ v ∣ 2 ) d x ′ , with ϕ ξ ( v ) = 1 1 + ξ ∣ v ∣ 1 + ξ , ξ > 0 ,

clearly j ξ ε ( ⋅ ) is convex and Gateaux differentiable with Gateaux derivative ( j ξ ) ′ defined on V div ε and given by

( ( j ξ ε ) ′ ( v ) , φ ) = ∫ ω k ε ( x ′ ) ∣ v ∣ 1 + ξ v . φ 1 + ξ d x ′ .

Since j ξ ε ( ⋅ ) is differentiable, adopting the classical arguments in [26], one can state that (13) is equivalent to

Find u ξ ε ( t ) ∈ L 2 ( 0 , T ; V div ε ) with ∂ t u ξ ε ( t ) ∈ L 2 ( 0 , T ; V div ε ) , such that for almost all t

(15) ( ρ ∂ t u ξ ε ( t ) , φ ) + μ A ( u ξ ε ( t ) , φ ) + ν A ( ∂ t u ξ ε ( t ) , φ ) + A ( ϕ ⋆ u ξ ε ( t ) , φ ) + ε ℬ ( u ξ ε ( t ) , u ξ ε ( t ) , φ ) + ( ( r ε + α ε ∣ u ξ ε ( t ) ∣ p − 2 ) u ξ ε ( t ) , φ ) + ( ( j ξ ε ) ′ ( u ε ( t ) ) , φ ) = ( f ε ( t ) , φ ) , ∀ φ ∈ V div ε .

By using Galerkin’s method, we show that there exists a unique solution u ξ ε ( t ) of this last approximate problem (see the works of Duvaut and Lions [26]. Then, the limit of u ξ ε ( t ) when ξ tends to zero is a solution of (13).□

5 Asymptotic analysis of the weak solution

For the asymptotic analysis of the problem (13), we introduce a small change of the variable z = x 3 ⁄ ε . For ( x ′ , x 3 ) in Ω ε , we have ( x ′ , z ) in

Ω = { ( x ′ , z ) ∈ R 3 : ( x ′ , 0 ) ∈ ω , 0 < z < h ( x ′ ) } ,

and we denote by ∂ Ω ¯ = ∂ Ω ¯ 1 ∪ ∂ Ω ¯ L ∪ ω ¯ its boundary.

Following this scaling, we define the following unknown functions in Ω

(16) u ˆ i ε ( x ′ , z , t ) = u i ε ( x ′ , x 3 , t ) , i = 1 , 2 , u ˆ 3 ε ( x ′ , z , t ) = ε − 1 u 3 ε ( x ′ , x 3 , t ) , π ˆ ε ( x ′ , z , t ) = ε 2 π ε ( x ′ , x 3 , t ) .

For the data of problem (13), it is assumed that they depend on ε as follows:

(17) f ˆ ( x ′ , z , t ) = ε 2 f ε ( x ′ , x 3 , t ) , k ˆ = ε k ε , α ˆ = ε 2 α ε , r ˆ = ε 2 r ε ,

and f ˆ , k ˆ , α ˆ , and r ˆ do not depend on ε .

We now introduce the functional framework on Ω as follows:

V = { φ ∈ H 1 ( Ω ) 3 : φ = 0 on ∂ Ω 1 ∪ ∂ Ω L and φ . n = 0 on ω } , V div = { φ ∈ V : div ( φ ) = 0 } , Π ( V ) = { φ ∈ H 1 ( Ω ) 2 : φ = ( φ 1 , φ 2 ) , φ i = 0 on ∂ Ω 1 ∪ ∂ Ω L , i = 1 , 2 } , L 0 2 ( Ω ) = φ ∈ L 2 ( Ω ) : ∫ Ω φ d x ′ d z = 0 .

Now, let

V z = { v = ( v 1 , v 2 ) ∈ L 2 ( Ω ) 2 : ∂ z v i ∈ L 2 ( Ω ) , i = 1 , 2 and v = 0 on ∂ Ω 1 } ,

be the Banach space with the norm

∥ v ∥ V z = ∑ i = 1 2 ( ∥ v i ∥ L 2 ( Ω ) 2 + ∥ ∂ z v i ∥ L 2 ( Ω ) 2 ) 1 2 .

From (16) and (17), we deduce that problem (13) is equivalent to the following variational inequality:

Find ( u ˆ ε , π ˆ ε ) ∈ L 2 ( 0 , T ; V div ) × L 2 ( 0 , T ; L 0 2 ( Ω ) ) , such that

(18) ε 2 ∑ i = 1 2 ( ρ ∂ t u ˆ i ε , φ ˆ i − u ˆ i ε ) + ε 4 ( ρ ∂ t u ˆ 3 ε , φ ˆ 3 − u ˆ 3 ε ) + μ A ˘ ( u ε , φ ˆ − u ˆ ε ) + ν A ˘ ( ∂ t u ˆ ε , φ ˆ − u ˆ ε ) A ˘ ( ϕ ⋆ u ˆ ε , φ ˆ − u ˆ ε ) + ε ℬ ˘ ( u ˆ ε , u ˆ ε , φ ˆ − u ˆ ε ) + ∑ i = 1 2 ( ( r ˆ + α ˆ ∣ u ˆ i ε ∣ p − 2 ) u ˆ i ε , φ ˆ i − u ˆ i ε ) + ε 2 ( ( r ˆ + α ˆ ε p − 2 ∣ u ˆ 3 ε ∣ p − 2 ) u ˆ 3 ε , φ ˆ 3 − u ˆ 3 ε ) − ( π ˆ ε , div ( φ ˆ ) ) + J ˆ ( φ ˆ ) − J ˆ ( u ˆ ε ) ≥ ∑ i = 1 2 ( f ˆ i , φ ˆ i − u ˆ i ε ) + ε ( f ˆ 3 , φ ˆ 3 − u ˆ 3 ε ) , ∀ φ ˆ ∈ V , u ˆ ε ( x ′ , z , 0 ) = u ˆ 0 , ∀ ( x ′ , z ) ∈ Ω ,

where

J ˆ ( φ ˆ ) = ∫ ω k ˆ ∣ φ ˆ ∣ d x ′ , A ˘ ( u ˆ ε , φ ˆ − u ˆ ε ) = ε 2 ∑ i , j = 1 2 ∫ Ω ( ∂ x j u ˆ i ε ) . ∂ x j ( φ ˆ i − u ˆ i ε ) d x ′ d z + ∑ i = 1 2 ∫ Ω [ ∂ z u ˆ i ε . ∂ z ( φ ˆ i − u ˆ i ε ) + ε 4 ∂ x i u ˆ 3 ε . ∂ x i ( φ ˆ 3 − u ˆ 3 ε ) ] d x ′ d z + ε 2 ∫ Ω ∂ z u ˆ 3 ε ∂ z ( φ ˆ 3 − u ˆ 3 ε ) d x ′ d z , ℬ ˘ ( u ˆ ε , u ˆ ε , φ ˆ − u ˆ ε ) = ε 2 ∑ i , j = 1 2 ∫ Ω u ˆ i ε . ∂ x i u ˆ j ε . ( φ ˆ j − u ˆ j ε ) d x ′ d z + ε 4 ∑ i = 1 2 ∫ Ω u ˆ i ε . ∂ x i u ˆ 3 ε . ( φ ˆ 3 − u ˆ 3 ε ) d x ′ d z + ε 2 ∑ i = 1 2 ∫ Ω u ˆ 3 ε . ∂ z u ˆ i ε . ( φ ˆ i − u ˆ i ε ) d x ′ d z + ε 4 ∑ i = 1 2 ∫ Ω u ˆ 3 ε . ∂ z u ˆ 3 ε . ( φ ˆ 3 − u ˆ 3 ε ) d x ′ d z ,

and

( π ˆ ε , div ( φ ˆ ) ) = ∑ j = 1 2 ∫ Ω π ˆ ε . ∂ x j φ ˆ j d x ′ d z + ∫ Ω π ˆ ε . ∂ z φ ˆ 3 d x ′ d z .

5.1 Uniform estimates

First, we will obtain estimates on u ˆ ε and ∂ t u ˆ ε .

Theorem 2

Let ( u ε , π ε ) ∈ L 2 ( 0 , T ; V div ε ) × L 2 ( 0 , T ; L 0 2 ( Ω ε ) ) be a solution to the problem (13), where k ε ∈ L + ∞ ( ω ) and u 0 ∈ L 2 ( p − 1 ) ( Ω ) 3 ∩ H 1 ( Ω ) 3 . There exists a constant c independent of ε such that

(19) ∑ i = 1 2 ( ∥ ∂ z u ˆ i ε ∥ L 2 ( Ω ) 2 + ∥ ε 2 ∂ x i u ˆ 3 ε ∥ L 2 ( Ω ) 2 ) + ∑ i , j = 1 2 ∥ ε ∂ x j u ˆ i ε ∥ L 2 ( Ω ) 2 + ∥ ε ∂ z u ˆ 3 ε ∥ L 2 ( Ω ) 2 + ∑ i = 1 2 ∥ u ˆ i ε ∥ L p ( 0 , T ; L p ( Ω ) ) p + ∥ ε u ˆ 3 ε ∥ L p ( 0 , T ; L p ( Ω ) ) p ≤ c ,

(20) ∑ i = 1 2 ( ∥ ∂ z ( ∂ t u ˆ i ε ) ∥ L 2 ( Ω ) 2 + ∥ ε 2 ∂ x i ( ∂ t u ˆ 3 ε ) ∥ L 2 ( Ω ) 2 ) + ∑ i , j = 1 2 ∥ ε ∂ x j ( ∂ t u ˆ i ε ) ∥ L 2 ( Ω ) 2 + ∥ ε ∂ z ( ∂ t u ˆ 3 ε ) ∥ L 2 ( Ω ) 2 + ∑ i = 1 2 ∥ ε ∂ t u ˆ i ε ∥ L 2 ( Ω ) 2 + ∥ ε 2 ∂ t u ˆ 3 ε ∥ L 2 ( Ω ) 2 ≤ c .

Proof

Choosing φ = ( 0 , 0 , 0 ) in (13), we have

( ρ ∂ t u ε , u ε ) + μ A ( u ε , u ε ) + ν A ( ∂ t u ε , u ε ) + ε ℬ ( u ε , u ε , u ε ) + A ( ϕ ⋆ u ε , u ε ) + ( ( r ε + α ε ∣ u ˆ ε ∣ p − 2 ) u ε , u ε ) + j ε ( u ε ) ≤ ( f ε , u ε ) ,

by using the identity ℬ ( u ε , u ε , u ε ) = 0 , and the fact that j ε ( u ε ) ≥ 0 , we deduce that

ρ 2 d d t ( ∥ u ε ( t ) ∥ L 2 ( Ω ε ) 3 2 + ν A ( u ε ( t ) , u ε ( t ) ) ) + μ A ( u ε ( t ) , u ε ( t ) ) + A ( ϕ ⋆ u ε ( t ) , u ε ( t ) ) + ( ( r ε + α ε ∣ u ˆ ε ( t ) ∣ p − 2 ) u ε ( t ) , u ε ( t ) ) ≤ ∥ f ε ( t ) ∥ L 2 ( Ω ε ) 3 ∥ u ε ( t ) ∥ L 2 ( Ω ε ) 3 .

