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Days-ahead water level forecasting using artificial neural networks for watersheds

  • 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

    {utμΔu+(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

    {utdiv[(μ0+μ1u2L2(Ω))Du]+(u)u+p=f,u=0,Du=u+uT, (2)

    and reduced to

    {ut(μ0+μ1u2L2(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 μ1=0. Lions [27] proved the existence and uniqueness of global weak solutions for the initial boundary value problem of the model (3) by using the Faedo-Galerkin method. In 2008, Dong and Jiang [10] studied the optimal upper and lower bounds of the decay for higher order derivatives of the solution of (3). Under some assumptions on the external force and initial data, the existence and structure of uniform attractors of the system (3) were proved in [9]. Recently, Yang and Feng et al [37] considered the pullback dynamics of (3), and proved the existence of minimal and unique family of pullback attractors, and also presented the finite fractal dimension of pullback attractors. In addition, the upper semi-continuity of pullback attractor was also studied in [37] when the perturbed external force disappears as parameter tends to zero.

    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 (τ,)×Ω, which can be described as

    {ut(μ0+μ1u2L2(Ω))Δu+(u(tρ(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),xΩ,uτ(s,x)=u(τ+s,x)=ϕ(s,x),s(h,0),xΩ, (4)

    here ΩR3 is a bounded domain with sufficiently smooth boundary Ω, u=(u1,u2,u3) is the velocity field of the fluid, p is the pressure, μ0>0 and μ1>0 are the kinematic viscosities of the fluid, f(x,t) is a generic external force, g(t,ut) is a external force with some hereditary characteristics and h>0 is a fixed positive constant. The function ut appeared in the delay term g(t,ut) is defined on (h,0) by the relation ut(s)=u(t+s), s(h,0). Assume that the delay function ρ in the convective term satisfies ρ(t)C1(R;[0,h]) and ρ(t)ρ<1 for all tR, where ρ is a constant.

    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 ρ in the convection term, hence, there are more difficulties to achieve well-posedness and pullback asymptotic compactness, which also require appropriate restriction on the delay function ρ.

    (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(C0(Ω))3,divu=0}

    and H is the closure of E in (L2(Ω))3 topology. The inner product and norm in H are represented by (,) and || respectively, which are defined as

    (u,v)=3i=1Ωui(x)vi(x)dx, |u|2=(u,u),u,v(L2(Ω))3.

    V is the closure of E in (H10(Ω))3 topology. The inner product and norm in V are represented by ((,)) and respectively, which are defined as

    ((u,v))=3i=1Ωui(x)vi(x)dx, u2=((u,u)),u,v(H10(Ω))3.

    Then we have u=u(L2(Ω))3=|u| for all uV, and it is easy to verify that H and V are Hilbert spaces. Let H and V be dual spaces of H and V respectively. Then there is V↪↪HHV, where the injections are dense and continuous. Let be the norm in V, , be the dual product in V and V, where is defined as

    f=supvV,v=1|f,v|,fV. (5)

    P denotes the Helmholz-Leray orthogonal projection from (L2(Ω))3 onto the space H (see [11,34]). We define A1:=PΔ as the Stokes operator on D(A1)=(H2(Ω))3V, then A1:VV satisfy A1u,v=((u,v)), and A1 is an isomorphism from V into V. There is A1u=supvV,v=1|A1u,v|=supvV,v=1|((u,v))|u, i.e., A11. Let {λi}i=1 be the eigenvalues of the operator A1 with Dirichlet boundary condition, which satisfies 0<λ1λ2. By the property of the Stokes operator, the corresponding eigenfunctions {ωi}i=1 form an orthonormal complete basis in H. In addition, at this time we have the following Poincaré inequality

    |u|21λ1u2,uV. (6)

    In order to deal with the nonlinear term μ1u2L2(Ω), we define the operator A2:VV as A2u:=μ1u2L2(Ω)Δu, which satisfies

    A2u,v=μ1u2L2(Ω)Δu,v=μ1u2((u,v)),u,vV. (7)

    It is easy to verify that A2uA2v,uv0 for any u,vV, that is, A2 is a monotone operator. We can obtain from (5) and (7) that

    A2u=supvV,v=1|A2u,v|=supvV,v=1μ1u2((u,v))μ1u3,uV. (8)

    We also introduce the bilinear operator

    B(u,v)=P((u)v),u,vV

    and trilinear operator

    b(u,v,ω)=(B(u,v),ω)=3i,j=1Ωuivjxiωjdx,u,v,ωV.

