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Medical robotic engineering selection based on square root neutrosophic normal interval-valued sets and their aggregated operators

  • We introduce the concepts of multiple attribute decision-making (MADM) using square root neutrosophic normal interval-valued sets (SRNSNIVS). The square root neutrosophic (SRNS), interval-valued NS, and neutrosophic normal interval-valued (NSNIV) sets are extensions of SRNSNIVS. A historical analysis of several aggregating operations is presented in this article. In this article, we discuss a novel idea for the square root NSNIV weighted averaging (SRNSNIVWA), NSNIV weighted geometric (SRNSNIVWG), generalized SRNSNIV weighted averaging (GSRNSNIVWA), and generalized SRNSNIV weighted geometric (GSRNSNIVWG). Examples are provided for the use of Euclidean distances and Hamming distances. Various algebraic operations will be applied to these sets in this communication. This results in more accurate models and is closed to an integer Δ. A medical robotics system is described as combining computer science and machine tool technology. There are five types of robotics such as Pharma robotics, Robotic-assisted biopsy, Antibacterial nano-materials, AI diagnostics, and AI epidemiology. A robotics system should be selected based on four criteria, including robot controller features, affordable off-line programming software, safety codes, and the manufacturer's experience and reputation. Using expert judgments and criteria, we will be able to decide which options are the most appropriate. Several of the proposed and current models are also compared in order to demonstrate the reliability and usefulness of the models under study. Additionally, the findings of the study are fascinating and intriguing.

    Citation: Murugan Palanikumar, Nasreen Kausar, Harish Garg, Aiyared Iampan, Seifedine Kadry, Mohamed Sharaf. Medical robotic engineering selection based on square root neutrosophic normal interval-valued sets and their aggregated operators[J]. AIMS Mathematics, 2023, 8(8): 17402-17432. doi: 10.3934/math.2023889

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  • We introduce the concepts of multiple attribute decision-making (MADM) using square root neutrosophic normal interval-valued sets (SRNSNIVS). The square root neutrosophic (SRNS), interval-valued NS, and neutrosophic normal interval-valued (NSNIV) sets are extensions of SRNSNIVS. A historical analysis of several aggregating operations is presented in this article. In this article, we discuss a novel idea for the square root NSNIV weighted averaging (SRNSNIVWA), NSNIV weighted geometric (SRNSNIVWG), generalized SRNSNIV weighted averaging (GSRNSNIVWA), and generalized SRNSNIV weighted geometric (GSRNSNIVWG). Examples are provided for the use of Euclidean distances and Hamming distances. Various algebraic operations will be applied to these sets in this communication. This results in more accurate models and is closed to an integer Δ. A medical robotics system is described as combining computer science and machine tool technology. There are five types of robotics such as Pharma robotics, Robotic-assisted biopsy, Antibacterial nano-materials, AI diagnostics, and AI epidemiology. A robotics system should be selected based on four criteria, including robot controller features, affordable off-line programming software, safety codes, and the manufacturer's experience and reputation. Using expert judgments and criteria, we will be able to decide which options are the most appropriate. Several of the proposed and current models are also compared in order to demonstrate the reliability and usefulness of the models under study. Additionally, the findings of the study are fascinating and intriguing.



    The three-dimensional Navier-Stokes equations with damping describe the flow when there exists resistance in the fluid motion. The damping is related to various physical phenomena, such as air drag, friction effects or relative motion, caused by internal friction of fluid and the limitation of flow channel interface leading to friction and collision between fluid particles and walls.

    This paper is concerned with the asymptotic behavior of the 3D Navier-Stokes equations with nonlinear damping for a viscous incompressible fluid on the torus Ω=3i=1(0,Li)R3 and tR+ in the space-periodic case:

    {tuνΔu+(u)u+α|u|β1u+p=f  in  Ω×R+,u=0  in  Ω×R+,u(x+Liei,t)=u(x,t),  i=1,2,3,u(x,t=0)=u0(x), (1.1)

    where the kinematic viscosity ν>0 and the external force f=f(x,t) are given in appropriate Sobolev space and the constant α>0 is a characteristic parameter of the elasticity for fluid flow, β1 is a fixed positive parameter which describes the increasing radio. The above system formed by the unknown three-component velocity field u=(u1,u2,u3) and the scalar pressure p represents the conservation law of momentum and mass.

