This article concerns the long time dynamics of a 2D incompress- ible Navier-Stokes equation with Rayleigh’s friction and distributed delay

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

DYNAMICS OF 2D NAVIER-STOKES EQUATIONS WITH RAYLEIGH’S FRICTION AND DISTRIBUTED DELAY

YADI WANG, XIN-GUANG YANG, XINGJIE YAN

Abstract. This article concerns the long time dynamics of a 2D incompress- ible Navier-Stokes equation with Rayleigh’s friction and distributed delay. Un- der appropriate assumptions on the external force and delay term, we obtain global well-posedness in new phase spaces with delay. Using uniform estimates and compact embedding, we obtain a global attractor.

1. Introduction

The Navier-Stokes equation is a well-known model to describe the essential law of fluid flow. Its asymptotic dynamics can be used to construct mathematical anal- ysis of turbulence for fluid flow, see for example [6, 7, 16, 28, 29, 31, 32, 33] and the references therein. The influence of the delay is originated from engineer and can be expressed by ordinary differential equation with delay terms such as con- trol feedback; see [17] and Hale and Lunel [13]. Time variable delay and memory terms arise in many fields, such as physics, chemistry, biology, economic phenom- ena, control theory and so on. Moreover, a delay term is a source of instability, which means that the research on asymptotic dynamics for dissipative evolutionary equations with delay is significant in engineer and mathematical analysis.

This article is concerned with asymptotic dynamics for the 2D Navier-Stokes equation with Rayleigh’s friction and distributed delay,

u_t−ν∆u+ (u· ∇)u+αu+∇p=f(x) + Z 0

−h

G(s, u(t+s))ds, (x, t)∈Ω×(τ,+∞),

divu= 0, (x, t)∈Ω×(τ,+∞),

u(t, x)|∂Ω=ϕ, ϕ·n= 0, (x, t)∈∂Ω×(τ,+∞), u(τ, x) =u₀(x), x∈Ω,

u(t, x) =φ(t−τ, x), (x, t)∈Ω×(τ−h, τ), h >0,

(1.1)

where Ω⊂R²is a bounded domain with smooth boundary,ν >0 andα >0 denote the viscosity and Ekman dissipative parameter respectively. In addition, u0 and φ(·) denote the initial data in timeτ and interval [−h,0] respectively. The terms f(x) and R0

−hG(s, u(t+s))ds be the autonomous and distributed delay external

2010Mathematics Subject Classification. 35Q30, 35B40, 35B41, 76D03, 76D05.

Key words and phrases. Navier-Stokes equations; distributed delay; Rayleigh’s friction.

c

2019 Texas State University.

Submitted June 21, 2018. Published June 12, 2019.

1

forces respectively. The Ekman damping αu denotes Rayleigh’s friction which is widely used in geophysical hydrodynamics such as oceanic models. Moreover, we assume ϕ ∈ L^∞(∂Ω) for the analysis of unknown velocity u= (u1(t, x), u2(t, x)) and pressurep=p(t, x).

Let us recall some known results for the dynamics and stability of the Navier- Stokes equation with delays.

(1) For Navier-Stokes models with finite continuous delays as constant or variable functions, such asF(t, z(t), z(t−ρ(t))) forρ(·)∈[−h,0], the global well-posedness and existence of pullback attractors have been studied in [8, 9, 11, 12, 14, 18, 21, 22, 23, 27]. If the delay belongs to infinite interval, which is called infinite continuous delay, such asF(t, z(t), z(t−ρ(t))) for ρ(·)∈(−∞,0], the pullback dynamics for Navier-Stokes equation has been investigated in [1, 10, 15, 19, 24].

(2) For Navier-Stokes system with finite distributed delay R0

−hω(s)b(t, s, z(t+ s))dsor infinite one R0

−∞ω(s)b(t, s, z(t+s))ds, we can see the pullback dynamics based on global existence of weak and strong solutions in [1, 2, 3, 4, 20], hereω(·) can be a function or constant.

(3) A comprehensive survey for the fluid flow model with delays, can be found in [5], which presentes also some open problems.

(4) The distributed delay has some similar form as memory, but the methods to deal the dynamics are different, especially the hypotheses on them, see [5, 15] and references therein.

Most of the above publications pay attentions to the pullback attractors for 2D Navier-Stokes equations or 3D modified systems, however there are fewer results on the forward dynamics, which is our objective here. The main results and features of this paper can be stated as following.

(I) Using background function (see [25, 26]), the inhomogeneous boundary sys- tem can be reduced to homogeneous problem, which is main feature for our problem.

Using Galerkin’s approximate procedure and compact argument, we can derive the existence of global weak solution for 2D Navier-Stokes equation with distributed delay in some new phase spaces.

(II) Since the distributed delay in (1.1) is defined in finite interval, for over- coming the uniqueness of global weak solution, we should assume that the kernel of distributed delay has Lipschitz continuous property, which guarantee that the solution generates a semigroup {S(t)} for τ ≤t ∈ R. By some estimates in the delay phase space, the absorbing set can be obtained. Moreover, the existence of global attractor also attained by using compact embedding.