    We define some functional Banach space as

    CH=C([h,0];H) with the norm ϕCH=sups[h,0]|ϕ|

    and some Lebesgue spaces on delayed interval as LpH=Lp(h,0;H) and LpV=Lp(h,0;V). The inner product and norm in L2H are defined as

    (u,v)L2H×L2H=0h(u(s),v(s))ds, u2L2H=0h|u(s)|2ds,u,vL2H.

    The inner product and norm in L2V are defined as

    (u,v)L2V×L2V=0h((u(s),v(s)))ds, u2L2V=0hu(s)2ds,u,vL2V.

    The following lemmas are used to prove the existence of weak solutions.

    Lemma 2.1. ([24,34]) The bilinear operator B(u,v) and trilinear operator b(u,v,ω) satisfy the properties

    {B(u,v)Cuv,u,vV,b(u,v,v)=0,u,vV,b(u,v,ω)=b(u,ω,v),u,v,ωV,|b(u,v,ω)|Cuvω,u,v,ωV,|b(u,v,ω)|C|u|14u34v|ω|14ω34,u,v,ωV.

    Lemma 2.2. ([41]) Suppose that A be a nonlinear monotone operator from a separable Banach space V to V satisfying conditions

    (i) For all vV,

    A(v)Cvp1,

    where 1<p< and C is a positive constant independent of v.

    (ii) (semi-continuous) for all u,v,ωV and λR,A(u+λv),ω is a continuous function of λ.

    If unLp([0,T],V) with 1<p< such that

    unuweakly in Lp([τ,T],V),A(un)ψweakly in Lp([τ,T],V)

    with 1p+1p=1, and

    ¯limnTτA(un),undtTτψ,udt,

    then

    ψ=A(u).

    The following assumptions on the external forces are imposed for our results.

    (Hg) Let the function g:R×CHH satisfies the following properties:

    (a) The function g(,ξ) is measurable for any ξCH and g(,0)0.

    (b) There exists a constant Lg>0 such that for all ξ,ηCH,

    |g(t,ξ)g(t,η)|LgξηCH,τt.

    (c) There exists a constant Cg>0 such that for all u,vC([τh,t];H),

    tτ|g(s,us)g(s,vs)|2dsC2gtτh|u(s)v(s)|2ds,τt.

    (Hf) The function fL43loc(R,V) satisfies that there exists some σ(0,μ0λ1) such that

    teσsf(s,)43Vds<,τt.

    Based on the previous definitions of operators P, A1 and A2, the system (4) can be written as the following abstract equivalent form

    {ut+μ0A1u+P(A2u+B(u(tρ(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),xΩ,uτ(s,x)=u(τ+s,x)=ϕ(s,x),s(h,0),xΩ.

    Definition 3.1. Let T>τ, assume that the initial data (uτ,ϕ)H×(CHL2V)MH, fL43loc(R,V) and g satisfies the assumption (Hg), a function u=u(t,x)C([τh,T];H)L4(τ,T;V) is called a weak solution to problem (4) if

    {t(u,v)+μ0(A1u,v)+(A2u,v)+b(u(tρ(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 vV in the sense of D(τ,T).

    Theorem 3.2. For any T>τ, if the initial data (uτ,ϕ)MH=H×(CHL2V), fL43</italic><italic>loc(R,V) and g satisfies the assumption (Hg), then problem (4) possesses a unique weak solution u(t,x)C([τh,T];H)L4(τ,T;V).

    Proof. Step 1. Local approximating sequence.

    From the property of the Stokes operator A1, we can see that the sequence of characteristic functions {ωi}i=1D(A1), satisfying A1ωi=λiωi, constitutes a complete orthogonal basis in H. Let Hm=span{ω1,,ωm}, and projection Pm:HHm is defined as

    Pmu=mi=1(u,ωi)ωi, uH.

