In this paper, we discuss the accuracy and the robustness of the mixed Virtual Element Methods when dealing with highly anisotropic diffusion problems. In particular, we analyze the performance of different approaches which are characterized by different sets of both boundary and internal degrees of freedom in the presence of a strong anisotropy of the diffusion tensor with constant or variable coefficients. A new definition of the boundary degrees of freedom is also proposed and tested.
Citation: Stefano Berrone, Stefano Scialò, Gioana Teora. The mixed virtual element discretization for highly-anisotropic problems: the role of the boundary degrees of freedom[J]. Mathematics in Engineering, 2023, 5(6): 1-32. doi: 10.3934/mine.2023099
[1] | Xin-Guang Yang, Lu Li, Xingjie Yan, Ling Ding . The structure and stability of pullback attractors for 3D Brinkman-Forchheimer equation with delay. Electronic Research Archive, 2020, 28(4): 1395-1418. doi: 10.3934/era.2020074 |
[2] | Shu Wang, Mengmeng Si, Rong Yang . Dynamics of stochastic 3D Brinkman-Forchheimer equations on unbounded domains. Electronic Research Archive, 2023, 31(2): 904-927. doi: 10.3934/era.2023045 |
[3] | Lingrui Zhang, Xue-zhi Li, Keqin Su . Dynamical behavior of Benjamin-Bona-Mahony system with finite distributed delay in 3D. Electronic Research Archive, 2023, 31(11): 6881-6897. doi: 10.3934/era.2023348 |
[4] | Pan Zhang, Lan Huang, Rui Lu, Xin-Guang Yang . Pullback dynamics of a 3D modified Navier-Stokes equations with double delays. Electronic Research Archive, 2021, 29(6): 4137-4157. doi: 10.3934/era.2021076 |
[5] | Jiwei Jia, Young-Ju Lee, Yue Feng, Zichan Wang, Zhongshu Zhao . Hybridized weak Galerkin finite element methods for Brinkman equations. Electronic Research Archive, 2021, 29(3): 2489-2516. doi: 10.3934/era.2020126 |
[6] | Wei Shi, Xinguang Yang, Xingjie Yan . Determination of the 3D Navier-Stokes equations with damping. Electronic Research Archive, 2022, 30(10): 3872-3886. doi: 10.3934/era.2022197 |
[7] | Keqin Su, Rong Yang . Pullback dynamics and robustness for the 3D Navier-Stokes-Voigt equations with memory. Electronic Research Archive, 2023, 31(2): 928-946. doi: 10.3934/era.2023046 |
[8] | Nisachon Kumankat, Kanognudge Wuttanachamsri . Well-posedness of generalized Stokes-Brinkman equations modeling moving solid phases. Electronic Research Archive, 2023, 31(3): 1641-1661. doi: 10.3934/era.2023085 |
[9] | Zhiqing Li, Wenbin Zhang, Yuanfei Li . Structural stability for Forchheimer fluid in a semi-infinite pipe. Electronic Research Archive, 2023, 31(3): 1466-1484. doi: 10.3934/era.2023074 |
[10] | Wenlong Sun . The boundedness and upper semicontinuity of the pullback attractors for a 2D micropolar fluid flows with delay. Electronic Research Archive, 2020, 28(3): 1343-1356. doi: 10.3934/era.2020071 |
In this paper, we discuss the accuracy and the robustness of the mixed Virtual Element Methods when dealing with highly anisotropic diffusion problems. In particular, we analyze the performance of different approaches which are characterized by different sets of both boundary and internal degrees of freedom in the presence of a strong anisotropy of the diffusion tensor with constant or variable coefficients. A new definition of the boundary degrees of freedom is also proposed and tested.
The delay effect originates from the boundary controllers in engineering. The dynamics of a system with boundary delay could be described mathematically by a differential equation with delay term subject to boundary value condition such as [20]. There are many results available in literatures on the well-posedness and pullback dynamics of fluid flow models with delays especially the 2D Navier-Stokes equations, which can be seen in [1], [2], [8] and references therein. Inspired by these works, in this paper, we study the stability of pullback attractors for 3D Brinkman-Forchheimer (BF) equation with delay, which is also a continuation of our previous work in [6]. The existence and structure of attractors are significant to understand the large time behavior of solutions for non-autonomous evolutionary equations. Furthermore, the asymptotic stability of trajectories inside invariant sets determines many important properties of trajectories. The 3D Brinkman-Forchheimer equation with delay is given below:
{∂u∂t−νΔu+αu+β|u|u+γ|u|2u+∇p=f(t,ut)+g(x,t),∇⋅u=0, u(t,x)|∂Ω=0,u|t=τ=uτ(x), x∈Ω,uτ(θ,x)=u(τ+θ,x)=ϕ(θ), θ∈(−h,0), h>0. | (1) |
Here,
(1). a general delay
or
(2). the special application of
f(t,ut)=F(u(t−ρ(t))) | (2) |
for a smooth function
The BF equation describes the conservation law of fluid flow in a porous medium that obeys the Darcy's law. The physical background of 3D BF model can be seen in [14], [9], [18], [19]. For the dynamic systems of problem
(a) For problem (1) with delay
(b) For problem (1) with special application of
(c) The asymptotic stability of trajectories inside pullback attractors is further research of the results established in [6]. However, the stability of pullback attractors for (1) with infinite delay is still unknown.
In this section, we give some notations and the equivalent abstract form of (1) in this section.
