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Research article

The porous cantilever beam as a model for spinal implants: Experimental, analytical and finite element analysis of dynamic properties


  • Investigation of the dynamic properties of implants is essential to ensure safety and compatibility with the host's natural spinal tissue. This paper presents a simplified model of a cantilever beam to investigate the effects of holes/pores on the structures. Free vibration test is one of the most effective methods to measure the dynamic response of a cantilever beam, such as natural frequency and damping ratio. In this study, the natural frequencies of cantilever beams made of polycarbonate (PC) containing various circular open holes were investigated numerically, analytically, and experimentally. The experimental data confirmed the accuracy of the natural frequencies of the cantilever beam with open holes calculated by finite element and analytical models. In addition, two finite element simulation methods, the dynamic explicit and modal dynamic methods, were applied to determine the damping ratios of cantilever beams with open holes. Finite element analysis accurately simulated the damped vibration behavior of cantilever beams with open holes when known material damping properties were applied. The damping behavior of cantilever beams with random pores was simulated, highlighting a completely different relationship between porosity, natural frequency and damping response. The latter highlights the potential of finite element methods to analyze the dynamic response of arbitrary and complex structures, towards improved implant design.

    Citation: Xiaoyu Du, Yijun Zhou, Lingzhen Li, Cecilia Persson, Stephen J. Ferguson. The porous cantilever beam as a model for spinal implants: Experimental, analytical and finite element analysis of dynamic properties[J]. Mathematical Biosciences and Engineering, 2023, 20(4): 6273-6293. doi: 10.3934/mbe.2023270

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  • Investigation of the dynamic properties of implants is essential to ensure safety and compatibility with the host's natural spinal tissue. This paper presents a simplified model of a cantilever beam to investigate the effects of holes/pores on the structures. Free vibration test is one of the most effective methods to measure the dynamic response of a cantilever beam, such as natural frequency and damping ratio. In this study, the natural frequencies of cantilever beams made of polycarbonate (PC) containing various circular open holes were investigated numerically, analytically, and experimentally. The experimental data confirmed the accuracy of the natural frequencies of the cantilever beam with open holes calculated by finite element and analytical models. In addition, two finite element simulation methods, the dynamic explicit and modal dynamic methods, were applied to determine the damping ratios of cantilever beams with open holes. Finite element analysis accurately simulated the damped vibration behavior of cantilever beams with open holes when known material damping properties were applied. The damping behavior of cantilever beams with random pores was simulated, highlighting a completely different relationship between porosity, natural frequency and damping response. The latter highlights the potential of finite element methods to analyze the dynamic response of arbitrary and complex structures, towards improved implant design.



    In this paper, we consider the following two-component Fornberg-Whitham (FW) system for a fluid

    {ut+uux=(12x)1x(ρu)ρt+(ρu)x=0(u,ρ)(0,x)=(u0,ρ0)(x) (1.1)

    where xT=R/2πZ, tR+. Here, u=u(x,t) is the horizontal velocity of the fluid, and ρ=ρ(x,t) is the height of the fluid surface above a horizontal bottom. This system was first proposed in [5], and local well-posedness and non-uniform dependence on the initial data were established in Sobolev spaces Hs(R)×Hs1(R) for s>32 in [11,12].

    Local well-posedness in Besov spaces Bsp,r(R)×Bs1p,r(R) of (1.1) was established in [4] for s>max{2+1p,52}. Besov spaces Bsp,r are a class of functions of interest in the study of nonlinear partial differential equations, as they are based on Sobolev spaces and introduce a measure of generalized Hölder regularity through the index r, along with the Sobolev index of differentiability s and the Lebesgue integrability index p. If s and p are fixed, the spaces Bsp,r grow larger with increasing r. In [4], the FW system was shown to be well-posed in the sense of Hadamard by establishing the existence and uniqueness of the solution to the system (1.1) and then proving continuity of the data-to-solution map when the initial data belong to Bsp,r(R)×Bs1p,r(R) for s>max{2+1p,52}.

    In this paper, our objective is to prove non-uniform dependence on periodic initial data for the two-component FW system (1.1) in Bsp,r(T)×Bs1p,r(T) for s>max{2+1p,52}. We work with periodic initial data, as that simplifies our choice of approximate solutions and the resulting estimates. Setting Λ=12x, we rewrite (1.1) as

    {ut+uux=Λ1x(ρu)ρt+uρx+ρux=0(u,ρ)(0,x)=(u0,ρ0)(x) (1.2)

    where xT=R/2πZ and tR+.

