Watersheds of tropical countries having only dry and wet seasons exhibit contrasting water level behaviour compared to countries having four seasons. With the changing climate, the ability to forecast the water level in watersheds enables decision-makers to come up with sound resource management interventions. This study presents a strategy for days-ahead water level forecasting models using an Artificial Neural Network (ANN) for watersheds by conducting data preparation of water level data captured from a Water Level Monitoring Station (WLMS) and two Automatic Rain Gauge (ARG) sensors divided into the two major seasons in the Philippines being implemented into multiple ANN models with different combinations of training algorithms, activation functions, and a number of hidden neurons. The implemented ANN model for the rainy season which is RPROP-Leaky ReLU produced a MAPE and RMSE of 6.731 and 0.00918, respectively, while the implemented ANN model for the dry season which is SCG-Leaky ReLU produced a MAPE and RMSE of 7.871 and 0.01045, respectively. By conducting appropriate water level data correction, data transformation, and ANN model implementation, the results of error computation and assessment shows the promising performance of ANN in days-ahead water level forecasting of watersheds among tropical countries.
Citation: Lemuel Clark Velasco, John Frail Bongat, Ched Castillon, Jezreil Laurente, Emily Tabanao. Days-ahead water level forecasting using artificial neural networks for watersheds[J]. Mathematical Biosciences and Engineering, 2023, 20(1): 758-774. doi: 10.3934/mbe.2023035
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Watersheds of tropical countries having only dry and wet seasons exhibit contrasting water level behaviour compared to countries having four seasons. With the changing climate, the ability to forecast the water level in watersheds enables decision-makers to come up with sound resource management interventions. This study presents a strategy for days-ahead water level forecasting models using an Artificial Neural Network (ANN) for watersheds by conducting data preparation of water level data captured from a Water Level Monitoring Station (WLMS) and two Automatic Rain Gauge (ARG) sensors divided into the two major seasons in the Philippines being implemented into multiple ANN models with different combinations of training algorithms, activation functions, and a number of hidden neurons. The implemented ANN model for the rainy season which is RPROP-Leaky ReLU produced a MAPE and RMSE of 6.731 and 0.00918, respectively, while the implemented ANN model for the dry season which is SCG-Leaky ReLU produced a MAPE and RMSE of 7.871 and 0.01045, respectively. By conducting appropriate water level data correction, data transformation, and ANN model implementation, the results of error computation and assessment shows the promising performance of ANN in days-ahead water level forecasting of watersheds among tropical countries.
The three dimensional incompressible Navier-Stokes equations is expressed as
{∂u∂t−μΔu+(u⋅∇)u+∇p=f,∇⋅u=0, | (1) |
which was proposed by Navier and Stokes respectively for the motion of incompressible viscous fluid with very small velocity gradient. Since last century, many mathematicians and physicists have studied the existence, uniqueness, regularity and long time behavior of Navier-Stokes equations deeply, and have obtained a series of significant results (see [1,3,19,21,24,29,33]). However, for the 3D Navier-Stokes model, the uniqueness of weak solution and the existence of strong solutions have not been solved as our best knowledge.
In the 1960s, Ladyzhenskaya [22,23] relaxed the limitation of small fluid velocity gradient, and derive several modified Navier-Stokes equations, one of which reads as
{ut−div[(μ0+μ1‖∇u‖2L2(Ω))Du]+(u⋅∇)u+∇p=f,∇⋅u=0,Du=∇u+∇uT, | (2) |
and reduced to
{∂u∂t−(μ0+μ1‖∇u‖2L2(Ω))Δu+(u⋅∇)u+∇p=f,∇⋅u=0 | (3) |
by Lions, see [27]. Note that it reduces into the classical Navier-Stokes equations (1) when
The Navier-Stokes equations with delays were firstly considered by Caraballo and Real in [5], then there are a lot of works concerning asymptotic behavior, stability, the existence of pullback attractors and the fractal dimensional of pullback attractors for time-delayed Navier-Stokes equations (see, e.g., [6,7,15,30,38]). It is worth to be pointed out that García-Luengo, Marín-Rubio and Real [15] obtained that the existence of pullback attractors for the 2D Navier-Stokes model with finite delay. Furthermore, the bounded fractal and Hausdorff dimension of the pullback attractors for 2D non-autonomous incompressible Navier-Stokes equations with delay was studied in [38]. The above work is to study the time delay which only exists in the external force. Later on, Planas et al [17,32] considered Navier-Stokes equations with double delays and proved the exponential stability of stationary solutions. For the double time-delayed 2D Navier-Stokes model, the existence of pullback attractors was proved in [12]. As far as we know, there are less results on the research for 3D modified Navier-Stokes equations with double time delays till we know now. Motivated by the results [12,15,37,39], this paper is concerned with the pullback dynamics for a three dimensional modified Navier-Stokes equation with double time delays defined on
{∂u∂t−(μ0+μ1‖∇u‖2L2(Ω))Δu+(u(t−ρ(t))⋅∇)u+∇p=f(x,t)+g(t,ut),∇⋅u=0,(t,x)∈(τ,∞)×Ω,u(t,x)|∂Ω=0,t∈(τ,∞),u(t,x)|t=τ=uτ(x),x∈Ω,uτ(s,x)=u(τ+s,x)=ϕ(s,x),s∈(−h,0),x∈Ω, | (4) |
here
The main features and results of this paper can be summarized as follows:
(Ⅰ) By using the Galerkin approximated technique and compact argument, we can prove the existence of global weak solution. Using the energy equation approach similar as in Ball [2], the pullback asymptotic compactness of the process can be shown, which leads to the existence of pullback attractors together with pullback dissipation of our problem. In addition, based on different universes, we can present the minimal family of pullback attractors on functional Banach space.