Now, integration from 0 to t , we obtain

ρ 2 ∥ u ε ( t ) ∥ L 2 ( Ω ε ) 3 2 + μ ∫ 0 t ∥ ∇ u ε ( s ) ∥ L 2 ( Ω ε ) 3 × 3 2 d s + ν ∥ ∇ u ε ( t ) ∥ L 2 ( Ω ε ) 2 + ∫ 0 t A ( ( ϕ ∗ u ε ) ( s ) , u ε ( s ) ) d s + α ε ∫ 0 t ∥ u ε ( s ) ∥ L p ( Ω ε ) 3 p d s ≤ ρ 2 ∥ u 0 ε ∥ L 2 ( Ω ε ) 3 2 + ν ∥ ∇ u 0 ε ∥ L 2 ( Ω ε ) 3 × 3 2 + ∫ 0 t ∥ f ε ( s ) ∥ L 2 ( Ω ε ) 3 ∥ u ε ( s ) ∥ L 2 ( Ω ε ) 3 d s ,

since ϕ ( t ) is a positive kernel, we obtain

(21) ρ 2 ∥ u ε ∥ L 2 ( Ω ε ) 3 2 + μ ∫ 0 t ∥ ∇ u ε ( s ) ∥ L 2 ( Ω ε ) 3 × 3 2 d s + ν ∥ ∇ u ε ∥ L 2 ( Ω ε ) 2 + α ε ∫ 0 t ∥ u ε ( s ) ∥ L p ( Ω ε ) 3 p d s ≤ ρ 2 ∥ u 0 ε ∥ L 2 ( Ω ε ) 3 2 + ν ∥ ∇ u 0 ε ∥ L 2 ( Ω ε ) 3 × 3 2 + ∫ 0 t ∥ f ε ( s ) ∥ L 2 ( Ω ε ) 3 ∥ u ε ( s ) ∥ L 2 ( Ω ε ) 3 d s .

On the other hand, we have by Poincaré’s inequality (1) and Young’s inequality (2)

∫ 0 t ∥ f ε ( s ) ∥ L 2 ( Ω ε ) 3 ∥ u ε ( s ) ∥ L 2 ( Ω ε ) 3 d s ≤ 1 μ ( ε h ¯ ) 2 ∫ 0 t ∥ f ε ( s ) ∥ L 2 ( Ω ε ) 3 2 d s + μ ∫ 0 t ∥ ∇ u ε ( s ) ∥ L 2 ( Ω ε ) 3 × 3 2 d s .

Thus, the inequality (21) becomes

(22) ν 2 ∥ ∇ u ε ∥ L 2 ( Ω ε ) 3 × 3 2 + α ε ∫ 0 t ∥ u ε ( s ) ∥ L p ( Ω ε ) 3 p d s ≤ 1 μ ( ε h ¯ ) 2 ∥ f ε ∥ L 2 ( 0 , T ; L 2 ( Ω ε ) 3 ) 2 + 1 2 ∥ u 0 ε ∥ L 2 ( Ω ε ) 3 2 + 1 2 ν ∥ ∇ u 0 ε ∥ L 2 ( Ω ε ) 3 × 3 2 .

Now, since

ε 2 ∥ f ε ∥ L 2 ( 0 , T ; L 2 ( Ω ε ) 3 ) 2 = ε − 1 ∥ f ˆ ∥ L 2 ( 0 , T ; L 2 ( Ω ) 3 ) 2 , ∥ ∂ x 3 u i ε ∥ L 2 ( Ω ε ) 2 = ε − 1 ∥ ∂ z u ˆ i ε ∥ L 2 ( Ω ) 2 , i = 1 , 2 ,

we multiply the inequality (22) by ε , and we deduce that

(23) ν 2 [ ε ∥ ∇ u ε ∥ L 2 ( Ω ε ) 3 × 3 2 ] + α ˆ 1 ε ∥ u ε ∥ L p ( 0 , T ; L p ( Ω ε ) 3 ) p ≤ A ,

where

A = 1 μ h ¯ 2 ∥ f ˆ ∥ L 2 ( 0 , T ; L 2 ( Ω ) 3 ) 2 + 1 2 α ˆ ∥ u ˆ 0 ∥ L 2 ( Ω ) 3 2 + 1 2 ν ∥ ∇ u ˆ 0 ∥ L 2 ( Ω ) 3 × 3 2 .

From (23), we find (19) after the change of variables.

To find the estimate (20), we differentiate (15), respect to t and we take φ = ∂ t u ξ ε , we obtain

ρ ( ∂ t 2 u ξ ε ( t ) , ∂ t u ξ ε ( t ) ) + μ A ( ∂ t u ξ ε ( t ) , ∂ t u ξ ε ( t ) ) + ν A ( ∂ t 2 u ξ ε ( t ) , ∂ t u ξ ε ( t ) ) + A ( ϕ ′ ∗ u ξ ε ( t ) , ∂ t u ξ ε ( t ) ) + ϕ ( 0 ) ( ∇ u ξ ε ( t ) , ∇ ∂ t u ξ ε ( t ) ) + ε ℬ ( ∂ t u ξ ε ( t ) , u ξ ε ( t ) , ∂ t u ξ ε ( t ) ) + ε ℬ ( u ξ ε ( t ) , ∂ t u ξ ε ( t ) , ∂ t u ξ ε ( t ) ) + ( ( r ε + ( p − 1 ) α ε ∣ u ξ ε ( t ) ∣ p − 2 ) ∂ t u ξ ε ( t ) , ∂ t u ξ ε ( t ) ) + ( ∂ t ( j ξ ε ) ′ ( u ξ ε ) , ∂ t u ξ ε ( t ) ) = ( ∂ t f ε ( t ) , ∂ t u ξ ε ( t ) ) ,

as ℬ ( u ξ ε ( t ) , ∂ t u ξ ε ( t ) , ∂ t u ξ ε ( t ) ) = 0 , and A ( ∂ t u ξ ε ( t ) , ∂ t u ξ ε ( t ) ) , ( ( r ε + ( p − 1 ) α ε ∣ u ξ ε ( t ) ∣ p − 2 ) ∂ t u ξ ε ( t ) , ∂ t u ξ ε ( t ) ) , and ( ( j ξ ε ) ′ ( u ξ ε ( t ) ) , ∂ t u ξ ε ( t ) ) are positive terms, we have

(24) ρ 2 d d t ( ∥ ∂ t u ξ ε ( t ) ∥ L 2 ( Ω ε ) 3 2 + ν ∥ ∇ ∂ t u ξ ε ( t ) ∥ L 2 ( Ω ε ) 3 × 3 2 ) + ∫ 0 t ( ϕ ′ ( t − s ) ∇ u ξ ε ( s ) , ∂ t u ξ ε ( t ) ) d s + ϕ ( 0 ) ( ∇ u ξ ε ( t ) , ∇ ( ∂ t u ξ ε ( t ) ) ) + ε ℬ ( ∂ t u ξ ε ( t ) , u ξ ε ( t ) , ∂ t u ξ ε ( t ) ) ≤ ∥ ∂ t f ε ( t ) ∥ L 2 ( Ω ε ) 3 ∥ ∂ t u ξ ε ( t ) ∥ L 2 ( Ω ε ) 3 .

By simple calculation, we have

(25) ∫ 0 t ϕ ′ ( t − s ) ∫ Ω ε ∇ u ε ( s ) . ∇ ∂ t u ε ( t ) d x d s = − 1 2 d d t ∫ 0 t ϕ ′ ( t − s ) ∥ ∇ u ε ( s ) − ∇ u ε ( t ) ∥ L 2 ( Ω ε ) 2 d s + ∫ 0 t ϕ ″ ( t − s ) ∥ ∇ u ε ( s ) − ∇ u ε ( t ) ∥ L 2 ( Ω ε ) 2 d s + 1 2 d d t ∫ 0 t ϕ ′ ( s ) d s . ∥ ∇ u ε ( t ) ∥ L 2 ( Ω ε ) 2 − 1 2 ϕ ′ ( t ) ∥ ∇ u ε ( t ) ∥ L 2 ( Ω ε ) 2 .

By combining (21) and (25), we obtain

ρ 2 d d t ∥ ∂ t u ξ ε ( t ) ∥ L 2 ( Ω ε ) 3 2 + ν A ( ∂ t u ξ ε ( t ) , ∂ t u ξ ε ( t ) ) + 1 2 ∫ 0 t ϕ ′ ( s ) d s . ∥ ∇ u ε ( t ) ∥ L 2 ( Ω ε ) 2 + ϕ ( 0 ) 2 ∥ ∇ ∂ t u ξ ε ( t ) ∥ L 2 ( Ω ε ) 3 2 − 1 2 ∫ 0 t ϕ ′ ( t − s ) ∥ ∇ u ε ( s ) − ∇ u ε ( t ) ∥ L 2 ( Ω ε ) 2 d s + ∫ 0 t ϕ ″ ( t − s ) ∥ ∇ u ε ( s ) − ∇ u ε ( t ) ∥ L 2 ( Ω ε ) 2 d s − 1 2 ϕ ′ ( t ) ∥ ∇ u ε ( t ) ∥ L 2 ( Ω ε ) 2 + ε ℬ ( ∂ t u ξ ε ( t ) , u ξ ε ( t ) , ∂ t u ξ ε ( t ) ) ≤ ∥ ∂ t f ε ( t ) ∥ L 2 ( Ω ε ) 3 . ∥ ∂ t u ξ ε ( t ) ∥ L 2 ( Ω ε ) 3 ,

and using the fact that ϕ ′ ( t ) < 0 and ϕ ″ ( t ) > 0 leads to

ρ 2 d d t ∥ ∂ t u ξ ε ( t ) ∥ L 2 ( Ω ε ) 3 2 + ( ν + ϕ ( 0 ) ) ∥ ∇ ∂ t u ξ ε ( t ) ∥ L 2 ( Ω ε ) 3 2 + 1 2 ϕ ( t ) . ∥ ∇ u ε ( t ) ∥ L 2 ( Ω ε ) 2 − 1 2 ϕ ( 0 ) . ∥ ∇ u 0 ε ∥ L 2 ( Ω ε ) 2 − 1 2 ∫ 0 t ϕ ′ ( t − s ) ∥ ∇ u ε ( s ) − ∇ u ε ( t ) ∥ L 2 ( Ω ε ) 2 d s + ε ℬ ( ∂ t u ξ ε ( t ) , u ξ ε ( t ) , ∂ t u ξ ε ( t ) ) ≤ ∥ ∂ t f ε ( t ) ∥ L 2 ( Ω ε ) 3 . ∥ ∂ t u ξ ε ( t ) ∥ L 2 ( Ω ε ) 3 .