    The paper is organized as follows: In Section 2, we recall the standard definitions and properties of Besov spaces, linear transport equations, the operator Λ, and the two-component FW system. In Section 3, we prove non-uniform dependence on initial data for the FW system (1.2) when the initial data belong to Bsp,r(T)×Bs1p,r(T) for s>max{2+1p,52}. For this proof, we use a technique previously seen in the study of non-uniform continuity of data-to-solution maps for other nonlinear PDEs, for instance in [6,7,8,10,12]. We construct two sequences of approximate solutions such that the initial data for these sequences converge to each other in Bsp,r(T)×Bs1p,r(T). Non-uniform dependence is then established by proving that the approximate and hence the exact solutions remain bounded away from each other for any positive time t>0. This idea was first explored by Kato in [9] to show that the data-to-solution map for Burgers' equation is not Hölder continuous in the Hs norm with s>3/2 for any Hölder exponent.

    This section is a review of relevant definitions and results on Besov spaces, linear transport equations, the operator Λ, and the two-component FW system (1.2). We begin by listing some useful notation to be used throughout Section 3.

    For any x,yR,

    xy denotes xαy for some constant α.

    xy denotes x=βy for some constant β.

    xy denotes xγy for some constant γ.

    We recall the construction of a dyadic partition of unity from [8]. Consider a smooth bump function χ such that suppχ=[43,43] and χ=1 on [34,34]. For ξ>0, set φ1(ξ)=χ, φ0(ξ)=χ(ξ2)χ(ξ) and φq(ξ)=φ0(2qξ). Then, suppφq=[342q,832q] and q1φq(ξ)=1. Using this partition, a Littlewood-Paley decomposition of any periodic distribution u is defined in [3] as follows:

    Definition 2.1 (Littlewood-Paley decomposition). For any uD(T) with the Fourier series u(x)=jZˆujeijx where ˆuj=12π2π0eijyu(y)dy, its Littlewood-Paley decomposition is given by u=q1Δqu, where Δqu are periodic dyadic blocks defined for all qZ as

    Δqu=jZφq(j)ˆujeijx.

    Using this Littlewood-Paley decomposition, Besov spaces on T are defined in [3] as follows:

    Definition 2.2 (Besov spaces). Let sR and p, r[1,]. Then the Besov spaces of functions are defined as

    Bsp,rBsp,r(T)={uD(T)|uBsp,r<},

    where

    uBsp,r={(q1(2sqΔquLp)r)1/rif1r<supq12sqΔquLpifr=.

    Following are some properties proved in [1, Section 2.8] and [3, Section 1.3] that facilitate the study of nonlinear partial differential equations in Besov spaces.

    Lemma 2.3. Let s,sjR for j=1,2 and 1p,r. Then the following properties hold:

    (1) Topological property: Bsp,r is a Banach space continuously embedded in D(T).

    (2) Algebraic property: For all s>0, Bsp,rL is a Banach algebra.

    (3) Interpolation: If fBs1p,rBs2p,r and θ[0,1], then fBθs1+(1θ)s2p,r and

    fBθs1+(1θ)s2p,rfθBs1p,rf1θBs2p,r.

    (4) Embedding: Bs1p,rBs2p,r whenever s1s2. In particular, Bsp,rBs1p,r for all sR.

    Remark on (2) in Lemma 2.3: When s>1p (or s1p and r=1), Bsp,rL. We will use the fact that for 0<s<1p, the result is still true as long as the functions are bounded.

    Given a linear transport equation, Proposition A.1 in [2] proves the following estimate for its solution size in Besov spaces:

    Proposition 2.4. Consider the linear transport equation

    {tf+vxf=Ff(x,0)=f0(x) (2.1)

    where f0Bsp,r(T), FL1((0,T);Bsp,r(T)) and v is such that xvL1((0,T);Bs1p,r(T)). Suppose fL((0,T);Bsp,r(T))C([0,T];D(T)) is a solution to (2.1). Let 1p,r. If either s1+1p, or s=1+1p and r=1, then for a positive constant C that depends on s, p, and r, we have

    f(t)Bsp,reCV(t)(f0Bsp,r+Ct0eCV(τ)F(τ)Bsp,rdτ)

    where

    V(t)=t0xv(τ)B1/pp,rLdτifs<1+1p

    and

    V(t)=t0xv(τ)Bs1p,rdτotherwise.

    For r<, fC([0,T],Bsp,r(T)), and if r=, then fC([0,T],Bsp,1(T)) for all s<s.