(Ⅱ) Since the topic in [16] only contains the delay on external force, the result in this paper is a further extension of our former result [16]. Moreover, the problem (4) has a delay function
(Ⅲ) The upper semi-continuity of pullback attractors as perturbed delay can not be shown easily similar as in [16] since the lack of regular estimate in appropriate phase spaces.
The structure of this paper is arranged as follows. In Section 2, we give the definitions of some usual functional spaces and operators. Moreover, Some lemmas used later are also given in the end of the section. In Section 3, we first give the needed assumptions on the external forces, then show the abstract equivalent form of the system (4), and establish the well-posedness of the global weak solutions for system (4). Finally, the existence of the minimal pullback attractors for the abstract non-autonomous system is showed in Section 4.
Denote
E:={u|u∈(C∞0(Ω))3,divu=0} |
and
(u,v)=3∑i=1∫Ωui(x)vi(x)dx, |u|2=(u,u),∀u,v∈(L2(Ω))3. |
((u,v))=3∑i=1∫Ω∇ui(x)∇vi(x)dx, ‖u‖2=((u,u)),∀u,v∈(H10(Ω))3. |
Then we have
‖f‖∗=supv∈V,‖v‖=1|⟨f,v⟩|,∀f∈V′. | (5) |
|u|2≤1λ1‖u‖2,∀u∈V. | (6) |
In order to deal with the nonlinear term
⟨A2u,v⟩=μ1‖∇u‖2L2(Ω)⟨−Δu,v⟩=μ1‖u‖2((u,v)),∀u,v∈V. | (7) |
It is easy to verify that
‖A2u‖∗=supv∈V,‖v‖=1|⟨A2u,v⟩|=supv∈V,‖v‖=1μ1‖u‖2((u,v))≤μ1‖u‖3,∀u∈V. | (8) |
We also introduce the bilinear operator
B(u,v)=P((u⋅∇)v),∀u,v∈V |
and trilinear operator
b(u,v,ω)=(B(u,v),ω)=3∑i,j=1∫Ωui∂vj∂xiωjdx,∀u,v,ω∈V. |
We define some functional Banach space as
CH=C([−h,0];H) with the norm ‖ϕ‖CH=sups∈[−h,0]|ϕ| |
and some Lebesgue spaces on delayed interval as
(u,v)L2H×L2H=∫0−h(u(s),v(s))ds, ‖u‖2L2H=∫0−h|u(s)|2ds,∀u,v∈L2H. |
The inner product and norm in
(u,v)L2V×L2V=∫0−h((u(s),v(s)))ds, ‖u‖2L2V=∫0−h‖u(s)‖2ds,∀u,v∈L2V. |
The following lemmas are used to prove the existence of weak solutions.