    Let um(t)=mi=1him(t)ωi be the approximated solutions satisfying the following Cauchy problem

    {(umt,ωi)+μ0(A1um,ωi)+(A2um,ωi)+b(um(tρ(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 {h1m(t),h2m(t),,hmm(t)}. According to Theorem A1 in the appendix of [5] for the Cauchy problem of functional differential equation, the system (9) possesses a local solution on the interval [τ,tm], that is, there exists a solution um(t)=mi=1him(t)ωi satisfying the approximation problem (9) on [τ,tm].

    Step 2. The priori estimates for {um} and {umt}.

    Multiplying (9) by him(t), and then summing from i=1 to i=m, in view of Young's inequality and the embedding VH, we obtain

    ddt|um(t)|2+2μ0um(t)2+2μ1um(t)4=2f(t),um(t)+2(g(t,umt),um(t))2f(t)um(t)+2μ0λ1|um(t)|2+12μ0λ1|g(t,umt)|2(2716μ1)13f(t)43+μ1um(t)4+2μ0um(t)2+12μ0λ1|g(t,umt)|2. (10)

    Integrating (10) with respect to the variable t from τ to t, using the definition of operator Pm, fL43loc(R,V) and the assumption (Hg), we have

    |um(t)|2+μ1tτum(s)4ds|uτm|2+(2716μ1)13tτf(s)43ds+12μ0λ1tτ|g(s,ums)|2ds|uτ|2+C2g2μ0λ1(ττh|um(s)|2ds+tτ|um(s)|2ds)+(2716μ1)13tτf(s)43dsC+C2g2μ0λ1tτ|um(s)|2ds.

    Thus, it follows from Gronwall's inequality that

    |um(t)|2+tτum(s)4dsC,τtT,

    which implies tm=T and

    {um(t)} is uniformly bounded in L(τ,T;H)L4(τ,T;V). (11)

    In particular, the sequence of functions {um(tρ(t))} is bounded in L2(τ,T;V). Furthermore, approximate equations (9) can be rewritten as

    umt=Pmf(t,x)+Pmg(t,umt)Pm(A2um)Pm(μ0A1um)PmB(um(tρ(t)),um). (12)

    We notice that PmL(V,V)1 and Pm=Pm, then PmL(V,V)1. By the assumption (Hg), in particular, we get

    tτ|g(s,ums)|2dsC2gtτh|um(s)|2dsC, (13)

    which implies that g(s,ums) is bounded in L43(τ,T;V).

    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

    tτB(um(tρ(t)),um)43dstτ(Cum(tρ(t)))43um43ds2C23tτum(tρ(t))2ds+13tτum4dsC, (14)

    i.e., B(um(tρ(t)),um) is bounded in L43(τ,T;V). By virtue of A11, it is easy to verify from (8) and (11) that

    A1um and A2um are bounded in L43(τ,T;V). (15)

    Combining (13)-(15), fL43loc(R,V) and (12), we can get

    {umt} is uniformly bounded in L43(τ,T;V). (16)

    Step 3. Compact results and strong convergence.

    Let

    W={u|uL4(τ,T;V);utL43(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}) and uL(τh,T;H)L4(τ,T;V) such that

    {umu strongly in L4(τ,T;H),g(t,umt)g(t,ut) strongly in L2(τ,T;H),um(tρ(t))u(tρ(t)) strongly in L2(τ,T;H),umu weakly * in L(τ,T;H),umu weakly in L4(τ,T;V),umtut weakly in L43(τ,T;V),A2(um)ψ weakly in L43(τ,T;V), (18)

    and

    um(τ)=Pmuτu(τ)=uτ in H,  ϕm(s)=Pmϕ(s)ϕ(s) in CHL2V.

    Next, we prove that

    umuin C([τ,T];H). (19)

    Let um and un be two solutions to the approximated system corresponding to initial value (uτm,ϕm) and (uτn,ϕn) respectively, and set ω(t)=um(t)un(t), then we can get

    ωt+μ0A1ω+A2umA2un+B(um(tρ(t)),um)B(un(tρ(t)),un)=g(t,umt)g(t,unt). (20)

    We observe the fact that

    B(um(tρ(t)),um)B(un(tρ(t)),un)=B(ω(tρ(t)),um)+B(un(tρ(t)),ω). (21)

    Multiplying (20) by ω(t), then integrating the resultant over Ω and applying (21), we obtain

    12t|ω|2+μ0ω2+(A2umA2un,umun)+b(ω(tρ(t)),um,ω)=(g(t,umt)g(t,unt),ω).