    Let Λ=12x; then, for any test function g, the Fourier transform of Λ1g is given by F(Λ1g)=11+ξ2ˆg(ξ). Moreover, for any sR, Λ1x is continuous from Bs1p,r to Bsp,r; that is, for all hBs1p,r, there exists a constant κ>0 depending on s, p, and r such that

    Λ1xhBsp,rκhBs1p,r. (2.2)

    The well-posedness of the two-component FW system (1.2) in Besov spaces was established on the real line in [4] with the following result:

    Theorem 2.5. Let s>max{2+1p,52}, p[1,], r[1,] and (u0,ρ0)Bsp,r(R)×Bs1p,r(R). Then the system (1.2) has a unique solution (u,ρ)C([0,T];Bsp,r(R)×Bs1p,r(R)), where the doubling time T is given by

    T=C(u0Bsp,r+ρ0Bs1p,r)2,

    with C being a constant that depends on s, p, and r, and the solution size is estimated as

    (u,ρ)Bsp,r×Bs1p,r2(u0Bsp,r+ρ0Bs1p,r).

    Moreover, the data-to-solution map is continuous.

    Since we work with Bsp,r(T)×Bs1p,r(T) in this paper, we state the following:

    Corollary 2.6. Theorem 2.5 holds when R is replaced by T.

    Proof. The existence of a solution to (1.2) is proved by altering the mollifier used to prove Theorem 2.5. This adaptation of the mollifier was done for the single Fornberg-Whitham equation in [7, Section 3.1]. Uniqueness and continuous dependence on periodic initial data for the system (1.2) are established by approximation arguments similar to those in [4, Sections 3.2 and 3.3].

    In this section, we establish non-uniform dependence on initial data in the periodic case for the two-component FW system (1.2) in Besov spaces.

    Theorem 3.1. Let s>max{2+1p,52} and r[1,]. The data-to-solution map (u0,ρ0)(u(t),ρ(t)) of the Cauchy problem (1.2) is not uniformly continuous from any bounded subset of Bsp,r(T)×Bs1p,r(T) into C([0,T];Bsp,r(T))×C([0,T];Bs1p,r(T)) where T is given by Theorem 2.5. In particular, there exist two sequences of solutions {(uω,n,ρω,n)} with ω=±1 such that the following hold:

    (i) limn(u1,n(0)u1,n(0)Bsp,r+ρ1,n(0)ρ1,n(0)Bs1p,r)=0.

    (ii) lim infn(u1,nu1,nBsp,r+ρ1,nρ1,nBs1p,r)|sint|.

    Proof. For nN, we consider two sequences of functions {(uω,n,ρω,n)} with ω=±1, defined by

    {uω,n=ωn+1nssin(nx+ωt)ρω,n=1n+1nssin(nx+ωt).

    We take initial data

    {u0ω,n=uω,n(0)=ωn+1nssinnxρ0ω,n=ρω,n(0)=1n+1nssinnx.

    Let the solutions to the FW system (1.2) with these initial data be denoted by (uω,n,ρω,n). At t=0, we have

    limn(u01,nu01,nBsp,r+ρ01,nρ01,nBs1p,r)=limn2n1Bsp,r=0,

    which proves part (i) of Theorem 3.1.

    To prove part (ii), first we estimate (u0ω,n,ρ0ω,n)Bγp,r×Bγ1p,r and (uω,n,ρω,n)Bγp,r×Bγ1p,r for any γ>0 and r<. Using the triangle inequality, we have

    (u0ω,n,ρ0ω,n)Bγp,r×Bγ1p,r2n1Bγp,r+nssinnxBγp,r+n1ssinnxBγ1p,r. (3.1)

    By Definition 2.2,

    sinnxBγp,r=(q12γqrΔqsinnxrLp)1r. (3.2)

    From Definition 2.1, as shown in the Appendix, we have Δqsin(nx)Lp=φq(n), where 0<φq(n)1 for all q such that 1ln(2)ln(38n)q1ln(2)ln(43n) and φq(n)=0 otherwise. Hence, (3.2) implies that for any γ>0,

    sin(nx)Bγp,r(1ln(2)ln(43n)q=1ln(2)ln(38n)(2q)γr)1r.

    As 2q43n for every term in the summation, from the above, we obtain

    sin(nx)Bγp,r(1ln(2)ln(43n)q=1ln(2)ln(38n)(43n)γr)1r=(1ln(2)[ln(43n)ln(38n)])1r(43n)γ=(1ln(2)ln(329))1r(43)γnγ=Cγnγ. (3.3)

    Here and in what follows, Cγ is a generic constant that depends only on γ for fixed p and r. Similarly, it follows that for any γ>0,

    cos(nx)Bγp,rCγnγ. (3.4)

    By Definition 2.1,

    Δqn1=φq(0)n1={n1ifq=10otherwise.