Lemma 2.1. ([24,34]) The bilinear operator
{‖B(u,v)‖∗≤C‖u‖‖v‖,∀u,v∈V,b(u,v,v)=0,∀u,v∈V,b(u,v,ω)=−b(u,ω,v),∀u,v,ω∈V,|b(u,v,ω)|≤C‖u‖‖v‖‖ω‖,∀u,v,ω∈V,|b(u,v,ω)|≤C|u|14‖u‖34‖v‖|ω|14‖ω‖34,∀u,v,ω∈V. |
Lemma 2.2. ([41]) Suppose that
(i) For all
‖A(v)‖∗≤C‖v‖p−1, |
where
(ii) (semi-continuous) for all
If
un⇀uweakly in Lp([τ,T],V),A(un)⇀ψweakly in Lp′([τ,T],V′) |
with
¯limn→∞∫Tτ⟨A(un),un⟩dt≤∫Tτ⟨ψ,u⟩dt, |
then
ψ=A(u). |
The following assumptions on the external forces are imposed for our results.
(a) The function
(b) There exists a constant
|g(t,ξ)−g(t,η)|≤Lg‖ξ−η‖CH,∀τ≤t. |
(c) There exists a constant
∫tτ|g(s,us)−g(s,vs)|2ds≤C2g∫tτ−h|u(s)−v(s)|2ds,∀τ≤t. |
∫t−∞eσs‖f(s,⋅)‖43V′ds<∞,∀τ≤t. |
Based on the previous definitions of operators
{∂u∂t+μ0A1u+P(A2u+B(u(t−ρ(t)),u))=Pf(x,t)+Pg(t,ut),(t,x)∈(τ,∞)×Ω,∇⋅u=0,(t,x)∈(τ,∞)×Ω,u(t,x)|∂Ω=0,t∈(τ,∞),u(t,x)|t=τ=uτ(x),x∈Ω,uτ(s,x)=u(τ+s,x)=ϕ(s,x),s∈(−h,0),x∈Ω. |
Definition 3.1. Let
{∂∂t(u,v)+μ0(A1u,v)+(A2u,v)+b(u(t−ρ(t)),u,v)=⟨f(t),v⟩+(g(t,ut),v),u(τ,x)=uτ(x),uτ(s,x)=u(τ+s,x)=ϕ(s,x),s∈(−h,0) |
holds for all
Theorem 3.2. For any
Proof. Step 1. Local approximating sequence.
From the property of the Stokes operator
Pmu=m∑i=1(u,ωi)ωi,∀ u∈H. |
Let
{(∂um∂t,ωi)+μ0(A1um,ωi)+(A2um,ωi)+b(um(t−ρ(t)),um,ωi)=⟨f(t),ωi⟩+(g(t,umt),ωi),um(τ,x)=uτm(x)=Pmuτ,umτ(s,x)=ϕm(s,x)=Pmϕ(s,x) for s∈[−h,0]. | (9) |
The problem (9) is equivalent to a system of functional differential equation with respect to the unknown variables
Step 2. The priori estimates for
Multiplying (9) by
ddt|um(t)|2+2μ0‖um(t)‖2+2μ1‖um(t)‖4=2⟨f(t),um(t)⟩+2(g(t,umt),um(t))≤2‖f(t)‖∗‖um(t)‖+2μ0λ1|um(t)|2+12μ0λ1|g(t,umt)|2≤(2716μ1)13‖f(t)‖43∗+μ1‖um(t)‖4+2μ0‖um(t)‖2+12μ0λ1|g(t,umt)|2. | (10) |
Integrating (10) with respect to the variable
|um(t)|2+μ1∫tτ‖um(s)‖4ds≤|uτm|2+(2716μ1)13∫tτ‖f(s)‖43∗ds+12μ0λ1∫tτ|g(s,ums)|2ds≤|uτ|2+C2g2μ0λ1(∫ττ−h|um(s)|2ds+∫tτ|um(s)|2ds)+(2716μ1)13∫tτ‖f(s)‖43∗ds≤C+C2g2μ0λ1∫tτ|um(s)|2ds. |
Thus, it follows from Gronwall's inequality that
|um(t)|2+∫tτ‖um(s)‖4ds≤C,τ≤t≤T, |
which implies
{um(t)} is uniformly bounded in L∞(τ,T;H)∩L4(τ,T;V). | (11) |
In particular, the sequence of functions
∂um∂t=Pmf(t,x)+Pmg(t,umt)−Pm(A2um)−Pm(μ0A1um)−PmB(um(t−ρ(t)),um). | (12) |
We notice that
∫tτ|g(s,ums)|2ds≤C2g∫tτ−h|um(s)|2ds≤C, | (13) |
which implies that
For the convective term with delay, the desired estimation can be obtained by using Lemma 2.1 and Young's inequality. Indeed, it holds that
∫tτ‖B(um(t−ρ(t)),um)‖43∗ds≤∫tτ(C‖um(t−ρ(t))‖)43‖um‖43ds≤2C23∫tτ‖um(t−ρ(t))‖2ds+13∫tτ‖um‖4ds≤C, | (14) |
i.e.,
A1um and A2um are bounded in L43(τ,T;V′). | (15) |
Combining (13)-(15),
{∂um∂t} is uniformly bounded in L43(τ,T;V′). | (16) |
Step 3. Compact results and strong convergence.