(III) At last, we also want to see the effect of Rayleigh’s friction and distributed delay on the dynamics for 2D Navier-Stokes equation. Comparing with the 2D Navier-Stokes equation with general external force, we can see that the Rayleigh’s friction effects the domain of absorbing set, hence the structure of attractors be- tween the above two problems is greatly different.

The plan of this article is the following. In Section 2, we derive the existence of continuous dependence global solution for our problem. The asymptotic compact- ness of semigroup and the global attractors are concluded in Section 3.

2. Global well-posedness

2.1. Notation. We setE :={u|u∈(C₀^∞(Ω))², divu= 0}, H is the closure of the setE in (L²(Ω))²topology,| · |2 and (·,·) denote the norm and inner product inH

respectively, i.e.,

(u, v) =

2

X

j=1

Z

Ω

uj(x)vj(x)dx, ∀u, v∈(L²(Ω))².

V is the closure of the setE in (H¹(Ω))²topology, and k · kand ((·,·)) denote the norm and inner product inV respectively, i.e.,

((u, v)) =

2

X

i,j=1

Z

Ω

∂uj

∂xi

∂vj

∂xi

dx, ∀u, v∈(H₀¹(Ω))².

k · k∗ is the norm inV⁰, andh·ibe the dual product betweenV andV⁰ or H.

The bilinear and trilinear operators are defined respectively as B(u, v) :=P((u· ∇)v), b(u, v, w) = (B(u, v), w) =

2

X

i,j=1

Z

Ω

ui

∂vj

∂xi

·wjdx which satisfies

b(u, v, v) = 0, b(u, v, w) =−b(u, w, v), (2.1) kb(u, v, w)k ≤C|u|^1/2kuk^1/2kvk|w|^1/2kwk^1/2, ∀u, v, w∈V. (2.2) Moreover, we define the function with delay as

ut=u(t+s), s∈(−h,0),

for any t ∈(τ, T) and the Bochner spaceL^p_H =L^p(−h,0;H) with 1 ≤p≤+∞, especiallyL²_H=L²(−h,0;H).

Also, we define two Banach spacesC_H=C([−h,0];H) andC_V =C([−h,0];V) with norms

kuk_CH = sup

θ∈[−h,0]

|u(t+θ)|, kuk_CV = sup

θ∈[−h,0]

ku(t+θ)k, respectively, which is our phase spaces in the sequel.

2.2. Abstract equivalent equation. Let ψ be the background function which satisfies

divψ= 0, x∈Ω, ψ=ϕ, x∈∂Ω, kψkL^∞ ≤ckϕk_L^∞_(∂Ω), u(τ, x) =u₀(x), x∈Ω,

|ψ|₂≤c⁰kϕk_L∞(∂Ω), kψk ≤c⁰⁰kϕkL^∞(∂Ω).

(2.3)

Denotingv=u−ψ, then (1.1) is translated into the following problem

∂v

∂t −ν∆v+ (v· ∇)ψ+ (ψ· ∇)v+αv+∇p= ¯f+gψ(vt), divv= 0,

v= 0, v(τ, x) =v0(x),

v(t, x) =φ(t−τ, x)−ψ(x) =η(t−τ, x),

(2.4)

here ¯f =f −αψ+ν∆ψ−(ψ· ∇)ψ,gψ(vt) =R0

−hG(s, v(t+s) +ψ)ds.

DefiningRu =B(u, ψ) +B(ψ, u), which is also continuous from V ×V to V⁰, hence the problem (2.4) can be written as the abstract functional equivalent form

v_t+νAv+αv+B(v) +R(v) =Pf¯+g_ψ(v_t), v(τ) =v0,

v(t) =η(t−τ).

(2.5) Next, we shall study well-posedness and dynamics of problem (2.5).

2.3. Assumptions. For the well-posedness and forward dynamics, we use the fol- lowing hypothesis.

(H1) G: [−h,0]×R²→R²is measurable;

(H2) G(s,0) = 0, s∈[−h,0];

(H3) there exists γ ∈L²(−h,0) such that|G(s, u)−G(s, v)|_R² ≤γ(s)|u−v|_R² which is also true for Ω⊂R²;

(H4) νλ1>2c1λ^1/2₁ kψk+ 4C_g²/α.

From (H1) and (H3) we have

|gψ(ξ)−gψ(η)|²₂≤ Z

Ω

Z 0

−h

|G(s, ξ(s))(x)−G(s, η(s))(x)|_R2ds2

dx

≤ Z

Ω

Z 0

−h

γ(s)|ξ(s)(x)−η(s)(x)|_R2ds2

dx

≤ Z

Ω

kγk²_L2(−h,0)

Z 0

−h

|ξ(s)(x)−η(s)(x)|_R²ds² dx

≤L_gkξ−ηk²_C

H

for anyξ, η∈CH, where Lg=hkγk²_L2(−h,0).