Denoting
By the Helmholz-Leray projection defined above, (1) can be transformed to the abstract equivalent form
{∂u∂t+νAu+P(αu+β|u|u+γ|u|2u)=Pf(t,ut)+Pg(t,x),u|∂Ω=0,u|t=τ=uτ(x),uτ(θ,x)=ϕ(θ,x) for θ∈(−h,0), | (3) |
then we show our results for (3) with
We also define some Banach spaces on delayed interval as
‖ϕ‖CH=supθ∈[−h,0]‖ϕ(θ)‖H, ‖ϕ‖CV=supθ∈[−h,0]‖ϕ(θ)‖V, |
respectively. The Lebesgue integrable spaces on delayed interval can be denoted as
Some assumptions on the external forces and parameters which will be imposed in our main results are the following:
‖f(t,ξ)−f(t,η)‖H≤Lf‖ξ−η‖CH, for ξ,η∈CH. |
∫tτ‖f(r,ur)−f(r,vr)‖2Hdr≤C2f∫tτ−h‖u(r)−v(r)‖2Hdr, for τ≤t. | (4) |
∫t−∞eηs‖g(s,⋅)‖2V′ds<∞. | (5) |
holds for any
Lemma 3.1. (The Gronwall inequality with differential form) Let
ddtm(t)≤v(t)m(t)+h(t), m(t=τ)=mτ, t≥τ. | (6) |
Then
m(t)≤mτe∫tτv(s)ds+∫tτh(s)e∫tsv(σ)dσds, t≥τ. | (7) |
In this part, we shall present some retarded integral inequalities from Li, Liu and Ju [5]. Consider the following retarded integral inequalities:
‖y(t)‖X≤E(t,τ)‖yτ‖X+∫tτK1(t,s)‖ys‖Xds+∫∞tK2(t,s)‖ys‖Xds+ρ, ∀ t≥τ, | (8) |
where
Let
κ(K1,K2)=supt≥τ(∫tτK1(t,s)ds+∫∞tK2(t,s)ds). |
We assume that
limt→+∞E(t+s,s)=0 | (9) |
uniformly with respect to
Lemma 3.2. (The retarded Gronwall inequality) Denoting
(1) If
‖yt‖X<μρ+ε, | (10) |
for
(2) If
‖yt‖X≤M‖y0‖Xe−λt+γρ, t≥τ | (11) |
for all bounded functions
(3) If
Proof. See Li, Liu and Ju [5].
Remark 1. (The special case:
The minimal family of pullback attractors will be stated here in preparation for our main result.
Lemma 3.3. (1) (See [7], [11]) Assume that
(|a|β−2a−|b|β−2b)⋅(a−b)≥γ0|a−b|β, |
where
(2) The following
|xq−yq|≤Cq(|x|q−1+|y|q−1)|x−y| |
for the integer
Theorem 3.4. Assume that the external forces
Proof. Step 1. Existence of local approximate solution.
By the property of the Stokes operator
Awi=λiwi, i=1,2,⋯. | (12) |
Let
{(∂tum,wj)+ν(∇um,∇wj)+(αum+β|um|um+γ|um|2um,wj)=(f(t,umt),wj)+⟨g,wj⟩,um(τ)=Pmuτ=uτm,umτ(θ,x)=Pmϕ(θ)=ϕm(θ) for θ∈[−h,0], | (13) |
Then it is easy to check that (13) is equivalent to an ordinary differential equations with unknown variable function
Step 2. Uniform estimates of approximate solutions.
Multiplying (13) by
12ddt‖um‖2H+ν‖um‖2V+α‖um‖2H+β‖um‖3L3(Ω)+γ‖um‖4L4(Ω)≤|(g(t)+f(s,umt(s)),um)|≤α‖um‖2H+ν2‖um‖2V+12ν‖g(t)‖2V′+14α‖f(t,umt)‖2H. | (14) |
Integrating in time, using the hypotheses on
‖um‖2H+ν∫tτ‖um‖2Vds+2β∫tτ‖um‖3L3(Ω)ds+2γ∫tτ‖um‖4L4(Ω)ds≤‖uτ‖2H+C2f4α∫0−h‖ϕ(s)‖2Hds+12ν∫tτ‖g(s)‖2V′ds+C2f4α∫tτ‖um‖2Hds. | (15) |
Using the Gronwall Lemma of integrable form, we conclude that
{um} is bounded in the spaceL∞(τ,T;H)∩L2(τ−h,T;V)∩L3(τ,T;L3(Ω))∩L4(τ,T;L4(Ω)). |
Step 3. Compact argument and passing to limit for deriving the global weak solutions.
In this step, we shall prove
dumdt=−νAum−αum−β|um|um−γ|um|2um+P(g(t)+f(t,umt) | (16) |
and assumptions
By virtue of the Aubin-Lions Lemma, we obtain that
{um(t)⇀u(t) weakly * in L∞(τ,T;H),um(t)→u(t) stongly in L2(τ,T;H),um(t)⇀u(t) weakly in L2(τ,T;V),dum/dt⇀du/dt weakly in L2(τ,T;V′),f(⋅,um⋅)⇀f(⋅,u⋅) weakly in L2(τ,T;H),um⇀u(t) weakly in L3(τ,T;L3(Ω)),um⇀u(t) weakly in L4(τ,T;L4(Ω)) | (17) |
which coincides with the initial data
For the purpose of passing to limit in (13), denoting
∫Tτ(β|um|um−β|u|u,wj)ds≤Cλ1β‖um‖4L4(τ,T;L4(Ω))‖um−u‖4L4(τ,T;L4(Ω))+Cβ‖um−u‖L∞(τ,T;H)‖u‖2L2(τ−h,T;H) |
and
∫Tτ(γ|um|2um−γ|u|2u,wj)ds≤Cγ‖um‖2L2(τ,T;V)‖um−u‖4L4(τ,T;L4(Ω))+Cγ‖um−u‖4L4(τ,T;L4(Ω))(‖u‖2L2(τ−h,T;V)+‖um‖4L4(τ,T;L4(Ω))) | (18) |
and the convergence of delayed external force
Thus, passing to the limit of (13), we conclude that
Proposition 1. Assume that the external forces
Proof. Taking inner product of (3) with
12ddt‖A1/2u‖2H+ν‖Au‖2H+α‖A1/2u‖2H+β∫Ω|u|u⋅Audx+γ∫Ω|u|2u⋅Audx=(f(t,ut),Au)+(g(t),Au). | (19) |
According to Lemma 3.3, the nonlinear terms have the following estimates
|β(|u|u,Au)|≤ν2‖Au‖2H+β4ν‖u‖4L4 | (20) |
and
γ∫Ω|u|2u⋅Audx=γ2∫Ω|∇(|u|2)|2dx+γ∫Ω|u|2|∇u|2dx | (21) |
and
(f(t,ut),Au)+(g(t),Au)≤12ν‖f(t,ut)‖2H+12ν‖g(t)‖2H+ν2‖Au‖2H, | (22) |
hence, we conclude that
ddt‖A1/2u‖2H+2α‖A1/2u‖2H+γ∫Ω|∇(|u|2)|2dx+2γ∫Ω|u|2|∇u|2dx≤β2ν‖u‖4L4+1ν‖f(t,ut)‖2H+1ν‖g(t)‖2H. | (23) |
Letting
‖A1/2u(t)‖2H+2α∫ts‖A1/2u(r)‖2Hdr≤‖A1/2u(s)‖2H+β2ν∫ts‖u(r)‖4L4dr+2ν∫ts‖f(r,ur)‖2Hdr+2ν∫ts‖g(r)‖2Hdr | (24) |
and
∫ts‖f(r,ur)‖2Hdr≤L2f‖ϕ(θ)‖2L2H+L2f∫ts‖u(r)‖2Hdr. | (25) |
Then integrating with
‖A1/2u(t)‖2H≤∫tt−1‖A1/2u(s)‖2Hds+β2ν∫tt−1‖u(r)‖4L4dr+2L2fν‖ϕ(θ)‖2L2H+2L2fν∫tτ‖u(r)‖2Hdr+2ν∫tt−1‖g(r)‖2Hdr≤C[‖ϕ‖2L2H+‖uτ‖2H]+C∫tτ‖g‖2Hds+2L2fνλ1∫tτ‖u(r)‖2Vdr, | (26) |
which means the uniform boundedness of the global weak solution
Proposition 2. Assume the hypotheses in Theorem 3.4 hold. Then the global weak solution
Proof. Using the same energy estimates as above, we can deduce the uniqueness easily, here we skip the details.