    Thanks to the assumption (Hg) and the monotonicity of operator A2, we can deduce from Lemma 2.1 and Young's inequality that

    12t|ω|2+μ0ω2|b(ω(tρ(t)),um,ω)|+(g(t,umt)g(t,unt),ω)C|ω(tρ(t))|14ω(tρ(t))34um|ω|14ω34+|g(t,umt)g(t,unt)||ω|Cωt12CHumω(tρ(t))34ω34+LgωtCH|ω|C44ε3ωt2CHum4+3ε4ω(tρ(t))ω+LgωtCH|ω|C44ε3ωt2CHum4+9ε232μ0ω(tρ(t))2+μ02ω2+LgωtCH|ω|. (22)

    Here ε is a positive number, which is determined later. Note the fact that |ω(s)|ωsCH. Integrating (22) with respect to time s on [τ,t], it holds that

    |ω(t)|2+μ0tτω(s)2dsC42ε3tτωs2CHum(s)4ds+9ε216μ0tτω(sρ(s))2ds+2Lgtτωs2CHds+|ω(τ)|2.

    With the help of a change of variable in the integral of ω(sρ(s)) and ρ(t)ρ<1, we can deduce by choosing ε2=16(1ρ)9μ20 that

    |ω(t)|2|ω(τ)|2+C42ε3tτωs2CHum(s)4ds+μ0ττhω(s)2ds+2Lgtτωs2CHdsωτ2CH+μ0ττhω(s)2ds+(C42ε3+2Lg)tτ(um(s)4+1)ωs2CHds.

    For simplicity, we set that C=(C42ε3+2Lg). If tτ+h, it holds that t+θτ for all θ[h,0] and

    |ω(t+θ)|2ωτ2CH+μ0ττhω(s)2ds+Ct+θτ(um(s)4+1)ωs2CHds.

    Consequently, it follows that for all tτ+h,

    umtunt2CHϕmϕn2CH+μ0ϕmϕn2L2V+Ctτ(um(s)4+1)umsuns2CHds, (23)

    which, by the Gronwall inequality, implies

    umtunt2CH(ϕmϕn2CH+μ0ϕmϕn2L2V)×etτC(um(s)4+1)ds. (24)

    Since ϕmϕ in CHL2V, (24) indicates that {um} is a Cauchy sequence in C([τh,T];H). Thus we complete the proof of (19).

    Step 4. Passing the limit and uniqueness.

    In order to pass the limit of (9), we need to discuss the convergence of nonlinear terms b(um,um,ωi) and A2um,ωi. Using the properties of trilinear operator b(u,v,ω), Lemma 2.1, and the fact that ωi is an eigenfunction of the Stokes operator, we obtain

    Tτ|b(um(tρ(t)),um,ωi)b(u(tρ(t)),u,ωi)|dsTτ|b(um(tρ(t))u(tρ(t)),ωi,um)|+|b(u(tρ(t)),ωi,umu)|dsCTτ|um(tρ(t))u(tρ(t))|14um(tρ(t))u(tρ(t))34|um|14um34ds+CTτ|u(tρ(t))|14u(tρ(t))34|umu|14umu34dsI1+I2. (25)

    We use the Hölder inequality to give

    {I1Cum(tρ(t))u(tρ(t))14L2(τ,T;H)um(tρ(t))u(tρ(t))34L2(τ,T;V)um14L2(τ,T;H)um34L2(τ,T;V),I2u(tρ(t))14L2(τ,T;H)u(tρ(t))34L2(τ,T;V)×umu14L2(τ,T;H)umu34L2(τ,T;V),

    which, together with (18)1, (18)3 and (25), yields that

    Tτ|b(um,um,ωi)b(u,u,ωi)|ds0,as m. (26)

    By a similar technique to the proof of Lemma 3.2 in [35], we can deduce

    ¯limnTτA2(um),umdtTτψ,udt. (27)

    Moreover, it is easy to verify that the operator A2 satisfies the conditions (i) and (ii) of Lemma 2.2. Combining (18)5, (18)7, (27) and Lemma 2.2, we get

    ψ=A2(u). (28)

    Now, passing to the limit of (9), by combining (18)-(19), (26) and (28), we can infer that u is indeed a weak solution to problem (4).

