Subrata Majumdar

ASYMPTOTIC BEHAVIOUR OF THE LINEARIZED COMPRESSIBLE BAROTROPIC NAVIER-STOKES SYSTEM WITH A TIME VARYING DELAY TERM IN THE BOUNDARY OR INTERNAL FEEDBACK

Keywords: 2020 Mathematics Subject Classification. 35B40, 35Q30, 93D15, 93D23 Compressible Navier-Stokes equation, time dependent delay, feedback stabilization, Lyapunov function approach

In this paper, we consider the linearized compressible barotropic Navier-Stokes system in a bounded interval (0, L) with a time-varying delay term acting in the Dirichlet boundary or internal feedback of the hyperbolic component. Assuming some suitable conditions on the time-dependent delay term and the coefficients of feedback (delayed or not), we study the exponential stability of the concerned hyperbolic-parabolic system. Due to the presence of the time-varying delay term, the corresponding spatial operator is also time-dependent. Using classical semigroup theory with Kato's variable norm approach, we first show the existence and uniqueness of the Navier-Stokes system with time delay, acting in the boundary or interior. Next, we prove the two stabilization results by means of interior delay feedback and boundary delay. In both cases, we establish the exponential stability results by introducing some suitable functional energy and using the Lyapunov function approach.

1. Introduction 1.1. Setting of the problem. Control and stability of fluid flow have been a significant topic of study and have numerous useful applications. Many researchers have been interested in the subject of the controllability of fluid flows, more so for incompressible flow (see [START_REF] Barbu | Tangential boundary stabilization of Navier-Stokes equations[END_REF], [START_REF] Fernández-Cara | Local exact controllability of the Navier-Stokes system[END_REF], [START_REF] Fursikov | Stabilization for the 3D Navier-Stokes system by feedback boundary control[END_REF], [START_REF] Raymond | Feedback boundary stabilization of the two-dimensional Navier-Stokes equations[END_REF], [START_REF] Raymond | Feedback boundary stabilization of the three-dimensional incompressible Navier-Stokes equations[END_REF], [START_REF] Vazquez | Control of turbulent and magnetohydrodynamic channel flows, Systems & Control: Foundations & Applications[END_REF]) than for compressible flow (see [START_REF] Collis | Numerical solution of optimal control problems governed by the compressible Navier-Stokes equations[END_REF], [START_REF]Optimal control of unsteady compressible viscous flows[END_REF]). The stability analysis of the linearized compressible Navier-Stokes system is of interest to us in this research.

The Navier-Stokes equations in Ω ⊂ R N for a compressible isothermal barotropic fluid consists of continuity equation and the momentum equation:

∂ t ζ(t, x) + div[ζ(t, x) v(t, x)] = 0,
ζ[∂ t v(t, x) + (v(t, x).∇)v(t, x)] = -∇p(t, x) + ν∆v(t, x) + (λ + ν)∇ (div v(t, x)) , ζ(t, x) and v(t, x) = (v 1 (t, x), v 2 (t, x), ..., v N (t, x)) denote fluid density and velocity vector in R N , p denotes the pressure and we assume that it satisfies the constitutive law p(t, x) = aζ γ (t, x) t > 0, x ∈ Ω for a > 0, γ ≥ 1.

The viscosity coefficients ν, λ are the constants satisfying the thermodynamic restrictions, ν > 0, λ + ν ≥ 0 (see [START_REF] Feireisl | Dynamics of viscous compressible fluids[END_REF] for more details). The second equation can be written component wise as follows

ζ (∂ t v i + v.∇v i ) = -∂ xi p + ν∆v i + (λ + ν)∂ xi [div v].
We write the Navier-Stokes equations for a viscous compressible barotropic fluid in a bounded subset Ω = (0, L), L < ∞ of the real line as

∂ t ζ(t, x) + (ζv) x (t, x) = 0, ζ(t, x)[∂ t v(t, x)+v(t, x)v x (t, x)] + (p(ζ)) x (t, x) -νv xx (t, x) = 0, p(t, x) = aζ γ (t, x) t > 0, x ∈ Ω, (1.1)
here ζ, v, p, ν are fluid density, velocity, pressure and viscosity like above. To get the linearized system around (Q 0 , V 0 ), Q 0 > 0, V 0 ≥ 0, we perform a change of variable:

σ(t, x) = ζ(t, x) -Q 0 , u(t, x) = v(t, x) -V 0 ,
and collect the system including first order terms of σ, u and this leads to the linearized compressible Navier-Stokes system around (Q 0 , V 0 ) as follows σ t + V 0 σ x + Q 0 u x = 0, (t, x) ∈ (0, ∞) × (0, L), u t -ν Q0 u xx + V 0 u x + aγQ γ-2 0 σ x = 0, (t, x) ∈ (0, ∞) × (0, L).

(1.2)

From the perspective of controllability and stabilization, the compressible Navier-stokes equations (1.2) is one of the fascinating topics of research. Lots of work have been done in this direction, and many results have yet to be explored, for example, the stabilizability of Navier-Stokes system (1.2) with Dirichlet boundary data σ(t, 0) = 0, u(t, 0) = u(t, L) = 0.

(1.3)

We refer to the works [START_REF] Chowdhury | Null controllability of the linearized compressible Navier Stokes system in one dimension[END_REF], [START_REF] Chowdhury | Null controllability of the linearized compressible Navier-Stokes equations using moment method[END_REF] [START_REF] Mitra | Largest space for the stabilizability of the linearized compressible Navier-Stokes system in one dimension[END_REF], where controllability and stabilization of the hyperbolic-parabolic coupled system (1.2) by means of interior control with periodic boundary data have been studied. In [START_REF] Beauchard | Null-controllability of linear parabolic transport systems[END_REF] and [START_REF] Koenig | Null-controllability of underactuated linear parabolic-transport systems with constant coefficients[END_REF], these null controllability results have been extended to more general transport-parabolic (d × d) systems with constant coefficients in one dimensional torus T. Null and approximate controllability of (1.2) with Dirichlet boundary data can be found in [START_REF] Bhandari | Boundary null-controllability of 1d linearized compressible navier-stokes system by one control force[END_REF] and [START_REF] Chowdhury | Approximate controllability for linearized compressible barotropic Navier-Stokes system in one and two dimensions[END_REF], respectively. More precisely, boundary null controllability result has been explored in [START_REF] Bhandari | Boundary null-controllability of 1d linearized compressible navier-stokes system by one control force[END_REF] with the control acting in the hyperbolic component (σ(t, 0) = q(t), u(t, 0) = u(t, L) = 0), whereas approximate controllability has been studied with localized interior control in [START_REF] Chowdhury | Approximate controllability for linearized compressible barotropic Navier-Stokes system in one and two dimensions[END_REF]. We must mention that in the work [START_REF] Mitra | Largest space for the stabilizability of the linearized compressible Navier-Stokes system in one dimension[END_REF], the authors studied the rapid feedback stabilization (exponential stabilization with arbitrary prescribed decay rate) of (1.2) with an interior control acting everywhere in the parabolic equation in periodic set-up. Boundary stabilizability result of (1.2) with V 0 = 0 with certain decay ω 0 has been studied in [START_REF] Arfaoui | Boundary stabilizability of the linearized viscous Saint-Venant system[END_REF], [START_REF] Chowdhury | Boundary stabilizability of the linearized compressible Navier-Stokes system in one dimension by backstepping approach[END_REF] and [START_REF] Chowdhury | Controllability and stabilizability of the linearized compressible Navier-Stokes system in one dimension[END_REF] in Dirichlet set up (u(t, 0) = u(t, 1) = 0) by spectral analysis and the method of backstepping, respectively.

Let us write the system (1.2) in the abstract formation for Dirichlet boundary conditions (1.3):

U(t) = AU(t), t > 0, U(0) = (σ 0 , u 0 ) T ,
where the operator A is defined as following:

   D(A) = {(σ, u) ∈ H 1 (0, L) × H 2 (0, L) ∩ H 1 0 (0, L) : σ(0) = 0}, A(σ, u) T = -V 0 σ x -Q 0 u x , ν Q0 u xx -V 0 u x -aγQ γ-2 0 σ x T .
The operator A is maximal dissipative in L 2 (0, L) × L 2 (0, L). Thus it generates a contraction semigroup {S(t)} t≥0 of continuous operator. Moreover, the semigroup is exponentially stable, that is for any U 0 = (σ 0 , u 0 ) T ∈ L 2 (0, L) × L 2 (0, L), there exist positive constants C, ω 0 such that

∥S(t)U 0 ∥ L 2 (0,L)×L 2 (0,L) ≤ Ce -ω0t ∥U 0 ∥ L 2 (0,L)×L 2 (0,L) , t ≥ 0,
see [START_REF] Girinon | Quelques problemes aux limites pour les equations de Navier-Stokes compressibles[END_REF] for more details. In this paper, we show that despite the time-varying delay acting in the boundary or interior of the hyperbolic component, the concerned system retains its asymptotic behaviour with the presence of an additional damping term acting in the same component. We explore the time-delayed phenomenon in the asymptotic analysis for the Navier-Stokes system (1.4). More precisely, we study the well-posedness and stability analysis of the linearized compressible barotropic Navier-Stokes system with the effect of delay depending on time acting in the boundary term or as interior feedback. In the next section, we discuss the stability problems which are the main concerns of this article.

1.2. Brief description about the problem. Let us take V 0 > 0 in (1.2). Introducing the change of variables

σ(t, x) → 1 ε σ t υ , x δ , u(t, x) → u t υ , x δ ,
with the following choices of ϵ, υ, δ > 0,

ε := aγQ γ-3 0 -1/2 , υ := Q 0 V 2 0 ν , δ := Q 0 V 0 ν ,
one can recast the system (1.2) into the following simplified version:

σ t + σ x + bu x = 0, (t, x) ∈ (0, ∞) × (0, δL), u t -u xx + u x + bσ x = 0 (t, x) ∈ (0, ∞) × (0, δL), (1.4
)

with b = Q0 V0 aγQ γ-3 0 1/2
. We utilize the advantage of this reduction of the number of system parameters in our computation by considering the system (1.4) in our analysis. To describe the introductory study about the problems, let us first consider the following system:

               σ t + σ x + bu x = 0, (t, x) ∈ (0, ∞) × (0, L), u t -u xx + u x + bσ x = 0, (t, x) ∈ (0, ∞) × (0, L), σ(t, 0) = ασ(t, L) + βσ(t -τ (t), L), u(t, 0) = u(t, L) = 0, t ∈ (0, ∞), σ(0, x) = σ 0 (x), u(0, x) = u 0 (x), x ∈ (0, L), σ(t -τ (0), L) = z 0 (t -τ (0)), 0 < t < τ (0), (1.5)
in which we assume that the time varying delay τ (t) satisfies the following properties

τ ∈ W 2,∞ [0, T ], for all T > 0, (1.6) 0 < τ 0 ≤ τ (t) ≤ M, for all t ≥ 0, (1.7)
τ (t) ≤ m < 1, for all t ≥ 0, where 0 ≤ m < 1.