To description of pullback attractors, the functional space
∫tτeηs‖f(s,us)‖2Hds<C2f∫tτ−heηs‖u(s)‖2Hds. | (27) |
for any
Proposition 3. For given
Lemma 3.5. Assume that
‖u(t)‖2H≤e−8ηCfα(t−τ)(‖uτ‖2H+Cf‖ϕ(r)‖2L2H)+e−8ηCfαtν−ηλ−1∫tτeηr‖g(r)‖2V′dr | (28) |
and
ν∫ts‖u(r)‖2Vdr≤‖u(s)‖2H+8Cfα‖us‖2L2H+1ν∫ts‖g(r)‖2V′dr+8Cfα∫ts‖u(r)‖2Hdr, | (29) |
β∫ts‖u(r)‖3L3(Ω)dr≤‖u(s)‖2H+8Cfα‖us‖2L2H+1ν∫ts‖g(r)‖2V′dr+8Cfα∫ts‖u(r)‖2Hdr, | (30) |
γ∫ts‖u(r)‖4L4(Ω)dr≤‖u(s)‖2H+8Cfα‖us‖2L2H+1ν∫ts‖g(r)‖2V′dr+8Cfα∫ts‖u(r)‖2Hdr. | (31) |
Proof. By the energy estimate of (1) and using Young's inequality, we arrive at
ddt‖u‖2H+2ν‖u‖2V+2α‖u‖2H+2β‖u‖3L3(Ω)+2γ‖u‖2L4(Ω)≤1ν−ηλ−1‖g‖2V′+(ν−ηλ−1)‖u‖2V+2α‖u‖2H+8α‖f(t,ut)‖2H, | (32) |
where
Multiplying the above inequality by
ddt(eηt‖u‖2H)+eηtνλ1‖u‖2H+2βeηt‖u‖3L3(Ω)+2γeηt‖u‖2L4(Ω)≤1ν−ηλ−1eηt‖g‖2V′+8Cfαeηt‖f(t,ut)‖2H. |
Thus integrating with respect to time variable, it yields
eηt‖u‖2H+νλ1∫tτeηr‖u(r)‖2Hdr≤eητ(‖uτ‖2H+Cf∫0−h‖ϕ(r)‖2Hdr)+1ν−ηλ−1∫tτeηr‖g(r)‖2V′dr+8Cfα∫tτeηr‖u(r)‖2Hdr | (33) |
and by the Gronwall Lemma, we can derive the estimate in our theorem.
Using the energy estimate of (1) again, we can check that
ddt‖u‖2H+2ν‖u‖2V+2α‖u‖2H+2β‖u‖3L3(Ω)+2γ‖u‖2L4(Ω)≤1ν‖g‖2V′+ν‖u‖2V+2α‖u‖2H+8α‖f(t,ut)‖2H, | (34) |
Integrating from
Based on Lemma 3.5, we can present the pullback dissipation based on the following universes for the tempered dynamics.
Definition 3.6. (Universe). (1) We will denote by
limτ→−∞(eητsup(ξ,ζ)∈D(τ)‖(ξ,ζ)‖2MH)=0. | (35) |
(2)
Remark 2. The universes
Proposition 4. (The
D0(t)=¯BH(0,ρH(t))×(¯BL2V(0,ρL2H(t))∩¯BCH(0,ρCH(t))) |
is the pullback
ρ2H(t)=1+e−8ηCfα(t−h)ν−ηλ−1∫t−∞eηr‖g(r)‖2V′dr,ρ2L2V(t)=1ν[1+‖uτ‖2H+8Cfα‖ϕ‖2L2H+‖g(r)‖2L2(t−h,t;V′)ν+8Cfhαρ2H(t)]. |
Moreover, the pullback
Proof. Using the estimates in Lemma 3.5, choosing any
‖u(t,τ;uτ,ϕ)‖2H≤ρ2H(t)=1+e−8ηCfα(t−h)ν−ηλ−1∫t−∞eηr‖g(r)‖2V′dr | (36) |
holds for any
Theorem 3.7. Assume that
Proof. Step 1. Weak convergence of the sequence
For arbitrary fixed
By using the similar energy estimate in Theorem 3.4 and technique in Proposition 4, there exists a pullback time
‖(un)′‖L2(t−h−1,t;V′)≤ν‖un‖L2(t−h−1,t;V)+αλ−11‖un‖L2(t−h−1,t;V)+β‖un‖L4(t−h−1,t;L4(Ω))+Cλ1,|Ω|γ‖un‖L2(t−h−1,t;V)+Cα‖f(t,unt)‖L2(t−h−1,t;H)+Cν‖g‖L2(t−h−1,t;V′). | (37) |
From the hypotheses
{un⇀u weakly * in L∞(t−3h−1,t;H),un⇀u weakly in L2(t−2h−1,t;V),(un)′⇀u′ weakly in L2(t−h−1,t;V′),um⇀u(t) weakly in L3(t−2h−1,t;L3(Ω)),um⇀u(t) weakly in L4(t−2h−1,t;L4(Ω)),un→u stongly in L2(t−h−1,t;H),un(s)→u(s) stongly in H, a.e. s∈(t−h−1,t). | (38) |
By Theorem 3.4, from the hypothesis on
f(⋅,un⋅)⇀f(⋅,u⋅) weakly in L2(t−h−1,t;H). | (39) |
Thus, from (38) and (39), we can conclude that
From the uniform bounded estimate of
un→u strongly in C([t−h−1,t];H). | (40) |
Therefore, we can conclude that
un(sn)⇀u(s) weakly in H | (41) |
for any
lim infn→∞‖un(sn)‖H≥‖u(s)‖H. | (42) |
Step 2. The strong convergence of corresponding sequences via energy equation method:
The asymptotic compactness of sequence
‖un(sn)−u(s)‖H→0 as n→+∞, | (43) |
which is equivalent to prove (42) combining with