    When the parameter α disappears, system (1.1) reduces to the 3D classic incompressible Navier-Stokes equations, whose well-posedness (see [1,2,3,4,5,6,7]) and dynamic systems (see [8,9]) have been extensively investigated based on non-uniqueness of weak solution and global existence of strong solution in bounded or periodic domain. Now for the more general case α>0, Cai and Jiu have proved that the Cauchy problem of the 3D Navier-Stokes equations with nonlinear damping term has global weak solution for β1, global strong solution for β72 and the uniqueness for 72β5 in [10]. Since the nonlinear damping term α|u|β1u leads to more regularity than the classical Navier-Stokes equations, the research on infinite dimensional dynamic systems for (1.1) are progressively improved, such as [11,12,13,14].

    Based on the development of 2D/3D Navier-Stokes equations, there have many related literatures paying attention to the determination and reduction of incompressible flow flows, for instance in [15,16], the authors give the upper bound of determining modes for the 2D Navier-Stokes equations in the periodic case.

    Inspired by [10,15,16,17,18], we consider the determination of trajectories for the damped Navier-Stokes model (1.1). The main results and features can be summarized as follows.

    (I) As shown in [10], the nonlinear damping term α|u|β1u resulting in more regularity, the system (1.1) possesses a global strong solution satisfying

    uL(0,T;˙Vper)L2(0,T;H2(Ω)3)L(0,T;Lβ+1(Ω)3), (1.2)

    which guarantees the research on asymptotic behavior can be achieved. However, more assumption on u and f are needed as

    Ωudx=Ωfdx=0,    tR+, (1.3)

    which implies Poincaré's inequality holds.

    (II) For the non-autonomous case, by using the estimates on Galerkin's approximated equation, the Mazur inequality and continuous embedding ˙VperL3(β1)2(Ω)3, the finite determining modes m has been presented via the restriction on generalized Grashof number Gr as mCG3r for 72β5.

    (III) For the autonomous case, the problem is called asymptotic determination if there exists finite Fourier functionals F={Fi} with i=1,2,,n, such that the trajectories inside global attractor can be determined. Based on the existence of global attractor in [14], we can proved that the autonomous system (1.1) is asymptotic determining if n>C(Lνλ1)32 is large enough, where L is defined in Section 3.4.

    The structure of this paper is organized as follows. In Section 2, the preliminaries and functional setting are stated. The main results are shown in Section 3, the further research and some comments are also presented in this part.

    Following the notation in [4,6,7,16], denote

    ˙Hper={uL2per(Ω)3|Ωudx=0,u=0}={u=kZ3{0}ˆuke2πikLx|ˆuk=ˆuk,kLˆuk=0,kZ3{0}|ˆuk|2<} (2.1)

    and

    ˙Vper={uH1per(Ω)3|Ωudx=0,u=0}={u=kZ3{0}ˆuke2πikLx|ˆuk=ˆuk,kLˆuk=0,kZ3{0}|kL|2|ˆuk|2<}, (2.2)

    where ˆuk denotes the kth Fourier coefficient, k=(k1,k2,k3)Z3{0} and L=(L1,L2,L3).

    It is easy to check that ˙Hper and ˙Vper are Hilbert spaces with the inner products

    (u,v)˙Hper=Ωu(x)v(x)dx,    (u,v)˙Vper=3i,j=1Ωuixjvixjdx (2.3)

    and the norms

    u˙Hper=(kZ3{0}|ˆuk|2)12,    u˙Vper=(kZ3{0}|kL|2|ˆuk|2)12 (2.4)

    respectively.

    In this part, the Helmholtz-Leray projector PL on the Lebesgue space ˙L2per(Ω) is defined via the Helmholtz-Weyl decomposition

    ˙L2per(Ω)3=˙HperG(Ω), (2.5)

    where the curl field ˙Hper is defined as (2.1) whose element satisfies the weakly divergence free condition

    <u,ϕ>=0  for every  u˙Hper,  ϕC0(Ω), (2.6)

    and the gradient field G(Ω) on the torus Ω is defined by

    G(Ω)={u˙L2per(Ω)3|u=g,  gH1(Ω)}. (2.7)

    The decomposition (2.5) means, every function u˙L2per(Ω)3 can be decomposed uniquely as

    u=h+g, (2.8)

    where the function h belongs to ˙Hper and the scalar function g belongs to H1(Ω) (see [6]).