(1.8)

Under the following condition on the damping parameters α, β:

|α| + |β| + m < 1, (1.9)
we prove that the functional energy associated to the system (1.5)

E(t) = 1 2 L 0 σ 2 (t, x)dx + 1 2 L 0 u 2 (t, x)dx + |β|τ (t) 2 1 0 σ 2 (t -τ (t)ρ, L)dρ, t ≥ 0 (1.10)
decays exponentially to zero. That is, there exist some positive constants C > 0, µ > 0 such that the following is the infinite time behaviour of the energy of the solution of (1.5):

E(t) ≤ CE(0)e -µt , ∀t > 0.
Next, we study the same issue for the linearized compressible Navier-Stokes equation with internal time varying delay

             σ t + σ x + bu x + a(x)σ(t, x) + c(x)σ(t -τ (t), x) = 0, (t, x) ∈ (0, ∞) × (0, L), u t -u xx + u x + bσ x = 0, (t, x) ∈ (0, ∞) × (0, L), u(t, 0) = u(t, L) = σ(t, 0) = 0, t ∈ (0, ∞), σ(0, x) = σ 0 (x), u(0, x) = u 0 (x), x ∈ (0, L), σ(t -τ (0), x) = z 0 (t -τ (0), x),
x ∈ (0, L), 0 < t < τ (0), (1.11) where a, c ∈ L ∞ (0, L) are two non-negative functions with supp c = ω ⊂ (0, L) and c(x) ≥ c 0 a.e. in ω, where c 0 is a positive number, (1.12)

a(x) ≥ k 0 + 2 -m 2(1 -m) c(x) a.e in ω,
where k 0 > 0.

(1.13)

Under the above assumptions on the damping functions a, c, we prove the exponential stability of the system by using the energy

E(t) = 1 2 L 0 σ 2 (t, x)dx + 1 2 L 0 u 2 (t, x)dx + τ (t) 2 ω 1 0 ξ(x)σ 2 (t -τ (t)ρ, x) dx dρ, t ≥ 0,
where ξ ∈ L ∞ (0, L), supp ξ = ω, a non negative function to be chosen later.

1.3. Bibliographical comments and motivation. Delay effects arise in many applications and practical problems and may destabilize an otherwise exponentially stable system, see [START_REF] Lagnese | Decay of solutions of wave equations in a bounded region with boundary dissipation[END_REF], [START_REF] Lagnese | Note on boundary stabilization of wave equations[END_REF], [START_REF] Datko | Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks[END_REF]. However, particular choices of delays may provide the exponential stability. Thus, it is important to explore the impact of the delay in the asymptotic behaviour of dynamical systems. Throughout years, exponential stability analysis of a dynamical systems governed by partial differential equations with time delay has gained immense interest among researchers. In [START_REF] Datko | An example on the effect of time delays in boundary feedback stabilization of wave equations[END_REF], R. F. Datko et al. have shown that an arbitrarily small time delay in the feedback control may destabilize a distributed system described by the wave equation.

Thereafter, numerous number of works have been performed on the stabilization of wave equations with time-delayed feedback controls [START_REF] Nicaise | Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks[END_REF], [START_REF]Stabilization of the wave equation with boundary or internal distributed delay[END_REF], [START_REF] Ammari | Feedback boundary stabilization of wave equations with interior delay[END_REF], [START_REF] Fragnelli | Stability of solutions to nonlinear wave equations with switching time delay[END_REF], [START_REF] Gugat | Boundary feedback stabilization by time delay for one-dimensional wave equations[END_REF], [START_REF]Stability of the wave equation with localized Kelvin-Voigt damping and boundary delay feedback[END_REF], [START_REF] Chentouf | Well posedness and asymptotic behavior of a wave equation with distributed time-delay and Neumann boundary conditions[END_REF], [START_REF]On the exponential and polynomial convergence for a delayed wave equation without displacement[END_REF], [START_REF] Ammari | Asymptotic behavior of a delayed wave equation without displacement term[END_REF]. Internal delay of an abstract wave equation with Kelvin-Voigt damping has been studied in [START_REF]Stability of an abstract-wave equation with delay and a Kelvin-Voigt damping[END_REF]. For semilinear wave type equation see [START_REF] Paolucci | Well-posedness and stability for semilinear wave-type equations with time delay[END_REF], nonlinear wave equation with switching delay see [START_REF] Fragnelli | Stability of solutions to nonlinear wave equations with switching time delay[END_REF]. In the constant time delay case, we refer to the other related works: [START_REF] Nicaise | Stabilization of the Schrödinger equation with a delay term in boundary feedback or internal feedback[END_REF] for Schrodinger equation, [START_REF] Baudouin | Two approaches for the stabilization of nonlinear KdV equation with boundary time-delay feedback[END_REF] for KdV equation with boundary time-delay, [START_REF] Valein | On the asymptotic stability of the korteweg-de vries equation with time-delayed internal feedback[END_REF] for KdV equation with interior delay feedback, [START_REF] Parada | Delayed stabilization of the Korteweg-de Vries equation on a star-shaped network[END_REF] KdV equation with star shaped network, [START_REF] Capistrano-Filho | Two stability results for the kawahara equation with a time-delayed boundary control[END_REF], [START_REF] Capistrano-Filho | Stabilization results for delayed fifth order kdv-type equation in a bounded domain[END_REF], and [START_REF] Chentouf | Well-posedness and exponential stability of the kawahara equation with a time-delayed localized damping[END_REF] for Kawahara equation with boundary and interior time delay feedback, respectively, [START_REF] Kang | Distributed stabilization of Korteweg-de Vries-Burgers equation in the presence of input delay[END_REF] for KdV-Burger equation, Kuramoto-Sivashinsky equation with the time delay in the nonlinear term [START_REF] Zhu | Asymptotic behavior of solutions for the time-delayed Kuramoto-Sivashinsky equation[END_REF], Benjamin-Bona-Mahony equation [START_REF] Feng | Asymptotic of solutions for the time-delayed equations of Benjamin-Bona-Mahony's type[END_REF], microbeam equation [START_REF] Feng | Exponential stabilization of a microbeam system with a boundary or distributed time delay[END_REF], other evolution equation with time delay feedback see [START_REF] Nicaise | Stabilization of second order evolution equations with unbounded feedback with delay[END_REF].

On the other hand, there is a rich literature regarding the stability of the dynamical system with timevarying delay feedback. In [START_REF] Nicaise | Stability of the heat and of the wave equations with boundary time-varying delays[END_REF], S. Nicaise et al. have studied the time-varying delay phenomenon for a parabolic and a hyperbolic model. More precisely, exponential stability has been studied for heat and wave equations in one dimension with boundary time-varying delays. This work motivates us to consider the time-dependent delay effect on the coupled hyperbolic-parabolic model, namely the linearized compressible Navier-Stokes system. To the best of the author's knowledge, there is no such result where the stability analysis of the linearized compressible Navier-Stokes system with time-dependent boundary or interior delay has been taken into account. This is indeed a motivation of this manuscript. Due to the presence of hyperbolic-parabolic coupling of two different orders of PDEs (transport-heat), the existence theory is not straightforward like heat or wave equations cases. We need to adopt a vanishing viscosity technique to establish the well-posedness of the time-dependent semigroup. Moreover, it is worth mentioning that, for the constant time delay case, there are some works related to the exponential decay analysis for coupled hyperbolic-parabolic systems with same orders spatial operators (heat-wave). More precisely, in the works [START_REF] Moulai Khatir | Well-posedness and exponential stability of a thermoelastic system with internal delay[END_REF] and [START_REF] Mustafa | Exponential decay in thermoelastic systems with internal distributed delay[END_REF], the authors have explored the exponential stabilization issue with internal delay acting in the first and second equation, respectively, for the system of linear thermoelasticity.

For the sake of completeness of the bibliographical study, let us mention some other related works for the time delayed system. In [START_REF] Nicaise | Exponential stability of the wave equation with boundary time-varying delay[END_REF], S. Nicaise et al. have extended their previous result [START_REF] Nicaise | Stability of the heat and of the wave equations with boundary time-varying delays[END_REF] for multi-dimensional wave system. Internal feedback stabilization with boundary delay for wave equation has been established [START_REF]Interior feedback stabilization of wave equations with time dependent delay[END_REF]. Recently, in [START_REF] Parada | Stability results for the KdV equation with time-varying delay[END_REF], Parada et al. have utilized the above mentioned works to produce stability analysis of KdV equation with interior and boundary time varying delay. Time-dependent delay phenomenon for second order evolution equation can be found in [START_REF] Fridman | Stabilization of second order evolution equations with unbounded feedback with time-dependent delay[END_REF]. It is worth mentioning that, in the works [START_REF] Caraballo | Navier-Stokes equations with delays[END_REF], [START_REF] Caraballo | Attractors for 2D-Navier-Stokes models with delays[END_REF], [START_REF]Asymptotic behaviour of two-dimensional Navier-Stokes equations with delays[END_REF], the authors have considered incompressible Navier-Stokes equations and studied its asymptotic behaviour with the distributed constant delay term (delay acting in the interior). Boundary time delay effect for the incompressible Navier-Stokes equations and higher dimensional compressible Navier-Stokes equations can be the objects of future research. 1.4. Organization. In the Section 2, we establish the existence and uniqueness of the solution of the linearized Navier-Stokes equation with boundary delay feedback. Exponential stability of the concerned system has been shown here, see Theorem 2.5. We study the well-posedness and stability analysis (Theorem 3.3) of (1.11), that is, the linearized Navier-Stokes system with internal delay in Section 3.

Exponential stability of Navier-Stokes with boundary delay

This section is devoted to the well-posedness and stability analysis of the linearized compressible Navier-Stokes system with a boundary delay feedback. At first, we show the existence and uniqueness of the Navier-Stokes system (1.5) with the presence of boundary delay term. We start by showing the wellposedness of the system (1.5) using semigroup theory and Kato's variable norm technique [START_REF] Kato | Linear and quasi-linear equations of evolution of hyperbolic type[END_REF].

2.1. Well-posedness. We will mainly follow the work [START_REF] Nicaise | Stability of the heat and of the wave equations with boundary time-varying delays[END_REF] to prove the well-posedness of the time varying system (1.5). Let us introduce a new variable to tackle the effect of the delay term

z(t, ρ) = σ(t -τ (t)ρ, L), ρ ∈ (0, 1) and t > 0.
It can be checked that z satisfies the following set of equations

     τ (t)z t + (1 -τ (t)ρ)z ρ = 0, t ∈ (0, ∞), z(t, 0) = σ(t, L), t ∈ (0, ∞), z(0, ρ) = z 0 (-τ (0)ρ),
ρ ∈ (0, 1).