lim supn→∞‖un(sn)‖H≤‖u(s)‖H | (44) |
for a sequence
Using the energy estimate to all
‖un(s2)‖2H+ν∫s2s1‖un(r)‖2Vdr+2β∫s2s1‖un(r)‖3L4(Ω)dr+2γ∫s2s1‖un(r)‖4L4(Ω)≤2C2fα∫s2s2‖unr‖2Hdr+8ν∫s2s1‖g(r)‖2V′dr | (45) |
and
‖u(s2)‖2H+ν∫s2s1‖u(r)‖2Vdr+2β∫s2s1‖u(r)‖3L4(Ω)dr+2γ∫s2s1‖u(r)‖4L4(Ω)≤2C2fα∫s2s2‖ur‖2Hdr+8ν∫s2s1‖g(r)‖2V′dr. | (46) |
Then, we define the functionals
Jn(s)=12‖un‖2H−∫st−h−1⟨g(r),un(r)⟩dr−∫st−h−1(f(r,unr),un(r))dr | (47) |
and
J(t)=12‖u(s)‖2H−∫st−h−1⟨g(r),u(r)⟩dr−∫st−h−1(f(r,ur),u(r))dr. | (48) |
Combining the convergence in (38), observing that
∫tt−h−1⟨g(r),un(r)⟩dr→2∫tt−h−1⟨g(r),u(r)⟩dr | (49) |
and
∫tt−h−1(f(r,unr),un(r))dr→2∫tt−h−1(f(r,ur),u(r))dr | (50) |
as
Jn(s)→J(s) a.e.s∈(t−h−1,t), | (51) |
i.e., for
|Jn(sk)−J(sk)|≤ε2. | (52) |
Since
|J(sk)−J(s)|≤ε2, | (53) |
Choosing
|Jn(sn)−J(s)|≤|Jn(sn)−J(sn)|+|J(sn)−J(s)|<ε. | (54) |
Therefore, for any
lim supn→∞Jn(sn)≤J(s), | (55) |
which implies
lim supn→∞‖un(sn)‖H≤‖u(s)‖H. | (56) |
we conclude the strong convergence
Step 3. The strong convergence:
Combining the energy estimates in (45) and (46), noting the energy functionals
‖un(s)‖L2(t−h,t;V)→‖u(s)‖L2(t−h,t;V). | (57) |
Hence jointing with the weak convergence in (38), we can derive that
Step 4. The
By using the results from Steps 2 to 4 and noting the definition of universe, we can conclude that the processes is
Remark 3. Using the similar technique, we can derive the processes
Theorem 3.8. Assume that
ADMHF(t)⊂ADMHη(t). | (58) |
Proof. From Proposition 3, we observe that the process
Based on the universes defined in Definition 3.6, the relation between
Definition 3.9. The pullback attractors is asymptotically stable if the trajectories inside attractor reduces to a single orbit as
Theorem 3.10. Assume that
G(t)≤K0, |
where
K0={[ν2λ1(2νλ1+α)]/[4C|Ω|β(L2fα2−L2fα1−2L2fα+1α)]}1/2, |
here
Proof. Let
u(τ+θ)|θ∈[−h,0]=ϕ(θ), u|t=τ=uτ | (59) |
and
v(τ+θ)|θ∈[−h,0]=˜ϕ(θ), v|t=τ=˜uτ | (60) |
respectively. Denoting
(u,ut)=U(t,τ)(uτ,φ) and (v,vt)=U(t,τ)(˜uτ,˜φ) | (61) |
as two trajectories inside the pullback attractors, letting
{∂w∂t+νAw+P(αw+β(|u|u−|v|v)+γ(|u|2u−|v|2v))=P(f(t,ut)−f(t,vt)),w|∂Ω=0,w(t=τ)=uτ−˜uτ,w(τ+θ)=ϕ(θ)−˜ϕ(θ), θ∈[−h,0]. | (62) |
Taking inner product of (62) with
γ(|u|2u−|v|2v,u−v)≥γγ0‖u−v‖4L4 | (63) |
and
12ddt‖w‖2H+ν‖w‖2V+α‖w‖2H+γγ0‖w‖4L4≤|β(|u|u−|v|v,w)|+|(f(t,ut)−f(t,vt),w)|≤β(∫Ω|u|2|w|dx+∫Ω|w||v|2dx)+α2‖w‖2H+L2f2α‖wt‖2H |
≤β(‖u‖2L4+‖v‖2L4)‖w‖2H+α2‖w‖2H+L2f2α‖wt‖2H≤C|Ω|β(‖u‖2V+‖v‖2V)‖w‖2H+α2‖w‖2H+L2f2α‖wt‖2H. | (64) |
Using the Poincaré inequality and Lemma 3.1, noting that if
2νλ1+α−2C|Ω|β(‖u‖2V+‖v‖2V)>0, | (65) |
then we can obtain
‖w‖2H≤e∫tτ[2C|Ω|β(‖u‖2V+‖v‖2V)−(2νλ1+α)]ds[‖uτ−˜uτ‖2H++L2fα∫tτe−∫ts[2νλ1+α−2C|Ω|β(‖u‖2V+‖v‖2V)]dσ‖wt‖2Hds]. | (66) |
Denoting
E(t,τ)=e−∫tτ[2νλ1+α−2C|Ω|β(‖u‖2V+‖v‖2V)]ds | (67) |
and
K1(t,s)=L2fαe−∫ts[2νλ1+α−2C|Ω|β(‖u‖2V+‖v‖2V)]dσ | (68) |
and
Θ=supt≥s≥τE(t,s), κ(K1,0)=supt≥τ∫tτK1(t,s)ds, | (69) |
by virtue of Lemma 3.2, choosing
‖wt‖2H≤M‖uτ−˜uτ‖2He−λ(t−τ). | (70) |
Substituting (70) into (64), using Lemma 3.1 again, we can conclude the following estimate