For any u, v∈C([−h, T];H), t > τ, there existsm0 ≥0 , we also have for any m∈[0, m0],

Z t

τ

e^ms|gψ(us)−gψ(vs)|²₂ds≤C_g² Z t

τ−h

e^ms|u(s)−v(s)|²₂ds, whereC_g²=kγk²_L2(−h,0)he^m0h.

2.4. Existence of a global weak solution.

Lemma 2.1 (Generalized Arzel`a-Ascoli Theorem [29]). Let {fγ(θ) : γ ∈ Γ} ⊂ C([τ−h, τ];X)is equicontinuous. Then for ∀θ∈[τ−h, τ], the sequence{fγ(θ) : γ∈Γ} is relatively compact in C([τ−h, τ];X).

Lemma 2.2 (Aubin-Lions Lemma [29, 32]). Let X ⊂⊂H ⊂Y be Banach spaces, andX is reflective. Ifunis a uniformly bounded sequence inL^p(τ, T;X), and there exists1< p <+∞such that ^du_dtⁿ is uniformly bounded inL^p(τ, T;Y), thenu_n has a strong convergence subsequence in C([τ, T];H).

Theorem 2.3. Assume that f ∈(L²(Ω))²,v0∈H, η∈L²_H, and (H1)–(H4) hold.

Then (2.4)possesses a unique solution v(t)satisfying v(t)∈L^∞(τ, T;H)∩L²(τ, T;V), dv

dt ∈L²(τ, T;V⁰).

Proof. Step 1: Approximate solution. Using Faedo-Galerkin method to find the approximation solutionvn(t) =Pn

j=0anj(t)wj to (2.4), whereanj(t) is to be deter- mined, we deduce thatvn(t) satisfies a ordinary differential equation

dvn

dt +νAvn+αvn+B(vn) +R(vn) =Pnf¯+gψ(vnt), (2.6)

vn(τ) =vn0, (2.7)

vn(t) =ηn(t−τ), t∈(τ−h, τ), (2.8) By the local existence theory for the ordinary differential equations, we can derive a local solution for problem (2.6).

Step 2: The priori estimate and compact argument. Multiplying (2.6) by e^mtv_n, we have

(dvn

dt , e^mtvn) +ν(Avn, e^mtvn) + (B(vn), e^mtvn) +R(vn, e^mtvn) +α(vn, e^mtvn)

=hPnf , e¯ ^mtvni+hgψ(vnt), e^mtvni.

(2.9) Noting that

(B(vn), e^mtvn) =e^mt(B(vn), vn) =e^mtb(vn, vn, vn) = 0, (2.10)

|R(vn, e^mtvn)|=e^mt|R(vn, vn)|

=e^mt|b(v_n, ψ, v_n)|+e^mt|b(ψ, v_n, v_n)|

=e^mt|b(vn, ψ, vn)|

≤c1e^mt|vn|2kvnkkψk,

(2.11)

|hPnf , e¯ ^mtv_ni|=|hf , P¯ _ne^mtv_ni|=|hf , e¯ ^mtv_ni| ≤e^mtkf¯k∗kvnk, (2.12)

|hgψ(vnt), e^mtvni| ≤e^mt|gψ(vnt)|2|vn|2, (2.13) we obtain

1 2

d(e^mt|vn|²₂)

dt +νe^mtkv_nk²+e^mtα|v_n|²₂

≤e^mtkf¯k_∗kvnk+e^mt|gψ(vnt)|2|vn|2+c1e^mt|vn|2kvnkkψk

≤e^mtνkvnk²

2 +kf¯k²_∗ 2ν

+e^mt|gψ(vnt)|²₂

α +α|vn|²₂

+c1e^mt|vn|2kvnkkψk, and

d(e^mt|vn|²₂) dt ≤e^mt

ν kf¯k²_∗+2e^mt

α |gψ(vnt)|²₂−e^mt

νλ1−2c1λ^1/2₁ kψk

|vn|²₂. (2.14) Choosing an appropriate parameterα >0 such thatνλ1>2c1λ^1/2₁ kψk+ 4C_g²/α, integrating (2.14) over [τ, t], we obtain

e^mt|vn(t)|²₂−e^mτ|vn0|²₂≤ Z t

τ

e^ms

ν kf¯k²_∗ds+ Z t

τ

2e^ms

α |gψ(vns)|²₂ds

− Z t

τ

e^ms

νλ1−2c1λ^1/2₁ kψk

|vn(s)|²₂ds.

(2.15)

Using the hypotheses (H3), we have Z t

τ

e^ms|gψ(vns)|²₂ds

≤C_g² Z t

τ−h

e^ms|vn(s) +ψ|²₂ds

≤C_g² Z t

τ

e^ms|vn(s) +ψ|²₂ds+C_g² Z τ

τ−h

e^ms|ηn(s) +ψ|²₂ds

≤2C_g² Z t

τ

e^ms|v_n(s)|²₂ds+ 2C_g² Z t

τ

e^ms|ψ|²₂ds+C_g² Z τ

τ−h

e^ms|φ_n|²₂ds

≤2C_g² Z t

τ

e^ms|vn(s)|²₂ds+2C_g²

m (e^mt−e^mτ)|ψ|²₂+C_g²e^mτ Z 0

−h

|φn|²₂ds.