    So, n1Bγp,r=(q12γqrΔqn1rLp)1r=2γn1. Using this and (3.3), it follows from (3.1) that

    (u0ω,n,ρ0ω,n)Bγp,r×Bγ1p,r21γn1+Cγnγns+Cγnγ1n1sCγmax{n1,nγs}. (3.5)

    Since (uω,n,ρω,n) is a phase shift of (u0ω,n,ρ0ω,n), we have

    (uω,n,ρω,n)Bγp,r×Bγ1p,rCγmax{n1,nγs}. (3.6)

    If r=, (3.5) and (3.6) follow immediately from Definition 2.2.

    We complete the proof of Theorem 3.1 by establishing (ii) for {(uω,n,ρω,n)}, taking advantage of the following lemma, whose proof follows the proof of Theorem 3.1. Lemma 3.2 establishes that for each n and ω, (uω,n,ρω,n) approximates (uω,n,ρω,n) in Bsp,r(T)×Bs1p,r(T) uniformly on [0,T] for some T>0.

    Lemma 3.2. Let Eω,n=(Eω,n1,Eω,n2) where Eω,n1=uω,nuω,n and Eω,n2=ρω,nρω,n, with ω=±1. Then for all t(0,T), where T is given by Theorem 2.5, Eω,n(t)Bsp,r×Bs1p,r=Eω,n1(t)Bsp,r+Eω,n2(t)Bs1p,r0 as n.

    We show that (u1,n,ρ1,n) and (u1,n,ρ1,n) stay bounded away from each other for any t>0. Since

    u1,nu1,nBsp,ru1,nu1,nBsp,ru1,nu1,nBsp,ru1,nu1,nBsp,r (3.7)

    and

    ρ1,nρ1,nBs1p,rρ1,nρ1,nBs1p,rρ1,nρ1,nBs1p,rρ1,nρ1,nBs1p,r, (3.8)

    adding (3.7) and (3.8) we obtain

    u1,nu1,nBsp,r+ρ1,nρ1,nBs1p,ru1,nu1,nBsp,r+ρ1,nρ1,nBs1p,rE1,n(t)Bsp,r×Bs1p,rE1,n(t)Bsp,r×Bs1p,rns(sin(nx+t)sin(nxt)Bsp,r+sin(nx+t)sin(nxt)Bs1p,r)2n1Bsp,rE1,n(t)Bsp,r×Bs1p,rE1,n(t)Bsp,r×Bs1p,r=2ns(cos(nx)Bsp,r|sin(t)|+cos(nx)Bs1p,r|sin(t)|)21γn1E1,n(t)Bsp,r×Bs1p,rE1,n(t)Bsp,r×Bs1p,r. (3.9)

    By Definition 2.2, if r=, we immediately have

    cos(nx)Bsp,rCsns, (3.10)

    where Cs is a constant that depends only on s for a given p. For 1r<, there is a similar estimate, whose proof is given in the Appendix. Also, by Lemma 3.2, we have Eω,n(t)Bsp,r×Bs1p,r0 for ω=±1, as n. Using this and (3.10), it follows from (3.9) that

    lim infn(u1,nu1,nBsp,r+ρ1,nρ1,nBs1p,r)2Cs(lim infn|sin(t)|+lim infnn1|sin(t)|)|sin(t)|>0.

    This proves part (ii) of Theorem 3.1 and completes the proof of non-uniform dependence on initial data for the two-component FW system (1.2) in Bsp,r(T)×Bs1p,r(T) for s>max{2+1p,52}.

    Now we prove Lemma 3.2.

    Proof. (Lemma 3.2) We show that Eω,n(t)Bγp,r×Bγ1p,r0 as n for any γ with max{s32,1+1p}<γ<s1, and then interpolate between such a γ and a value δ>s. Recall that Eω,n1=uω,nuω,n and Eω,n2=ρω,nρω,n. It can be seen that Eω,n1 and Eω,n2 vanish at t=0 and that they satisfy the equations

    {tEω,n1+uω,nxEω,n1=Eω,n1xuω,n+Λ1x(Eω,n2Eω,n1)R1tEω,n2+uω,nxEω,n2=Eω,n2xuω,nρω,nxEω,n1Eω,n1xρω,nR2. (3.11)

    Here, R1 and R2 are the approximate solutions for the FW system, that is,

    {R1=tuω,n+uω,nxuω,nΛ1x(ρω,nuω,n)R2=tρω,n+x(ρω,nuω,n).

    ● Estimate for R1Bγp,r: Using the definitions of uω,n and ρω,n, we have

    R1=tuω,n+uω,nxuω,nΛ1x(ρω,nuω,n)=12n2s1sin(2(nx+ωt)).