‖w‖2H≤‖uτ−˜uτ‖2He−∫tτ[2νλ1+α−2C|Ω|β(‖u‖2V+‖v‖2V)]ds+L2fαM‖uτ−˜uτ‖2He−λ(t−τ)∫tτe−∫ts[2νλ1+α−2C|Ω|β(‖u‖2V+‖v‖2V)]dσds. | (71) |
From (70) and (71), if we fixed
2νλ1+α>2C|Ω|β⟨‖u‖2V+‖v‖2V⟩≤t, | (72) |
where
⟨h⟩≤t=lim supτ→−∞1t−τ∫tτh(r)dr. | (73) |
Since
12ddt‖u‖2H+ν‖A1/2u‖2H+α‖u‖2H+β‖u‖3L3+γ‖u‖4L4≤α‖u‖2H+12α[‖f(t,ut)‖2H+‖g‖2H]≤α‖u‖2H+L2f2α‖ut‖2H+12α‖g‖2H. | (74) |
Using the Poincaré inequality and Lemma 3.1, then we can obtain
‖u‖2H≤e−2νλ1(t−τ)‖uτ‖2H++L2fα∫tτe−2νλ1(t−s)‖us‖2Hds+1α∫tτe−2νλ1(t−s)‖g‖2Hds. | (75) |
Denoting
E(t,τ)=e−2νλ1(t−τ) | (76) |
and
K1(t,s)=L2fαe−2νλ1(t−s) | (77) |
and
ρ=1α∫tτe−2νλ1(t−s)‖g‖2Hds, | (78) |
letting
Θ=supt≥s≥τE(t,s), κ(K1,0)=supt≥τ∫tτK1(t,s)ds, | (79) |
by virtue of Lemma 3.2, choosing
‖ut‖2H≤ˆM‖uτ‖2He−λ(t−τ)+2−L2fα1−2L2fα∫tτe−2νλ1(t−s)‖g‖2Hds≤ˆM‖uτ‖2He−λ(t−τ)+2−L2fα1−2L2fα∫tτ‖g‖2Hds. | (80) |
Substituting (80) into (75), using Lemma 3.1 again, we can conclude the following estimate
‖u‖2H≤C‖uτ‖2He−λ(t−τ)+(L2fα2−L2fα1−2L2fα+1α)∫tτ‖g‖2Hds. | (81) |
Integrating (74) from
‖u‖2H+2ν∫tτ‖u‖2Vds+2β∫tτ‖u‖3L3ds+2γ∫tτ‖u‖4L4ds≤[1α‖ϕ‖2L2H+‖uτ‖2H]+L2fα∫tτ‖ut(s)‖2Hds+1α∫tτ‖g‖2Hds. | (82) |
By the estimate of (80) and (81), we derive
∫tτ‖u(r)‖4L4dr≤C[1α‖ϕ‖2L2H+‖uτ‖2H]+(L2fα2−L2fα1−2L2fα+1α)∫tτ‖g‖2Hds | (83) |
and
∫tτ‖u(r)‖2Vdr≤C[1α‖ϕ‖2L2H+‖uτ‖2H]+(L2fα2−L2fα1−2L2fα+1α)∫tτ‖g‖2Hds. | (84) |
Combining (72), (73) with (84), we conclude that
\begin{eqnarray} && \langle \|u\|^2_V+\|v\|^2_V\rangle|_{\leq t}\leq 2 \Big(\frac{L_f^2}{\alpha}\frac{2-\frac{L^2_f}{\alpha}}{1-\frac{2L^2_f}{\alpha}}+\frac{1}{\alpha}\Big)\langle \|g\|^2_H\rangle|_{\leq t} \end{eqnarray} | (85) |
and hence the asymptotic stability holds provided that
\begin{eqnarray} 4C_{|\Omega|}\beta \Big(\frac{L_f^2}{\alpha}\frac{2-\frac{L^2_f}{\alpha}}{1-\frac{2L^2_f}{\alpha}}+\frac{1}{\alpha}\Big)\langle \|g\|^2_H\rangle|_{\leq t} \leq 2\nu\lambda_1+\alpha. \end{eqnarray} | (86) |
If we define the generalized Grashof number as
\begin{eqnarray} G(t)\leq \Big\{(2\nu\lambda_1+\alpha)\Big/\Big[4C_{|\Omega|}\nu^2\beta\lambda_1 \Big(\frac{L_f^2}{\alpha}\frac{2-\frac{L^2_f}{\alpha}}{1-\frac{2L^2_f}{\alpha}}+\frac{1}{\alpha}\Big)\Big]\Big\}^{1/2} = K_0, \end{eqnarray} | (87) |
which completes the proof for our first result.
Remark 4. Theorem 3.10 is a further research for the existence of pullback attractor in [6].
We first state some hypothesis on the external forces and sub-linear operator.
\Big|\frac{d\rho}{dt}\Big|\leq\rho^{\ast} < 1, \ \ \forall t\geq 0. |
\begin{eqnarray} \|F(y)\|^2_H\leq a(t)\|y\|^2_H+b(t), \ \ \forall t\geq\tau, y\in H. \end{eqnarray} | (88) |
\begin{eqnarray} \|F(u)-F(v)\|_H\leq L(R)\kappa^\frac{1}{2}(t)\|u-v\|_H, \ u,v\in H. \end{eqnarray} | (89) |
holds for
\begin{eqnarray} \int^{t}_{-\infty}e^{ms}\|g(s,\cdot)\|^{2}_Hds < \infty, \ \ \forall t\in\mathbb{R}. \end{eqnarray} | (90) |
\begin{eqnarray} \frac{\nu}{2}-\frac{\|a\|_{L^q_{loc}(\mathbb{R})}}{1-\rho^\ast} > 0. \end{eqnarray} | (91) |
In this part, the well-posedness and pullback attractors for problem (1) with sub-linear operator will be stated for our discussion in sequel.