Then, by integrating this inequality over ( 0 , t ) , we find

(26) ρ 2 ∥ ∂ t u ξ ε ∥ L 2 ( Ω ε ) 3 2 + ( ν + ϕ ( 0 ) ) ∥ ∇ ∂ t u ξ ε ∥ L 2 ( Ω ε ) 3 × 3 2 − 1 2 ∫ 0 t ϕ ′ ( t − s ) ∥ ∇ u ε ( s ) − ∇ u ε ( t ) ∥ L 2 ( Ω ε ) 2 d s ≤ ( ε h ¯ ) 2 ν ∫ 0 t ∥ ∂ t f ε ( s ) ∥ L 2 ( Ω ε ) 3 2 d s + ν ∫ 0 t ∥ ∇ ∂ t u ξ ε ( s ) ∥ L 2 ( Ω ε ) 3 × 3 2 d s + 1 2 ϕ ( 0 ) . ∥ ∇ u 0 ∥ L 2 ( Ω ε ) 2 + ρ 2 ∥ ∂ t u ξ ε ( 0 ) ∥ L 2 ( Ω ε ) 3 2 + ( ν + ϕ ( 0 ) ) ∥ ∇ ∂ t u ξ ε ( 0 ) ∥ L 2 ( Ω ε ) 3 × 3 2 + ∫ 0 t ε ∣ ℬ ( ∂ t u ξ ε ( s ) , u ξ ε ( s ) , ∂ t u ξ ε ( s ) ) ∣ d s .

Next, we estimate ∣ ℬ ( ∂ t u ξ ε ( s ) , u ξ ε ( s ) , ∂ t u ξ ε ( s ) ) ∣ using Cauchy-Schwartz and Sobolev’s inequalities, and we obtain

ε ∫ 0 t ∣ ℬ ( ∂ t u ξ ε ( s ) , u ξ ε ( s ) , ∂ t u ξ ε ( s ) ) ∣ d s ≤ ε ∫ 0 t ∥ ∂ t u ξ ε ( s ) ∥ L 3 ( Ω ε ) 3 ∥ ∇ u ξ ε ( s ) ∥ L 2 ( Ω ε ) 3 ∥ ∂ t u ξ ε ( s ) ∥ L 6 ( Ω ε ) 3 d s ≤ c s ∫ 0 t ε ∥ ∇ u ξ ε ( s ) ∥ L 2 ( Ω ε ) 3 × 3 ∥ ∇ ∂ t u ξ ε ( s ) ∥ L 2 ( Ω ε ) 3 × 3 2 d s ,

By the aforementioned estimates, the inequality (26) leads to

(27) ρ 2 ∥ ∂ t u ξ ε ∥ L 2 ( Ω ε ) 3 2 + ( ν + ϕ ( 0 ) ) ∥ ∇ ∂ t u ξ ε ∥ L 2 ( Ω ε ) 3 × 3 2 ≤ ( ε h ¯ ) 2 ν ∫ 0 t ∥ ∂ t f ε ( s ) ∥ L 2 ( Ω ε ) 3 2 d s + ∫ 0 t ( ν + c s ε ∥ ∇ u ξ ε ( s ) ∥ L 2 ( Ω ε ) 3 × 3 ) ∥ ∇ ∂ t u ξ ε ( s ) ∥ L 2 ( Ω ε ) 3 × 3 2 d s + ρ 2 + ν + ϕ ( 0 ) [ ∥ ∂ t u ξ ε ( 0 ) ∥ L 2 ( Ω ε ) 3 2 + ∥ ∇ ∂ t u ξ ε ( 0 ) ∥ L 2 ( Ω ε ) 3 × 3 2 ] .

It remains for us to estimate ∂ t u ξ ε ( 0 ) . From (14) and (15), we deduce

ρ ( ∂ t u ξ ε ( 0 ) , φ ) + ν A ( ∂ t u ξ ε ( 0 ) , φ ) + μ A ( u ξ ε ( 0 ) , φ ) + ε ℬ ( u ξ ε ( 0 ) , u ξ ε ( 0 ) , φ ) + α ε ( ∣ u ξ ε ( 0 ) ∣ p − 2 u ξ ε ( 0 ) , φ ) = ( f ε ( 0 ) , φ ) , for all φ ∈ H 0 1 ( Ω ε ) 3 .

By using Cauchy-Schwarz’s and Poincaré’s inequalities (1), we obtain

∣ ρ ( ∂ t u ξ ε ( 0 ) , φ ) + ν A ( ∂ t u ξ ε ( 0 ) , φ ) ∣ ≤ μ ∥ ∇ u 0 ε ∥ L 2 ( Ω ε ) 3 × 3 ∥ ∇ φ ∥ L 2 ( Ω ε ) 3 × 3 + c s ε ∥ ∇ u 0 ε ∥ L 2 ( Ω ε ) 3 × 3 2 ∥ ∇ φ ∥ L 2 ( Ω ε ) 3 × 3 + α ˆ h ¯ ε ∫ Ω ∣ u ˆ 0 ∣ 2 ( p − 1 ) d x ′ d z 1 2 ∥ ∇ φ ∥ L 2 ( Ω ε ) 3 + ε h ¯ ∥ f ε ( 0 ) ∥ L 2 ( Ω ε ) 3 ∥ ∇ φ ∥ L 2 ( Ω ε ) 3 × 3 .

Therefore,

∣ ⟨ ∂ t u ξ ε ( 0 ) , φ ⟩ H 1 ( Ω ε ) 3 ∣ ≤ ( μ ∥ ∇ u 0 ε ∥ L 2 ( Ω ε ) 3 × 3 + α ˆ h ¯ 2 ∥ ∇ u 0 ε ∥ L 2 ( Ω ε ) 3 × 3 + c s ε ∥ ∇ u 0 ε ∥ L 2 ( Ω ε ) 3 × 3 2 ) ∥ φ ∥ H 1 ( Ω ε ) 3 + α ˆ h ¯ ε ∫ Ω ∣ u ˆ 0 ∣ 2 ( p − 1 ) d x ′ d z 1 2 + ε h ¯ ∥ f ε ( 0 ) ∥ L 2 ( Ω ε ) 3 ∥ φ ∥ H 1 ( Ω ε ) 3 .

We multiply this inequality by ε , we obtain

ε ∥ ∂ t u ξ ε ( 0 ) ∥ H 1 ( Ω ε ) 3 ≤ c 0 ,

where c 0 = ( μ + α ˆ h ¯ 2 ) ∥ u ˆ 0 ∥ H 1 ( Ω ) 3 + c s ( α ˆ h ¯ + 1 ) ∥ u ˆ 0 ∥ H 1 ( Ω ) 3 2 + h ¯ ∥ f ˆ ( 0 ) ∥ L 2 ( Ω ) 3 is a constant independent of ε .

Now, passing to the limit in (27) when ξ tends to zero, we deduce that

ρ 2 ∥ ∂ t u ε ∥ L 2 ( Ω ε ) 3 2 + ( ν + ϕ ( 0 ) ) ∥ ∇ ∂ t u ε ∥ L 2 ( Ω ε ) 3 × 3 2 ≤ ∫ 0 t 1 2 ∥ ∂ t u ε ( s ) ∥ L 2 ( Ω ε ) 3 2 + ( ν + ϕ ( 0 ) + c s ε ∥ ∇ u ε ( s ) ∥ L 2 ( Ω ε ) 3 × 3 ) ∥ ∇ ∂ t u ε ( s ) ∥ L 2 ( Ω ε ) 3 × 3 2 d s + ( ε h ¯ ) 2 ν ∥ ∂ t f ε ∥ L 2 ( 0 , T ; L 2 ( Ω ε ) 3 ) 2 + ρ 1 2 + ν + ϕ ( 0 ) ( c 0 ) 2 .

We multiply this inequality by ε and use the fact that ε ∥ ∇ u ε ( s ) ∥ L 2 ( Ω ε ) 3 × 3 ≤ c 1 , and we find

ε 1 2 ∥ ∂ t u ε ∥ L 2 ( Ω ε ) 3 2 + ν ∥ ∇ ∂ t u ε ∥ L 2 ( Ω ε ) 3 × 3 2 ≤ ∫ 0 t ε 1 2 ∥ ∂ t u ε ( s ) ∥ L 2 ( Ω ε ) 3 2 + ( ν + ϕ ( 0 ) + c 1 . c s ) ∥ ∇ ∂ t u ε ( s ) ∥ L 2 ( Ω ε ) 3 × 3 2 d s + B ,

where B = h ¯ 2 ν ∥ ∂ t f ˆ ∥ L 2 ( 0 , T ; L 2 ( Ω ) 3 ) 2 + ρ 1 2 + ν + ϕ ( 0 ) ( c 0 ′ ) 2 is a constant independent of ε .

By using the Gronwall lemma, we complete the proof.□

Now, we are looking for a priori estimates on the pressure π ˆ ε . For this, we need to establish the following result.

Theorem 3

Under the hypotheses of Theorem 2, there exists a constant c independent of ε such that

(28) ∥ ∂ x 1 π ˆ ε ∥ L 2 ( 0 , T ; H − 1 ( Ω ) ) ≤ c ,

(29) ∥ ∂ x 2 π ˆ ε ∥ L 2 ( 0 , T ; H − 1 ( Ω ) ) ≤ c ,

(30) ∥ ∂ z π ˆ ε ∥ L 2 ( 0 , T ; H − 1 ( Ω ) ) ≤ ε . c .

Proof

Let ψ in L 2 ( 0 , T ; H 0 1 ( Ω ε ) 3 ) , putting in (13), φ = u ε + ψ , we deduce

( π ε , div ( ψ ) ) = ρ ( ∂ t u ε , ψ ) + μ A ( u ε , ψ ) + ν A ( ∂ t u ε , ψ ) + A ( ϕ ⋆ u ε , ψ ) + ε ℬ ( u ε , u ε , ψ ) + ( r ε + α ε ∣ u ε ∣ p − 2 u ε , ψ ) + ( f ε , ψ ) .

After using Cauchy-Schwarz, Poincaré, Sobolev’s, and Young’s inequalities, one obtains

(31) ∫ 0 t ( π ε , div ( ψ ) ) d s ≤ μ + r ˆ h ¯ 2 + ∫ 0 t ϕ ( s ) d s 2 ∫ 0 t ∥ ∇ u ε ( s ) ∥ L 2 ( Ω ε ) 3 × 3 2 + ( ( ρ h ¯ ) 2 + ν ) ∫ 0 t ∥ ∇ ∂ t u ε ( s ) ∥ L 2 ( Ω ε ) 3 × 3 2 d s + ( ε h ¯ ) 2 μ + ν ∫ 0 t ∥ f ε ( s ) ∥ L 2 ( Ω ε ) 3 2 d s + ( 1 + 2 μ + 2 ν + r ˆ h ¯ 2 ) ∫ 0 t ∥ ∇ ψ ( s ) ∥ L 2 ( Ω ε ) 3 × 3 2 d s + c s ∫ 0 t ε ∥ ∇ u ε ( s ) ∥ L 2 ( Ω ε ) 3 × 3 2 ∥ ∇ ψ ( s ) ∥ L 2 ( Ω ε ) 3 × 3 d s + α ε ∫ 0 t ( ∣ u ε ( s ) ∣ p − 2 u ε ( s ) , ψ ( s ) ) d s .

The last term of (31) can be estimated by

α ε ∫ 0 t ∫ Ω ε ∣ ∣ u ε ( s ) ∣ p − 2 u ε ( s ) ψ ∣ d x ′ d x 3 d s ≤ α ˆ ε 2 ∫ 0 t ∫ Ω ε ∣ u ε ( s ) ∣ p d x ′ d x 3 1 q ∫ Ω ε ∣ ψ ∣ p d x ′ d x 3 1 p d s , 1 p + 1 q = 1 ≤ α ˆ ε ∫ 0 t 1 ε ∥ u ε ( s ) ∥ L p ( Ω ε ) 3 p d s + α ˆ ε 2 ∫ 0 t ∥ ψ ( s ) ∥ L p ( Ω ε ) 3 p d s ≤ 1 ε c ″ + α ˆ ε 2 ∫ 0 t ∥ ψ ( s ) ∥ L p ( Ω ε ) 3 p d s .