    Now, the Helmholtz-Leray projector PL on the torus Ω is defined as

    PL:˙L2per(Ω)3˙Hper,  (2.9)

    where u˙L2per(Ω)3 and h˙Hper, i.e., PLu=h for u˙L2per(Ω)3.

    The Stokes operator A is defined as

    Au=PL(Δu)=PLΔu,  for any  uH2(Ω)3, (2.10)

    with noting that A=Δ in the three-dimensional periodic case (see [4]). Since A is a positive operator, we can deduce that the eigenvalues λi of the operator A are positive and satisfy

    0<λ1λ2λi<  and  limiλi=. (2.11)

    Hence, for u belongs to ˙Hper, the Poincaré inequality

    u2˙Hper1λ1u2˙Vper,   for all  u˙Vper (2.12)

    holds.

    The bilinear and trilinear operators are defined as

    B(u,v)=PL(u)v  and  b(u,v,w)=(PL(u)v,w)˙Hper, (2.13)

    which satisfy

    b(u,v,w)=b(u,w,v)  and  b(u,v,v)=0,  for any  u,v,w˙Vper. (2.14)

    Lemma 2.1. (The Ladyzhenskaya inequality) For u defined on the tours ΩR3, the following estimates

    u(x)L4per(Ω)3u14˙Hperu34˙Vper, (2.15)
    u(x)Lper(Ω)3u12˙VperAu12˙Hper. (2.16)

    hold.

    Proof. See, e.g., [6] for more detail.

    Lemma 2.2. (The Mazur inequality) Let p0 is an arbitrary non-negative constant, one can deduce the following inequality for any x, yR

    2p|xy|p+1|x|x|py|y|p|(p+1)|xy|(|x|p+|y|p). (2.17)

    Proof. See, e.g., Kuang [19].

    Lemma 2.3. (The generalized Gronwall inequality) Let η=η(t) and ϕ=ϕ(t) be locally integrable real-valued functions on [0,) that satisfy the following conditions for some T>0:

    lim inft1Tt+Ttη(τ)dτ>0,lim supt1Tt+Ttη(τ)dτ<,limt1Tt+Ttϕ+(τ)dτ=0, (2.18)

    where η(t)=max{η(t),0} and ϕ+(t)=max{ϕ(t),0}. Suppose that γ=γ(t) is an absolutely continuous nonnegative function on [0,) that satisfies the following inequality almost everywhere on [0,):

    dγdt+ηγϕ. (2.19)

    Then γ0, as t0.

    Proof. See, the detailed proof in [4,16].

    Recalling the Galerkin decomposition u=i=1ˆuiωi associated with the eigenfunctions ωi of the Stokes operator, we denote the first m Fourier modes Pmu and residual modes Qmu as follows

    Pmu(x,t)=mi=1ˆuiωi,  Qmu(x,t)=i=m+1ˆuiωi, (3.1)

    which satisfy

    (Pmu(x,t),Qmu(x,t))˙Hper=0, (3.2)

    according to orthogonal properties of eigenfunctions ωi(i=1,2,,). Moreover, the Poincaré-Wirtinger inequality with respect to the function Qmu with zero space average under the periodic case

    Qmu(x,t)2˙Hper1λm+1Qmu(x,t)2˙Vper (3.3)

    and the inverse Poincaré-Wirtinger inequality for Pmu

    Pmu(x,t)2˙VperλmPmu(x,t)2˙Hper (3.4)

    are true.

    The Grashof number Gr is a dimensionless number for fluid dynamics whose approximate is the ratio of the buoyant to viscous forces acting on a fluid. Following [4,16], we define the Grashof number with regard to the first eigenvalues of the Stokes operator λ1, fluid viscosity ν and external force term f.

    Definition 3.1. (The Grashof number) In the three-dimensional space, we define the Grashof number by

    Gr=Fν2λ1, (3.5)

    where F2=lim supt1Tt+Ttf(s)2˙Hperds with F>0.