Assume that the initial date
\begin{equation} \begin{cases} u(t)+\int^t_\tau P(\nu Au+\alpha u+\beta|u|u+\gamma |u|^2u)ds &\\ \quad = u(\tau) +\int^t_\tau P\Big(F\big(u(s-\rho(s))\big)+g(s,x)\Big)ds,& \\ w|_{\partial\Omega} = 0,& \\ u(t = \tau) = u_{\tau},&\\ u(\tau+t) = \phi(t),\ t\in [-h,0],& \end{cases} \end{equation} | (92) |
which possesses a global mild solution as the following theorem.
Theorem 4.1. Assume that the external forces
\begin{eqnarray} &&\|u(t)\|^{2}_H+2\nu\int^t_{\tau}\|u(s)\|^{2}_Vds+2\alpha \int^t_{\tau}\|u(s)\|^{2}_Hds\\ &&+2\beta\int^t_{\tau}\|u(s)\|^{3}_{\bf{L}^3}ds+2\gamma\int^t_{\tau}\|u(s)\|^{4}_{\bf{L}^4}ds\\ & = &\|u_{\tau}\|^{2}_H+2\int^t_{\tau}\Big[\big(F(u(s-\rho(s))),u(s)\big)+2(g(s,x),u(s))\Big]ds. \end{eqnarray} | (93) |
Moreover, we can define a continuous process
Proof. Using the Galerkin method and compact argument as in Section 3.3, we can easily derive the result.
After obtaining the existence of the global well-posedness, we establish the existence of the pullback attractors to (1) with sub-linear operator.
Theorem 4.2. (The pullback attractors in
Proof. Using the similar technique as in Section3.3, we can obtain the existence of pullback attractors, here we skip the details.
Theorem 4.3. We assume that the external forces
Then the trajectories inside pullback attractors
\begin{equation} \mathit{\mbox{G}}(t)\leq \tilde{K}_0, \end{equation} | (94) |
where
\begin{equation} \tilde{K}_0 = \Big\{(2\nu\lambda_1+\alpha)\Big/\Big[2C_{|\Omega|}\beta\nu\lambda_1 (\frac{1}{\alpha^2(1-\rho^*)}+\|\tilde{a}(t)\|_{L^1})\Big]\Big\}^{1/2} > 0,\nonumber \end{equation} |
here
Proof. Step 1. The inequality for asymptotic stability of trajectories.
Let
\begin{eqnarray} u(\theta+\tau)|_{\theta\in [-h,0]} = \phi(\theta)|_{\theta\in[-h,0]},\ \ u|_{t = \tau} = u_{\tau} \end{eqnarray} | (95) |
and
\begin{eqnarray} v(\theta+\tau)|_{\theta\in [-h,0]} = \tilde{\phi}(\theta)|_{\theta\in[-h,0]},\ \ v|_{t = \tau} = \tilde{u}_{\tau} \end{eqnarray} | (96) |
respectively, then
\begin{eqnarray} (u,u_t) = (U(t,\tau)u_{\tau},U(t,\tau)\phi),\ \ (v,v_t) = (U(t,\tau)\tilde{u}_{\tau},U(t,\tau)\tilde{\phi}). \end{eqnarray} | (97) |
If we denote
\begin{equation} \begin{cases} \frac{\partial w}{\partial t}+\nu A w +P\Big(\alpha w+\beta(|u|u-|v|v)+\gamma (|u|^2u-|v|^2v)\Big)&\\ \quad = P\Big(F\big(u(t-\rho(t))\big)-F\big(v(t-\rho(t))\big)\Big),& \\ w|_{\partial\Omega} = 0,& \\ w(t = \tau) = u_{\tau}-\tilde{u}_{\tau},&\\ w(\tau+\theta) = \phi(\theta)-\tilde{\phi}(\theta),\ \theta\in [-h,0].& \end{cases} \end{equation} | (98) |
Multiplying (98) with
\begin{eqnarray} &&\frac{1}{2}\frac{d}{d t}\|w\|^2_H+\nu\|w\|^2_V+\alpha \|w\|_H^2+\gamma \gamma_0 \|w\|^4_{\bf{L}^4}\\ &\leq& |\beta(|u|u-|v|v,w)|+\Big|\Big(F\big(u(t-\rho(t))\big)-F\big(v(t-\rho(t))\big),w\Big)\Big|\\ &\leq& C_{|\Omega|}\beta(\|u\|^2_{V}+\|u\|^2_{V})\|w\|^2_{H}+\frac{\alpha}{2}\|w\|^2_H\\ &&+\frac{1}{\alpha}\|F\big(u(t-\rho(t))\big)-F\big(v(t-\rho(t))\big)\|^2_H\\ &\leq& C_{|\Omega|}\beta(\|u\|^2_{V}+\|u\|^2_{V})\|w\|^2_{H}+\frac{\alpha}{2}\|w\|^2_H\\ &&+\frac{L^2(R)\kappa(t)}{\alpha}\|w(t-\rho(t))\|^2_H. \end{eqnarray} | (99) |
Using the Poincaré inequality and Lemma 3.1, noting that if
\begin{eqnarray} 2\nu\lambda_1+\alpha-2C_{|\Omega|}\beta(\|u\|^2_V+\|v\|^2_V) > 0, \end{eqnarray} | (100) |
then we can obtain
\begin{eqnarray} \|w\|^2_H&\leq& e^{\int^t_{\tau}[2C_{|\Omega|}\beta(\|u\|^2_V+\|v\|^2_V)-(2\nu\lambda_1+\alpha)]ds}\Big[\|u_{\tau}-\tilde{u}_{\tau}\|^2_H+\\ &&+\frac{L^2(R)\|\kappa(t)\|_{L^{\infty}}}{\alpha}\int^t_{\tau}e^{-\int^t_{s}[2\nu\lambda_1+\alpha-2C_{|\Omega|}\beta(\|u\|^2_V+\|v\|^2_V)]d\sigma}\\ && \quad \times\|w(t-\rho(t))\|^2_Hds\Big]. \end{eqnarray} | (101) |
Denoting
\begin{eqnarray} E(t,\tau) = e^{-\int^t_{\tau}[2\nu\lambda_1+\alpha-2C_{|\Omega|}\beta(\|u\|^2_V+\|v\|^2_V)]ds} \end{eqnarray} | (102) |
and
\begin{eqnarray} K_1(t,s) = \frac{L^2(R)\|\kappa(t)\|_{L^{\infty}}}{\alpha}e^{-\int^t_{s}[2\nu\lambda_1+\alpha-2C_{|\Omega|}\beta(\|u\|^2_V+\|v\|^2_V)]d\sigma} \end{eqnarray} | (103) |
and
\begin{eqnarray} \Theta = \sup\limits_{t\geq s\geq \tau}E(t,s),\ \ \ \ \kappa(K_1,0) = \sup\limits_{t\geq\tau}\int^t_{\tau}K_1(t,s)ds, \end{eqnarray} | (104) |