Let
W={u|u∈L4(τ,T;V);∂u∂t∈L43(τ,T;V′)}. |
Now, applying the Aubin-Lions Lemma, we can derive
W↪↪L4(τ,T;H). | (17) |
By using (11) and (15)-(17), we deduce that there exist a subsequence (still denote it by
{um→u strongly in L4(τ,T;H),g(t,umt)→g(t,ut) strongly in L2(τ,T;H),um(t−ρ(t))→u(t−ρ(t)) strongly in L2(τ,T;H),um⇀u weakly * in L∞(τ,T;H),um⇀u weakly in L4(τ,T;V),∂um∂t⇀∂u∂t weakly in L43(τ,T;V′),A2(um)⇀ψ weakly in L43(τ,T;V′), | (18) |
and
um(τ)=Pmuτ→u(τ)=uτ in H, ϕm(s)=Pmϕ(s)→ϕ(s) in CH∩L2V. |
Next, we prove that
um→uin C([τ,T];H). | (19) |
Let
∂ω∂t+μ0A1ω+A2um−A2un+B(um(t−ρ(t)),um)−B(un(t−ρ(t)),un)=g(t,umt)−g(t,unt). | (20) |
We observe the fact that
B(um(t−ρ(t)),um)−B(un(t−ρ(t)),un)=B(ω(t−ρ(t)),um)+B(un(t−ρ(t)),ω). | (21) |
Multiplying (20) by
12∂∂t|ω|2+μ0‖ω‖2+(A2um−A2un,um−un)+b(ω(t−ρ(t)),um,ω)=(g(t,umt)−g(t,unt),ω). |
Thanks to the assumption
12∂∂t|ω|2+μ0‖ω‖2≤|b(ω(t−ρ(t)),um,ω)|+(g(t,umt)−g(t,unt),ω)≤C|ω(t−ρ(t))|14‖ω(t−ρ(t))‖34‖um‖|ω|14‖ω‖34+|g(t,umt)−g(t,unt)||ω|≤C‖ωt‖12CH‖um‖‖ω(t−ρ(t))‖34‖ω‖34+Lg‖ωt‖CH|ω|≤C44ε3‖ωt‖2CH‖um‖4+3ε4‖ω(t−ρ(t))‖‖ω‖+Lg‖ωt‖CH|ω|≤C44ε3‖ωt‖2CH‖um‖4+9ε232μ0‖ω(t−ρ(t))‖2+μ02‖ω‖2+Lg‖ωt‖CH|ω|. | (22) |
Here
|ω(t)|2+μ0∫tτ‖ω(s)‖2ds≤C42ε3∫tτ‖ωs‖2CH‖um(s)‖4ds+9ε216μ0∫tτ‖ω(s−ρ(s))‖2ds+2Lg∫tτ‖ωs‖2CHds+|ω(τ)|2. |
With the help of a change of variable in the integral of
|ω(t)|2≤|ω(τ)|2+C42ε3∫tτ‖ωs‖2CH‖um(s)‖4ds+μ0∫ττ−h‖ω(s)‖2ds+2Lg∫tτ‖ωs‖2CHds≤‖ωτ‖2CH+μ0∫ττ−h‖ω(s)‖2ds+(C42ε3+2Lg)∫tτ(‖um(s)‖4+1)‖ωs‖2CHds. |
For simplicity, we set that
|ω(t+θ)|2≤‖ωτ‖2CH+μ0∫ττ−h‖ω(s)‖2ds+C′∫t+θτ(‖um(s)‖4+1)‖ωs‖2CHds. |
Consequently, it follows that for all
‖umt−unt‖2CH≤‖ϕm−ϕn‖2CH+μ0‖ϕm−ϕn‖2L2V+C′∫tτ(‖um(s)‖4+1)‖ums−uns‖2CHds, | (23) |
which, by the Gronwall inequality, implies
‖umt−unt‖2CH≤(‖ϕm−ϕn‖2CH+μ0‖ϕm−ϕn‖2L2V)×e∫tτC′(‖um(s)‖4+1)ds. | (24) |
Since
Step 4. Passing the limit and uniqueness.