Finally, we obtain

(32) ∫ 0 t ( π ε , div ( ψ ) ) d s ≤ μ + r ˆ h ¯ 2 + ∫ 0 t ϕ ( s ) d s 2 ∫ 0 t ∥ ∇ u ε ( s ) ∥ L 2 ( Ω ε ) 3 × 3 2 + ( ( ρ h ¯ ) 2 + ν ) ∫ 0 t ∥ ∇ ∂ t u ε ( s ) ∥ L 2 ( Ω ε ) 3 × 3 2 d s + ( ε h ¯ ) 2 μ + ν ∫ 0 t ∥ f ε ( s ) ∥ L 2 ( Ω ε ) 3 2 d s + ( 1 + 2 μ + 2 ν + c s ) ∫ 0 t ∥ ∇ ψ ( s ) ∥ L 2 ( Ω ε ) 3 × 3 2 d s + c s ∫ 0 t ε ∥ ∇ u ε ( s ) ∥ L 2 ( Ω ε ) 3 × 3 4 d s + 1 ε c ″ + α ˆ ε 2 ∫ 0 t ∥ ψ ( s ) ∥ L p ( Ω ε ) 3 p d s .

Now, we choose ψ = ( ψ 1 , 0 , 0 ) , ψ = ( 0 , ψ 2 , 0 ) , and ψ = ( 0 , 0 , ψ 3 ) , respectively, in (32), then change variables, to obtain (28)–(30).□

5.2 The limit problem

The objective of this paragraph is to find the limit behavior of the sequences ( u ˆ ε ) and ( π ˆ ε ) as ε → 0 . We will demonstrate the weak convergence of these sequences and give the limit problem that characterizes these limits.

Lemma 2

Under the hypotheses of Theorems 2 and 3, there exists u i ∗ ∈ L 2 ( 0 , T ; V z ) , i = 1 , 2 and π ∗ ∈ L 2 ( 0 , T ; L 0 2 ( Ω ) ) such that

(33) u ˆ i ε ⇀ u i ∗ , i = 1 , 2 , weakly in L 2 ( 0 , T ; V z ) ∂ t u ˆ i ε ⇀ ∂ t u i ∗ , i = 1 , 2 , weakly i n L 2 ( 0 , T ; V z ) ,

(34) ∣ u ˆ i ε ∣ p − 2 u ˆ i ε ⇀ ∣ u i ∗ ∣ p − 2 u i ∗ , i = 1 , 2 , weakly i n L q ( 0 , T ; L q ( Ω ) ) ,

(35) ε ∂ x j u ˆ i ε ⇀ 0 , ε ∂ x j ( ∂ t u ˆ i ε ) ⇀ 0 , i , j = 1 , 2 , weakly i n L 2 ( 0 , T ; L 2 ( Ω ) ) ,

(36) ε 2 ∂ x i u ˆ 3 ε ⇀ 0 , ε ∂ x i ( ∂ t u ˆ 3 ε ) ⇀ 0 , i = 1 , 2 , weakly i n L 2 ( 0 , T ; L 2 ( Ω ) ) ,

(37) ε ∂ z u ˆ 3 ε ⇀ 0 , ε ∂ z ( ∂ t u ˆ 3 ε ) ⇀ 0 , weakly i n L 2 ( 0 , T ; L 2 ( Ω ) ) ,

(38) ε u ˆ 3 ε ⇀ 0 , ε ∂ t u ˆ 3 ε ⇀ 0 , weakly i n L 2 ( 0 , T ; L 2 ( Ω ) ) ,

(39) π ˆ ε ⇀ π ∗ , weakly i n L 2 ( 0 , T ; L 0 2 ( Ω ) ) .

Proof

By Theorem 2, there exists a constant C T independent of ε such that

∥ ∂ z u ˆ i ε ∥ L 2 ( 0 , T ; L 2 ( Ω ) ) 2 ≤ C T , ∥ ∂ z ( ∂ t u ˆ i ε ) ∥ L 2 ( 0 , T ; L 2 ( Ω ) ) 2 ≤ C T , i = 1 , 2 .

Thus, by using these estimates with Poincaré’s inequality in the domain Ω ( ∥ u ˆ i ε ∥ L 2 ( 0 , T ; L 2 ( Ω ) ) ≤ h ¯ ∥ ∂ z u ˆ i ε ∥ L 2 ( 0 , T ; L 2 ( Ω ) ) ), we obtain (33). We obtain (35)–(36) according to (19), (20), and (33). For (34), we have

∫ 0 T ∫ Ω ( ∣ ∣ u ˆ i ε ∣ p − 2 u ˆ i ε ∣ ) q d x ′ d z d s ≤ ∫ 0 T ∥ u ˆ i ε ( s ) ∥ L p ( Ω ) p d s ≤ C ,

so ∣ u ˆ i ε ∣ p − 2 u ˆ i ε ⇀ ∣ u i ∗ ∣ p − 2 u i ∗ , i = 1 , 2 a.e in L q ( 0 , T ; L q ( Ω ) ) . Because div ( u ˆ ε ) = 0 , by (20) and with a particular choice of the test function, we obtain (37) and (38) [19]. To find (39), we use (28)–(30) and the following inequality:

□ ∥ π ˆ ε ∥ L 2 ( 0 , T ; L 2 ( Ω ) ) ≤ c ′ ∥ ∇ π ˆ ε ∥ L 2 ( 0 , T ; H − 1 ( Ω ) 3 ) .

Theorem 4

The weak limit ( u ∗ , π ∗ ) satisfies the following properties:

(40) π ∗ ( x ′ , z , t ) = π ∗ ( x ′ , t ) , a.e in Ω ,

(41) ∫ Ω π ∗ ( ∂ x 1 u 1 ∗ + ∂ x 2 u 2 ∗ ) d x ′ d z = 0 .

Also ( u ∗ , π ∗ ) is characterized as solution of the variational inequality and the limit problem

(42) ∑ i = 1 2 ∫ Ω μ ∂ z u i ∗ . ∂ z ( φ ˆ i − u i ∗ ) d x ′ d z + ∑ i = 1 2 ∫ Ω ν ∂ z ( ∂ t u i ∗ ) . ∂ z ( φ ˆ i − u i ∗ ) d x ′ d z + ∑ i = 1 2 ∫ Ω ∫ 0 t ϕ ( t − s ) ∂ z u i ∗ ( s ) d s . ∂ z ( φ ˆ i − u i ∗ ) d x ′ d z + ∑ i = 1 2 ∫ Ω ( r ˆ + α ˆ ∣ u i ∗ ∣ p − 2 ) u i ∗ . ( φ ˆ i − u i ∗ ) d x ′ d z − ∫ Ω π ∗ ( ∂ x 1 φ ˆ 1 + ∂ x 2 φ ˆ 2 ) d x ′ d z + ∫ ω k ˆ ( ∣ φ ˆ ∣ − ∣ u ∗ ∣ ) d x ′ ≥ ∑ i = 1 2 ∫ Ω f ˆ i . ( φ ˆ i − u i ∗ ) d x ′ d z , ∀ φ ˆ ∈ Π ( V ) ,

(43) − ∂ z 2 [ μ u i ∗ + ν ∂ t u i ∗ + ϕ ⋆ u i ∗ ] ( t ) + ( r ˆ + α ˆ ∣ u i ∗ ∣ p − 2 ) u i ∗ ( t ) + ∂ x i π ∗ ( t ) = f ˆ i ( t ) , i = 1 , 2 a.e on Ω × ( 0 , T ) .

Proof

By choosing φ i = u ˆ i ε ( i = 1 , 2 ) , φ 3 = u ˆ 3 ε ± ψ , where ψ ∈ L 2 ( 0 , T ; H 0 1 ( Ω ) ) in (18), and using the convergence results (33)–(39), we deduce

∫ 0 t ( π ∗ ( s ) , ∂ z ψ ( s ) ) d s = − ∫ 0 t ∫ Ω ε ∂ π ∗ ( s ) ∂ z ψ ( s ) d x ′ d z d s = 0 , ∀ ψ ∈ L 2 ( 0 , T ; H 0 1 ( Ω ) ) .

Therefore π ∗ depends only on x ′ .

Let θ ∈ D ( ] 0 , T [ × ω ) , and by using the fact div ( u ˆ ε ) = 0 in Ω , we have

∫ 0 t ∫ Ω θ ( x ′ , s ) ( ∂ x 1 u ˆ 1 ε + ∂ x 2 u ˆ 2 ε + ∂ z u ˆ 3 ε ) d x ′ d z d s = ∫ 0 t ∫ Ω θ ( x ′ , s ) ( ∂ x 1 u ˆ 1 ε + ∂ x 2 u ˆ 2 ε ) d x ′ d z d s = 0 .

As u ˆ i ε ⇀ u i ∗ , i = 1 , 2 in L 2 ( 0 , T ; V z ) , we obtain

(44) ∫ 0 t ∫ Ω θ ( x ′ , s ) ( ∂ x 1 u 1 ∗ + ∂ x 2 u 2 ∗ ) d x ′ d z d s = 0 .

Now, using (40), π ∗ is now in L 2 ( 0 , T ; L 2 ( w ) ) , and this means that there exists ( θ m ) in D ( ] 0 , T [ × ω ) such that θ m → π ∗ in L 2 ( 0 , T ; L 2 ( w ) ) , and then from (44), we obtain (41) when m → ∞ .

By passing the limit in (18), using the result of Lemma 2, and the fact that J ˆ ( ⋅ ) is convex and lower semi-continuous, we obtain

∑ i = 1 2 ∫ Ω μ ∂ z u i ∗ . ∂ z ( φ ˆ i − u i ∗ ) d x ′ d z + ∑ i = 1 2 ∫ Ω ν ∂ z ( ∂ t u i ∗ ) . ∂ z ( φ ˆ i − u i ∗ ) d x ′ d z ∑ i = 1 2 ∫ Ω ∫ 0 t ϕ ( t − s ) ⋆ ∂ z u i ∗ ( s ) d s . ∂ z ( φ ˆ i − u i ∗ ) d x ′ d z + ∑ i = 1 2 ∫ Ω ( r ˆ + α ˆ ∣ u i ∗ ∣ p − 2 ) u i ∗ ( φ ˆ i − u i ∗ ) d x ′ d z − ∫ Ω π ∗ ( ∂ x 1 φ ˆ 1 + ∂ x 2 φ ˆ 2 ) d x ′ d z + ∫ ω k ˆ ( ∣ φ ˆ ∣ − ∣ u ∗ ∣ ) d x ′ ≥ ∑ i = 1 2 ∫ Ω f ˆ i . ( φ ˆ i − u i ∗ ) d x ′ d z , ∀ φ ˆ ∈ Π ( V ) .