by virtue of Lemma 3.2, choosing
\begin{eqnarray} \|w(t-\rho(t))\|^2_H&\leq& \tilde{M}\|u_{\tau}-\tilde{u}_{\tau}\|^2_H e^{-\tilde{\lambda} (t-\tau)}. \end{eqnarray} | (105) |
Substituting (105) into (99), using Lemma 3.1 again, we can conclude the following estimate
\begin{eqnarray} \|w\|^2_H &\leq& \|u_{\tau}-\tilde{u}_{\tau}\|^2_He^{-\int^t_{\tau}[2\nu\lambda_1+\alpha-2C_{|\Omega|}\beta(\|u\|^2_V+\|v\|^2_V)]ds}\\ &&+\frac{L^2(R)\|\kappa(t)\|_{L^{\infty}}}{\alpha}\tilde{M}\|u_{\tau}-\tilde{u}_{\tau}\|^2_H e^{-\tilde{\lambda} (t-\tau)}\\ && \quad \times\int^t_{\tau}e^{-\int^t_{s}[2\nu\lambda_1+\alpha-2C_{|\Omega|}\beta(\|u\|^2_V+\|v\|^2_V)]d\sigma}ds. \end{eqnarray} | (106) |
From the result in last section, we can find that the pullback attractors is asymptotically stable as
\begin{equation} 2\nu\lambda_1+\alpha > 2C_{|\Omega|}\beta \langle \|u\|^2_V+\|v\|^2_V\rangle_{\leq t}. \end{equation} | (107) |
Step 2.Some energy estimate for (1) with sub-linear operator.
Multiplying (3) with
\begin{eqnarray} &&\frac{1}{2}\frac{d}{dt}\|u\|^2_H+\nu\|A^{1/2}u\|^2_H+\alpha\|u\|^2_H+\beta \|u\|^3_{\bf{L}^3}+\gamma \|u\|^4_{\bf{L}^4}\\ &\leq&\alpha \|u\|^2_H+\frac{1}{2\alpha}\Big[\|f\big(t,u(t-\rho(t))\big)\|^2_H+\|g\|^2_{H}\Big]. \end{eqnarray} | (108) |
Moreover, let
\begin{equation} d\theta = (1-\rho'(s))ds,\ a(t)\rightarrow \tilde{a}(\bar{t})\in L^p(\tau,T), \end{equation} | (109) |
which means
\begin{align} &\int^t_\tau\|f(s,u(s-\rho(s)))\|^2_Hds\\ \leq&\int^t_\tau a(s)\|u(s-\rho(s))\|^2_Hds+\int^T_\tau b(s)ds\\ \leq& \ \dfrac{1}{1-\rho^*}\int_{\tau-\rho(\tau)}^{t-\rho(t)} \tilde{a}(s)\|u(s)\|^2_Hds+\int^t_\tau b(s)ds\\ \leq& \ \dfrac{1}{1-\rho^*}\left(\int_{-\rho(\tau)}^{0}\tilde{a}(t+\tau)\|\phi(t)\|^2_Hdt +\int^t_\tau\tilde{a}(s)\|u(s)\|^2_Hds\right)+\int^t_\tau b(s)ds\\ \leq& \dfrac{1}{1-\rho^*}\left(\|\phi(t)\|^2_{L^{2q}_{H}}\|\tilde{a}\|_{L^q(\tau-h,\tau)} +\int^t_\tau\tilde{a}(s)\|u(s)\|^2_Hds\right)+\int^t_\tau b(s)ds, \end{align} | (110) |
Integrating (108) with time variable from
\begin{eqnarray} &&\|u\|^2_H+2\nu\int^t_{\tau}\|u\|^2_Vds+2\beta\int^t_{\tau} \|u\|^3_{\bf{L}^3}ds+2\gamma \int^t_{\tau}\|u\|^4_{\bf{L}^4}ds\\ &\leq& \dfrac{\|\tilde{a}\|_{L^q(\tau-h,\tau)}}{\alpha(1-\rho^*)}\|\phi(t)\|^2_{L^{2q}_{H}}+\|u_{\tau}\|^2_H+\frac{1}{\alpha(1-\rho^*)}\int^{t}_{\tau}\tilde{a}(s)\|u(s)\|^2_Hds\\ &&+\frac{1}{\alpha}\int^t_{\tau}\|g\|^2_{H}ds+\frac{1}{\alpha}\int^t_\tau b(s)ds, \end{eqnarray} | (111) |
then we can achieve that
\begin{eqnarray} \|u(t)\|^2_H&\leq& \Big[\dfrac{\|\tilde{a}\|_{L^q(\tau-h,\tau)}}{\alpha(1-\rho^*)}\|\phi(t)\|^2_{L^{2q}_{H}}+\|u_{\tau}\|^2_H\Big]e^{-\chi_{\sigma}(t,\tau)}\\ &&+\frac{1}{\alpha}\int^t_{\tau}\|g\|^2_{H}e^{-\chi_{\sigma}(t,s)}ds+\frac{1}{\alpha}\int^t_{\tau}b(s)e^{-\chi_{\sigma}(t,s)}ds, \end{eqnarray} | (112) |
where the new variable index
\begin{eqnarray} \chi_{\sigma}(t,s) = (2\nu\lambda_1-\sigma)(t-s)-\frac{1}{\alpha(1-\rho^*)}\int^t_{s}\tilde{a}(r)dr, \end{eqnarray} | (113) |
which satisfies the relations
\begin{eqnarray} \chi_{\sigma}(0,t)-\chi_{\sigma}(0,s) = -\chi_{\sigma}(t,s) \end{eqnarray} | (114) |
and
\begin{eqnarray} \chi_{\sigma}(0,r)\leq \chi_{\sigma}(0,t)+\Big(2\nu\lambda_1-\delta\Big)h,\ \ \mbox{if}\ 2\nu\lambda_1+\alpha-\delta > 0 \end{eqnarray} | (115) |
for
Moreover, using the variable index introduced above, we can conclude that
\begin{eqnarray} &&2\nu\int^t_{\tau}\|u(r)\|^2_{V}dr\\ &\leq& \dfrac{\|\tilde{a}\|_{L^q(\tau-h,\tau)}}{\alpha(1-\rho^*)}\|\phi(t)\|^2_{L^{2q}_{H}}+\|u_{\tau}\|^2_H\\ &&+\frac{1}{\alpha}\int^t_{\tau}\|g\|^2_{H}ds+\frac{1}{\alpha}\int^t_\tau b(s)ds\\ &&+\frac{1}{\alpha(1-\rho^*)}\Big[\dfrac{\|\tilde{a}\|_{L^q(\tau-h,\tau)}}{\alpha(1-\rho^*)}\|\phi(t)\|^2_{L^{2q}_{H}}+\|u_{\tau}\|^2_H\Big]\int^t_{\tau}\tilde{a}(s)e^{-\chi_{\sigma}(s,\tau)}ds\\ &&+\frac{1}{\alpha^2(1-\rho^*)}\int^t_{\tau}\|g(s)\|^2_{H}ds\int^t_{\tau}\tilde{a}(s)ds+\frac{\|b\|_{L^1(\tau,T)}}{\alpha^2(1-\rho^*)}\int^t_{\tau}\tilde{a}(s)ds. \end{eqnarray} | (116) |
Step 3. The sufficient condition for asymptotic stability of trajectories inside pullback attractors.