(2.1)

Thus combining (1.5) and (2.1), we write the full system as follows:

                                 σ t + σ x + bu x = 0, (t, x) ∈ (0, ∞) × (0, L), u t -u xx + u x + bσ x = 0, (t, x) ∈ (0, ∞) × (0, L), τ (t)z t + (1 -τ (t)ρ)z ρ = 0, (t, ρ) ∈ (0, ∞) × (0, 1), u(t, 0) = u(t, L) = 0, t ∈ (0, ∞), σ(t, 0) = ασ(t, L) + βσ(t -τ (t), L), t ∈ (0, ∞), z(t, 0) = σ(t, L), t ∈ (0, ∞), σ(0, x) = σ 0 (x), u(0, x) = u 0 (x), x ∈ (0, L), z(0, ρ) = z 0 (-τ (0)ρ), ρ ∈ (0, 1), σ(t -τ (0), L) = z 0 (t -τ (0)), 0 < t < τ (0). (2.2)
Now we will show the existence and uniqueness of the system (2.2) in a semigroup theory framework. Let us first write this system in infinite dimensional ODE set up. Denote

U = (σ, u, z) T , U 0 = (σ 0 , u 0 , z 0 (-τ (0)•)) T .
Thereafter, one can recast the above system as the following abstract ODE:

U(t) = A(t)U(t), t > 0, U(0) = U 0 , (2.3)
where the time dependent operator A(t) can be written as

A(t)U =   -σ x -bu x u xx -u x -bσ x -1-τ (t)ρ τ (t) z ρ   , (2.4)
with the domain

D(A(t)) = (σ, u, z) ∈ H 1 (0, L) × H 2 (0, L) ∩ H 1 0 (0, L) × H 1 (0, 1) : z(0) = σ(L), σ(0) = ασ(L) + βz(1) . (2.5) Clearly D(A(t)) = D(A(0)), ∀ t > 0.
Let us introduce the Hilbert space

H = L 2 (0, L) × L 2 (0, L) × L 2 (0, 1),
equipped with the following inner product

  σ 1 u 1 z 1   ,   σ 2 u 2 z 2   H = L 0 σ 1 σ 2 dx + L 0 u 1 u 2 dx + 1 0 z 1 z 2 dρ. (2.6)
Now, using the variable norm theory of Kato [START_REF]Abstract differential equations and nonlinear mixed problems[END_REF], [START_REF] Kato | Nonlinear semigroups and evolution equations[END_REF], [START_REF] Kato | Linear and quasi-linear equations of evolution of hyperbolic type[END_REF], we will show the existence and uniqueness of the above abstract system with time dependent operator. Similar types of analysis has been done in many works, see [START_REF] Nicaise | Stability of the heat and of the wave equations with boundary time-varying delays[END_REF], [START_REF] Nicaise | Exponential stability of the wave equation with boundary time-varying delay[END_REF], [START_REF] Kong | Exponential stability for magnetic effected piezoelectric beams with time-varying delay and time-dependent weights[END_REF], [START_REF] Parada | Stability results for the KdV equation with time-varying delay[END_REF] for instances. The main idea of the existence-uniqueness theory is to show that the triplet {A, H, D(A(0)} with A = {A(t) : t ∈ [0, T ]}, for some fixed T > 0 forms a constant domain system. More precisely, the following theorem of Kato (see [START_REF] Kato | Linear and quasi-linear equations of evolution of hyperbolic type[END_REF]Theorem 1.9]) is enough to serve our purpose:

Theorem 2.1. Let us assume that

• D(A(0)) is dense in H, • D(A(t)) = D(A(0)), ∀t > 0,
• A(t) generates a strongly continuous semigroup on H for all t ∈ [0, T ] and the family A = {A(t) : t ∈ [0, T ]} is stable with stability constant C and ω 0 independent of t (i.e. the semigroup {S t (s)} s≥0 generated by A(t)) satisfies ∥S t (s)u∥ H ≤ Ce ω0s ∥u∥ H , for all u ∈ H and s ≥ 0),

• ∂ t A(t) belongs to L ∞ * ([0, T ]; B(D(A(0)), H))
, the space of equivalent classes of essentially bounded, strongly measurable functions from [0, T ] into the set B(D(A(0)), H)) of bounded operators from D(A(0)) into H. Then, the system (2.3) has a unique solution U ∈ C([0, T ]; D(A(0))) ∩ C 1 ([0, T ]; H), when the initial data lies in D(A(0)).

Henceforth, we need to verify all the assumptions of the above Theorem 2.1 with the operator (A(t), D(A(0))) defined in (2.4)-(2.5) and this will ensure that the system (2.3) has a unique solution.

First, we define the following time dependent inner product on H:

  σ 1 u 1 z 1   ,   σ 2 u 2 z 2   t = L 0 σ 1 σ 2 dx + L 0 u 1 u 2 dx + τ (t)|β| 1 0 z 1 z 2 dρ.
It can be shown that the norm ∥•∥ H and ∥•∥ t are equivalent. Indeed,

min(1, |β|τ 0 ) ∥(σ, u, z)∥ H ≤ ∥(σ, u, z)∥ t ≤ (1 + |β|M ) ∥(σ, u, z)∥ H . (2.7)
Now, we are in the position of proving the well-posedness of our system (2.3):

Theorem 2.2. For any

U 0 = (σ 0 , u 0 , z 0 (-τ (0)•)) T ∈ H, there exists a unique solution U ∈ C([0, ∞); H) of the Navier-Stokes system (2.3). Moreover, if U 0 ∈ D(A(0)), then the solution U ∈ C([0, ∞); D(A(0))) ∩ C 1 ([0, ∞); H).
Proof. We prove this result by showing all the assumptions of the Theorem 2.1 hold for the system (2.3).

• Straightforward analysis will show that

D(A(0)) is dense in H. Indeed, let (f, g, w) T ∈ H such that (f, g, w) T ∈ D(A(0)) ⊥ .
Thus by definition, we have

L 0 f (x)σ(x)dx + L 0 g(x)u(x)dx + 1 0 z(ρ)w(ρ)dρ = 0, for all (σ, u, z) ∈ D(A(0)).
Let us take σ = 0, u = 0 and z ∈ C ∞ c (0, 1). As (0, 0, z) T ∈ D(A(0)), we have

1 0 z(ρ)w(ρ)dρ = 0.
By density argument, we obtain w = 0. In a similar manner taking

z = 0, σ = 0, u ∈ C ∞ c (0, L) or z = 0, u = 0, σ ∈ C ∞ c (0, L), we can prove that f = g = 0. Therefore we deduce that D(A(0)) ⊥ = {0}. Thus we have D(A(0)) is dense in H.
• By definition, the second conditions of Theorem 2.1 holds.

• Hereinafter, we show that the family A = {A(t), t ∈ [0, T ]} is stable. Let us denote Φ = (σ, u, z) T .

It can be shown that

∥Φ∥ t ∥Φ∥ s ≤ e ĉ 2τ 0 |t-s| , ∀ t, s ∈ [0, T ], (2.8)
where ĉ is a positive constant. Indeed for all t, s ∈ [0, T ], we have

∥Φ∥ 2 t -∥Φ∥ 2 s e ĉ τ 0 |t-s| = (1 -e ĉ τ 0 |t-s| ) L 0 u 2 (x)dx + L 0 σ 2 (x)dx + τ (t) -e ĉ τ 0 |t-s| τ (s) |β| 1 0 z 2 (ρ)dρ. Note that (1 -e ĉ τ 0 |t-s| ) ≤ 0. Furthermore, τ (t) -e ĉ τ 0
|t-s| τ (s) is also non positive for some constant ĉ > 0. Indeed, as τ ∈ W 2,∞ [0, T ], ∀ T > 0, by mean value theorem

τ (t) = τ (s) + (t -s)τ ′ (a), a ∈ (s, t),
and therefore

τ (t) τ (s) ≤ 1 + ĉ τ 0 |t -s| ≤ e ĉ τ 0 |t-s| .
Thus we get the estimate (2.8).

• Next, we will show that A(t) is an infinitesimal generator of a contraction semigroup. To prove this, we first establish that the operator A(t) is maximal dissipative up to some bounded perturbation. First we evaluate ⟨A(t)U, U⟩ t for some fixed t. Let us take U = (σ, u, z) T ∈ D(A(t)). Then we have

⟨A(t)U, U⟩ t =   -σ x -bu x u xx -u x -bσ x -1-τ (t)ρ τ (t) z ρ   ,   σ u z   t .
Performing integration by parts and using the boundary conditions we obtain

⟨A(t)U, U⟩ t = - L 0 σσ x dx -b L 0 σu x dx - L 0 uu x dx + L 0 uu xx dx -b L 0 uσ x dx + |β|τ (t) 1 0 τ (t)ρ -1 τ (t) zz ρ dρ = 1 2 -σ 2 (L) + σ 2 (0) - |β| τ (t) 2 1 0 z 2 dρ + |β|( τ (t) -1) 2 z 2 (1) + |β| 2 z 2 (0) - L 0 u 2 x dx = 1 2 (ασ(L) + βz(1)) 2 - 1 2 σ 2 (L) - |β| τ (t) 2 1 0 z 2 dρ + |β|( τ (t) -1) 2 z 2 (1) + |β| 2 σ 2 (L) - L 0 u 2 x dx.
Let us define the following function

k(t) = ( τ (t) 2 + 1) 1/2 2τ (t) , t ∈ [0, T ].
(2.9)

Thus we further have,

⟨A(t)U, U⟩ t -k(t) ⟨U, U⟩ t = 1 2 α 2 σ 2 (L) + 2αβσ(L)z(1) + β 2 z 2 (1) + 1 2 (|β| -1)σ 2 (L) - L 0 u 2 x dx - |β| τ (t) 2 + k(t) |β|τ (t) 2 1 0 z 2 dρ -k(t) L 0 σ 2 dx -k(t) L 0 u 2 dx + |β|( τ (t) -1) 2 z 2 (1) ≤ 1 2 α 2 σ 2 (L) + 2αβσ(L)z(1) + β 2 z 2 (1) + 1 2 (|β| -1)σ 2 (L) + |β|(m -1) 2 z 2 (1) = 1 2 ⟨BΨ, Ψ⟩ R 2 ,
where Ψ = (σ(L), z(1)) T and

B =   α 2 -1 + |β| αβ αβ β 2 + |β|(m -1)   . (2.10)
To prove

⟨A(t)U, U⟩ t -k(t) ⟨U, U⟩ t ≤ 0,
it is enough to show that B is negative definite matrix. Indeed, the following lemma will prove it. This implies that A(t) = A(t) -k(t)I is a dissipative operator with the inner product ⟨•, •⟩ t .

Lemma 2.3. Under the assumption (1.9), the matrix B defined in (2.10) is negative definite.

Proof. Indeed, thanks to (1.9), we compute:

trace(B) =α 2 -1 + |β| + β 2 + |β|(m -1) <|α| + |β| -1 + |β|m < 0.
In particular α 2 -1 + β < 0. Also,

det(B) =α 2 |β|(m -1) -β 2 -|β|(m -1) + |β| 3 + β 2 (m -1) =|β| α 2 (m -1) -m + 1 -2|β| + β 2 + |β|m =|β| α 2 m -α 2 + (1 -|β|)(1 -|β| -m) > 0.
As B is a symmetric matrix, we write B = Therefore B is negative definite. □

Next, we prove that for all t ∈ [0, T ], A(t) is maximal. Thus we need to prove that for some λ > 0, R(λI -A(t)) = H.