(2.16)

Combining (2.15) and (2.16), we have e^mt|v_n(t)|²₂−e^mτ|v_n0|²₂

≤e^mt

mνkf¯k²_∗+4C_g²

αme^mt|ψ|²₂+4C_g² α

Z t

τ

e^ms|vn(s)|²₂ds +2C_g²e^mτ

α Z 0

−h

|φn|²₂ds−4C_g²e^mτ

αm |ψ|²−(νλ₁−2c₁λ^1/2₁ kψk) Z t

τ

e^ms|vn(s)|²₂ds

=e^mtkf¯k²_∗ mν +4C_g²

mα|ψ|²₂

+2C_g²e^mτ α

Z 0

−h

|φn|²₂ds

−4C_g²e^mτ

αm |ψ|²₂−

νλ₁−2c₁λ^1/2₁ kψk −4C_g²/αZ t τ

e^ms|vn(s)|²₂ds, which implies

|vn(t)|²₂≤ kf¯k²_∗ mν +4C_g²

mα|ψ|²₂ +e^−mt2C_g²e^mτ

α Z 0

−h

|φn|²₂ds−4C_g²e^mτ

mα |ψ|²₂+e^mτ|vn0|²₂

≤ kf¯k²_∗ mν +4C_g²

mα|ψ|²₂+2C_g² α

Z 0

−h

|φ_n|²₂ds+|v_n0|²₂

≤ kf¯k²_∗

mν +4C_g²c⁰²

mα kϕk²_L∞(∂Ω)+2C_g² α

Z 0

−h

|φn|²₂ds+|vn0|²₂:=K.

(2.17)

It is sufficient to showv_n(t)∈L^∞(τ, T;H)∩L²(τ, T;V) in the following by some estimates. Multiplying (2.6) byvn, we obtain

1 2

d|v_n|²₂

dt +ν(Av_n, v_n) +b(v_n, v_n, v_n) +R(v_n, v_n) =hPnf , v¯ _ni+hgψ(v_nt), v_ni, which yields

1 2

d|vn|²₂

dt +νkvnk²+α|vn|²₂

≤ kf¯k∗kvnk+|gψ(vnt)|2|vn|2+c1|vn|2kvnkkψk

≤νkvnk²

2 +kf¯k²_∗

2ν +|gψ(vnt)|²₂

α +α|v_n|²₂+c₁λ^−1/2₁ kv_nk²kψk,

which implies d|vn|²₂

dt ≤kf¯k_∗ ν + 2

α|gψ(vnt)|²₂−(ν−2c1λ^−1/2₁ kψk)kvnk². (2.18) Integrating (2.18) over [t, t+ 1], we obtain

|vn(t+ 1)|²₂− |vn(t)|²₂+ (ν−2c1λ^−1/2₁ kψk) Z t+1

t

kvnk²ds

≤ kf¯k²_∗ ν +2

α Z t+1

t

|gψ(vns)|²₂ds.

From the H¨older inequality and hypotheses (H3), we derive that Z t+1

t

|gψ(vns)|²₂ds≤ Z t+1

t

Z 0

−h

|G(s, vn(r+s) +ψ)|²₂dr ds

≤ Z t+1

t

Z 0

−h

|γ(s)|²|vn(r+s) +ψ)|²₂dr ds

≤ kγk²_L2(−h,0)

Z t+1

t

Z 0

−h

|v_n(r+s) +ψ|²₂dr ds and

Z t+1

t

Z 0

−h

|vn(r+s) +ψ|²₂dr ds= Z t+1

t−h

Z 0

−h

|vn(k) +ψ|²₂dr dk.

Noting thatv(k) dependents only onk, it follows that Z t+1

t

Z 0

−h

|vn(r+s) +ψ|²₂dr ds= Z t+1

t−h

Z 0

−h

|vn(k) +ψ|²₂dr dk

= Z t+1

t−h

|vn(k) +ψ|²₂dk and

Z t+1

t

|gψ(vns)|²₂ds≤hkγk²_L2(−h,0)

Z t+1

t−h

|vns+ψ|²₂ds

=C_g² Z t+1

t−h

|vns+ψ|²₂ds . Then

Z t+1

t−h

|v_ns+ψ|²₂ds= Z t

t−h

|v_ns+ψ|²₂ds+ Z t+1

t

|v_ns+ψ|²₂ds.