    Considering two solenoidal vector fields u(x,t), v(x,t) and two scalar functions p(x,t), q(x,t) respectively satisfying 3D Navier-Stokes equations with the damping term sharing the same periodic boundary condition

    {tuνΔu+(u)u+α|u|β1u+p=f  in  Ω×R+,u=0  in  Ω×R+ (3.6)

    and

    {tvνΔv+(v)v+α|v|β1v+q=g  in  Ω×R+,v=0  in  Ω×R+, (3.7)

    where f=f(x,t) and g=g(x,t) are the corresponding external force terms for the above two systems, then we can derive the appropriate evolution equations for u and v by using the Helmholtz-Weyl decomposition PL as

    dudt+νAu+B(u,u)+αPL|u|β1u=PLf  in  Ω×R+ (3.8)

    and

    dvdt+νAu+B(v,v)+αPL|v|β1v=PLg  in  Ω×R+ (3.9)

    respectively.

    Theorem 3.2. Suppose that β1, T>0, the initial value u0˙Hper and the external force term f˙Hper. Then there exists a global weak solution of the initial boundary value problem (1.1) such that

    uL(0,T;˙Hper)L2(0,T;˙Vper)Lβ+1(0,T;Lβ+1(Ω)3). (3.10)

    Moreover, suppose that 72β5, u0˙VperLβ+1(Ω)3, there exists a unique global strong solution to system (1.1) satisfying

    uL(0,T;˙Vper)L2(0,T;H2(Ω)3)L(0,T;Lβ+1(Ω)3). (3.11)

    Proof. Here by using the Galerkin approximated approach and localized technique to achieve a priori estimate, then by virtue of compact argument and limiting process, we can obtain the desired results, we skip the detail in this part. The proof is similar as the existence of global weak and strong solution in Rn can be seen in Cai and Jiu [10], and the bounded domain in Song and Hou [14], except some minor revision for our problem with periodic boundary.

    Theorem 3.3. Assume u is the strong solution in Theorem 3.2, then for any t,T>0, u satisfies the following estimate

    t+TtAu2˙HperdτCν2t+Ttf2˙Hperdτ, (3.12)

    when T is large enough.

    Proof. Multiplying (3.8) by Au and integrate over Ω, we obtain

    12ddtu2˙Vper+νAu2˙Hper+αΩ|u|β1|u|2dx+α(β1)4Ω|u|β3||u|2|2dx   =(f,Au)˙Hperb(u,u,Au). (3.13)

    The Hölder inequality, the Young and Gagliardo-Nirenberg inequalities result in

    |b(u,u,Au)|C(Ω|uu|2dx)12Au˙Hperν8Au2˙Hper+CνΩ|uu|2dxν8Au2˙Hper+Cνu2Lβ+1(Ω)3u2L2(β+1)β1(Ω)3ν8Au2˙Hper+Cνu2Lβ+1(Ω)3Au2(11β)β+7˙Hperu4(β2)β+7Lβ+1(Ω)3ν8Au2˙Hper+Cνu6(β+1)β+7Lβ+1(Ω)3Au2(11β)β+7˙Hperν4Au2˙Hper+Cν3u2(β+1)Lβ+1(Ω)3, (3.14)

    and

    |(f,Au)˙Hper|Cνf2˙Hper+ν4Au2˙Hper (3.15)

    for 72β5.

    Substituting (3.14) and (3.15) into (3.13) for any t, t0R+, we derive

    νtt0Au2˙HperdτCνtt0f2˙Hperdτ+Cν3tt0u2(β+1)Lβ+1(Ω)3dτ+u(t0)2˙Vper. (3.16)

    Then, the uniform boundedness of u(t)˙Vper and u(t)Lβ+1(Ω)3 results in the following estimate

    tt0Au2˙HperdτCν2tt0f2˙Hperdτ (3.17)

    for sufficiently large constant C which is bigger then the one in (3.16), which leads to (3.12), the proof is complete.

    Theorem 3.4. Assume that 72β5, u is the global strong solution in Theorem 3.2. Then the first m modes are determining of (1.1), i.e.,

    PmuPmv˙Hper0,  as  t, (3.18)

    implies

    uv˙Hper0,  as  t (3.19)

    provided that mR+ satisfies

    mCG3r, (3.20)

    where C is the constant only depending on λ1, and Gr is the Grashof number (see Definition 3.1).