Let us first assume that (f, g, h) T ∈ H. We need to find U = (σ, u, z) T ∈ D(A(t)) such that (λI -A(t))U = (f, g, h) T , which is equivalent to find U = (σ, u, z) T ∈ D(A(t)) such that

                     λσ + σ x + bu x = f, λu -u xx + u x + bσ x = g, λz + 1-τ (t)ρ τ (t) z ρ = h, u(0) = u(L) = 0, σ(0) = ασ(L) + βz(1), z(0) = σ(L).
(2.11)

We mainly follow the work [START_REF] Nicaise | Stability of the heat and of the wave equations with boundary time-varying delays[END_REF] to solve the above problem. If we are able to find u and σ with proper regularity then the third equation of the system (2.11) will give the solution z as follows:

z(ρ) = σ(L)e -λτ (t)ρ + τ (t)e -λτ (t)ρ ρ 0 e λτ (t)s h(s) ds, if τ (t) = 0, σ(L)e λ τ (t) τ (t) ln(1-τ (t)ρ) + e λ τ (t) τ (t) ln(1-τ (t)ρ) ρ 0 e -λ τ (t) τ (t) ln(1-τ (t)s) h(s)τ (t) (1-τ (t)s) ds, if τ (t) ̸ = 0.
Putting ρ = 1, we write z(1) = σ(L)S 0 + S h , where

S 0 = e -λτ (t) , if τ (t) = 0, e λ τ (t) τ (t) ln(1-τ (t)) , if τ (t) ̸ = 0,
and

S h = τ (t)e -λτ (t) 1 0 e λτ (t)s h(s) ds, if τ (t) = 0, e λ τ (t) τ (t) ln(1-τ (t)) 1 0 e -λ τ (t) τ (t) ln(1-τ (t)s) h(s)τ (t)
(1-τ (t)s) ds if τ (t) ̸ = 0. Thus, (σ, u) satisfies the following system

         λσ + σ x + bu x = f, λu -u xx + u x + bσ x = g, u(0) = u(L) = 0, σ(0) = (α + βS 0 )σ(L) + βS h .
(2.12) Let us consider a constant function ψ 0 defined by (with fixed t)

ψ 0 (x) = βS h (t) (1 -α -βS 0 ) ,
and we assume that α = α + βS 0 (t). Let us denote η = σ -ψ 0 . Then, we can see that (η, u) satisfy the following equations

         λη + η x + bu x = f , λu -u xx + u x + bη x = g, u(0) = u(L) = 0, η(0) = αη(L), (2.13)
where f = f -λψ 0 ∈ L 2 (0, L). Thus the problem (2.13) boils down to showing that there exists (η, u)

∈ D = H 1 (0, L) × H 2 (0, L) ∩ H 1 0 (0, L) : η(0) = αη(L) satisfying (2.13
). We solve this problem by using vanishing viscosity approach. Let ϵ > 0. Instead of solving directly the above problem (2.13), we first deal with the following regularized problem

λη -ϵη xx + η x + bu x = f , λu -u xx + bη x + u x = g, (2.14)
with the following boundary conditions

η(0) = αη(L), αη x (0) = η x (L), u(0) = 0, u(L) = 0.
Let us proceed through the following steps.

Step 1. We consider the space V, given by

V = (η, u) ∈ H 1 (0, L) × H 1 (0, L) : η(0) = αη(L), u(0) = 0, u(L) = 0 .
Using Lax-Milgram theorem, we first prove that the system (2.14) has a unique solution in V. Let us define the operator B :

V × V → R by B η u , ψ φ = λ L 0 ηψdx + ϵ L 0 η x ψ x dx + b L 0 u x ψdx + L 0 η x ψdx + λ L 0 uφdx + L 0 u x φ x dx + b L 0 η x φdx + L 0 u x φdx,
for all η u , ψ φ ∈ V. Then, one can show that B is continuous and coercive. Thus, by Lax-Milgram theorem, for every ϵ > 0, there exists a unique solution (η ϵ , u ϵ ) ∈ V such that

B η ϵ u ϵ , ψ φ = G ψ φ , ∀ ψ φ ∈ V, (2.15)
where G : V → R is the linear functional given by

G ψ φ := L 0 f ψdx + L 0 gφdx.
Let us take ψ 0 ∈ V in (2.15), so that we obtain

λ L 0 η ϵ ψdx + ϵ L 0 η ϵ x ψ x dx + b L 0 u ϵ x ψdx + L 0 η ϵ x ψdx = L 0 f ψdx. (2.16)
Similarly, by taking 0

φ ∈ V, we get λ L 0 u ϵ φdx + L 0 u ϵ x φ x dx + b L 0 u ϵ x φdx + b L 0 η ϵ x φdx = L 0 gφdx. (2.17)
These shows that the equations (2.14) are satisfied in the sense of distribution. Consequently it gives the regularity of η ϵ xx and u ϵ xx in L 2 (0, L). Thus the equation (2.14) is satisfied in the strong sense. Now multiplying the first equation of (2.14) by ψ ∈ H 1 (0, L) with ψ(0) = αψ(L), and using integration by parts we obtain

λ L 0 η ϵ ψdx + ϵ L 0 η ϵ x ψ x dx-ϵ (η ϵ x (L)ψ(L) -η ϵ x (0)ψ(0)) + b L 0 u ϵ x ψdx + L 0 η ϵ x ψdx = L 0 f ψdx. (2.18)
Comparing the above equation with (2.16), we obtain η ϵ x (L) = αη ϵ x (0). Step 2. Now, from (2.15), observe that

B η ϵ u ϵ , η ϵ u ϵ = G η ϵ u ϵ .
which yields, applying integration by parts and Young's inequality

λ L 0 |η ϵ | 2 dx + ϵ L 0 |η ϵ x | 2 dx + 1 2 (1 -α 2 )|η ϵ (L)| 2 + λ L 0 |u ϵ | 2 dx + L 0 |u ϵ x | 2 dx ≤ λ 2 L 0 |η ϵ | 2 dx + λ 2 L 0 |u ϵ | 2 dx + C L 0 f 2 dx + L 0 g 2 dx
Note that |α| < 1, by (1.9) and the definition of α. This shows that (u ϵ ) ϵ≥0 is bounded in H 1 (0, L), (η ϵ ) ϵ≥0 is bounded in L 2 (0, L) and ( √ ϵη ϵ x ) ϵ≥0 is bounded in L 2 (0, L). Since the spaces H 1 (0, L) and L 2 (0, L) are reflexive, there exist subsequences, still denoted by (u ϵ ) ϵ≥0 , (η ϵ ) ϵ≥0 , and functions η ∈ L 2 (0, L) and u ∈ H 1 (0, L), l ∈ R such that u ϵ ⇀ u in H 1 (0, L), and η ϵ ⇀ η in L 2 (0, L), η ϵ (L) → l.

(2.19) Furthermore, we have

L 0 |ϵη ϵ x | 2 dx = ϵ L 0 | √ ϵη ϵ x | 2 dx → 0, as ϵ → 0. (2.20)
Multiplying the first equation of (2.14) by ψ ∈ C ∞ c (0, L) and applying integration by parts, we get,

λ L 0 η ϵ ψdx + ϵ L 0 η ϵ x ψ x dx + b L 0 u ϵ x ψdx - L 0 η ϵ ψ x = L 0 f ψdx, ∀ψ ∈ C ∞ c (0, L).
Then, passing to the limit ϵ → 0, we obtain

λ L 0 ηψdx + b L 0 u x ψdx - L 0 ηψ x dx = L 0 f ψdx,
and the above relation is true ∀ψ ∈ C ∞ c (0, L). As a consequence,

η x + λη + bu x = f , (2.21)
in the sense of distribution and therefore η x = f -bu x -λη ∈ L 2 (0, L); in other words, η ∈ H 1 (0, L).

Step 3. We now show u(0) = u(L) = 0. Since the inclusion map i :

H 1 (0, L) → C 0 [0, L] is compact and u ϵ ⇀ u in H 1 (0, L), we obtain u ϵ → u in C 0 [0, L].
Thus, (u ϵ (0), u ϵ (L)) → (u(0), u(L)). Since u ϵ (0) = u ϵ (L) = 0 for all ϵ > 0, we have

u(0) = u(L) = 0.
Similarly from the second equation of (2.14), one can deduce that

λu -u xx + u x + bη x = g, (2.22)
in the sense of distribution and therefore u xx ∈ L 2 (0, L), that is u ∈ H 2 (0, L). Now replacing ψ from the equation of (2.16) by φ ∈ C ∞ c (0, L) and using (2.19), (2.20) and (2.21), we have

lim ϵ→0 L 0 η ϵ x φ = L 0 η ϵ x φ. (2.23)
By density argument, we can say that η ϵ x ⇀ η x in L 2 (0, L). Similarly using the equations (2.17) and (2.22) we have u ϵ ⇀ u in H 2 (0, L). We now show η(0) = αη(L). Let us first denote the space W = (η, u) ∈ H 1 (0, L) × H 2 (0, L) : u(0) = 0, u(L) = 0 .

(2.24)

W is a Hilbert space with graph norm and D is a closed subspace of W. Hence the weak closure coincides with the strong closure of D in W. In our analysis, we have actually shown that (η, u) lies in the weak closure of D as (η ϵ , u ϵ ) ∈ D. Thus it follows that (η, u) ∈ D. In particular, we have η(0) = αη(L).

Therefore we have proved that A(t) is maximal operator for any fixed t ∈ [0, T ]. As k(t) > 0 defined in (2.9), it can be shown that the time dependent operator λI -A = (λ + k(t)) I -A is also surjective for some λ > 0. This shows that A is maximal. Hence A(t) generates a strongly continuous semigroup of contraction on H. Also by (2.8), A = { A(t) : t ∈ [0, T ]} is a stable family of generators in H with suitably constants independent of t.

• We need to establish the last assumption of Theorem 2.1, that is,

∂ t A(t) ∈ L ∞ * ([0, T ]; B(D(A(0)), H)). First we compute ∂ t A(t)U =   0 0 τ (t)τ (t)ρ-τ (t)( τ (t)ρ-1) τ 2 (t) z ρ   . (2.25)
It can be shown that τ (t)τ (t)ρ-τ (t)( τ (t)ρ-1)

τ 2 (t)
z ρ is bounded on [0, T ] by the properties of τ ( see (1.6), (1.7), (1.8)). Next from (2.9), we have

k(t) = τ (t) τ (t) 2τ (t) (1 + τ 2 (t)) 1/2 - τ (t) 1 + τ 2 (t) 1/2 2τ 2 (t) (2.26)
and it is bounded in [0, T ] for any T > 0. Thus combining (2.25) and (2.26), we finally have:

∂ t A(t) ∈ L ∞ * ([0, T ]; B(D(A(0)), H)).
Thanks to the Theorem 2.1, there exists unique solution U ∈ C([0, ∞); D(A(0))) ∩ C 1 ([0, ∞); H) of the following abstract system

∂ t U(t) = A(t) U(t), U(0) = U 0 .
Let us denote U(t) = e t 0 k(s)ds U(t). It can be shown that this U(t) is the unique solution of (2.3). Because, on differentiation we have

∂ t U(t) =k(t)e t 0 k(s)ds U(t) + e t 0 k(s)ds ∂ t U(t) =e t 0 k(s)ds k(t) U(t) + A(t) U(t) =e t 0 k(s)ds A(t) U(t) =A(t)U(t).
Thus the proof is completed. □ 2.2. Boundary stability result. In this section, we prove one of the main results regarding the boundary stability of the system (1.5). It relies on the choice of suitable Lyapunov function and crucial energy estimate.