Sincev(t, x) =φ(t−τ, x)−ψ(x) for arbitrary (t, x)∈(τ−h, τ)×Ω, it yields Z t

t−h

|v_ns+ψ|²₂ds= Z 0

−h

|φ_n(s)−ψ+ψ|²₂ds= Z 0

−h

|φ_n|²₂ds, hence, we have

Z t+1

t

|vns+ψ|²₂ds≤2 Z t+1

t

|vns|²₂ds+ 2 Z t+1

t

|ψ|²₂ds

= 2 Z t+1

t

|vns|²₂ds+ 2|ψ|²₂,

and

Z t+1

t

|vns+ψ|²₂ds≤2λ⁻¹₁ Z t+1

t

kvnsk²ds+ 2c⁰²kϕk²_L∞(∂Ω). Thus we conclude that

Z t+1

t

|gψ(vns)|²₂ds≤C_g² 2λ⁻¹₁

Z t+1

t

kvnk²ds+ 2c⁰²kϕk²_L∞(∂Ω)+ Z 0

−h

|φn|²₂ds , which implies

(ν−2c₁λ^−1/2₁ kψk) Z t+1

t

kv_nk²ds

≤kf¯k²_∗

ν +K+2C_g² α

2λ⁻¹₁

Z t+1

t

kvnk²ds+ 2c⁰²kϕk²_L∞(∂Ω)+ Z 0

−h

|φn|²₂ds , and

ν−2c₁λ^−1/2₁ kψk − 4C_g²λ⁻¹₁ α

Z t+1

t

kvnk²ds

≤kfk¯ ²_∗

ν +K+4C_g²c⁰²

α kϕk²_L∞(∂Ω)+2C_g² α

Z 0

−h

|φn|²₂ds, i.e.,

Z t+1

t

kvnk²ds≤K⁰, (2.19)

where

K⁰=

kfk¯²_∗

ν +K+^4C

2 gc⁰²

α kϕk²_L∞(∂Ω)+^2C

2 g

α

R0

−h|φ_n|²₂ds ν−2c1λ^−1/2₁ kψk − ^4Cg2λ

−1 1

α

.

By the above estimates, we conclude that vn(t) ∈ L^∞(τ, T;H)∩L²(τ, T;V).

From Lemma 2.1, there exists a subsequence (relabeled asvn(t) without confusion) such that

v_n →^∗vtextinL^∞(τ, T;H), v_n→vtextinL²(τ, T;V), i.e.,v∈L^∞(τ, T;H)∩L²(τ, T;V). Since

dvn

dt =−νAvn−B(vn)−R(vn)−αvn+Pnf¯+gψ(vnt), andvn ∈L²(τ, T;V), we haveνAvn, αvn, gψ(vnt)∈L²(τ, T;V⁰) and

k(PnB(vn), vn)k²_L2(0,T;V^∗)≤ Z T

0

k(B(vn, vn)k²_∗ds

= Z T

0

k(vn· ∇)vnk²_∗ds

≤c5

Z T

0

|vn|²₂kvnk²ds

≤c₅kv_nk²_L∞(0,T;H)kv_nk²_L2(0,T;H). i.e.,PnB(vn)∈L²(τ, T;V⁰).

Passing to the limit asn→+∞, we conclude that

v_n→v inL²(τ, T;H), v_n(τ) =P_nv_n0→v(τ) =v₀,

which implies ^dv_dtⁿ ∈L²(τ, T;V⁰). Using Lemma 2.2, we can derive the existence of a strong convergent subsequence which is the solution for our problem.

Step 3: The uniqueness and continuous dependence on initial data. Assume that v1andv2are two solutions to the system (2.6)–(2.8), and denotew=v1−v2, then wsatisfies

dw

dt −νAw+B(v1, v1)−B(v2, v2) +R(w) +αw=gψ(v1t)−gψ(v2t).

Noting that

B(v1, v1)−B(v2, v2) =B(v1−v2, v1)−B(v2, v1−v2)

=B(w, v1) +B(v2, w), we have

dw

dt −νAw+B(w, v1)−B(v2, w) +R(w) +αw=gψ(v1t)−gψ(v2t). (2.20) Multiplying (2.20) bye^mtw, we have

1 2

d(e^mt|w|²₂)

dt + (νAw, e^mtw) + (B(w, v1), e^mtw) + (B(v2, w), e^mtw) + (Rw, e^mtw) + (αw, e^mtw)

=hg_ψ(v_1t)−g_ψ(v_2t), e^mtwi

Using (2.1)-(2.2) and the H¨older inequality, we derive 1

2

d(e^mt|w|²)

dt + (νAw, e^mtw)

≤ |e^mtb(w, v₁, w)|+|e^mt(Rw, w)|+e^mthg_ψ(v_1t)−g_ψ(v_2t), wi

≤e^mt(c2|w|2kwkkv1k+c1|w|2kwkkψk+|gψ(v1t)−gψ(v2t)|2|w|2)

≤e^mtν

2kwk²+ c²₂

2ν|w|²₂kv1k²

+e^mtν

2kwk²+ c²₁

2ν|w|²₂kψk² +e^mt|gψ(v1t)−gψ(v2t)|²₂

2 +|w|²₂ 2

, i.e.,

d(e^mt|w|²₂)