Now, by choosing the aforementioned variational inequality,

φ ˆ i = u i ∗ ± w i , where w i ∈ H 0 1 ( Ω ) , i = 1 , 2 ,

We find

∑ i = 1 2 ∫ Ω [ μ ∂ z u i ∗ + ν ∂ z ( ∂ t u i ∗ ) + ϕ ⋆ ∂ z u i ∗ ] . ∂ z w i d x ′ d z + ∑ i = 1 2 ∫ Ω ( r ˆ + α ˆ ∣ u i ∗ ∣ p − 2 ) u i ∗ . w i d x ′ d z − ∑ i = 1 2 ∫ Ω π ∗ ∂ x i w i d x ′ d z = ∑ i = 1 2 ∫ Ω f ˆ i w i d x ′ d z .

Using the Green’s formula and choosing w 1 = 0 and w 2 ∈ H 0 1 ( Ω ) , then w 2 = 0 and w 1 ∈ H 0 1 ( Ω ) , we obtain

− ∫ Ω ∂ z [ μ ∂ z u i ∗ ( t ) + ν ∂ z ( ∂ t u i ∗ ( t ) ) + ϕ ⋆ ∂ z u i ∗ ( t ) ] w i d x ′ d z + ∑ i = 1 2 ∫ Ω ( r ˆ + α ˆ ∣ u i ∗ ∣ p − 2 ) u i ∗ ( t ) w i d x ′ d z + ∫ Ω ∂ x i π ∗ ( t ) w i d x ′ d z = ∫ Ω f ˆ i ( t ) w i d x ′ d z , ∀ w i ∈ H 0 1 ( Ω ) , i = 1 , 2 .

Thus

(45) − ∂ z 2 [ μ u i ∗ ( t ) + ν ∂ t u i ∗ ( t ) + ϕ ⋆ u i ∗ ( t ) ] + ( r ˆ + α ˆ ∣ u i ∗ ∣ p − 2 ) u i ∗ ( t ) + ∂ x i π ∗ ( t ) = f ˆ i ( t ) , in H − 1 ( Ω ) , for i = 1 , 2 .

As f ˆ i ( t ) ∈ L 2 ( Ω ) , then (45) is valid in L 2 ( Ω ) .□

5.3 A specific Reynolds equation and the limit boundary conditions

Theorem 5

Let

τ ∗ ( x ′ , t ) = ∂ z u ∗ ( x ′ , 0 , t ) and ℓ ∗ ( x ′ , t ) = u ∗ ( x ′ , 0 , t ) ,

with the traces of the velocity u ∗ on ω . Under the assumptions of Lemma 3.5, the solution u ∗ and π ∗ satisfy the weak generalized of Reynolds equation:

(46) ∫ ω h 3 ( x ′ ) 12 ∇ π ∗ − h ( x ′ ) 2 ( μ ℓ ∗ + ν ∂ t ℓ ∗ + ϕ ⋆ ℓ ∗ ) + ∫ 0 h ( μ u ∗ + ν ∂ t u ∗ + ϕ ⋆ u ∗ ) ( x ′ , ζ , t ) d ζ + F ˜ + U ˜ ∇ ψ d x ′ = 0 , ∀ ψ ∈ H 1 ( ω ) 2 ,

with

F ˜ ( x ′ , h , t ) = ∫ 0 h F ( x ′ , z , t ) d z − h 2 F ( x ′ , h , t ) , U ˜ ( x , h , t ) = − ∫ 0 h U ( x , z , t ) d z + α ˆ h 2 U ( x , h , t ) , F ( x ′ , z , t ) = ∫ 0 z ∫ 0 ζ f ˆ ( x ′ , η , t ) d η d ζ , U ( x , z , t ) = ∫ 0 z ∫ 0 ζ ( r ˆ + α ˆ ∣ u ∗ ∣ p − 2 ) u ∗ ( x , η , t ) d η d ζ .

Moreover, the traces τ ∗ and ℓ ∗ satisfy the following inequality:

(47) ∫ ω k ˆ ( ∣ ψ + ℓ ∗ ( t ) ∣ − ∣ ℓ ∗ ( t ) ∣ ) d x ′ − ∫ ω ( μ τ ∗ ( t ) + ν ∂ t τ ∗ ( t ) + ϕ ⋆ τ ∗ ( t ) ) ψ d x ′ ≥ 0 , ∀ ψ ∈ L 2 ( ω ) 2 ,

and the following limit form of the Tresca boundary conditions

(48) μ τ ∗ ( t ) + ν ∂ t τ ∗ ( t ) + ϕ ⋆ τ ∗ ( t ) < k ˆ ⇒ ℓ ∗ ( t ) = 0 , μ τ ∗ ( t ) + ν ∂ t τ ∗ ( t ) + ϕ ⋆ τ ∗ ( t ) = k ˆ ⇒ ∃ λ > 0 : ℓ ∗ ( t ) = λ μ τ ∗ ( t ) + ν ∂ t τ ∗ ( t ) + ϕ ⋆ τ ∗ ( t ) a.e o n ω × ] 0 , T [ .

Proof

To prove (46), we integrate (43) over ( 0 , z ) , and we note that

− μ ∂ z u i ∗ − ν ∂ z ( ∂ t u i ∗ ) − ϕ ⋆ ∂ z u i ∗ + μ ∂ z u i ∗ ( x ′ , 0 , t ) + ν ∂ z ( ∂ t u i ∗ ) ( x ′ , 0 , t ) + ϕ ⋆ ∂ z u i ∗ ( x ′ , 0 , t ) + z ∂ x i π ∗ = − ∫ 0 z ( r ˆ + α ˆ ∣ u i ∗ ∣ p − 2 ) u i ∗ ( x ′ , η , t ) d η + ∫ 0 z f ˆ i ( x ′ , η , t ) d η .

By integrating for the second time between 0 and z , we obtain

(49) [ μ u i ∗ + ν ∂ t u i ∗ + ϕ ⋆ u i ∗ ] ( x ′ , z , t ) = μ z τ i ∗ + ν z ∂ t τ i ∗ + ϕ ⋆ τ i ∗ + μ ℓ i ∗ + ν ∂ t ℓ i ∗ + ϕ ⋆ ℓ i ∗ + z 2 2 ∂ x i π ∗ + ∫ 0 z ∫ 0 ζ ( r ˆ + α ˆ ∣ u i ∗ ∣ p − 2 ) u i ∗ ( x ′ , η , t ) d η d ζ − ∫ 0 z ∫ 0 ζ f ˆ i ( x ′ , η , t ) d η d ζ .

We replace z by h ( x ′ ) , and hence,

(50) h ( x ′ ) [ μ τ i ∗ + ν ∂ t τ i ∗ + ϕ ⋆ τ i ∗ ] + μ ℓ i ∗ + ν ∂ t ℓ i ∗ + ϕ ⋆ ℓ i ∗ + h ( x ′ ) 2 2 ∂ x i π ∗ + ∫ 0 z ∫ 0 ζ ( r ˆ + α ˆ ∣ u i ∗ ∣ p − 2 ) u i ∗ ( x ′ , η , t ) d η d ζ = ∫ 0 h ∫ 0 ζ f i ( x ′ , η , t ) d η d ζ .

By integrating (49) from 0 to h ( x ′ ) , we obtain

(51) μ ∫ 0 h u i ∗ ( x ′ , ζ , t ) d ζ + ν ∫ 0 h ∂ t u i ∗ ( x ′ , ζ , t ) d ζ + ∫ 0 h ϕ ⋆ u i ∗ ( x ′ , ζ , t ) d ζ = h ( x ′ ) [ μ ℓ i ∗ + ν ∂ t ℓ i ∗ + ϕ ⋆ ℓ i ∗ ] + h ( x ′ ) 2 2 [ μ τ i ∗ + ν ∂ t τ i ∗ + ϕ ⋆ τ i ∗ ] + h ( x ′ ) 3 6 ∂ x i π ∗ + ∫ 0 h ∫ 0 z ∫ 0 ζ ( r ˆ + α ˆ ∣ u i ∗ ∣ p − 2 ) u i ∗ ( x ′ , η , t ) d η d ζ d z − ∫ 0 h ∫ 0 z ∫ 0 ζ f ˆ i ( x ′ , η , t ) d η d ζ d z .

From (50) and (51), we deduce (46).

By choosing in (42) φ = ( u 1 ∗ + ψ 1 , u 2 ∗ + ψ 2 ) , where ψ ∈ Π ( V ) 2 , we obtain

∫ ω k ˆ ( ∣ ψ + ℓ ∗ ∣ − ∣ ℓ ∗ ∣ ) d x ′ ≥ − ∑ i = 1 2 ∫ Ω [ μ ∂ z u i ∗ ( t ) + ν ∂ z ( ∂ t u i ∗ ( t ) ) + ϕ ⋆ ∂ z u i ∗ ( t ) ] ∂ z ψ i d x ′ d z − ∑ i = 1 2 ∫ Ω ( r ˆ + α ˆ ∣ u i ∗ ∣ p − 2 ) u i ∗ . ψ i d x ′ d z + ∑ i = 1 2 ∫ Ω π ∗ ( ∂ x i ψ i ) d x ′ d z + ∑ i = 1 2 ∫ Ω f ˆ i . ψ i d x ′ d z .

By using now the Green formula, equality (43), and the fact that ψ i = 0 on ∂ Ω 1 ∪ ∂ Ω L , we obtain

∫ ω k ˆ ( ∣ ψ + ℓ ∗ ∣ − ∣ ℓ ∗ ∣ ) d x ′ − ∫ ω ( μ τ ∗ ( t ) + ν ∂ t τ ∗ ( t ) + ϕ ⋆ τ ∗ ( t ) ) ψ ≥ 0 , ∀ ψ ∈ Π ( V ) 2 .

This inequality remain valid for any ψ ∈ D ( ω ) 2 , and by density of D ( ω ) 2 in L 2 ( ω ) 2 , it also remain valid for any ψ ∈ L 2 ( ω ) 2 .

For (48), we choose ψ = ± ℓ ∗ in (47), and then we follow the same techniques of [19].□

5.4 Uniqueness

Theorem 6

The solution ( u ∗ , π ∗ ) of the limit problems (42) and (43) is unique in L 2 ( 0 , T ; V z ) × L 2 ( 0 , T ; L 0 2 ( ω ) ) .

Proof

Let us suppose that there exist two solutions ( u ∗ 1 , π ∗ 1 ) and ( u ∗ 2 , π ∗ 2 ) of the limit problems (42) and (43), and we take φ ˆ = u ∗ 2 and φ ˆ = u ∗ 1 from (42), and by summing the two inequalities, we obtain

∑ i = 1 2 μ ∫ Ω ( ∂ z u i ∗ 1 − ∂ z u i ∗ 2 ) . ( ∂ z u i ∗ 1 − ∂ z u i ∗ 2 ) d x ′ d z + ν ∫ Ω ( ∂ z ( ∂ t u i ∗ 1 ) − ∂ z ( ∂ t u i ∗ 2 ) ) . ( ∂ z u i ∗ 1 − ∂ z u i ∗ 2 ) d x ′ d z + ∫ Ω ϕ ⋆ ( ∂ z u i ∗ 1 − ∂ z u i ∗ 2 ) . ( ∂ z u i ∗ 1 − ∂ z u i ∗ 2 ) d x ′ d z + α ˆ ∑ i = 1 2 ∫ Ω [ ( r ˆ + α ˆ ∣ u i ∗ 1 ∣ p − 2 ) u i ∗ 1 − ( r ˆ + α ˆ ∣ u i ∗ 2 ∣ p − 2 ) u i ∗ 2 ] ( u i ∗ 1 − u i ∗ 2 ) d x ′ d z ≤ 0 .