2.2.1. Energy decay. We recall the energy functional defined in (1.10)

E(t) = 1 2 L 0 σ 2 (t, x)dx + 1 2 L 0 u 2 (t, x)dx + |β|τ (t) 2 1 0 σ 2 (t -τ (t)ρ, L)dρ, t ≥ 0. (2.27)
The choice of this energy functional is classical when we deal with the time-varying delay effect through the boundary, see [START_REF] Nicaise | Stability of the heat and of the wave equations with boundary time-varying delays[END_REF]. Below we prove that this energy (2.27) is non-increasing.

Proposition 2.4. Let us assume that (1.6), (1.7), (1.8) and (1.9) hold. Then for all regular solution of the system (1.5), the energy E in (2.27) is non-increasing and satisfies:

Ė(t) ≤ 1 2 B σ(t, L) σ (t -τ (t), L) , σ(t, L) σ (t -τ (t), L) R 2 - L 0 u 2 x ≤ 0.
Proof. Indeed on differentiation (2.27) with respect to t, we have:

Ė(t) = L 0 σσ t dx+ L 0 uu t dx+ |β| τ (t) 2 1 0 σ 2 (t-τ (t)ρ, L)dρ+|β|τ (t) 1 0 σ(t-τ (t)ρ, L)∂ t σ(t-τ (t)ρ, L)dρ.
(2.28) Plugging the expressions for σ t , u t from (1.5) and applying integration by parts, we have:

L 0 σσ t dx + L 0 uu t dx = 1 2 α 2 σ 2 (t, L) + 2αβσ(t, L)σ (t -τ (t), L) + β 2 σ 2 (t -τ (t), L) - 1 2 σ 2 (t, L) - 1 0 u 2 x dx. (2.29)
Thanks to the identity

-τ (t)∂ t σ(t -τ (t)ρ, L) = (1 -τ (t)ρ)∂ ρ σ(t -τ (t)ρ, L)
and then applying integration by parts, we have

|β|τ (t) 1 0 σ(t -τ (t)ρ, L)∂ t σ(t -τ (t)ρ, L)dρ = -|β| 1 0 (1 -τ (t)ρ)σ(t -τ (t)ρ, L)∂ ρ σ(t -τ (t)ρ, L)dρ = - |β| τ (t) 2 1 0 σ 2 (t -τ (t)ρ, L)dρ + 1 2 |β|σ 2 (t, L) + |β|( τ (t) -1) 2 σ 2 (t -τ (t), L) . (2.30)
Thus combining (2.29) and (2.30), we obtain from (2.28),

Ė(t) = 1 2 α 2 σ 2 (t, L) + 2αβσ(t, L)σ (t -τ (t), L) + β 2 σ 2 (t -τ (t), L) + 1 2 (|β| -1)σ 2 (t, L) + |β|( τ (t) -1) 2 σ 2 (t -τ (t), L) - 1 0 u 2 x dx. (2.31)
Recall the negative definite matrix B in (2.10)

B =   α 2 -1 + |β| αβ αβ β 2 + |β|(m -1)   .
From (2.31), we have further,

Ė(t) ≤ 1 2 B σ(t, L) σ (t -τ (t), L) , σ(t, L) σ (t -τ (t), L) R 2 - 1 0 u 2 x ≤ 0.

□

This result regarding the energy E is not sufficient to address the exponential stability issues of the studied system. In the next section we answer this question by choosing appropriate Lyapunov function.

Exponential stability.

Let us take γ 1 , γ 2 ∈ (0, 1) are positive constants, small enough, to be chosen later. Consider the Lyapunov functional

E(t) = E(t) + γ 1 E 1 (t) + γ 2 E 2 (t), t > 0, γ 1 , γ 2 ∈ (0, 1), (2.32)
where

E 1 (t) = 1 2
L 0 e -λx σ 2 (t, x)dx, for some λ > 0.

(2.33)

E 2 (t) = τ (t) 2 1 0 (1 -ρ)σ 2 (t -τ (t)ρ, L)dρ. (2.34)
It is easy to note that E(t) ≤ E(t), for all t ≥ 0. On the other hand, it follows that

γ 1 E 1 (t) + γ 2 E 2 (t) ≤ γ 1 2 L 0 σ 2 (t, x)dx + γ 2 τ (t) 2 1 0 σ 2 (t -τ (t)ρ, L)dρ ≤ γ 1 2 L 0 σ 2 (t, x)dx + γ 2 |β|τ (t) 2|β| 1 0 σ 2 (t -τ (t)ρ, L)dρ ≤ max γ 1 , γ 2 |β| E(t),
that is, .35) This ensures that the Lyapunov E is equivalent to the energy E.

E(t) ≤ E(t) ≤ 1 + max γ 1 , γ 2 |β| E(t), for all t ≥ 0. ( 2
Theorem 2.5. Let us assume that (1.6), (1.7), (1.8) and (1.9) hold. Let us also assume that U 0 = (σ 0 , u 0 , z 0 (-τ (0)•)) T ∈ H. Then the energy E defined in (2.27) decays exponentially, that is there exist positive constants C, µ such that the following happens

E(t) ≤ CE(0)e -µt , ∀t > 0,
where for γ 1 , γ 2 ∈ (0, 1) small enough,

C ≤ 1 + max γ 1 , γ 2 |β| and µ ≤ min π 2 L 2 1 - bγ 1 2 , γ 2 (1 -m) 2M (γ 2 + |β|) , γ 1 (λ -b) 2(γ 1 + e λL )
, λ > b.

Proof. First we assume that the solution of the system is sufficiently smooth by taking U 0 = (σ 0 , u 0 , z 0 (-τ (0)•)) T ∈ D(A(0)). We perform a differentiation on (2.33) and write

Ė1 (t) = L 0 e -λx σσ t dx = - L 0 e -λx σσ x dx -b L 0 e -λx σu x dx = - λ 2 L 0 e -λx σ 2 dx - e -λL 2 σ 2 (t, L) + 1 2 σ 2 (t, 0) -b L 0 e -λx σu x dx. (2.36)
Differentiating both sides of (2.34),we have

Ė2 (t) = τ (t) 2 1 0 (1 -ρ)σ 2 (t -τ (t)ρ, L)dρ + τ (t) 1 0 (1 -ρ)σ(t -τ (t)ρ, L)∂ t σ(t -τ (t)ρ, L)dρ = τ (t) 2 1 0 (1 -ρ)σ 2 (t -τ (t)ρ, L)dρ - 1 0 (1 -ρ)(1 -τ (t)ρ)σ(t -τ (t)ρ, L)∂ ρ σ(t -τ (t)ρ, L)dρ.
(2.37)

In the last term of (2.37), we have used the identity -τ (t

)∂ t σ(t -τ (t)ρ, L) = (1 -τ (t)ρ)∂ ρ σ(t -τ (t)ρ, L).
Performing integration by parts for the last term of the above identity we have

Ė2 (t) = τ (t) 2 1 0 (1 -ρ)σ 2 (t -τ (t)ρ, L)dρ - τ (t) 2 1 0 (1 -ρ)σ 2 (t -τ (t)ρ, L)dρ - 1 2 1 0 (1 -τ (t)ρ)σ 2 (t -τ (t)ρ, L) dρ + 1 2 σ 2 (t, L).
Thus we have finally

Ė2 (t) = - 1 2 1 0 (1 -τ (t)ρ)σ 2 (t -τ (t)ρ, L) + 1 2 σ 2 (t, L). (2.38)
Using (2.31), (2.36) and (2.38), for some µ > 0 we have

Ė(t) + 2µE(t)≤ 1 2 B σ(t, L) σ (t -τ (t), L) , σ(t, L) σ (t -τ (t), L) R 2 - λγ 1 2 L 0 e -λx σ 2 dx - γ 1 e -λL 2 σ 2 (t, L) + γ 1 2 σ 2 (t, 0) -bγ 1 L 0 e -λx σu x dx - L 0 u 2 x dx - γ 2 2 1 0 (1 -τ (t)ρ)σ 2 (t -τ (t)ρ, L) dρ + γ 2 2 σ 2 (t, L) + µ L 0 σ 2 dx + µ L 0 u 2 dx + µ|β|τ (t) 1 0 σ 2 (t -τ (t)ρ, L)dρ + µγ 1 L 0 e -λx σ 2 (t, x) dx + γ 2 µτ (t) 1 0 (1 -ρ)σ 2 (t -τ (t)ρ, L)dρ ≤ 1 2 B σ(t, L) σ (t -τ (t), L) , σ(t, L) σ (t -τ (t), L) + γ 1 2 σ 2 (t, 0) + γ 2 2 σ 2 (t, L) - λγ 1 2 L 0 e -λx σ 2 dx -bγ 1 L 0 e -λx σu x dx - L 0 u 2 x dx + µ L 0 σ 2 dx + µ L 0 u 2 dx + µγ 1 L 0 e -λx σ 2 (t, x) dx + µ|β|M + γ 2 µM - γ 2 2 (1 -m) 1 0 σ 2 (t -τ (t)ρ, L)dρ. (2.39)
Let us simplify the first three terms of the right hand side of (2.39). Now,

1 2 B σ(t, L) σ (t -τ (t), L) , σ(t, L) σ (t -τ (t), L) R 2 + γ 1 2 σ 2 (t, 0) + γ 2 2 σ 2 (t, L) = 1 2 B σ(t, L) σ (t -τ (t), L) , σ(t, L) σ (t -τ (t), L) R 2 + γ 1 2 (ασ(t, L) + βz(t, 1)) 2 + γ 2 2 σ 2 (t, L) = 1 2 (B + B γ1 γ2 ) σ(t, L) σ (t -τ (t), L) , σ(t, L) σ (t -τ (t), L) R 2
, where

B γ1 γ2 =   α 2 γ 1 + γ 2 αβγ 1 αβγ 1 β 2 γ 1   = γ 1   α 2 αβ αβ β 2   + γ 2   1 0 0 0   .
Since B is negative definite and trace and determinant map is continuous, we can find γ 1 , γ 2 small enough such that B + B γ1 γ2 is negative definite. Thanks to the Poincaré inequality with best constant, we have

µ L 0 u 2 (x) dx ≤ µL 2 π 2 L 0 u 2 x (x) dx, as u ∈ H 1 0 (0, 1). (2.40)
Using Young's inequality we estimate the following:

bγ 1 L 0 e -λx σu x dx ≤ bγ 1 2 L 0 e -λx σ 2 dx + bγ 1 2 L 0 e -λx u 2 x dx. (2.41)
Thus combining (2.39), (2.40) and (2.41), we have :

Ė(t) + 2µE(t) ≤ µL 2 π 2 -1 + bγ 1 2 J1 L 0 u 2 x (t, x)dx + µ|β|M + γ 2 µM - γ 2 2 (1 -m) J2 1 0 σ 2 (t -τ (t)ρ, L)dρ + - λγ 1 2 + bγ 1 2 + µe λL + µγ 1 J3 L 0 e -λx σ 2 (t, x)dx.
Now we are at the position of choosing our parameters involved in the Lyapunov function to achieve the required estimate: Ė(t) + 2µE(t) ≤ 0.