    Proof. Let w=uv. Then w satisfies

    (dwdt,ωi)+ν(Aw,ωi)+b(w,u,ωi)+b(v,w,ωi)+α(|u|β1u|v|β1v,ωi)˙Hper    =(fg,ωi)˙Hper (3.21)

    in distribution sense.

    Since Qmu=i=m+1ˆui(t)ωi, Pmu and Qmu are orthogonal, it follows that

    12ddtQmw2˙Hper+νQmw2˙Vper+b(w,u,Qmw)+b(v,w,Qmw)+α(|u|β1u|v|β1v,Qmw)˙Hper=(f(x,t)g(x,t),Qmw)˙Hper. (3.22)

    Based on the monotonicity of the nonlinear term, we can get

       α(|u|β1u|v|β1v,Qmw)˙Hper=α(|u|β1u|v|β1v,w)˙Hperα(|u|β1u|v|β1v,Pmw)˙Hperα(|u|β1u|v|β1v,Pmw)˙Hper, (3.23)

    which leads to the following inequality

       12ddtQmw2˙Hper+νQmw2˙Vper|b(w,u,Qmw)|+|b(v,w,Qmw)|+|α(|u|β1u|v|β1v,Pmw)˙Hper|   +|(f(x,t)g(x,t),Qmw)˙Hper|. (3.24)

    Next, the estimates for every term in (3.24) will be proceed for applying the generalized Gronwall inequality (Lemma 2.3) with ξ=Qmw˙Hper.

    Noting that u=Pmu+Qmu and the properties of trilinear operators (2.14), we write

    |b(w,u,Qmw)||b(Pmw,Qmw,u)|+|b(Qmw,u,Qmw)|=:b1+b2 (3.25)

    and

    |b(v,w,Qmw)|=|b(v,Pmw,Qmw)|=|b(v,Qmw,Pmw)|=:b3. (3.26)

    The Hölder and Young inequalities for bi(i=1,2,3), and Lemma 2.1 result in

    b1CPmwL4per(Ω)3Qmw˙VperuL4per(Ω)3CPmw14˙HperPmw34˙VperQmw˙Vperu14˙Hperu34˙Vper, (3.27)
    b2CQmw˙HperQmw˙VperuL(Ω)3ν2Qmw2˙Vper+C2νQmw2˙HperAu2˙Hper

    and

    b3CvL4per(Ω)3Qmw˙VperPmwL4per(Ω)3Cv14˙Hperv34˙VperQmw˙VperPmw14˙HperPmw34˙Vper, (3.28)

    where C represents different variable-independent constants.

    Noting that 72β5, we have the following embedding

    ˙VperL3(β1)2(Ω)3 (3.29)

    by the Sobolev theorem. For the damping term in (3.24), applying (3.29), Mazur's inequality (Lemma 2.2), Hölder's inequality, Young's inequality and ˙VperL6(Ω)3, we obtain

       |α(|u|β1u|v|β1v,Pmw)˙Hper|αβΩ|w|((|u|β1+|v|β1)|Pmw|dxαβwL6(Ω)3(uβ1L3(β1)2(Ω)3+vβ1L3(β1)2(Ω)3)PmwL6(Ω)3Cαβw˙Vper(uβ1˙Vper+vβ1˙Vper)Pmw˙Vper.. (3.30)

    For the remaining external force term in (3.24), the Cauchy-Schwarz inequality results in

    |(f(t)g(t),Qmw)˙Hper|f(t)g(t)L2(Ω)Qmw˙Hper. (3.31)

    Combining (3.22), (3.27), (3.28), (3.30) and (3.31), we conclude

    ddtQmw2˙Hper+νQmw2˙VperCνQmw2˙HperAu2˙HperCPmw14˙HperPmw34˙VperQmw˙Vperu14˙Hperu34˙Vper+Cv14˙Hperv34˙VperQmw˙VperPmw14˙HperPmw34˙Vper+Cαβw˙Vper(uβ1˙Vper+vβ1˙Vper)Pmw˙Vper+f(t)g(t)˙HperQmw˙Hper, (3.32)

    which can be rewritten in the form

    dγ(t)dt+η(t)γ(t)ϕ(t) (3.33)

    from (3.3) and (3.4) and the notations

    γ(t)=Qmw2˙Hper,η(t)=νλm+1CνAu2˙Hper,ϕ(t)=CPmw14˙HperPmw34˙VperQmw˙Vperu14˙Hperu34˙Vper          +CPmw˙Hper(λ34mv14˙Hperv34˙VperQmw˙Vper          +αβλmw˙Vper(uβ1˙Vper+vβ1˙Vper))          +f(t)g(t)˙HperQmw˙Hper. (3.34)