    Last, we consider the uniqueness of solutions for our problem. Let u,v be two weak solutions with the same initial (uτ,ϕ) and set ω=uv. By a similar technique as proving (23), we can obtain that for t[τ,T],

    utvt2CHCtτ(um(s)4+1)usvs2CHds.

    By Gronwall's inequality, it yields ω(t)0, i.e., the solution is unique. Similarly, we can also verify that the solution is continuously dependent on the initial value.

    In this section, we shall obtain the existence of pullback attractors for the process associated to (4).

    Let (X,dX) be a given metric space, and we denote R2d={(t,τ)R2:τt}.

    A process on X is a mapping U(,) such that R2d×X(t,τ,x)U(t,τ)xX with U(τ,τ)x=x for any (τ,x)R×X, and U(t,r)U(r,τ)x=U(t,τ)x for any τrt and all xX. A process U(,) is said to be continuous if for any pair τt, U(t,τ):XX is continuous. A process U(,) is said to be closed if for any τt, and any sequence {xn}X, if xnxX and U(t,τ)xnyX, then U(t,τ)x=y. Clearly, every continuous process is closed.

    Let P(X) be the family of all nonempty subsets of X, and consider a family of nonempty sets parameterized in time ˆD0={D0(t):tR}P(X). Let D be a nonempty class of families parameterized in time ˆD={D(t):tR}P(X). The class D will be called a universe in P(X).

    Definition 4.1. (1) A process U(,) on X is said to be pullback ˆD0-asymptotically compact if for any tR and any sequences {τn}(,t] and {xn}X satisfying τn and xnD0(τn) for all n, the sequence {U(t,τn)xn} is relatively compact in X.

    (2) Further, we say that a process U(,) on X is pullback D-asymptotically compact if it is pullback ˆD-asymptotically compact for all ˆDD.

    Definition 4.2. ˆD0={D0(t):tR}P(X) is pullback D-absorbing for the process U(,) on X if for any tR and any ˆD={D(t):tR}D, there exists a τ0(t,ˆD)t such that

    U(t,τ)D(τ)D0(t),ττ0(t,ˆD).

    Observe from the above definition that ˆD0 does not belong necessarily to the class D.

    Denote

    Λ(ˆD0,t):=st¯τsU(t,τ)D0(τ)X,tR,

    where ¯{}X is the closure in X. We denote by distX(X1,X2) the Hausdorff semi-distance in X between two sets X1 and X2, defined as

    distX(X1,X2)=supxX1infyX2dX(x,y),X1,X2X.

    In order to get our result, we need to use a classical theorem in [14,15].

    Theorem 4.3. Consider a closed process U:R2d×XX, a universe DP(X), and a family ˆD0={D0(t):tR}P(X) which is pullback D-absorbing for U, and assume also that U is pullback ˆD0-asymptotically compact. Then, the family AD={AD(t):tR} is a family of pullback D-attractors which is defined by

    AD(t)=¯ˆDDΛ(ˆD,t)X,tR,

    and has the following properties:

    (a) for any tR, the set AD(t) is a nonempty compact subset of X, and AD(t)Λ(ˆD0,t).

    (b)AD is pullback D-attracting, i.e.,

    limτdistX(U(t,τ)D(τ),AD(t))=0,ˆDD, tR.

    (c)AD is invariant, i.e., U(t,τ)AD(τ)=AD(t), τt.

    (d) if ˆD0D, then AD(t)=Λ(ˆD0,t)¯D0(t)X, tR.

    Moreover, family AD is minimal in the sense that if ˆC={C(t):tR}P(X) is a family of closed sets such that for any ˆD={D(t):tR}D,

    limτdistX(U(t,τ)D(τ),C(t))=0,

    then AD(t)C(t).

    Remark 1. If ˆD0D, D0(t) is closed for any tR and D is inclusion-closed, then ADD and AD is unique family in D that satisfies (a), (b) and (c) above.

    Let DXF be the universe of fixed nonempty bounded subsets of X, i.e., the class of all families ˆD of the form ˆD={D(t)=D:tR} with D a fixed nonempty bounded subset of X.