(2.42)

• To have J 1 ≤ 0 we need to take µ ≤ π 2 L 2 (1 -bγ1 2 )
and also there is a restriction on the small parameter γ 1 used in the Lyapunov expression as

γ 1 ≤ 2 b . • J 2 ≤ 0 iff µ ≤ γ2(1-m) 2M (γ2+|β|) . • If we take λ > b and µ < γ1(λ-b)
2(γ1+e λL ) , then we have J 3 ≤ 0. By the conditions of the Theorem 2.5 we have (2.42). Therefore using the inequality (2.35) we have

E(t) ≤ 1 + max γ 1 , γ 2 |β| E(0)e -2µt , ∀ t > 0.
A standard density argument allows to extend the desired result to U 0 ∈ H. □

Exponential Stability of NSE with time varying internal delay

In this section, we study the stability analysis of the Navier-Stokes equations (1.11) with internal timedelay feedback. As before, we first study the well-posedness of the internal time varying delay system. Then using Lyapunov approach, we prove the exponential stability of the system.

3.1.

Well-posedness of the problem. This section is devoted to the well-posedness of the above system (1.11). As the boundary delay case, let us introduce the variable z(t, x, ρ) = σ(t -τ (t)ρ, x), x ∈ ω, ρ ∈ (0, 1), t > 0.

Then the system (1.11) reduces to the following:

                             σ t + σ x + bu x + a(x)σ(t, x) + c(x)σ(t -τ (t), x) = 0, (t, x) ∈ (0, ∞) × (0, L), u t -u xx + u x + bσ x = 0, (t, x) ∈ (0, ∞) × (0, L), τ (t)z t + (1 -τ (t)ρ)z ρ = 0, x ∈ ω, (t, ρ) ∈ (0, ∞) × (0, 1), u(t, 0) = u(t, L) = σ(t, 0) = 0, t ∈ (0, ∞), z(t, x, 0) = σ(t, x), x ∈ ω, t ∈ (0, ∞), z(0, x, ρ) = z 0 (-τ (0)ρ, x), x ∈ ω, ρ ∈ (0, 1), σ(0, x) = σ 0 (x), u(0, x) = u 0 (x), x ∈ (0, L), σ(t -τ (0), x) = z 0 (t -τ (0), x),
x ∈ (0, L), 0 < t < τ (0).

(3.1)

Let us set U = (σ, u, z) T , U 0 = (σ 0 , u 0 , z 0 (-τ (0)•, •)) T . Thus we write the above system as:

U(t) = A(t)U(t), t > 0, U(0) = U 0 , (3.2)
where the time dependent operator A(t) can be written as

A(t)U =   -σ x -bu x -aσ -cz e (•, 1) u xx -u x -bσ x -1-τ (t)ρ τ (t) z ρ   ,
where z e is the zero extension of z outside ω, with the domain

D(A(t)) = (σ, u, z) ∈ H 1 (0, L) × H 2 (0, L) ∩ H 1 0 (0, L) × L 2 (ω, H 1 (0, 1)) : z(x, 0) = σ| ω (x), σ(0) = 0 .
Clearly D(A(t)) = D(A(0)), ∀t > 0. Let us introduce the Hilbert space H = L 2 (0, L) × L 2 (0, L) × L 2 (ω × (0, 1)) induced with the following inner product depending on time

  σ 1 u 1 z 1   ,   σ 2 u 2 z 2   t = L 0 σ 1 σ 2 dx + L 0 u 1 u 2 dx + τ (t) ω 1 0 ξ(x)z 1 z 2 dx dρ, (3.3)
where ξ is a non-negative function in L ∞ (0, L) such that supp ξ = supp c = ω and it satisfies following condition 1 1 -m c(x) + k 0 ≤ ξ(x) ≤ 2a(x) -c(x) -k 0 a.e. in ω. (3.4) By the definition of the inner product (3.3) we further have:

min(1, τ 0 c 0 ) ∥(σ, u, z)∥ 2 H ≤ ∥(σ, u, z)∥ 2 t ≤ 1 + 2M ∥a∥ L ∞ (0,L) ∥(σ, u, z)∥ 2 H , (3.5)
where ∥.∥ H is the usual norm on H. Now we are ready to prove the well-posedness result for the linear system (1.11) using time dependent semigroup theory.

Theorem 3.1. Let a, c ∈ L ∞ (0, L) be two non-negative functions with (1.12), (1.13) and we also assume (1.6), (1.7), (1.8) and (3.4). For any U 0 = (σ 0 , u 0 , z 0 (-τ (0)•, •))

T ∈ H, there exists a unique solution

U ∈ C([0, ∞); H) of (3.2). Moreover, if U 0 ∈ D(A(0)), then U ∈ C([0, ∞); D(A(0))) ∩ C 1 ([0, ∞); H).
Proof. To prove the existence-uniqueness of the solution of (3.2), we follow similar kind of argument as the boundary time delay case. Since the analysis is standard, we only give a sketch of the proof. At first, it is elementary to check that D(A(0)) is dense in H and D(A(t)) = D(A(0)), ∀t > 0. Let us take U = (σ, u, z) T ∈ D(A(t)). We compute

⟨A(t)U, U⟩ t = - L 0 σσ x dx -b L 0 σu x dx + L 0 uu xx dx - L 0 uu x dx -b L 0 σu x dx - L 0 a(x)σ 2 dx - ω c(x)σ(x)z(x, 1)dx + τ (t) ω 1 0 ξ(x) τ (t)ρ -1 τ (t) zz ρ dρdx = 1 2 -σ 2 (L) + σ 2 (0) - L 0 u 2 x dx - L 0 a(x)σ 2 dx - ω c(x)σ(x)z e (x, 1) dx + 1 2 ω ξ(x)(-1 + τ (t))z 2 (x, 1) dx + 1 2 ω ξ(x)z 2 (x, 0) dx - τ (t) 2 ω 1 0 ξ(x)z 2 dρdx.
Applying the inequality ab ≤ a 2 2 + b 2 2 for the term ω c(x)σ(x)z(x, 1), we further obtain

⟨A(t)U, U⟩ t ≤ ω -a(x) + c(x) 2 + ξ(x) 2 σ 2 (x)dx + 1 2 ω (c(x) + ξ(x)( τ (t) -1)) z 2 (x, 1) dx - τ (t) 2 ω 1 0 ξ(x)z 2 (x, ρ) dρdx.
Thanks to (3.4), we have -a(x)

+ c(x) 2 + ξ(x) 2 < 0 and (c(x) + ξ(x)( τ (t) -1)) < c(x) + ξ(x)(m -1) < 0, a.e. in ω. Let us recall the function k(t) = (1+ τ 2 (t)) 1/2 2τ (t)
. Therefore one can easily show that the time dependent operator A(t) = A(t) -k(t)I is dissipative.

Next, we will compute ⟨A(t) * U, U⟩ t , where A(t) * , the adjoint of the operator A(t) * is defined by:

A(t) * U =    σ x + bu x -aσ + ξz e (•, 0) u xx + u x + bσ x 1-τ (t)ρ τ (t) z ρ -τ (t) τ (t) z    ,
with the domain:

D(A(t) * ) = (σ, u, z) ∈ H 1 (0, L) × H 2 (0, L) ∩ H 1 0 (0, L) × L 2 (ω, H 1 (0, 1))| σ(L) = 0, z(x, 1) = -c(x) ξ(x)(1 -τ (t))
σ| ω (x) .

Let us consider U = (σ, u, z) T ∈ D(A(t) * ). Thus applying integration by parts, we obtain

⟨A(t) * U, U⟩ t = L 0 σσ x dx + b L 0 σu x dx + L 0 uu xx dx + L 0 uu x dx + b L 0 σu x dx - L 0 a(x)σ 2 dx - ω c(x)σ(x)z(x, 0) dx + τ (t) ω 1 0 ξ(x) -τ (t)ρ + 1 τ (t) zz ρ dx dρ -τ (t) ω 1 0 ξ(x) τ (t) τ (t) z 2 (x, ρ)dx dρ = 1 2 (σ 2 (L) -σ 2 (0)) - L 0 u 2 x dx - ω a(x)σ 2 dx - ω 1 0 c(x)σ(x)z(x, 0) dx + 1 2 ω 1 0 τ (t)ξ(x)z 2 (x, ρ) dρ dx + 1 2 ω ξ(x)(-1 + τ (t))z 2 (x, 1) dx - 1 2 ω ξ(x)z 2 (x, 0) dx -τ (t) ω 1 0 ξ(x)z 2 (x, ρ)dρdx.
Utilizing the boundary data, we derive

⟨A(t) * U, U⟩ t = - 1 2 σ 2 (0) - L 0 u 2 x dx - ω a(x)σ 2 dx - ω 1 0 c(x)σ(x)z(x, 0) dx - 1 2 ω 1 0 τ (t)ξ(x)z 2 (x, ρ) dx + 1 2 ω c 2 (x) ξ(x)(1 -τ (t)) σ 2 (x) dx - 1 2 ω ξ(x)z 2 (x, 0) dx ≤ 1 2 ω -2a(x) + ξ(x) + c 2 (x) ξ(x)(1 -τ (t)) σ 2 (x) - 1 2 ω 1 0 τ (t)ξ(x)z 2 (x, ρ)dxdρ.
Thanks to the inequality (3.4) and the upper bound condition (1.8) of τ (t), we have

c 2 (x) ξ(x)(1 -τ (t)) ≤ c(x) ξ(x) c(x) (1 -m) ≤ c(x), a.e. in ω.
Again by (3.4), we further obtain -2a(x)

+ ξ(x) + c 2 (x) ξ(x)(1-τ (t)) ≤ -k 0 < 0, a.
e. in ω. Therefore as in the previous case, straightforward computations show that the time dependent operator A(t) * = A(t) * -k(t)I is dissipative. It can be shown that A(t) is a densely defined closed linear operator. Henceforth A(t) generates a C 0 semigroup of contraction on H (see [START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF]).