In order to pass the limit of (9), we need to discuss the convergence of nonlinear terms
∫Tτ|b(um(t−ρ(t)),um,ωi)−b(u(t−ρ(t)),u,ωi)|ds≤∫Tτ|b(um(t−ρ(t))−u(t−ρ(t)),ωi,um)|+|b(u(t−ρ(t)),ωi,um−u)|ds≤C∫Tτ|um(t−ρ(t))−u(t−ρ(t))|14‖um(t−ρ(t))−u(t−ρ(t))‖34|um|14‖um‖34ds+C∫Tτ|u(t−ρ(t))|14‖u(t−ρ(t))‖34|um−u|14‖um−u‖34ds≜I1+I2. | (25) |
We use the Hölder inequality to give
{I1≤C‖um(t−ρ(t))−u(t−ρ(t))‖14L2(τ,T;H)‖um(t−ρ(t))−u(t−ρ(t))‖34L2(τ,T;V)‖um‖14L2(τ,T;H)‖um‖34L2(τ,T;V),I2≤‖u(t−ρ(t))‖14L2(τ,T;H)‖u(t−ρ(t))‖34L2(τ,T;V)×‖um−u‖14L2(τ,T;H)‖um−u‖34L2(τ,T;V), |
which, together with
∫Tτ|b(um,um,ωi)−b(u,u,ωi)|ds→0,as m→∞. | (26) |
By a similar technique to the proof of Lemma 3.2 in [35], we can deduce
¯limn→∞∫Tτ⟨A2(um),um⟩dt≤∫Tτ⟨ψ,u⟩dt. | (27) |
Moreover, it is easy to verify that the operator
ψ=A2(u). | (28) |
Now, passing to the limit of (9), by combining (18)-(19), (26) and (28), we can infer that
Last, we consider the uniqueness of solutions for our problem. Let
‖ut−vt‖2CH≤C′∫tτ(‖um(s)‖4+1)‖us−vs‖2CHds. |
By Gronwall's inequality, it yields
In this section, we shall obtain the existence of pullback attractors for the process associated to (4).
Let
A process on
Let
Definition 4.1. (1) A process
(2) Further, we say that a process
Definition 4.2.
U(t,τ)D(τ)⊂D0(t),∀τ≤τ0(t,ˆD). |
Observe from the above definition that
Denote
Λ(ˆD0,t):=⋂s≤t¯⋃τ≤sU(t,τ)D0(τ)X,∀t∈R, |
where
distX(X1,X2)=supx∈X1infy∈X2dX(x,y),∀X1,X2⊂X. |
In order to get our result, we need to use a classical theorem in [14,15].
Theorem 4.3. Consider a closed process
AD(t)=¯⋃ˆD∈DΛ(ˆD,t)X,t∈R, |
and has the following properties:
(a) for any
(b)
limτ→−∞distX(U(t,τ)D(τ),AD(t))=0,∀ˆD∈D, t∈R. |
(c)
(d) if
Moreover, family
limτ→−∞distX(U(t,τ)D(τ),C(t))=0, |
then
Remark 1. If
Let
Lemma 4.4. ([14]) Under the assumptions of Theorem 4.3, if the universe
ADXF⊂AD,∀t∈R. |
In view of Theorem 3.2, here we take the phase space
Proposition 1. Consider given
In order to prove the existence of pullback attractors for the process
∫tτeσs|g(s,us)|2ds<C2g∫tτ−heσs|u(s)|2ds,∀t≤T. |
Lemma 4.5.