    The uniform boundedness of u, v, w in ˙Hper, ˙Vper, Lβ+1(Ω)3 from Theorem 3.2 together with convergences Pmw˙Hper and f(t)g(t)L2(Ω) yield

    ϕ(t)0  as  t (3.35)

    provided that (2.18) is true. Hence, (2.18) is verified by

    lim inft1Tt+Ttη(τ)dτ=νλm+1lim supt1Tt+TtCνAu2˙Hperdτνλm+1CF2ν3>0 (3.36)

    via the estimate

    lim supt1Tt+TtAu2˙HperdτCF2ν2 (3.37)

    from Theorem 3.3.

    According to λm=Cλ1m23, and the definition of Grashof number Gr (see Definition 3.1), we conclude that

    m>CG3r. (3.38)

    The proof is complete.

    Remark 3.1. In Theorem 3.4, we give the determining modes for weak solution of system (1.1) when the strong solution exists. However, if there is only the existence of weak solution, the determining modes for system (1.1) is still open.

    In this part, we consider the 3D damped Navier-Stokes equations with autonomous force f(x) as

    {tuνΔu+(u)u+α|u|β1u+p=f(x)  in  Ω×R+,u=0  in  Ω×R+,u(x+Liei,t)=u(x,t),  i=1,2,3,u(x,t=0)=u0. (3.39)

    Similar as the non-autonomous case above, the equivalent abstract form of (3.39) can be given by

    dudt+νAu+B(u,u)+αPL|u|β1u=PLf(x)  in  Ω×R+, (3.40)

    where PL still denotes the Helmholtz-Leray projector. Based on the well-posedness of (3.40), Li et al. study the existence of a finite dimensional global attractor as following.

    Theorem 3.5. Assume that f˙Hper and u0˙Vper, Then the semigroup {Lt}t0 generated by problem (3.39) possesses a ˙Vper-global attractor A.

    Proof. See, e.g., Li et al. [12], which studied the damped Navier-Stokes equations in the non-slip boundary condition. We can similarly get the dynamical system to the problem (3.39) on the periodic boundary condition.

    Consider the system F={F1,F2,,Fn} of linear functionals generated by the corresponding Fourier modes Fn(u)=:(u,ωn), where ωn denotes the eigenfunctions of the Stokes operator. In what follows, we will prove the asymptotic determination for (3.39) with autonomous external force f(x) according to Theorem 3.5, namely, the system F is determining if n is large enough.

    Theorem 3.6. Let 72β5 and n satisfy the inequality

    n>C(Lνλ1)32, (3.41)

    where L=L(ν,α,β,uL(0,T;˙Vper),uL2(0,T;H2(Ω)3),uL(0,T;Lβ+1(Ω)3)) is a dimensionless constant, λ1 denotes the primary eigenvalue. Then the system F of the first n Fourier modes is asymptotically determining for the dynamical system generated by (3.39).

    Proof. Let u(t) and v(t) be two trajectories inside the finite dimensional global attractor A in Theorem 3.5. Denote w(t)=u(t)v(t), then it is easy to check that w(t) satisfies

    dwdt+νAw=B(v,v)B(u,u)+αPL|v|β1vαPL|u|β1u  in  Ω×R+. (3.42)

    Next, multiplying (3.42) by Aw and integrating over Ω, noting that PL is symmetric and (2.14), we obtain

    12ddtw2˙Vper+νAw2˙Hper     |b(w,u,Aw)|+|b(v,w,Aw)|+α|(|u|β1u|v|β1v,Aw)|. (3.43)

    The Hölder inequality, Young's inequality and Lemma 2.1 yield the estimates

    |b(w,u,Aw)|CwLper(Ω)3u˙VperAw˙HperCw12˙Vperu˙VperAw32˙Hperν6Aw2˙Hper+C1(ν)w2˙Vperu4˙Vper (3.44)

    and

    |b(v,w,Aw)|CvLper(Ω)3w˙VperAw˙HperCv12˙VperAv12˙Hperw˙VperAw˙Hperν6Aw2˙Hper+C2(ν)v˙VperAv˙Hperw2˙Vper, (3.45)

    where C1(ν)=729C32ν3 and C2(ν)=3C2ν depend on the dimensionless constant ν only.