The last term is monotone, and ϕ ( ⋅ ) is positive kernel, and thus, we find

∥ ∂ z ( u ∗ 1 − u ∗ 2 ) ∥ L 2 ( Ω ) 2 2 + 1 2 d d t ∥ ∂ z ( u ∗ 1 − u ∗ 2 ) ∥ L 2 ( Ω ) 2 2 ≤ 0 .

By integrating this inequality over ( 0 , t ) , we obtain

∥ ∂ z ( u ∗ 1 − u ∗ 2 ) ∥ L 2 ( 0 , T ; L 2 ( Ω ) ) 2 2 ≤ 0 .

Now, by Poincaré’s inequality, we find

∥ u ∗ 1 − u ∗ 2 ∥ L 2 ( 0 , T ; V z ) = 0 .

The uniqueness of the π ∗ in L 2 ( 0 , T ; L 0 2 ( ω ) ) is inferred from (46). First, we obtain

∫ 0 t ∫ ω h 3 12 ∇ ( π ∗ 1 ( s ) − π ∗ 2 ( s ) ) ∇ ψ d x ′ d s = 0 .

Then we take ψ = π ∗ 1 − π ∗ 2 , and using Poincaré’s inequality, we obtain

π ∗ 1 = π ∗ 2 , a.e in ω × ] 0 , T [ .

This ends the proof of the uniqueness.□

6 Conclusion

This article investigates the asymptotic behavior of solutions to the Navier-Stokes-Voigt-Brinkman-Forchheimer equations in a thin domain Ω ε ⊂ R 3 , incorporating a memory term and the Tresca friction law. By using the functional theory, we analyze how the solution evolves as one dimension of the domain tends to zero. We first introduce the problem statement and derive the associated variational formulation, and we mentioned the existence and uniqueness of weak solutions for the governing mechanical system. Then, we obtained a priori estimates for the velocity field and pressure, which hold independently of the parameter ε . Furthermore, we derived the specific Reynolds equation linked to variational inequalities and prove the uniqueness of the limit problem. These results contribute to a deeper understanding of the fluid flow behavior in thin domains, with significant applications in fluid mechanics and material science.

Acknowledgements

The researchers would like to thank the Deanship of Graduate Studies and Scientific Research at Qassim University for financial support (QU-APC-2024-9/1).

  1. Funding information: The researchers would like to thank the Deanship of Graduate Studies and Scientific Research at Qassim University for financial support (QU-APC-2024-9/1).

  2. Author contributions: All authors contributed equally to the writing of this article. All authors read and approved the final manuscript.

  3. Conflict of interest: Prof. Salah Boulaaras is a member of the Editorial Board of Demonstratio Mathematica but was not involved in the review process of this article.

  4. Data availability statement: No data were used to support this study.

References

[1] A. P. Oskolkov, The uniqueness and solvability in the large of boundary value problems for the equations of motion of aqueous solutions of polymers, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 38 (1973), 98–136. Search in Google Scholar

[2] M. A. Ebrahimi and E. Lunasin, The Navier-Stokes-Voight model for image inpainting. IMA J. Appl. Math. 78 (2013), no. 3, 869–894. 10.1093/imamat/hxr069Search in Google Scholar

[3] C. T. Anh and N. V. Thanh, On the existence and long-time behavior of solutions to stochastic three-dimensional Navier-Stokes-Voigt equations, Stochastics 91 (2019), 485–513, DOI: https://doi.org/10.1080/17442508.2018.1551396. 10.1080/17442508.2018.1551396Search in Google Scholar

[4] M. C. Zelati and C. G. Gal, Singular limits of Voigt models in fluid dynamics, J. Math. Fluid Mech. 17 (2015), 233–259. 10.1007/s00021-015-0201-1Search in Google Scholar

[5] C. T. Anh and P. T. Trang, Decay rate of solutions to the 3D Navier-Stokes-Voigt equations in Hm space, Appl. Math. Lett. 61 (2016), 1–7. 10.1016/j.aml.2016.04.015Search in Google Scholar

[6] W. J. Layton and L. G. Rebholz, On relaxation times in the Navier-Stokes-Voigt model, Int. J. Comput. Fluid Dyn. 27 (2013), 184–187. 10.1080/10618562.2013.766328Search in Google Scholar

[7] D. A. Nield, The limitations of the Brinkman-Forchheimer equation in modeling flow in a saturated porous medium and at an interface, Int. J. Heat Fluid Flow 12 (1991), no. 3, 269–272. 10.1016/0142-727X(91)90062-ZSearch in Google Scholar

[8] R. C. Givler and S. A. Altobelli. A determination of the effective viscosity for the Brinkman-Forchheimer flow model, J. Fluid Mech. 258 (1994), 355–370. 10.1017/S0022112094003368Search in Google Scholar

[9] J. K. Djoko and P. A. Razafimandimby, Analysis of the Brinkman-Forchheimer equations with slip boundary conditions, Appl. Anal. 93 (2014), no. 7, 1477–1494. 10.1080/00036811.2013.836183Search in Google Scholar

[10] T. T. Leeee, Asymptotic behavior of solutions to 3D convective Brinkman-Forchheimer equations with finite delays, Commun. Korean Math. Soc. 36 (2021), no. 3, 527–548. Search in Google Scholar

[11] X. Cai and Q. Jiu, Weak and strong solutions for the incompressible Navier-Stokes equations with damping, J. Math. Anal. Appl. 343 (2008), 799–809. 10.1016/j.jmaa.2008.01.041Search in Google Scholar

[12] F. Huang and R. Pan, Convergence rate for compressible Euler equations with damping and vacuum, Arch. Ration. Mech. Anal. 166 (2003), 359–376. 10.1007/s00205-002-0234-5Search in Google Scholar

[13] C. T. Anh, D. T. P. Thanh, and N. D. Toan, Global attractors for nonclassical diffusion equations with hereditary memory and a new class of nonlinearities, Ann. Polon. Math. 119 (2017), no. 1, 1–21. 10.4064/ap4015-2-2017Search in Google Scholar

[14] F. Di Plinio, A. Giorgini, V. Pata, and R. Temam, Navier-Stokes-Voigt equations with memory in 3D lacking instantaneous kinematic viscosity, J. Nonlinear Sci. 28 (2018), no. 2, 653–686. 10.1007/s00332-017-9422-1Search in Google Scholar

[15] D. T. P. Thanh and N. D. Toan, Uniform attractors of 3D Navier-Stokes-Voigt equations with memory and singularly oscillating external forces. Evolut. Equ. Control Theory 10 (2021), no. 1, 1. 10.3934/eect.2020039Search in Google Scholar

[16] D. Iftimie, G. Raugel, and G. R. Sell, Navier-Stokes equations in thin 3D domains with Navier boundary conditions, Indiana Univ. Math. J. 56 (2007), no. 3, 1083–1156. 10.1512/iumj.2007.56.2834Search in Google Scholar

[17] R. Temam and M. Ziane, Navier-Stokes equations in thin spherical domains, Optimization methods in partial differential equations (South Hadley, MA, 1996), Contemporary Mathematics, vol. 209, American Mathematical Society, Providence, RI, 1997, pp. 281–314. 10.1090/conm/209/02772Search in Google Scholar

[18] M. Dilmi, A. Benseghir, M. Dilmi, and H. Benseridi, Calculation of Reynolds equation for the generalized non-Newtonian fluids and its asymptotic behavior in a thin domain. Georgian Math. J. 31 (2024), no. 2, 229–242. 10.1515/gmj-2023-2090Search in Google Scholar

[19] G. Bayada and M. Boukrouche, On a free boundary problem for Reynolds equation derived from the Stokes system with Tresca boundary conditions, J. Math. Anal. Appl. 382 (2003), 212–231. 10.1016/S0022-247X(03)00140-9Search in Google Scholar

[20] M. Dilmi, M. Dilmi, and H. Benseridi, Study of generalized Stokes operator in a thin domain with friction law (case p<2), Math. Methods Appl. Sci. 41 (2018), 18. 10.1002/mma.4876Search in Google Scholar

[21] H. Benseridi and M. Dilmi, Some inequalities and asymptotic behavior of a dynamic problem of linear elasticity, Georgian Math. J. 20 (2013), no. 1, 25–41. 10.1515/gmj-2013-0004Search in Google Scholar

[22] M. Dilmi, M. Dilmi, and H. Benseridi, Asymptotic behavior for the elasticity system with a nonlinear dissipative term, Rend. Istit. Mat. Univ. Trieste 51 (2019), 41–60. Search in Google Scholar

[23] M. Dilmi, M. Dilmi, and H. Benseridi, A 3D-2D asymptotic analysis of viscoelastic problem with nonlinear dissipative and source terms, Math. Meth. Appl. Sci. 42 (2019), no. 18, 6505–6521, DOI: https://doi.org/10.1002/mma.5755. 10.1002/mma.5755Search in Google Scholar

[24] M. T. Mohan, On the three dimensional Kelvin-Voigt fluids: Global solvability, exponential stability and exact controllability of Galerkin approximations, Evol. Equ. Control Theory 9 (2020), 301–339. 10.3934/eect.2020007Search in Google Scholar

[25] V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff International Publishing, Leiden, 1976. 10.1007/978-94-010-1537-0Search in Google Scholar

[26] G. Duvaut and J. L. Lions, Les Inéquations en Mécanique et en Physique, Dunod, Paris, 1972. Search in Google Scholar
Received: 2024-06-09
Revised: 2024-10-21
Accepted: 2024-10-30
Published Online: 2024-11-27