|u(t)|2≤Ce−σ(t−τ)(|uτ|2+C2gμ0λ1‖ϕ‖L2H)+C(2716μ1)13e−σt∫tτeσs‖f(s)‖43∗ds, | (29) |
μ0∫ts‖u(r)‖2dr≤|u(s)|2+C2gμ0λ1‖us‖L2H+C2gμ0λ1∫ts|u(r)|2dr+(2716μ1)13∫ts‖f(r)‖43∗dr, | (30) |
and
μ1∫ts‖u(r)‖4dr≤|u(s)|2+C2gμ0λ1‖us‖L2H+C2gμ0λ1∫ts|u(r)|2dr+(2716μ1)13∫ts‖f(r)‖43∗dr. | (31) |
Proof. Multiplying (4) by
12ddt|u(t)|2+μ0‖u(t)‖2+μ1‖u(t)‖4=⟨f(t),u(t)⟩+(g(t,ut),u(t))≤‖f(t)‖∗‖u(t)‖+|g(t,ut)||u(t)|≤12(2716μ1)13‖f(t)‖43∗+μ12‖u(t)‖4+12μ0λ1|g(t,ut(t))|2+μ0λ12|u(t)|2. | (32) |
Now multiplying (32) by 2
ddt(eσt|u(t)|2)=σeσt|u(t)|2+eσtddt|u(t)|2≤(σ+μ0λ1)eσt|u(t)|2−2μ0‖u(t)‖2eσt+(2716μ1)13eσt‖f(t)‖43∗+eσtμ0λ1|g(t,ut)|2 |
≤(σλ1−μ0)‖u(t)‖2eσt+(2716μ1)13eσt‖f(t)‖43∗+eσtμ0λ1|g(t,ut)|2. | (33) |
We integrate (33) over the interval
eσt|u(t)|2≤eστ|uτ|2+1μ0λ1∫tτeσs|g(s,us)|2ds+(2716μ1)13∫tτeσs‖f(s)‖43∗ds≤eστ|uτ|2+C2gμ0λ1∫tτ−teσs|u(s))|2ds+(2716μ1)13∫tτeσs‖f(s)‖43∗ds≤eστ|uτ|2+C2gμ0λ1(eστ∫0−h|ϕ(s)|2ds+∫tτeσs|u(s)|2ds)+(2716μ1)13∫tτeσs‖f(s)‖43∗ds≤eστ(|uτ|2+C2gμ0λ1∫0−h|ϕ(s)|2ds)+C2gμ0λ1∫tτeσs|u(s)|2ds+(2716μ1)13∫tτeσs‖f(s)‖43∗ds. | (34) |
On account of the assumption
eσt|u(t)|2≤Ceστ(|uτ|2+C2gμ0λ1∫0−h|ϕ(s)|2ds)+C(2716μ1)13∫tτeσs‖f(s)‖43∗ds, |
which means
|u(t)|2≤Ce−σ(t−τ)(|uτ|2+C2gμ0λ1‖ϕ‖L2H)+C(2716μ1)13e−σt∫tτeσs‖f(s)‖43∗ds,∀t≥τ. |
Consequently, the estimation (29) is proved. Thanks to (32) and (6), we obtain
ddt|u(t)|2+μ0‖u(t)‖2+μ1‖u(t)‖4≤1μ0λ1|g(t,ut)|2+(2716μ1)13‖f(t)‖43∗. | (35) |
Integrating (35) over the interval
|u(r)|2+μ0∫ts‖u(r)‖2dr+μ1∫ts‖u(r)‖4dr≤|u(s)|2+(2716μ1)13∫ts‖f(r)‖43∗dr+C2gμ0λ1(∫0−h|us(r)|2dr+∫ts|u(r)|2dr). | (36) |
Thus, (30) and (31) are obtained immediately from (36). Now we complete the proof.
Definition 4.6. (Universe) We will denote by
limτ→−∞(eστsup(ξ,ϕ)∈D(τ)‖(ξ,ϕ)‖2MH)=0. |
Remark 2. According to the above definition and the notation
Based on the above universe and Lemma 4.5, we can present the pullback dissipation in
Proposition 2. Suppose that
D0(t)=¯BH(0,RH)×(¯BCH(0,RCH)∩¯BL2V(0,RL2V)) |
is pullback
R2H(t)=1+C⋅(2716μ1)13e−σ(t−2h)∫t−∞eσs‖f(s)‖43∗ds,R2L2V(t)=1μ0[(1+2C2ghμ0λ1)R2H+(2716μ1)13‖f(r)‖L43(t−h,t;V′)]. |
Proof. Fix
|u(t;τ,uτ,ϕ)|2≤1+C⋅(2716μ1)13e−σt∫t−∞eσs‖f(s)‖43∗ds≤R2H(t),∀ˆD∈DMHσ(t) |
holds for all
In particular, we observe that
To use Theorem 4.3, we also need to establish the asymptotically compact of the process. We give the following result.