Proceeding with similar argument as boundary delay feedback cases, one can prove (2.8). And thus the family A = { A(t) : t ∈ [0, T ]} is stable with stability constants C and ω 0 independent of t. Hence, the first three conditions of Theorem 2.1 hold. Finally, one can also show that ∂

t A(t) ∈ L ∞ * ([0, T ]; B(D(A(0)), H)). Thanks to Theorem 2.1, the following ODE ˙ U(t) = A(t) U(t), t > 0, U(0) = U 0 , (3.6)
has a unique solution

U ∈ C([0, ∞); D(A(0))) ∩ C 1 ([0, ∞); H), provided U 0 ∈ D(A(0)). Thus U(t) = e t 0 k(s)ds U(t) solves the ODE (3.2). □ 3.
2. Stability analysis. In this section, we prove the exponential stability result for the system (1.11).

Let us first consider the energy for the system (1.11)

E(t) = 1 2 L 0 σ 2 (t, x)dx + 1 2 L 0 u 2 (t, x)dx + τ (t) 2 ω 1 0 ξ(x)σ 2 (t -τ (t)ρ, x)dρ, t ≥ 0. (3.7)
Our first goal is to show that this energy is a decreasing function of time.

Proposition 3.2. Let a, c ∈ L ∞ (0, L) be two non-negative functions with (1.12), (1.13) and we also assume that (1.6), (1.7), (1.8) and (3.4) hold. Then for all regular solutions of the system (1.11), the energy E in (3.7) is non-increasing and satisfies:

Ė(t) < 0.
Proof. Differentiating E with respect to t and using the equation (1.11) and applying integration by parts successively, we get,

Ė(t) = L 0 σσ t dx + L 0 uu t dx + τ (t) 2 ω 1 0 ξ(x)σ 2 (t -τ (t)ρ, x)dρ + τ (t) ω 1 0 ξ(x)σ(t -τ (t)ρ, x)∂ t σ(t -τ (t)ρ, x)dρ dx = - 1 2 σ 2 (L) - L 0 u 2 x dx - 1 0 a(x)σ 2 dx - ω c(x)σ(x)σ(t -τ (t), x) dx - 1 2 ω ξ(x)(1 -τ (t))σ 2 (t -τ (t), x)dx + 1 2 ω ξ(x)σ 2 (t, x) dx
Here, we have inserted the identity -τ

(t)∂ t σ(t -τ (t)ρ, x) = (1 -τ (t)ρ)∂ ρ σ(t -τ (t)ρ, x), x ∈ ω in the term τ (t) ω 1 0 ξ(x)σ(t -τ (t)ρ, x)∂ t σ(t -τ (t)ρ, x)dρ dx
and performing integration by parts we derived

τ (t) ω 1 0 ξ(x)σ(t -τ (t)ρ, x)∂ t σ(t -τ (t)ρ, x)dρdx = - τ (t) 2 ω 1 0 ξ(x)σ 2 (t -τ (t)ρ, x)dρdx - 1 2 ω ξ(x)(1 -τ (t))σ 2 (t -τ (t), x)dx + 1 2 ω ξ(x)σ 2 (t, x)dx
On simplification, we obtain

Ė(t) ≤ 1 2 ω (-2a(x) + ξ(x) + c(x)) σ 2 (t, x)dx + 1 2 ω (c(x) -(1 -m)ξ(x)) σ 2 (t -τ (t), x)dx - L 0 u 2 x dx.
Thanks to the inequality (3.4), we obtain Ė(t) < 0. □ Now, we will establish that the energy E decays exponentially towards the origin. Let us consider the Lyapunov functional

E(t) = E(t) + γ 1 E 1 (t) + γ 2 E 2 (t), t > 0, γ 1 , γ 2 ∈ (0, 1),
where

E 1 (t) = 1 2
L 0 e -λx σ 2 (t, x)dx, for some λ > 0.

(3.8)

E 2 (t) = τ (t) 2 ω 1 0 (1 -ρ)σ 2 (t -τ (t)ρ, x)dxdρ. (3.9)
It is easy to note that E(t) ≤ E(t), for all t ≥ 0. On the other hand, it follows that .10) This ensures that the Lyapunov E is equivalent to the energy E. Now, we prove our exponential stability result for interior delay case.

γ 1 E 1 (t) + γ 2 E 2 (t) ≤ γ 1 2 L 0 σ 2 (t, x)dx + γ 2 τ (t) 2 ω 1 0 σ 2 (t -τ (t)ρ, x)dx dρ ≤ γ 1 2 L 0 σ 2 (t, x)dx + γ 2 τ (t) 2c 0 ω 1 0 ξ(x)σ 2 (t -τ (t)ρ, x) dx dρ ≤ max γ 1 , γ 2 c 0 E(t), that is, E(t) ≤ E(t) ≤ 1 + max γ 1 , γ 2 c 0 E(t), for all t ≥ 0. ( 3
Theorem 3.3. Let a, c ∈ L ∞ (0, L) be two non-negative functions with (1.12), (1.13) and we also assume that (1.6), (1.7), (1.8), (1.13) and (3.4) hold. Let us also assume that U 0 = (σ 0 , u 0 , z 0 (-τ (0)•, •)) T ∈ H.

Then the energy E defined in (3.7) decays exponentially, that is there exist positive constants C, µ such that the following happens E(t) ≤ CE(0)e -µt , ∀t > 0,

where for γ 1 , γ 2 ∈ (0, 1) small enough,

C ≤ 1 + max γ 1 , γ 2 c 0 and µ ≤ min π 2 L 2 1 - bγ 1 2 , γ 2 (1 -m) 2M (γ 2 + ∥ξ∥ L ∞ (0,L) ) , γ 1 (λ -b) 2(γ 1 + e λL )
, λ > b.

Proof. Analogous to boundary feedback case, here we first assume that the solution of the system (1. (1 -ρ)σ(t -τ (t)ρ, x)∂ t σ(t -τ (t)ρ, x)dxdρ (3.12)

Thanks to the identity -τ (t)∂ t σ(t -τ (t)ρ, x) = (1 -τ (t)ρ)∂ ρ σ(t -τ (t)ρ, x), x ∈ ω, we write the last term of the above identity as follows: Thus to prove Ė(t) + 2µE(t) < 0, we need to choose the Lyapunov parameters γ 1 , γ 2 and the decay µ in such a way that I 1 , I 2 , I 3 , I 4 , I 5 are negative.

τ (t)
• As (-2a(x) + ξ(x) + c(x)) < 0, c(x) -(1 -m)ξ(x) < 0 a.e. on ω, we can choose γ 1 , γ 2 small enough to get I 1 , I 2 < 0. In fact, the inequality (3.4) ensures that the following choices serve our purpose: • Clearly, if we consider µ ≤

γ 1 ≤ inf ω 2a (
(1-m)γ2

2M (∥ξ∥ L ∞ (0,L) +γ2) , then I 3 ≤ 0.

• To prove that I 4 < 0, that is, µ < γ1(λ+a(x)-b) 2(e λL +γ1) , it is enough to consider that µ < γ 1 (λ -b) 2(e λL + γ 1 ) , λ > b.

• To have I 5 < 0 we need to take µ < π 2 L 2 (1 -bγ1 2 ) and also there is a restriction on the small parameter γ 1 used in the Lyapunov expression as γ 1 < 2 b . This completes the Theorem 3.3. □

Conclusion

In this article, we have investigated the time-dependent delay effect in the asymptotic behaviour of the linearized compressible Navier-Stokes equations. We established two well-posedness results of the concerned system with the time delay term acting in the boundary and in the interior of the hyperbolic component. We have explored the exponential stability results of the Navier-Stokes system under some conditions on the damping parameters and delayed terms as well. In this work, we have assumed that the delay function is strictly positive; see (1.7). An interesting question arises that, can we demonstrate our stability analysis for the degenerate delay case, like [START_REF] Nicaise | Exponential stability of the wave equation with boundary time-varying delay[END_REF]? We will address this question in the near future. It is reasonable to examine the impact of the time dependent delay acting in the parabolic component of the linearized Navier-Stokes system. Also, it will be interesting to explore these stability results in other related fluid models, such as linearized compressible Navier-Stokes equations in the case of Creeping flow [START_REF]Boundary controllability and stabilizability of a coupled first-order hyperbolic-elliptic system[END_REF] or linearized compressible Navier-Stokes system with Maxwell's law [START_REF] Ahamed | Some controllability results for linearized compressible Navier-Stokes system with Maxwell's law[END_REF], and [START_REF] Ahamed | Controllability and stabilizability of the linearized compressible navier-stokes system with maxwell's law[END_REF].

  b 11 b 12 b 12 b 22 . 2 =b 11 x + b 12 b 11 y 2 + b 22 b 11 -b 2 12 b 11 y 2

 1222222 Here b 11 < 0, b 11 b 22 -b 2 12 > 0. Taking x = (x, y) ∈ R 2 \ {(0, 0)}, we compute ⟨Bx, x⟩ R 2 =b 11 x 2 + 2b 12 xy + b 22 y < 0.

- λ 2 L 0 e -λx σ 2 dx - e -λL 2 σ 2 (e- λ 2 L 0 e -λx σ 2 dx -b L 0 e( 1

 2022001 [START_REF] Bhandari | Boundary null-controllability of 1d linearized compressible navier-stokes system by one control force[END_REF] is sufficiently smooth by taking U 0 = (σ 0 , u 0 , z 0 (-τ (0)•, •)) T ∈ D(A(0)). A standard density argument allows to extend the desired result to U 0 ∈ H.Differentiating (3.8) with respect to t, we haveĖ1 (t) = L 0 e -λx σσ t dx = -L 0 e -λx σσ x dx -b L 0 e -λx σu x dx -L 0 a(x)e -λx σ 2 dxω e -λx c(x)σ(t -τ, x)σ(t, x) dx = -λx σu x dx -L 0 a(x)e -λx σ 2 dx ω e -λx c(x)σ(t -τ, x)σ(t, x) dx ≤ -λx σu x dx -L 0 a(x)e -λx σ 2 dxω e -λx c(x)σ(t -τ, x)σ(t, x) dx.-ρ)σ 2 (t -τ (t)ρ, x)dxdρ + τ (t)

ω 1 0( 1 -- ω 1 0( 1 -- 1 2 ω 1 0( 1 -( 1 - 1 2 ω σ 2 ω 1 0( 1 ( 2 L 0 -γ 1 λ 1 I4e -λx σ 2 dx + - 1

 1111111121120111 ρ)σ(t -τ (t)ρ, x)∂ t σ(t -τ (t)ρ, x)dxdρ = ρ)(1 -τ (t)ρ)σ(t -τ (t)ρ, x)∂ ρ σ(t -τ (t)ρ, x)dxdρ = τ (t)ρ)σ 2 (t -τ (t)ρ, x)dxdρ -ρ)σ 2 (t -τ (t)ρ, x)dxdρ + (t, x)dx.The above identity along with (3.12) providesĖ2 (t) = -1 2 -τ (t)ρ)σ 2 (t -τ (t)ρ, x)dx + ω σ 2 (t, x)dx.(3.13)Combining (3.11) and (3.13) and using Young's inequality, we proceedĖ(t) + 2µE(t) ≤ 1 2 ω (-2a(x) + ξ(x) + c(x) + γ 1 c(x) + γ 2 ) x) -(1 -m)ξ(x) + γ 1 c(x)) I2 σ 2 (t -τ (t), x)dx -2µM ξ(x) -(1 -m)γ 2 + M 2µγ 2 ) I3 σ 2 (t -τ (t)ρ, x)dx + 1 -γ 1 a(x) + 2µe λL + 2γ 1 µ + bγ

  x) -ξ(x) -c(x) c(x) , (1 -m)ξ(x) -c(x) c(x) , γ 2 ≤ inf ω 2a(x) -ξ(x) -c(x) -γ 1 c(x) .

Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai-400076 Email address: sm18rs016@iiserkol.ac.in, subratamajumdar634@gmail.com

Acknowledgments

This work is supported by Department of Atomic Energy and National Board for Higher Mathematics (Grant No. 0203/16(21)/2018-R&D-II/10708).

Conflict of Interest

The author declare that he has no conflict of interest. ORCID https://orcid.org/0000-0001-6724-6943


Mathieu Renouf

Hong-Phong Cao

Frédéric Dubois

Impact of interaction laws and particle modelling in Discrete Element Simulations

Keywords: Discrete Element, contact law, Finite Element PACS: 68.60.Dv, 46.55.+d

Among mechanical systems, a large number presents naturally a divided feature: granular media, masonries, metal at the grain level, etc. Some of them, due to external loading, may evolve from a continuous state to a, globally or locally, divided one: fracture, wear, cracks, etc. When numerical studies of such media are performed, Discrete Elements Methods (DEMs) appear as most appropriate tool to face to different problems.

Classically, DEMs are based on some hypotheses concerning contact area, deformations and contact interactions which are in some circumstance questionable, especially when they have a strong influence on the macroscopic behaviour of the media.

Using the Contact Dynamics framework, the paper presents how classical hypothesis could be extended to avoid numerical effects. A reflection is proposed taking into account both physical and numerical aspects. Static and dynamic configuration have been used to illustrate the paper purposes.

INTRODUCTION

Among mechanical systems, a large number presents naturally a divided feature: granular media, masonries, ceramics, metal, etc. Some of them, due to external loading, may evolve from a continuous state to a, globally or locally, divided one: fracture, wear, cracks, etc. When numerical studies of such media are performed, Discrete Elements Methods (DEM) appear as an appropriate tool to face the various difficulties of the modelling.

Classically, DEMs are based on some hypotheses concerning contact area, bulk models and interaction laws which are in some circumstance restrictive and questionable, especially when they have a strong influence on the macroscopic behaviour of the media.

Based on the Contact Dynamics (CD) framework, the paper presents how classical hypothesis can be extended to avoid some DEM restrictions. A reflection is proposed to improve both physical and numerical aspects. The section presents the headlines of the mechanical framework used in this paper. Various static and dynamic configuration have been studied to illustrate our purpose.

MECHANICAL MODEL

The method used in this paper relies on the (Non Smooth) Contact Dynamics frameworks developed by Moreau and Jean [START_REF] Moreau | Unilateral contact and dry friction in finite freedom dynamics[END_REF]2].

Considering a multi-contact problem, the approach leads to a reformulation of the classical equation of dynamics in term of contact unknowns written in the contact frame : the local impulse p and the relative velocity u. Over a given time interval [t i ,t i+1 [, starting from a the balance of momentum, the problem is written as a set of transfer equations and interaction laws :

Wp i+1 -u i+1 = -u f ree ContactLaw[p i+1 , u i+1 ] ( 1
)
where W is the Delassus operator that models the dynamics of the solids at the contact points. The right-handside of the first equation of the system (1) represents the free relative velocity, velocity of bodies free of contact forces. The second equation closes the system (1), it assumes that each contact law must be satisfied by each couple (u i+1 , p i+1 ).

The Delassus operator is equal to H * M-1 H where H is a linear mapping between the local unknown and the global one while M-1 is the inverse of the mass matrix.

Considering rigid bodies, one obtains:

M = M u f ree = M-1 (h(1 -θ )F ext i + hθ F ext i+1 ) (2)
where θ is the time integrator parameter, h the time step and F ext the external forces.

   M = M + hθ C + h 2 θ 2 K u f ree = M-1 (h(1 -θ )F ext i + hθ F ext i+1 + hF vol (q i + θ v i , v i+1 )) (3)
where C and K represents respectively the viscosity and the stiffness matrices and F vol represents the internal forces of the deformable bodies.

To close the system an interaction law, which must be satisfied by each couple (u i+1 , p i+1 ), must be defined. In the present study we consider only unilateral contact with dry Coulomb's friction. The unilateral contact means that when contact occurs (gap is equal to zero) the normal impulse is positive and when the contact vanishes (gap is positive) it is equal to zero. As proposed by Jean [2], this Signorini condition may be written as follow (subscript i + 1 is intentionally omitted) :

g ini /h + u n ≥ 0 p n ≥ 0 (g ini /h + u n ).p n = 0, (4)
where the index n denotes the normal component of the various quantities, g ini is the distance between objects at the beginning of the time step and h the time step. One advantage of this writing is that you are able to implicitly settle that the gap is equal to zero at the end of time step if contact occurs. One drawback is that this writing means a fully plastic shock law. An alternative writing was proposed by Moreau [3], it is summarized by the so-called velocity Signorini condition i f g ini ≤ 0. ũn ≥ 0 p n ≥ 0 ũn .p n = 0, (5) otherwise p n = 0 (6)

The main advantage of this writing is that it implicitly embed a collision law based on the Newton restitution law through ũn = u n,i+1 + e n u n,i . The main drawback comes from the fact that you may recover a negative gap. It is important to note that the inelastic shock is not rigorously equal to the restitution law with e n equal to 0.

The dry Coulomb's friction law enforce that the friction force lies in the Coulomb's cone (||p t || ≤ µ p n , µ friction coefficient), and a sliding occurs only when you reach the cone (||p t || = µ p n ) and its direction is opposed to the friction force. We can summarize previous explanations by the following relation:

||p t || ≤ µ p n ||u t || = 0 → p t = -µ p n u t ||u t || (7)

NUMERICAL RESULTS

In this paper, all the results are obtained using LMGC90 software developed by Dubois and Jean ([4]). Our simulations concern the biaxial compression and shear of a 2D sample made of circular particles either rigid or deformable. We consider the two different contacts law presented before: plastic Signorini Coulomb (IQS), restitution Signorini Coulomb (RST).

Rigid bodies

We consider a sample of 2000 rigid bodies with randomly generated radius between 8 mm -16 mm (R max /R min = 2) in an area bounded contained by four rigid frictionless walls (see figure 1). In these simulations we consider rigid particles made of steel with a specific mass equal to 7800 kg/m 3 .

FIGURE 1. Geometry of the sample and boundary condition. Case a: applied force in Y direction and applied velocity in X direction. Case b: applied force in Y direction and applied rotation velocity around Z direction

The loads are applied in two phases: isotropic compression followed by a biaxial or a shear load. During the first phase F y = F x , and during the second phase F y = F, where F is the confining force reached at the end of the first phase. The initial configuration of the second phase is the same for both loads (see figure 1) :

• case (a): walls 1,3 are fixed, a force F equal to 1000 kN is applied on the wall 2 and velocity V of 0.01 m/s is applied on the wall 4 (F,V are linear functions of time so that the force F and the velocity V reach their maximum values at t = 1 s). • case (b): wall 1 is fixed, a force F equal to 1000 kN is applied on the wall 2 and rotation velocity W z of 0.04 rad/s is applied on walls 3 and 4.

We have conducted our tests with different contact laws: plastic Signorini Coulomb law (IQS) and restitution Signorini Coulomb's law (RST). With the RST contact law, we have tested two different value of normal restitution e n : 0 and 0.9.

The walls do not interact. The contact between disks and walls is without friction. Gravity forces are neglected. The time step is equal to 1.10 -3 s and the simulation time is equal to 4 s.

Since contact between walls and particles is frictionless, the principal directions of stresses and strains coincide with the coordinate axes x and y (see figure 1). Therefore principal strains were calculated directly from wall displacements, whereas principal stresses were obtained as the sum of the normal forces applied by disks on surrounding walls divided by the wall length (and disks thickness !?).

FIGURE 2. Velocity of particles and contact force network at the end of the load for case a and b

Figures (2) show the velocity field and the contact force network between particles. When we apply a load (force or velocity) on the walls it is supported by the particles through the contact network.

The orientation and the amplitude distribution of the contact force depends on the external load, the physical parameters and the history. One can see in the figure (3) the compacity evolution with respect to the increasing external load. One can observe in the figure (3) that :

• case a. The evolutions are more or less the same for the 3 contact laws. The small variations around the mean value are triggered by arch breaking, dilatation induced by shear, etc. • case b. The evolutions are less similar and more noisy. This may be explained by the different behavior the contact law act. One can observe that the IQS law evolves the more regularly due to its dissipative and fully implicit properties.

When using the CD approach one needs to manage the precision of the computation which depends on the time step, the converging tolerance of the contact solver (Non Linear Gauss-Seidel in the present work), etc. Relevant informations to compare the results are the necessary computational effort represented by the number of iterations and the precision of the results represented by the violation. We can see in the figure (4) the influence of the numerical model on the average number of interation. One can notice that with the contact law RST-0.9 the results are more smooth. One can observe in the figures (5 and 6) the mean violation evolution for the different contacts laws. The median radius of the particles is around 10 -2 m which means that for RST law the violation is less than 0.05% of the radius and for IQS law it is less than 10 -3 % of the radius. • Left :the angular distribution of the normal contact forces at the end of the simulation. We have obtain a good condordance between diffirent measured quantities. The orientations distributions of contact normale obtained by the two model are identical. we obtain a constant probability, almost independent of the orientation feature when compress a multi disks without gravity. • Right : the number of contacts as a function of the normal contact force normalized by its mean value at the end of the simulation.

In both cases we obtain very similar results.

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Deformable bodies

In this part, we have considered the same problems as before (see ), replacing the rigid particles by deformable ones (see figure 9). The material is considered to stay elastic with a Young modulus E = 320GPa and a Poisson coefficient ν = 0.33. Each particle is meshed with quadrilateral elements. The time step h is equal to 10 -4 s.

CONCLUSION

The Contact Dynamics framework allows to use various interaction laws and bulk models. The compared interaction laws allow to obtain similar results, from a macroscopic physical point of view, in various situations. Depending on the required numerical precision or physical precision one is able to choose or switch from one model to the other.

FIGURE 3 .

 3 FIGURE 3. Compacity evolution for case a and b

FIGURE 4 .

 4 FIGURE 4. Average number of interations with the different contact laws IQS, RST-0.0 and RST-0.9 for case a (right) and case b (left)

FIGURE 5 .FIGURE 6 .

 56 FIGURE 5. Evolution of the mean violation for case a

FIGURE 9 .

 9 FIGURE 9. Velocity field and contact force network of the meshed deformable particles

Considering deformable bodies, in the case of linear case, one obtains: Impact of interaction laws and particle modelling in Discrete Element Simulations December 29, 2008

Impact of interaction laws and particle modelling in Discrete Element SimulationsDecember 29, 2008