dt ≤e^mthc²₁

νkψk²+c²₂

νkv₁k²+ 1

|w|²₂+|g_ψ(v_1t)−g_ψ(v_2t)|²₂i

. (2.21) Integrating (2.21) over [τ, t], we obtain

e^mt|w(t)|²₂−e^mτ|w(0)|²₂

≤ Z t

τ

e^msc²₁

ν kψk²+c²₂

ν kv₁k²+ 1

|w|²ds+ Z t

τ

e^ms|g_ψ(v_1t)−g_ψ(v_2t)|²₂ds

≤ Z t

τ

e^msc²₁

ν kψk²+c²₂

ν kv1k²+ 1

|w|²₂ds+C_g² Z t

τ−h

e^ms|v1(s)−v2(s)|²₂ds

≤ Z t

τ

c²₁

ν kψk²+c²₂

ν kv1k²+ 1

e^ms|w|²₂ds +C_g²Z τ

τ−h

e^ms|v1(s)−v2(s)|²₂ds+ Z t

τ

e^ms|v1(s)−v2(s)|²₂ds

≤e^mt Z t

τ

c²₁

ν kψk²+c²₂

ν kv1k²+ 1

|w|²₂ds +C_g²

e^mτ Z τ

τ−h

|v1(s)−v₂(s)|²₂ds+e^mt Z t

τ

|v1(s)−v₂(s)|²₂ds , and

|w(t)|²− |w(0)|²₂≤ Z t

τ

c²₁

νkψk²+c²₂

νkv1k²+ 1

|w|²₂ds +C_g²Z 0

−h

|v1(r)−v2(r)|²₂dr+ Z t

τ

|v1(r)−v2(r)|²₂dr . It follows that

|w(t)|²₂≤ |w(0)|²₂+C_g²kη1−η₂k²_L2 H

+ Z t

0

c²₁

νkψk²+c²₂

νkv1k²+C_g²+ 1

|w|²ds, by the Gronwall inequality, we conclude that

|w(t)|²₂≤

|w(0)|²+C_g²kη1−η2k²_L2 H

e^R0t(

c2 1 νkψk²+^c

2 2

νkv₁k²+C_g²+1)|w|²₂ds, which leads to the uniqueness and continuous dependence on initial data for our

global weak solution.

3. Long-time asymptotic dynamics

3.1. Existence of absorbing set. In this subsection, from Theorem 2.3, we see that the global weak solution generates a continuous semigroup S(t)(v₀, η) = v_t(·; (v₀, η)) for any (v₀, η)∈H×L²_H which satisfies

k(v0, η)k²_H×L2

H =|v0|²₂+ Z 0

−h

|η(s)|²₂ds.

Theorem 3.1. Assume thatf ∈(L²(Ω))²,(v₀, η)∈H×L²_H, and(H1)–(H4)hold.

Then the semigroup S(t)possesses an absorbing ball inC_H for the system (2.4).

Proof. LetD ⊂H×L²_H be any bounded set with radiusd for (v₀, η)∈D which satisfies

|v0|²₂+kηk²_L2

H ≤d². (3.1)

Multiplying (2.5) bye^mtv, we obtain (dv

dt, e^mtv) +ν(Av, e^mtv) + (B(v), e^mtv) +R(v, e^mtv) +α(v, e^mtv)

=hPnf , e¯ ^mtvi+hgψ(vt), e^mtvi.

Noting that

(B(v), e^mtv) =e^mt(B(v), v) =e^mtb(v, v, v) = 0,

|R(v, e^mtv)|=e^mt|R(v, v)|=e^mt|b(v, ψ, v)|+e^mt|b(ψ, v, v)|

=e^mt|b(v, ψ, v)|

≤c1e^mt|v|2kvkkψk,

|hPnf , e¯ ^mtvi|=|hf , P¯ ne^mtvi|=|hf , e¯ ^mtvi| ≤e^mtkf¯k∗kvk, and

|hgψ(v_t), e^mtvi|=e^mt|gψ(v_t)|2|v|2, (3.2)

we obtain 1 2

d(e^mt|v|²₂)

dt +νe^mtkvk²+e^mtα|v|²₂

≤e^mtkf¯k∗kvk+e^mt|gψ(vt)|2|v|2+c1e^mt|v|2kvkkψk

≤e^mtνkvk²

2 +kf¯k²_∗ 2ν

+e^mt|g_ψ(v_t)|²₂

α +α|v|²₂

+c₁e^mt|v|₂kvkkψk, and

d(e^mt|v|²₂) dt ≤ e^mt

ν kf¯k²_∗+2e^mt

α |gψ(vt)|²₂−e^mt

νλ1−2c1λ^1/2₁ kψk

|v|²₂ (3.3) from the Poincar´e inequality, where α > 0 is an appropriate constant satisfying νλ₁>2c₁λ^1/2₁ kψk+ 4C_g²/α.

Integrating (3.3) over [τ, t], we obtain e^mt|v(t)|²₂−e^mτ|v(0)|²₂

≤ Z t

τ

e^ms

ν kf¯k²_∗ds+ Z t

τ

2e^ms

α |gψ(vs)|²₂ds− Z t

τ

e^ms νλ1−2c1λ^1/2₁ kψk

|v|²₂ds

≤ 1

mν(e^mt−e^mτ)kf¯k²_∗+2 α

Z t

τ

e^ms|g_ψ(v_s)|²₂ds

− Z t

τ

e^ms(νλ1−2c1λ^1/2₁ kψk)|v|²₂ds.