Theorem 4.7. Assume that
Proof. Fix a value
In the same way as Proposition 2, by using the estimations in Lemma 4.5, we obtain that there exists a pullback time
‖(un)′‖L43(t−2h−1,t;V′)≤μ0‖un‖L43(t−2h−1,t;V)+μ1‖un‖3L4(t−2h−1,t;V)+‖f‖L43(t−2h−1,t;V′)+‖(un(t−ρ(t))⋅∇)un‖L43(t−2h−1,t;V′)+C‖g(t,unt)‖L2(t−2h−1,t;H). |
By applying a similar technique as proving (16) in Theorem 3.2, we derive from the assumption
{un⇀u weakly * in L∞(t−4h−1,t;H),un⇀u weakly in L4(t−2h−1,t;V),(un)′⇀u′ weakly in L43(t−2h−1,t;V′),un→u strongly in L4(t−2h−1,t;H),un(s)→u(s) strongly in H, a.e. s∈(t−2h−1,t),g(⋅,unt)→g(⋅,ut) strongly in L2(t−h−1,t;H). | (37) |
From the above convergences, we derive that
By using the same technique as proving (19) in Theorem 3.2, we can get
un→u strongly in C([t−h−1,t];H). |
Consequently, we obtain that for any sequence
un(sn)⇀u(s∗) weakly in H. | (38) |
Now, our goal is to obtain that
\begin{align*} |u^n(s_n) - u(s_*)|\rightarrow0 \text{ as } n\rightarrow +\infty. \end{align*} |
First of all, we can conclude from the weak convergence (38) that
\begin{align} | u(s_*)|\leq \liminf\limits_{n\rightarrow \infty} |u^n(s_n)|. \end{align} | (39) |
Furthermore, in view of the energy equality (32), we infer that for all
\begin{align} &\frac{1}{2}|y(s_2)|^2+\mu_0\int_{s_1}^{s_2} \|y(r)\|^2dr+\mu_1\int_{s_1}^{s_2}\|y(r) \|^4dr \\ & = \frac{1}{2}|y(s_1)|^2 + \int_{s_1}^{s_2}\langle f(r),y(r)\rangle dr +\int_{s_1}^{s_2}(g(r,y_r),y(r))dr, \end{align} | (40) |
where
\begin{align*} \mathcal{J}(s) = \frac{1}{2}|u(s)|^2- \int_{t-h-1}^{s}\langle f(r),u(r)\rangle dr -\int_{t-h-1}^{s}(g(r,u_r),u(r))dr \end{align*} |
and
\begin{align*} \mathcal{J}_n(s) = \frac{1}{2}|u^n(s)|^2- \int_{t-h-1}^{s}\langle f(r),u^n(r)\rangle dr -\int_{t-h-1}^{s}(g(r,u^n_r),u^n(r))dr. \end{align*} |
It is clear that
\begin{align*} \mathcal{J}_n(s)\rightarrow \mathcal{J}(s) \quad a.e. \quad s \in (t-h-1,t). \end{align*} |
Therefore, it is possible to choose a sequence
\begin{align*} \lim\limits_{n\rightarrow \infty}\mathcal{J}_n(s_k) = \mathcal{J}(s_k), \ \forall k. \end{align*} |
Since
\begin{align} |\mathcal{J}(s_k)-\mathcal{J}(s_*)| < \frac \varepsilon 2, \quad \forall k \geq k_\varepsilon. \end{align} | (41) |
Because
\begin{align} s_{k_\epsilon} \leq s_n,\ |\mathcal{J}_n(s_{k_\epsilon})-\mathcal{J}(s_{k_\epsilon})| < \frac \varepsilon 2, \quad \forall n \geq n(k_\varepsilon). \end{align} | (42) |
According to the non-increasing property of all
\begin{align*} \mathcal{J}_n(s_n)-\mathcal{J}(s_*) &\leq \mathcal{J}_n(s_{k_\epsilon})-\mathcal{J}(s_*) \notag\\ &\leq |\mathcal{J}_n(s_{k_\epsilon})-\mathcal{J}(s_*)| \notag\\ &\leq |\mathcal{J}_n(s_{k_\epsilon})-\mathcal{J}(s_{k_\epsilon})| +|\mathcal{J}(s_{k_\epsilon})-\mathcal{J}(s_*)| < \varepsilon. \end{align*} |
Because of the arbitrariness of
\begin{align} \limsup\limits_{n\rightarrow \infty} |u^n(s_n)| \leq | u(s_*)|. \end{align} | (43) |
Therefore, combining (38), (39) and (43), we conclude that
\begin{align} u^n(s_n)\rightarrow u(s_*) \text{ strongly in } \ C([t-h,t];H). \end{align} | (44) |
By using again the energy equality (40) satisfied by
\begin{align*} \|u^n\|_{L^2(t-h,t;V)}\rightarrow \|u\|_{L^2(t-h,t;V)}, \end{align*} |
which, together with the weak convergence already proved in (37), gives
\begin{align} u^n(s_n)\rightarrow u(s_*) \text{ strongly in } \ L^2(t-h,t;V). \end{align} | (45) |
Combining (44) with (45), we derive that the process is pullback
In this subsection, by using the results obtained in subsection 4.2 and subsection 4.3, we shall establish the main result of the paper as follows.