© 2024 the author(s), published by De Gruyter

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

Articles in the same Issue

  1. Regular Articles
  2. On the p-fractional Schrödinger-Kirchhoff equations with electromagnetic fields and the Hardy-Littlewood-Sobolev nonlinearity
  3. L-Fuzzy fixed point results in ℱ -metric spaces with applications
  4. Solutions of a coupled system of hybrid boundary value problems with Riesz-Caputo derivative
  5. Nonparametric methods of statistical inference for double-censored data with applications
  6. LADM procedure to find the analytical solutions of the nonlinear fractional dynamics of partial integro-differential equations
  7. Existence of projected solutions for quasi-variational hemivariational inequality
  8. Spectral collocation method for convection-diffusion equation
  9. New local fractional Hermite-Hadamard-type and Ostrowski-type inequalities with generalized Mittag-Leffler kernel for generalized h-preinvex functions
  10. On the asymptotics of eigenvalues for a Sturm-Liouville problem with symmetric single-well potential
  11. On exact rate of convergence of row sequences of multipoint Hermite-Padé approximants
  12. The essential norm of bounded diagonal infinite matrices acting on Banach sequence spaces
  13. Decay rate of the solutions to the Cauchy problem of the Lord Shulman thermoelastic Timoshenko model with distributed delay
  14. Enhancing the accuracy and efficiency of two uniformly convergent numerical solvers for singularly perturbed parabolic convection–diffusion–reaction problems with two small parameters
  15. An inertial shrinking projection self-adaptive algorithm for solving split variational inclusion problems and fixed point problems in Banach spaces
  16. An equation for complex fractional diffusion created by the Struve function with a T-symmetric univalent solution
  17. On the existence and Ulam-Hyers stability for implicit fractional differential equation via fractional integral-type boundary conditions
  18. Some properties of a class of holomorphic functions associated with tangent function
  19. The existence of multiple solutions for a class of upper critical Choquard equation in a bounded domain
  20. On the continuity in q of the family of the limit q-Durrmeyer operators
  21. Results on solutions of several systems of the product type complex partial differential difference equations
  22. On Berezin norm and Berezin number inequalities for sum of operators
  23. Geometric invariants properties of osculating curves under conformal transformation in Euclidean space ℝ3
  24. On a generalization of the Opial inequality
  25. A novel numerical approach to solutions of fractional Bagley-Torvik equation fitted with a fractional integral boundary condition
  26. Holomorphic curves into projective spaces with some special hypersurfaces
  27. On Periodic solutions for implicit nonlinear Caputo tempered fractional differential problems
  28. Approximation of complex q-Beta-Baskakov-Szász-Stancu operators in compact disk
  29. Existence and regularity of solutions for non-autonomous integrodifferential evolution equations involving nonlocal conditions
  30. Jordan left derivations in infinite matrix rings
  31. Nonlinear nonlocal elliptic problems in ℝ3: existence results and qualitative properties
  32. Invariant means and lacunary sequence spaces of order (α, β)
  33. Novel results for two families of multivalued dominated mappings satisfying generalized nonlinear contractive inequalities and applications
  34. Global in time well-posedness of a three-dimensional periodic regularized Boussinesq system
  35. Existence of solutions for nonlinear problems involving mixed fractional derivatives with p(x)-Laplacian operator
  36. Some applications and maximum principles for multi-term time-space fractional parabolic Monge-Ampère equation
  37. On three-dimensional q-Riordan arrays
  38. Some aspects of normal curve on smooth surface under isometry
  39. Mittag-Leffler-Hyers-Ulam stability for a first- and second-order nonlinear differential equations using Fourier transform
  40. Topological structure of the solution sets to non-autonomous evolution inclusions driven by measures on the half-line
  41. Remark on the Daugavet property for complex Banach spaces
  42. Decreasing and complete monotonicity of functions defined by derivatives of completely monotonic function involving trigamma function
  43. Uniqueness of meromorphic functions concerning small functions and derivatives-differences
  44. Asymptotic approximations of Apostol-Frobenius-Euler polynomials of order α in terms of hyperbolic functions
  45. Hyers-Ulam stability of Davison functional equation on restricted domains
  46. Involvement of three successive fractional derivatives in a system of pantograph equations and studying the existence solution and MLU stability
  47. Composition of some positive linear integral operators
  48. On bivariate fractal interpolation for countable data and associated nonlinear fractal operator
  49. Generalized result on the global existence of positive solutions for a parabolic reaction-diffusion model with an m × m diffusion matrix
  50. Online makespan minimization for MapReduce scheduling on multiple parallel machines
  51. The sequential Henstock-Kurzweil delta integral on time scales
  52. On a discrete version of Fejér inequality for α-convex sequences without symmetry condition
  53. Existence of three solutions for two quasilinear Laplacian systems on graphs
  54. Embeddings of anisotropic Sobolev spaces into spaces of anisotropic Hölder-continuous functions
  55. Nilpotent perturbations of m-isometric and m-symmetric tensor products of commuting d-tuples of operators
  56. Characterizations of transcendental entire solutions of trinomial partial differential-difference equations in ℂ2#
  57. Fractional Sturm-Liouville operators on compact star graphs
  58. Exact controllability for nonlinear thermoviscoelastic plate problem
  59. Improved modified gradient-based iterative algorithm and its relaxed version for the complex conjugate and transpose Sylvester matrix equations
  60. Superposition operator problems of Hölder-Lipschitz spaces
  61. A note on λ-analogue of Lah numbers and λ-analogue of r-Lah numbers
  62. Ground state solutions and multiple positive solutions for nonhomogeneous Kirchhoff equation with Berestycki-Lions type conditions
  63. A note on 1-semi-greedy bases in p-Banach spaces with 0 < p ≤ 1
  64. Fixed point results for generalized convex orbital Lipschitz operators
  65. Asymptotic model for the propagation of surface waves on a rotating magnetoelastic half-space
  66. Multiplicity of k-convex solutions for a singular k-Hessian system
  67. Poisson C*-algebra derivations in Poisson C*-algebras
  68. Signal recovery and polynomiographic visualization of modified Noor iteration of operators with property (E)
  69. Approximations to precisely localized supports of solutions for non-linear parabolic p-Laplacian problems
  70. Solving nonlinear fractional differential equations by common fixed point results for a pair of (α, Θ)-type contractions in metric spaces
  71. Pseudo compact almost automorphic solutions to a family of delay differential equations
  72. Periodic measures of fractional stochastic discrete wave equations with nonlinear noise
  73. Asymptotic study of a nonlinear elliptic boundary Steklov problem on a nanostructure
  74. Cramer's rule for a class of coupled Sylvester commutative quaternion matrix equations
  75. Quantitative estimates for perturbed sampling Kantorovich operators in Orlicz spaces
  76. Review Articles
  77. Penalty method for unilateral contact problem with Coulomb's friction in time-fractional derivatives
  78. Differential sandwich theorems for p-valent analytic functions associated with a generalization of the integral operator
  79. Special Issue on Development of Fuzzy Sets and Their Extensions - Part II
  80. Higher-order circular intuitionistic fuzzy time series forecasting methodology: Application of stock change index
  81. Binary relations applied to the fuzzy substructures of quantales under rough environment
  82. Algorithm selection model based on fuzzy multi-criteria decision in big data information mining
  83. A new machine learning approach based on spatial fuzzy data correlation for recognizing sports activities
  84. Benchmarking the efficiency of distribution warehouses using a four-phase integrated PCA-DEA-improved fuzzy SWARA-CoCoSo model for sustainable distribution
  85. Special Issue on Application of Fractional Calculus: Mathematical Modeling and Control - Part II
  86. A study on a type of degenerate poly-Dedekind sums
  87. Efficient scheme for a category of variable-order optimal control problems based on the sixth-kind Chebyshev polynomials
  88. Special Issue on Mathematics for Artificial intelligence and Artificial intelligence for Mathematics
  89. Toward automated hail disaster weather recognition based on spatio-temporal sequence of radar images
  90. The shortest-path and bee colony optimization algorithms for traffic control at single intersection with NetworkX application
  91. Neural network quaternion-based controller for port-Hamiltonian system
  92. Matching ontologies with kernel principle component analysis and evolutionary algorithm
  93. Survey on machine vision-based intelligent water quality monitoring techniques in water treatment plant: Fish activity behavior recognition-based schemes and applications
  94. Artificial intelligence-driven tone recognition of Guzheng: A linear prediction approach
  95. Transformer learning-based neural network algorithms for identification and detection of electronic bullying in social media
  96. Squirrel search algorithm-support vector machine: Assessing civil engineering budgeting course using an SSA-optimized SVM model
  97. Special Issue on International E-Conference on Mathematical and Statistical Sciences - Part I
  98. Some fixed point results on ultrametric spaces endowed with a graph
  99. On the generalized Mellin integral operators
  100. On existence and multiplicity of solutions for a biharmonic problem with weights via Ricceri's theorem
  101. Approximation process of a positive linear operator of hypergeometric type
  102. On Kantorovich variant of Brass-Stancu operators
  103. A higher-dimensional categorical perspective on 2-crossed modules
  104. Special Issue on Some Integral Inequalities, Integral Equations, and Applications - Part I
  105. On parameterized inequalities for fractional multiplicative integrals
  106. On inverse source term for heat equation with memory term
  107. On Fejér-type inequalities for generalized trigonometrically and hyperbolic k-convex functions
  108. New extensions related to Fejér-type inequalities for GA-convex functions
  109. Derivation of Hermite-Hadamard-Jensen-Mercer conticrete inequalities for Atangana-Baleanu fractional integrals by means of majorization
  110. Some Hardy's inequalities on conformable fractional calculus
  111. The novel quadratic phase Fourier S-transform and associated uncertainty principles in the quaternion setting
  112. Special Issue on Recent Developments in Fixed-Point Theory and Applications - Part I
  113. A novel iterative process for numerical reckoning of fixed points via generalized nonlinear mappings with qualitative study
  114. Some new fixed point theorems of α-partially nonexpansive mappings
  115. Generalized Yosida inclusion problem involving multi-valued operator with XOR operation
  116. Periodic and fixed points for mappings in extended b-gauge spaces equipped with a graph
  117. Convergence of Peaceman-Rachford splitting method with Bregman distance for three-block nonconvex nonseparable optimization
  118. Topological structure of the solution sets to neutral evolution inclusions driven by measures
  119. (α, F)-Geraghty-type generalized F-contractions on non-Archimedean fuzzy metric-unlike spaces
  120. Solvability of infinite system of integral equations of Hammerstein type in three variables in tempering sequence spaces c 0 β and ℓ 1 β
  121. Special Issue on Nonlinear Evolution Equations and Their Applications - Part I
  122. Fuzzy fractional delay integro-differential equation with the generalized Atangana-Baleanu fractional derivative
  123. Klein-Gordon potential in characteristic coordinates
  124. Asymptotic analysis of Leray solution for the incompressible NSE with damping
  125. Special Issue on Blow-up Phenomena in Nonlinear Equations of Mathematical Physics - Part I
  126. Long time decay of incompressible convective Brinkman-Forchheimer in L2(ℝ3)
  127. Numerical solution of general order Emden-Fowler-type Pantograph delay differential equations
  128. Global smooth solution to the n-dimensional liquid crystal equations with fractional dissipation
  129. Spectral properties for a system of Dirac equations with nonlinear dependence on the spectral parameter
  130. A memory-type thermoelastic laminated beam with structural damping and microtemperature effects: Well-posedness and general decay
  131. The asymptotic behavior for the Navier-Stokes-Voigt-Brinkman-Forchheimer equations with memory and Tresca friction in a thin domain
  132. Absence of global solutions to wave equations with structural damping and nonlinear memory
  133. Special Issue on Differential Equations and Numerical Analysis - Part I
  134. Vanishing viscosity limit for a one-dimensional viscous conservation law in the presence of two noninteracting shocks
  135. Limiting dynamics for stochastic complex Ginzburg-Landau systems with time-varying delays on unbounded thin domains
  136. A comparison of two nonconforming finite element methods for linear three-field poroelasticity
https://doi.org/10.1515/dema-2024-0084
Keywords for this article
asymptotic behavior; Navier-Stokes-Voigt-Brinkman-Forchheimer equations; Reynolds equation; mathematical model; nonlinear systems; Tresca friction law
Creative Commons
BY 4.0