    Lemma 4.4. ([14]) Under the assumptions of Theorem 4.3, if the universe D contains the universe DXF, then both attractors, ADXF and AD, exist, and the following relation holds:

    ADXFAD,tR.

    In view of Theorem 3.2, here we take the phase space MH=H×(CHL2V) equipped with the norm (ξ,ϕ)MH=ξH+ϕCH+ϕL2V for (ξ,ϕ)MH. It is easy to verify the following proposition.

    Proposition 1. Consider given g:R×CHH and fL43loc(R;V) satisfying assumptions (Hg) and (Hf). Then the solution of problem (1.1) generates a bipara-metric family of mappings U(t,τ):MHMH by U(t,τ)(uτ,ϕ)=(u(t;τ,uτ,ϕ),ut(;τ,uτ,ϕ)), which is a continuous process.

    In order to prove the existence of pullback attractors for the process U, we need the following assumption:

    (H1) For every uL2(τh,T;V), there exists a value σ(0,μ0λ1) which is independent of u such that

    tτeσs|g(s,us)|2ds<C2gtτheσs|u(s)|2ds,tT.

    Lemma 4.5. g:R×CHH and fL43loc(R;V) satisfy assumptions (Hg), (H1) and (Hf). Then, for any (uτ,ϕ)MH, there exists a value σ(0,μ0λ1) such that the solution u of (4) holds the estimates

    |u(t)|2Ceσ(tτ)(|uτ|2+C2gμ0λ1ϕL2H)+C(2716μ1)13eσttτeσsf(s)43ds, (29)
    μ0tsu(r)2dr|u(s)|2+C2gμ0λ1usL2H+C2gμ0λ1ts|u(r)|2dr+(2716μ1)13tsf(r)43dr, (30)

    and

    μ1tsu(r)4dr|u(s)|2+C2gμ0λ1usL2H+C2gμ0λ1ts|u(r)|2dr+(2716μ1)13tsf(r)43dr. (31)

    Proof. Multiplying (4) by u and integrating the resultant over Ω, using integration by parts and Young's inequality, we have

    12ddt|u(t)|2+μ0u(t)2+μ1u(t)4=f(t),u(t)+(g(t,ut),u(t))f(t)u(t)+|g(t,ut)||u(t)|12(2716μ1)13f(t)43+μ12u(t)4+12μ0λ1|g(t,ut(t))|2+μ0λ12|u(t)|2. (32)

    Now multiplying (32) by 2eσt and applying Poincaré's inequality (6), one has

    ddt(eσt|u(t)|2)=σeσt|u(t)|2+eσtddt|u(t)|2(σ+μ0λ1)eσt|u(t)|22μ0u(t)2eσt+(2716μ1)13eσtf(t)43+eσtμ0λ1|g(t,ut)|2
    (σλ1μ0)u(t)2eσt+(2716μ1)13eσtf(t)43+eσtμ0λ1|g(t,ut)|2. (33)

    We integrate (33) over the interval [τ,t] with respect to t, use σ(0,μ0λ1) and assumption (H1), to conclude

    eσt|u(t)|2eστ|uτ|2+1μ0λ1tτeσs|g(s,us)|2ds+(2716μ1)13tτeσsf(s)43dseστ|uτ|2+C2gμ0λ1tτteσs|u(s))|2ds+(2716μ1)13tτeσsf(s)43dseστ|uτ|2+C2gμ0λ1(eστ0h|ϕ(s)|2ds+tτeσs|u(s)|2ds)+(2716μ1)13tτeσsf(s)43dseστ(|uτ|2+C2gμ0λ10h|ϕ(s)|2ds)+C2gμ0λ1tτeσs|u(s)|2ds+(2716μ1)13tτeσsf(s)43ds. (34)

    On account of the assumption (Hf), we apply the Gronwall inequality to (34) to derive

    eσt|u(t)|2Ceστ(|uτ|2+C2gμ0λ10h|ϕ(s)|2ds)+C(2716μ1)13tτeσsf(s)43ds,

    which means

    |u(t)|2Ceσ(tτ)(|uτ|2+C2gμ0λ1ϕL2H)+C(2716μ1)13eσttτeσsf(s)43ds,tτ.