Theorem 4.8. Assume that
\mathcal{A}_{\mathcal{D}_F^{M_H}} = \{\mathcal{A}_{\mathcal{D}_F^{M_H}} (t) : t \in \mathbb{R}\} |
and the minimal pullback
\mathcal{A}_{\mathcal{D}_\sigma^{M_H}} = \{\mathcal{A}_{\mathcal{D}_\sigma^{M_H}} (t) : t \in \mathbb{R}\}, |
for the process defined in Proposition 1. The family
\begin{align} \mathcal{A}_{\mathcal{D}_F^{M_H}}(t) \subset \mathcal{A}_{\mathcal{D}_\sigma^{M_H}}(t) \subset D_0(t), \quad \forall t \in \mathbb{R}. \end{align} | (46) |
Moreover, the pullback attractor
Proof. From Proposition 1, we observe that the process
\begin{align} \mathcal{A}_{\mathcal{D}_F^{M_H}}(t) \subset \mathcal{A}_{\mathcal{D}_\sigma^{M_H}}(t), \quad \forall t \in \mathbb{R}. \end{align} | (47) |
Since
\begin{align*} \mathcal{A}_{\mathcal{D}_\sigma^{M_H}}(t) \subset D_0(t), \quad \forall t \in \mathbb{R}, \end{align*} |
which, along with (47), gives (46). The proof is complete.
In the Appendix of Ladyzhenskaya [23], the classical incompressible Navier-Srokes equations is approximated by using a class of regular Navier-Stokes systems which are described as
\begin{equation} \begin{cases} u_t-\nu_0 \mbox{div}\Big[(1+\varepsilon \hat{u}^2)\mathcal{D}u\Big] + (u\cdot \nabla)u+\nabla p = f(t,x), & \\ \nabla \cdot u = 0,&\\ \mathcal{D}u = \nabla u+\nabla u^T,\ \hat{u}^2 = \|\mathcal{D}u\|^2_{L^2}, & \end{cases} \end{equation} | (48) |
and its special case (2), which reflects the physical phenomena that
\begin{equation} \begin{cases} u_t- \mbox{div}\Big[(\nu_0+\nu_1 \|\mathcal{D}u\|^{p-2}_{L^2(\Omega)})\mathcal{D}u\Big] + (u\cdot \nabla)u+\nabla p = f(t,x), & \\ \nabla \cdot u = 0,\ \mathcal{D}u = \nabla u+\nabla u^T.& \end{cases} \end{equation} | (49) |
However, even for these systems, the uniqueness and stability are still open questions when Reynold number is large. To overcome this difficulty and simplify Ladyzhenskaya models, Lions [27] replaced
\begin{equation} \begin{cases} u_t-\nu_0 \Delta u-\nu_1 {\sum^{n}_{i = 1}}\frac{\partial}{\partial x_i}(|\nabla u|^{p-1}\frac{\partial u}{\partial x_i}) + (u\cdot \nabla)u+\nabla p = f(t,x), & \\ \nabla \cdot u = 0& \end{cases} \end{equation} | (50) |
and
\begin{equation} \begin{cases} u_t-\nu {\sum^{n}_{i = 1}}\frac{\partial}{\partial x_i}(|\nabla u|^{p-1}\frac{\partial u}{\partial x_i}) + (u\cdot \nabla)u+\nabla p = f(t,x), & \\ \nabla \cdot u = 0.& \end{cases} \end{equation} | (51) |
Lions [27] proved the existence of weak solutions to systems (50) and (51) for
The authors are grateful to the referees for their helpful suggestions which improved the presentation of this paper. Lan Huang was partially supported by the NSFC (No. 11501199 and No. 11871212) and the Young Key Teachers Project in Higher Vocational Colleges of Henan Province (No. 2020GZGG109). Xin-Guang Yang was partially supported by the Fund of Young Backbone Teachers in Henan Province (No. 2018GGJS039), Incubation Fund Project of Henan Normal University (No. 2020PL17) and Henan Overseas Expertise Introduction Center for Discipline Innovation (No. CXJD2020003).
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