Articles in the same Issue

  1. Regular Articles
  2. On the p-fractional Schrödinger-Kirchhoff equations with electromagnetic fields and the Hardy-Littlewood-Sobolev nonlinearity
  3. L-Fuzzy fixed point results in ℱ -metric spaces with applications
  4. Solutions of a coupled system of hybrid boundary value problems with Riesz-Caputo derivative
  5. Nonparametric methods of statistical inference for double-censored data with applications
  6. LADM procedure to find the analytical solutions of the nonlinear fractional dynamics of partial integro-differential equations
  7. Existence of projected solutions for quasi-variational hemivariational inequality
  8. Spectral collocation method for convection-diffusion equation
  9. New local fractional Hermite-Hadamard-type and Ostrowski-type inequalities with generalized Mittag-Leffler kernel for generalized h-preinvex functions
  10. On the asymptotics of eigenvalues for a Sturm-Liouville problem with symmetric single-well potential
  11. On exact rate of convergence of row sequences of multipoint Hermite-Padé approximants
  12. The essential norm of bounded diagonal infinite matrices acting on Banach sequence spaces
  13. Decay rate of the solutions to the Cauchy problem of the Lord Shulman thermoelastic Timoshenko model with distributed delay
  14. Enhancing the accuracy and efficiency of two uniformly convergent numerical solvers for singularly perturbed parabolic convection–diffusion–reaction problems with two small parameters
  15. An inertial shrinking projection self-adaptive algorithm for solving split variational inclusion problems and fixed point problems in Banach spaces
  16. An equation for complex fractional diffusion created by the Struve function with a T-symmetric univalent solution
  17. On the existence and Ulam-Hyers stability for implicit fractional differential equation via fractional integral-type boundary conditions
  18. Some properties of a class of holomorphic functions associated with tangent function
  19. The existence of multiple solutions for a class of upper critical Choquard equation in a bounded domain
  20. On the continuity in q of the family of the limit q-Durrmeyer operators
  21. Results on solutions of several systems of the product type complex partial differential difference equations
  22. On Berezin norm and Berezin number inequalities for sum of operators
  23. Geometric invariants properties of osculating curves under conformal transformation in Euclidean space ℝ3
  24. On a generalization of the Opial inequality
  25. A novel numerical approach to solutions of fractional Bagley-Torvik equation fitted with a fractional integral boundary condition
  26. Holomorphic curves into projective spaces with some special hypersurfaces
  27. On Periodic solutions for implicit nonlinear Caputo tempered fractional differential problems
  28. Approximation of complex q-Beta-Baskakov-Szász-Stancu operators in compact disk
  29. Existence and regularity of solutions for non-autonomous integrodifferential evolution equations involving nonlocal conditions
  30. Jordan left derivations in infinite matrix rings
  31. Nonlinear nonlocal elliptic problems in ℝ3: existence results and qualitative properties
  32. Invariant means and lacunary sequence spaces of order (α, β)
  33. Novel results for two families of multivalued dominated mappings satisfying generalized nonlinear contractive inequalities and applications
  34. Global in time well-posedness of a three-dimensional periodic regularized Boussinesq system
  35. Existence of solutions for nonlinear problems involving mixed fractional derivatives with p(x)-Laplacian operator
  36. Some applications and maximum principles for multi-term time-space fractional parabolic Monge-Ampère equation
  37. On three-dimensional q-Riordan arrays
  38. Some aspects of normal curve on smooth surface under isometry
  39. Mittag-Leffler-Hyers-Ulam stability for a first- and second-order nonlinear differential equations using Fourier transform
  40. Topological structure of the solution sets to non-autonomous evolution inclusions driven by measures on the half-line
  41. Remark on the Daugavet property for complex Banach spaces
  42. Decreasing and complete monotonicity of functions defined by derivatives of completely monotonic function involving trigamma function
  43. Uniqueness of meromorphic functions concerning small functions and derivatives-differences
  44. Asymptotic approximations of Apostol-Frobenius-Euler polynomials of order α in terms of hyperbolic functions
  45. Hyers-Ulam stability of Davison functional equation on restricted domains
  46. Involvement of three successive fractional derivatives in a system of pantograph equations and studying the existence solution and MLU stability
  47. Composition of some positive linear integral operators
  48. On bivariate fractal interpolation for countable data and associated nonlinear fractal operator
  49. Generalized result on the global existence of positive solutions for a parabolic reaction-diffusion model with an m × m diffusion matrix
  50. Online makespan minimization for MapReduce scheduling on multiple parallel machines
  51. The sequential Henstock-Kurzweil delta integral on time scales
  52. On a discrete version of Fejér inequality for α-convex sequences without symmetry condition
  53. Existence of three solutions for two quasilinear Laplacian systems on graphs
  54. Embeddings of anisotropic Sobolev spaces into spaces of anisotropic Hölder-continuous functions
  55. Nilpotent perturbations of m-isometric and m-symmetric tensor products of commuting d-tuples of operators
  56. Characterizations of transcendental entire solutions of trinomial partial differential-difference equations in ℂ2#
  57. Fractional Sturm-Liouville operators on compact star graphs
  58. Exact controllability for nonlinear thermoviscoelastic plate problem
  59. Improved modified gradient-based iterative algorithm and its relaxed version for the complex conjugate and transpose Sylvester matrix equations
  60. Superposition operator problems of Hölder-Lipschitz spaces
  61. A note on λ-analogue of Lah numbers and λ-analogue of r-Lah numbers
  62. Ground state solutions and multiple positive solutions for nonhomogeneous Kirchhoff equation with Berestycki-Lions type conditions
  63. A note on 1-semi-greedy bases in p-Banach spaces with 0 < p ≤ 1
  64. Fixed point results for generalized convex orbital Lipschitz operators
  65. Asymptotic model for the propagation of surface waves on a rotating magnetoelastic half-space
  66. Multiplicity of k-convex solutions for a singular k-Hessian system
  67. Poisson C*-algebra derivations in Poisson C*-algebras
  68. Signal recovery and polynomiographic visualization of modified Noor iteration of operators with property (E)
  69. Approximations to precisely localized supports of solutions for non-linear parabolic p-Laplacian problems
  70. Solving nonlinear fractional differential equations by common fixed point results for a pair of (α, Θ)-type contractions in metric spaces
  71. Pseudo compact almost automorphic solutions to a family of delay differential equations
  72. Periodic measures of fractional stochastic discrete wave equations with nonlinear noise
  73. Asymptotic study of a nonlinear elliptic boundary Steklov problem on a nanostructure
  74. Cramer's rule for a class of coupled Sylvester commutative quaternion matrix equations
  75. Quantitative estimates for perturbed sampling Kantorovich operators in Orlicz spaces
  76. Review Articles
  77. Penalty method for unilateral contact problem with Coulomb's friction in time-fractional derivatives
  78. Differential sandwich theorems for p-valent analytic functions associated with a generalization of the integral operator
  79. Special Issue on Development of Fuzzy Sets and Their Extensions - Part II
  80. Higher-order circular intuitionistic fuzzy time series forecasting methodology: Application of stock change index
  81. Binary relations applied to the fuzzy substructures of quantales under rough environment
  82. Algorithm selection model based on fuzzy multi-criteria decision in big data information mining
  83. A new machine learning approach based on spatial fuzzy data correlation for recognizing sports activities
  84. Benchmarking the efficiency of distribution warehouses using a four-phase integrated PCA-DEA-improved fuzzy SWARA-CoCoSo model for sustainable distribution
  85. Special Issue on Application of Fractional Calculus: Mathematical Modeling and Control - Part II
  86. A study on a type of degenerate poly-Dedekind sums
  87. Efficient scheme for a category of variable-order optimal control problems based on the sixth-kind Chebyshev polynomials
  88. Special Issue on Mathematics for Artificial intelligence and Artificial intelligence for Mathematics
  89. Toward automated hail disaster weather recognition based on spatio-temporal sequence of radar images
  90. The shortest-path and bee colony optimization algorithms for traffic control at single intersection with NetworkX application
  91. Neural network quaternion-based controller for port-Hamiltonian system
  92. Matching ontologies with kernel principle component analysis and evolutionary algorithm
  93. Survey on machine vision-based intelligent water quality monitoring techniques in water treatment plant: Fish activity behavior recognition-based schemes and applications
  94. Artificial intelligence-driven tone recognition of Guzheng: A linear prediction approach
  95. Transformer learning-based neural network algorithms for identification and detection of electronic bullying in social media
  96. Squirrel search algorithm-support vector machine: Assessing civil engineering budgeting course using an SSA-optimized SVM model
  97. Special Issue on International E-Conference on Mathematical and Statistical Sciences - Part I
  98. Some fixed point results on ultrametric spaces endowed with a graph
  99. On the generalized Mellin integral operators
  100. On existence and multiplicity of solutions for a biharmonic problem with weights via Ricceri's theorem
  101. Approximation process of a positive linear operator of hypergeometric type
  102. On Kantorovich variant of Brass-Stancu operators
  103. A higher-dimensional categorical perspective on 2-crossed modules
  104. Special Issue on Some Integral Inequalities, Integral Equations, and Applications - Part I
  105. On parameterized inequalities for fractional multiplicative integrals
  106. On inverse source term for heat equation with memory term
  107. On Fejér-type inequalities for generalized trigonometrically and hyperbolic k-convex functions
  108. New extensions related to Fejér-type inequalities for GA-convex functions
  109. Derivation of Hermite-Hadamard-Jensen-Mercer conticrete inequalities for Atangana-Baleanu fractional integrals by means of majorization
  110. Some Hardy's inequalities on conformable fractional calculus
  111. The novel quadratic phase Fourier S-transform and associated uncertainty principles in the quaternion setting
  112. Special Issue on Recent Developments in Fixed-Point Theory and Applications - Part I
  113. A novel iterative process for numerical reckoning of fixed points via generalized nonlinear mappings with qualitative study
  114. Some new fixed point theorems of α-partially nonexpansive mappings
  115. Generalized Yosida inclusion problem involving multi-valued operator with XOR operation
  116. Periodic and fixed points for mappings in extended b-gauge spaces equipped with a graph
  117. Convergence of Peaceman-Rachford splitting method with Bregman distance for three-block nonconvex nonseparable optimization
  118. Topological structure of the solution sets to neutral evolution inclusions driven by measures
  119. (α, F)-Geraghty-type generalized F-contractions on non-Archimedean fuzzy metric-unlike spaces
  120. Solvability of infinite system of integral equations of Hammerstein type in three variables in tempering sequence spaces c 0 β and ℓ 1 β
  121. Special Issue on Nonlinear Evolution Equations and Their Applications - Part I
  122. Fuzzy fractional delay integro-differential equation with the generalized Atangana-Baleanu fractional derivative
  123. Klein-Gordon potential in characteristic coordinates
  124. Asymptotic analysis of Leray solution for the incompressible NSE with damping
  125. Special Issue on Blow-up Phenomena in Nonlinear Equations of Mathematical Physics - Part I
  126. Long time decay of incompressible convective Brinkman-Forchheimer in L2(ℝ3)
  127. Numerical solution of general order Emden-Fowler-type Pantograph delay differential equations
  128. Global smooth solution to the n-dimensional liquid crystal equations with fractional dissipation
  129. Spectral properties for a system of Dirac equations with nonlinear dependence on the spectral parameter
  130. A memory-type thermoelastic laminated beam with structural damping and microtemperature effects: Well-posedness and general decay
  131. The asymptotic behavior for the Navier-Stokes-Voigt-Brinkman-Forchheimer equations with memory and Tresca friction in a thin domain
  132. Absence of global solutions to wave equations with structural damping and nonlinear memory
  133. Special Issue on Differential Equations and Numerical Analysis - Part I
  134. Vanishing viscosity limit for a one-dimensional viscous conservation law in the presence of two noninteracting shocks
  135. Limiting dynamics for stochastic complex Ginzburg-Landau systems with time-varying delays on unbounded thin domains
  136. A comparison of two nonconforming finite element methods for linear three-field poroelasticity
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 13.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/dema-2024-0084/html
Scroll to top button