    Consequently, the estimation (29) is proved. Thanks to (32) and (6), we obtain

    ddt|u(t)|2+μ0u(t)2+μ1u(t)41μ0λ1|g(t,ut)|2+(2716μ1)13f(t)43. (35)

    Integrating (35) over the interval (s,t), and using the assumptions on g, we can get

    |u(r)|2+μ0tsu(r)2dr+μ1tsu(r)4dr|u(s)|2+(2716μ1)13tsf(r)43dr+C2gμ0λ1(0h|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 DMHσ the class of all families of nonempty subsets ˆD={D(t):tR}P(MH) such that

    limτ(eστsup(ξ,ϕ)D(τ)(ξ,ϕ)2MH)=0.

    Remark 2. According to the above definition and the notation DXF, it is obvious that DMHFDMHσ and that both are inclusion-closed.

    Based on the above universe and Lemma 4.5, we can present the pullback dissipation in MH. Let ¯BX(0,R) be the closed ball with zero as the center and R as the radius in X.

    Proposition 2. Suppose that g:R×CHH and fL43loc(R;V) satisfy assumptions (Hg), (Hf) and (H1). Then, the family ˆD0={D0(t):tR}MH is defined by

    D0(t)=¯BH(0,RH)×(¯BCH(0,RCH)¯BL2V(0,RL2V))

    is pullback DMHσ-absorbing for the process U(t,τ) on MH and ˆD0DMHσ, where

    R2H(t)=1+C(2716μ1)13eσ(t2h)teσsf(s)43ds,R2L2V(t)=1μ0[(1+2C2ghμ0λ1)R2H+(2716μ1)13f(r)L43(th,t;V)].

    Proof. Fix tR, we derive from (29) that there exists a pullback time τ(ˆD,t)t2h such that

    |u(t;τ,uτ,ϕ)|21+C(2716μ1)13eσtteσsf(s)43dsR2H(t),ˆDDMHσ(t)

    holds for all tτ with ττ(ˆD,t) and (uτ,ϕ)D(τ).

    In particular, we observe that utCHR2H(t). Now, putting s=th in the estimate (30) and using (29), we deduce immediately that ut2L2VR2L2V(t). Moreover, in view of the above estimates, (Hf) and the definition of universe, we can see clearly the fact that ˆD0 belongs to DMHσ. Therefore, the proof is done.

    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 g:R×CHH and fL43loc(R;V) satisfy assumptions (Hg) and (H1). Then, the process U(t,τ) defined in proposition 1 is pullback DMHσ-asymptotically compact.

    Proof. Fix a value tR and consider a family ˆDDMHσ, let {τn}(,t] with τn and {(uτn,ϕn}) with {(uτn,ϕn)}D(τn) be two sequences for all n, then we denote {(un,unt)}ˆD as a sequence with un()=u(;τn,uτn,ϕn).

    In the same way as Proposition 2, by using the estimations in Lemma 4.5, we obtain that there exists a pullback time τ1(ˆD,t)t4h1 such that the subsequence {un:τnτ1(ˆD,t)} is bounded in L(t4h1,t;H)L4(t3h1,t;V). By virtue of the equation (12) and the embedding theorem, we have that

    (un)L43(t2h1,t;V)μ0unL43(t2h1,t;V)+μ1un3L4(t2h1,t;V)+fL43(t2h1,t;V)+(un(tρ(t)))unL43(t2h1,t;V)+Cg(t,unt)L2(t2h1,t;H).

    By applying a similar technique as proving (16) in Theorem 3.2, we derive from the assumption (Hg) that {(un)} is uniformly bounded in L43(t2h1,t;V). Thanks to the Aubin-Lions Lemma, the assumptions on g and the diagonal procedure, there exists a subsequence (still denote it by {un}) and a function uL(t4h1,t;H)L4(t3h1,t;V) such that

    {unu weakly * in L(t4h1,t;H),unu weakly in L4(t2h1,t;V),(un)u weakly in L43(t2h1,t;V),unu strongly in L4(t2h1,t;H),un(s)u(s) strongly in H,  a.e. s(t2h1,t),g(,unt)g(,ut) strongly in L2(th1,t;H). (37)

    From the above convergences, we derive that uC([t2h1,t];H) is a weak solution for problem (1.1) in the interval (th1,t) with uth1 as initial data.