(3.4) Using hypothesis (H3), we have

Z t

τ

e^ms|gψ(v_s)|²₂ds≤2C_g² Z t

τ

e^ms|v(s)|²₂ds+2C_g²

m (e^mt−e^mτ)|ψ|²₂ +C_g²e^mτ

Z 0

−h

|η(s) +ψ|²₂ds.

(3.5)

From (3.4)–(3.5), we conclude that e^mt|v(t)|²₂−e^mτ|v(0)|²₂

≤ e^mt

mνkf¯k²_∗+4C_g²

αme^mt|ψ|²₂+4C_g² α

Z t

τ

e^ms|v(s)|²₂ds+2C_g²e^mτ α

Z 0

−h

|η(s) +ψ|²₂ds

−(νλ1−2c1λ^1/2₁ kψk) Z t

τ

e^ms|v(s)|²₂ds

=e^mtkf¯k²_∗ mν +4C_g²

mα|ψ|²₂

+2C_g²e^mτ α

Z 0

−h

|η(s) +ψ|²₂ds

−

νλ1−2c1λ^1/2₁ kψk −4C_g²/αZ t τ

e^ms|v(s)|²₂ds.

Choosing an appropriate constant such that νλ1−2c1λ^1/2₁ kψk −4C_g²/α ≥ 0, we obtain

|v(t)|²₂≤ kf¯k²_∗ mν +4C_g²

mα|ψ|²₂+e^−mt2C_g²e^mτ α

Z 0

−h

|η(s) +ψ|²₂ds+e^mτ|v0|²₂ ,

where Z 0

−h

|η(s) +ψ|²₂ds≤2 Z 0

−h

|η(s)|²₂ds+ 2 Z 0

−h

|ψ|²₂ds= 2h|η(s)|²₂+ 2h|ψ|²₂. From (3.1), we have

|v0|²₂+4C_g²he^mτ

α |η(s)|²₂≤

1 + 4C_g²he^mτ α

d², i.e.,

|v(t)|²≤kf¯k²_∗

mν +4C_g²c⁰²

mα kϕk²_L∞(∂Ω)+e^−mth4C_g²he^mτ

α |ψ|²+ (1 +4C_g²he^mτ α )d²i

. Hence, fort > handθ∈[−h,0], we have

|v(t+θ)|²₂−kf¯k²_∗

mν +4C_g²c⁰²

mα kϕk²_L∞(∂Ω)

≤e^−m(t+θ)h4C_g²he^mτ

α |ψ|²₂+ (1 + 4C_g²he^mτ α )d²i

≤e^−mte^mhh4C_g²c⁰²he^mτ

α kϕk²_L∞(∂Ω)+ (1 +4C_g²he^mτ α )d²i and

kvtk²_CH −kf¯k²_∗

mν +4C_g²c⁰²

mα kϕk²_L∞(∂Ω)

≤e^−mte^mhh4C_g²c⁰²he^mτ

α kϕk²_L∞(∂Ω)+ (1 +4C_g²he^mτ α )d²i

. If we take

e^−mte^mh≤ αkf¯k²_∗+ 4νC_g²c⁰²kϕk²_L∞(∂Ω)

4mνhC_g²c⁰²e^mτkϕk²_L∞(∂Ω)+mν(α+ 4hC_g²e^mτ)d², i.e.,

t≥TH= 1 mln

mνh

4hC_g²c⁰²e^mτkϕk²_L∞(∂Ω)+ (α+ 4hC_g²e^mτ)d²i αkf¯k²_∗+ 4νC_g²c⁰²kϕk²_L∞(∂Ω)

, and denoting

ρ²_H = 2( 1

mνkf¯k²_∗+2C_g² α |ψ|²₂), then it is sufficient to show that

kvk²_CH ≤ρ²_H (3.6)

for any (v₀, η) ∈ D ⊂ H×L²_H, where B_H(0, ρ_H) denotes an absorbing ball with center 0 and radiusρ_H inC_H, the proof is complete.

Theorem 3.2. Assume f¯∈ (L²(Ω))², (v0, η) ∈ H ×L²_H, and (H1)–(H4) hold.

Then the semigroup S(t)possesses an absorbing ball inCV to system (2.4).

Proof. Multiplying (2.5) byv, we obtain 1

2 d|v|²₂

dt +ν(Av, v) +b(v, v, v) +R(v, v) =hP_nf , vi¯ +hg_ψ(v_nt), vi,

hence, 1 2

d|v|²₂

dt +νkvk²+α|v|²₂

≤ kf¯k∗kvk+|gψ(vnt)||v|2+c1|v|2kvkkψk

≤ νkvk²

2 +kf¯k²_∗

2ν +|g_ψ(v_nt)|²₂

α +α|v|²₂+c₁λ^−1/2₁ kvk²kψk, which implies

d|v|²₂

dt ≤ kf¯k∗

ν + 2

α|gψ(vt)|²₂−(ν−2c1λ^−1/2₁ kψk)kvk². (3.7) Integrating over [t, t+ 1], we obtain

|v(t+ 1)|²₂− |v(t)|²₂+ (ν−2c₁λ^−1/2₁ kψk) Z t+1

t

kvk²ds

≤ kfk¯ ²_∗ ν + 2

α Z t+1

t

|gψ(v_s)|²₂ds.