Combining (107) with (116), we conclude that
\begin{eqnarray} &&2C_{|\Omega|}\beta\langle \|u\|^2_V+\|v\|^2_V\rangle|_{\leq t}\\ &\leq& \frac{2C_{|\Omega|}\beta}{\nu}\Big[\Big(\frac{1}{\alpha^2(1-\rho^*)}+\int^t_{\tau}\tilde{a}(s)ds\Big)\langle \|g(t)\|_H^2\rangle|_{\leq t} \\ &&+\frac{1}{\alpha}\langle \|b(t)\|_{L^1}\rangle|_{\leq t}+\frac{\|b\|_{L^1(\tau,T)}}{\alpha^2(1-\rho^*)}\langle \|\tilde{a}(t)\|_{L^1}\rangle|_{\leq t}\Big]. \end{eqnarray} | (117) |
and hence the asymptotic stability holds provided that
\begin{eqnarray} &&\Big(\frac{1}{\alpha^2(1-\rho^*)}+\|\tilde{a}(t)\|_{L^1}\Big)\langle \|g(t)\|_H^2\rangle|_{\leq t}+\frac{1}{\alpha}\langle \|b(t)\|_{L^1}\rangle|_{\leq t}+\frac{\|b\|_{L^1(\tau,T)}}{\alpha^2(1-\rho^*)}\langle \|\tilde{a}(t)\|_{L^1}\rangle|_{\leq t}\\ &&\leq \frac{\nu(2\nu\lambda_1+\alpha)}{2C_{|\Omega|}\beta}. \end{eqnarray} | (118) |
If we define the generalized Grashof number as
\begin{eqnarray} G(t)\leq \Big\{(2\nu\lambda_1+\alpha)\Big/\Big[2C_{|\Omega|}\beta\nu\lambda_1 (\frac{1}{\alpha^2(1-\rho^*)}+\|\tilde{a}(t)\|_{L^1})\Big]\Big\}^{1/2} = \tilde{K}_0, \end{eqnarray} | (119) |
which completes the proof for our first result.
Remark 5. If we denote
\begin{eqnarray} \limsup\limits_{\tau\rightarrow -\infty}\frac{1}{t-\tau}\int^t_{\tau}b(r)dr = b_0\in [0,+\infty) \end{eqnarray} | (120) |
and
\begin{eqnarray} \limsup\limits_{\tau\rightarrow -\infty}\frac{1}{t-\tau}\int^t_{\tau}\tilde{a}(r)dr = \tilde{a}_0\in [0,+\infty), \end{eqnarray} | (121) |
such that there exists some
\begin{eqnarray} \frac{\nu(2\nu\lambda_1+\alpha)}{2C_{|\Omega|}\beta} > \frac{b_0}{\alpha}+\frac{\|b\|_{L^1(\tau,T)}\tilde{\alpha}_0}{\alpha^2(1-\rho^*)}+\delta \end{eqnarray} | (122) |
holds. Then more precise sufficient condition for the asymptotic stability of pullback attractors is
\begin{eqnarray} G(t)\leq \Big[\frac{\frac{\nu(2\nu\lambda_1+\alpha)}{2C_{|\Omega|}\beta}-\frac{b_0}{\alpha}-\frac{\|b\|_{L^1(\tau,T)}\tilde{\alpha}_0}{\alpha^2(1-\rho^*)}}{\nu^2\lambda_1(\frac{1}{\alpha^2(1-\rho^*)}+\|\tilde{a}(t)\|_{L^1})}\Big]^{1/2} \end{eqnarray} | (123) |
which has smaller upper boundedness than (119).
The structure and stability of 3D BF equations with delay are investigated in this paper. A future research in the pullback dynamics of (1) is to study the geometric property of pullback attractors, such as the fractal dimension.
Xin-Guang Yang was partially supported by the Fund of Young Backbone Teacher in Henan Province (No. 2018GGJS039) and Henan Overseas Expertise Introduction Center for Discipline Innovation (No. CXJD2020003). Xinjie Yan was partly supported by Excellent Innovation Team Project of "Analysis Theory of Partial Differential Equations" in China University of Mining and Technology (No. 2020QN003). Ling Ding was partly supported by NSFC of China (Grant No. 1196302).
The authors want to express their most sincere thanks to refrees for the improvement of this manuscript. The authors also want to thank Professors Tomás Caraballo (Universidad de Sevilla), Desheng Li (Tianjin University) and Shubin Wang (Zhengzhou University) for fruitful discussion on this subject.
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