    By using the same technique as proving (19) in Theorem 3.2, we can get

    unu strongly in  C([th1,t];H).

    Consequently, we obtain that for any sequence {sn}[th1,t] with sns,

    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 t-h-1 \leq s_1 \leq s_2 \leq t ,

    \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 y can be u and all u^n . Hence we can define the continuous functions on the interval [t-h-1,t] as

    \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 \mathcal{J} and \mathcal{J}_n are non-increasing functions. In addition, the convergences (37) indicates 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 \{s_k\} \subset (t-h-1,t) satisfying that \lim\limits_{k\rightarrow \infty}s_k = s_* and

    \begin{align*} \lim\limits_{n\rightarrow \infty}\mathcal{J}_n(s_k) = \mathcal{J}(s_k), \ \forall k. \end{align*}

    Since \mathcal{J}(s) is continuous, for \forall \varepsilon>0 , there exists k_\varepsilon \in \mathbb{N} such that

    \begin{align} |\mathcal{J}(s_k)-\mathcal{J}(s_*)| < \frac \varepsilon 2, \quad \forall k \geq k_\varepsilon. \end{align} (41)

    Because \mathcal{J}_n(s) is uniformly continuous with respect to time s , there exists n(k_\varepsilon) such that

    \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 \mathcal{J}_n and (41)-(42), we derive that for all n \geq n(k_\varepsilon) ,

    \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 \varepsilon , we have that \limsup\limits_{n\rightarrow \infty} \mathcal{J}_n(s_n) \leq \mathcal{J}(s_*) , which, by virtue of (37) again, implies

    \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 u and u^n , the convergences in (37) and (44), we can deduce that

    \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 \mathcal{D}_\sigma^{M_H} -asymptotically compact. Thus we finish the proof.

    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 g:\mathbb{R}\times C_H \rightarrow H satisfies (H_g) and f \in L_{loc}^{\frac{4}{3}}(\mathbb{R};V^\prime) fulfills conditions (H_f) and (H_1) . Then, there exist the minimal pullback \mathcal{D}_F^{M_H} -attractor

    \mathcal{A}_{\mathcal{D}_F^{M_H}} = \{\mathcal{A}_{\mathcal{D}_F^{M_H}} (t) : t \in \mathbb{R}\}

    and the minimal pullback \mathcal{D}_\sigma^{M_H} -attractor

    \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 \mathcal{A}_{\mathcal{D}_\sigma^{M_H}} \in \mathcal{D}_\sigma^{M_H} and the following relation holds:

    \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 \mathcal{A}_{\mathcal{D}_\sigma^{M_H}} is unique (in the sense of Remark 1).

    Proof. From Proposition 1, we observe that the process U is continuous in M_H . Furthermore, we can also obtain that there exists a pullback absorbing family \widehat{D}_0 \in \mathcal{D}_\sigma^{M_H} from Proposition 2 and the process U is pullback \mathcal{D}_\sigma^{M_H} -asymptotically compact from Theorem 4.7. Consequently, by using Theorem 4.3 and Lemma 4.4, we derive that the pullback attractors \mathcal{A}_{\mathcal{D}_\sigma^{M_H}} and \mathcal{A}_{\mathcal{D}_F^{M_H}} exist and

    \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 \widehat{D}_0 \in \mathcal{D}_\sigma^{M_H}, D_0 (t) is closed for any t \in \mathbb{R} and \mathcal{D}_\sigma^{M_H} is inclusion-closed, we conclude from Remark 1 that \mathcal{A}_{\mathcal{D}_\sigma^{M_H}} belongs to \mathcal{D}_\sigma^{M_H} and \mathcal{A}_{\mathcal{D}_\sigma^{M_H}} is unique. Moreover, in view of the property (d) in Theorem 4.3, we can get

    \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 \|\nabla u(x,t)\|_{L^2(\Omega)} should not be too large or infinite. In this line of work, Smagorinsky in 1960s proposed a similar approximating equation, known as Ladyzhenskaya-Smagorinsky model

    \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 \mathcal{D}u by \nabla u , and thus deduced another two systems as

    \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 p \geq 1 + 2n/(n+2) and the uniqueness of solutions to system (50) for p \geq (n+2)/2 . However, the uniqueness of the system (51) is still an open problem.

    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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