(3.8)

Noting that Z t+1

t

|gψ(vs)|²₂ds

≤C_g² Z t+1

t−h

|v+ψ|²₂ds

≤2C_g²d²+ 2hC_g²c⁰²kϕk²_L∞(∂Ω)+ 2C_g²c⁰²kϕk²_L∞(∂Ω)+ 2C_g²λ⁻¹₁ Z t+1

t

kvk²ds.

(3.9)

we can derive

|v(t+ 1)|²₂− |v(t)|²₂+

ν−2c1λ^−1/2₁ kψkZ t+1 t

kvk²ds

≤ kf¯k_∗ ν + 2

α

2C_g²d²+ 2hC_g²c⁰²kϕk²_L∞(∂Ω)+ 2C_g²c⁰²kϕk²_L∞(∂Ω)

+ 2C_g²λ⁻¹₁ Z t+1

t

kvk²ds from (3.8)–(3.9), i.e.,

ν−2c1λ^−1/2₁ kψk −4C_g²λ⁻¹₁ α

Z t+1

t

kvk²ds

≤ kf¯k_∗

ν +4C_g²d²

α +4C_g²c⁰²

α (1 +h)kϕk²_L∞(∂Ω)+K, whereK is defined as in (2.17).

From the above estimates, we deduce that Z t+1

t

kvk²≤I_v, (3.10)

where

Iv =

kfk¯∗

ν +^4C

2 gd² α +^4C

2 gc⁰²

α (1 +h)kϕk²_L∞(∂Ω)+K ν−2c₁λ^−1/2₁ kψk −^4Cg2λ

−1 1

α

. (3.11)

Multiplying (2.5) byAv, we have 1

2 dkvk²

dt +ν(Av, Av) +B(v, Av) +R(v, Av) + (αv, Av) =hf , Avi¯ +hgψ(vt), Avi, i.e.,

1 2

dkvk²

dt +ν|Av|²₂+αkvk²

≤ |hf , Avi|¯ +|hgψ(vt), Avi|+|b(v, v, Av)|+|R(v, Av)|.

(3.12) Noting that

|hf , Avi|¯ +|hgψ(vt), Avi| ≤ |f¯|2|Av|2+|gψ(vt)|2|Av|2

≤ ν

6|Av|²₂+ 3

2ν|f|¯²₂+ν

6|Av|²₂+ 3

2ν|gψ(v_t)|²₂

= ν

3|Av|²₂+ 3

2ν|f|¯²₂+ 3

2ν|gψ(v_t)|²₂,

(3.13)

|b(v, v, Av)| ≤c1|v|L^∞kvk|Av|2

≤C|v|^1/2₂ |Av|^1/2₂ kvk|Av|2

≤ν

3|Av|²₂+C

ν|v|²₂kvk⁴,

(3.14)

|R(v, v, Av)|

≤ |b(v, ψ, Av)|+|b(ψ, v, Av)|

≤c₃|v|^1/2₂ |Av|^3/2₂ kϕkL^∞(∂Ω)+c₄kϕkL^∞(∂Ω)kvk|Av|2

≤ ν

3|Av|²₂+ (6

ν)³c⁴₃|v|²₂kϕk⁴_L∞(∂Ω)+ 3

2νc²₄kϕk²_L∞(∂Ω)kvk²,

(3.15)

and using (3.13)-(3.15) in (3.12), we obtain 1

2 dkvk²

dt +ν|Av|²₂+αkvk²

≤ |f¯|₂|Av|₂+|g_ψ(v_t)|₂|Av|₂

≤ν

3|Av|²₂+ 3

2ν(|f¯|²₂+|gψ(vt)|²₂) +ν

3|Av|²₂+ (3

ν)³c⁴₁|v|²₂kvk⁴ +ν

3|Av|²₂+ (6

ν)³c⁴₃|v|²₂kϕk⁴_L∞(∂Ω)+ 3

2νc²₄kϕk²_L∞(∂Ω)kvk²

=ν|Av|²₂+ 3

2ν(|f|¯²₂+|gψ(vt)|²₂) + (3

ν)³c⁴₁|v|²₂kvk⁴ + (6

ν)³c⁴₃|v|²₂kϕk⁴_L∞(∂Ω)+ 3

2νc²₄kϕk²_L∞(∂Ω)kvk². Hence, the above estimate yields

1 2

dkvk²

dt +αkvk²≤ 3

2ν(|f¯|²₂+|gψ(v_t)|²₂) + (6

ν)³c⁴₃|v|²₂kϕk⁴_L∞(∂Ω)

+ (3

ν)³c⁴₁|v|²₂kvk⁴+3c²₄

2ν kϕk²_L∞(∂Ω)kvk², i.e.,

dkvk² dt ≤ 3

ν(|f¯|²₂+|g_ψ(v_t)|²₂) + 2(6

ν)³c⁴₃|v|²₂kϕk⁴_L∞(∂Ω)