    For the damping term in (3.43), by using Hölder's and Young's inequalities, we obtain

       α|(|u|β1u|v|β1v,Aw)|C(Ω|α|u|β1uα|v|β1v|2dx)12Aw˙Hperν6Aw2˙Hper+C2(ν)Ω|α|u|β1uα|v|β1v|2dx. (3.46)

    Hence, Mazur's inequality, ˙Vper˙L6per(Ω) and Hölder's inequality result in

       Ω|α|u|β1uα|v|β1v|2dxCΩ(|u|β1|w|+||u|β1|v|β1||v|)2dxCΩ(|u|2(β1)|w|2dx+CΩ(|u|β2+|v|β2)2|v|2|w|2dxCu2(β1)L3(β1)(Ω)3w2L6(Ω)3+C(u2(β2)L6(β2)(Ω)3+v2(β2)L6(β2)(Ω)3)      v2L6(Ω)3w2L6(Ω)3Cu2(β1)L3(β1)(Ω)3w2˙Vper+C(u2(β2)L6(β2)(Ω)3+v2(β2)L6(β2)(Ω)3)      v|2˙Vperw|2˙Vper. (3.47)

    Substituting (3.44)–(3.47) into (3.43), we conclude

       ddtw2˙Vper+νAw2˙Hper2w2˙Vper(C1u4˙Vper+C2v˙VperAv˙Hper+C3u2(β1)L3(β1)(Ω)3   +C3(u2(β2)L6(β2)(Ω)3+v2(β2)L6(β2)(Ω)3)v|2˙Vper), (3.48)

    where C3=C3(ν,α,β) is a dimensionless constant.

    Owing to 72<β<5, we have

    t0u2(β1)L3(β1)(Ω)3dτCu2(β+1)2β+7L(0,t;Lβ+1(Ω)3)u8(β2)β+7L2(0,T;H2(Ω)3)t3(5β)β+7,t0u2(β2)L6(β2)(Ω)3dτCu2(β1)3β+7L(0,t;Lβ+1(Ω)3)u2(5β13)β+7L2(0,T;H2(Ω)3)t4(5β)β+7 (3.49)

    are all bounded for any 0<t<.

    Denote

    L=2(C1u4˙Vper+C2v˙VperAv˙Hper+C3u2(β1)L3(β1)(Ω)3   +C3(u2(β2)L6(β2)   (Ω)3+v2(β2)L6(β2)   (Ω)3)v|2˙Vper), (3.50)

    which is a finite dimensionless constant because of (3.11). Then there exists an n large enough such that \nu\lambda_{n+1} > L holds, the estimate (3.48) can be reduced to

    \begin{eqnarray} \begin{array}{ll} \frac{d}{dt}\|w^*\|_{\dot{V}_{per}}^2+(\nu\lambda_{n+1}-L)\|w^*\|_{\dot{V}_{per}}^2\leq0, \end{array} \end{eqnarray} (3.51)

    which yields

    \begin{eqnarray} \begin{array}{ll} \|w^*(t)\|_{\dot{V}_{per}}^2\leq e^{-C'(t-s)}\|w^*(s)\|_{\dot{V}_{per}}^2, \ \ s\leq t \end{array} \end{eqnarray} (3.52)

    for some positive constant C' .

    Since u^* and v^* belong to the global attractor \mathcal{A} , we have \|w^*(t)\|_{\dot{V}_{per}}\to 0 as t\to\infty . Hence, the system \mathcal{F} is asymptotically determining for the dynamical system of (3.39) when n > C(\frac{L}{\nu\lambda_1})^{\frac{3}{2}} because of \lambda_{n+1} = C\lambda_{1}n^{\frac{2}{3}} . Therefore, the proof is completed.

    In this section, we discuss some open interrelated issues that might be studied in the prospective investigations and research.

    (I) The determining modes in the periodic case is presented in this paper, which can be proved similarly in the whole space case \mathbb{R}^3 (where the boundary conditions can be considered as |u|\to 0 as |x|\to \infty ) by utilizing the relevant conclusions in [10], but the situation turns out to be quite different in the Dirichlet condition, due to the fact

    \begin{align} A = -P_L\Delta\neq-\Delta, \end{align} (3.53)

    which implies b(u, u, Au)\neq0 in the Dirichlet boundary condition. So the main ideas and difficulties of the subject is summarized as follows

    \begin{eqnarray} \left\{\begin{array}{ll} \partial_{t}u-\nu\Delta u+(u\cdot \nabla)u+\alpha|u|^{\beta-1}u+\nabla p = f\ \ \text{in}\ \ \widetilde{\Omega}\times\mathbb{R}^+, \\ \nabla\cdot u = 0\ \ \text{in}\ \ \widetilde{\Omega}\times\mathbb{R}^+, \\ u(x, t) = 0\ \ \text{on} \ \ \partial\widetilde{\Omega}, \\ u(x, t = 0) = u_0(x), \end{array}\right. \end{eqnarray} (3.54)

    where \widetilde{\Omega} denotes an open bounded region in three dimensions. The determination and reduction for our problem defined on bounded domain is our objective in future.

    (II) For the generalized fluid flow, Ladyzhenskaya proposed a revised version of Ladyzhenskaya-type Navier-Stokes model in the 1960s, where the surrounding flow problem was considered

    \begin{eqnarray} \left\{\begin{array}{ll} \partial_{t}u-\nabla\cdot[(\nu_0+\nu_1\|Du\|_{L^2}^{q-2})Du]+(u\cdot \nabla)u+\nabla p = f, \\ \nabla\cdot u = 0, \\ \end{array}\right. \end{eqnarray} (3.55)

    where Du = \frac{1}{2}(\nabla u+\nabla u^T) , Guo and Zhu studied the partial regularity of the distribution solution of the initial boundary value problem in the three-dimensional case of the model (see [20]). In order to overcome the difficulties of the model, Lions proposed a new class of polished Navier-Stokes equations (see [5]) with the establishment of the monotonicity method

    \begin{eqnarray} \left\{\begin{array}{ll} \partial_{t}u-\nu_0\Delta u-\nu_1\sum\limits_{i = 1}^n\frac{\partial}{\partial x_i}(|\nabla u|^{q-1}\frac{\partial u}{\partial x_i})+(u\cdot \nabla)u+\nabla p = f, \\ \nabla\cdot u = 0.\\ \end{array}\right. \end{eqnarray} (3.56)

    Exploiting the technique of this work, it is actually possible to study the above models (3.55) and (3.56) in the whole space \mathbb{R}^3 , torus \mathbb{T}^3 , and bounded smooth region \Omega respectively, the main difficulty for dealing with their determining modes lies in the unknown spectral relationship of the corresponding operator.

    (III) A further meaningful research is concerned with finite dimensional reduction of the Navier-Stokes equations with damping (1.1). If we can get the Lipschitz property

    \begin{eqnarray} \begin{array}{ll} \|h(u)-h(v)\|_{(\dot{H}_{per})'}\leq L\|u-v\|_{\dot{H}_{per}}, \ \ \forall u, v\in\dot{H}_{per}, \end{array} \end{eqnarray} (3.57)

    where h(u) denotes (u\cdot \nabla)u+\alpha|u|^{\beta-1}u , and

    \begin{eqnarray} \begin{array}{ll} L < \lambda_{N+1}, \end{array} \end{eqnarray} (3.58)

    then the first N Fourier modes is asymptotically determining for the dynamic system of (1.1) (see [21]), which leads to the reduction of (1.1) and even the existence of inertial manifold.

    Xin-Guang Yang was partially supported by Cultivation Fund of Henan Normal University (No. 2020PL17), Henan Overseas Expertise Introduction Center for Discipline Innovation (No. CXJD2020003), Key project of Henan Education Department (No. 22A110011). Wei Shi was partially supported by Henan Normal University Postgraduate Research and Innovation Project (No. YL202020).

    The authors declare there is no conflict of interest.



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