    Then by (3.3),

    R1Bγp,rCγnγ2s+1.

    ● Estimate for R2Bγ1p,r: Using the definitions of uω,n and ρω,n,

    R2=tρω,n+x(ρω,nuω,n)=1nscos(nx+ωt)+1n2s1sin(2(nx+ωt)).

    So from (3.3) and (3.4), it follows that

    R2Bγ1p,rCγ(nsnγ1+n12snγ1)Cγnγs1.

    Therefore,

    R1Bγp,r+R2Bγ1p,rnγs1. (3.12)

    Since Eω,n1(t) and Eω,n2(t) satisfy the linear transport equations (3.11), to estimate the error Eω,n(t)Bγp,r×Bγ1p,r, we apply Proposition 2.4 to obtain

    Eω,n1(t)Bγp,rK1eK1V1(t)t0eK1V1(τ)F1(τ)Bγp,rdτ (3.13)

    and

    Eω,n2(t)Bγ1p,rK2eK2V2(t)t0eK2V2(τ)F2(τ)Bγ1p,rdτ (3.14)

    where K1, K2 are positive constants depending on γ and

    F1(t)=Eω,n1xuω,n+Λ1x(Eω,n2Eω,n1)R1, (3.15)
    F2(t)=Eω,n2xuω,nρω,nxEω,n1Eω,n1xρω,nR2. (3.16)
    V1(t)=t0xuω,n(τ)Bγ1p,rdτ,
    V2(t)={t0xuω,n(τ)B1/pp,rLdτifγ<2+1pt0xuω,n(τ)Bγ2p,rdτotherwise.

    Since max{s32,1+1p}<γ<s1, we have

    V1(t)nγstn1t andV2(t)Ct0uω,n(τ)Bγp,rdτ (3.17)

    for some constant C that depends on γ, p, and r. By Theorem 2.5 and Eq (3.5), it follows that

    V2(t)2Ct0(u0ω,n,ρ0ω,n)Bγp,r×Bγ1p,rdτn1t. (3.18)

    Let K=max{K1,K2}. Using (3.17) and (3.18), we combine (3.13) and (3.14) to obtain

    Eω,n1(t)Bγp,r+Eω,n2(t)Bγ1p,rt0eK(tτ)/n(F1(τ)Bγp,r+F2(τ)Bγ1p,r)dτ. (3.19)

    ● Estimate for F1(τ)Bγp,r: From (3.15), as Bγp,r is a Banach algebra, we have

    F1Bγp,rEω,n1Bγp,rxuω,nBγp,r+Λ1x(Eω,n2Eω,n1)Bγp,r+R1Bγp,rEω,n1Bγp,ruω,nBγ+1p,r+Λ1x(Eω,n2Eω,n1)Bγp,r+R1Bγp,r. (3.20)

    From (2.2),

    Λ1x(Eω,n2Eω,n1)Bγp,rκEω,n2Eω,n1Bγ1p,rM(Eω,n1Bγp,r+Eω,n2Bγ1p,r) (3.21)

    where M is a constant depending on γ,p, and r. By Theorem 2.5, we have

    uω,nBγ+1p,r2(u0ω,n,ρ0ω,n)Bγ+1p,r×Bγp,r,

    so by (3.5), uω,nBγ+1p,r2Cγmax{n1,nγ+1s}. As γ>max{s32,1+1p},

    uω,nBγ+1p,rnγ+1s. (3.22)

    Using (3.21) and (3.22), from (3.20), we obtain

    F1(τ)Bγp,r(M+nγ+1s)Eω,n1(τ)Bγp,r+MEω,n2(τ)Bγ1p,r+R1(τ)Bγp,r. (3.23)

    ● Estimate for F2(τ)Bγ1p,r: We may use the algebra property, item (2) of Lemma 2.3, for Bγ1p,r since γ1>max{s52,1p}>0 and the functions we are dealing with are bounded. Then, from (3.16),

    F2Bγ1p,rEω,n2Bγ1p,rxuω,nBγ1p,r+ρω,nBγ1p,rxEω,n1Bγ1p,r+xρω,nBγ1p,rEω,n1Bγ1p,r+R2Bγ1p,rn1Eω,n1Bγp,r+Eω,n2Bγ1p,ruω,nBγp,r+R2Bγ1p,r. (3.24)

    By Corollary 2.6, uω,nBγp,r2(u0ω,n,ρ0ω,n)Bγp,r×Bγ1p,r, which implies

    uω,nBγp,r2Cγmax{n1,nγs}

    by (3.5). As γ<s1, uω,nBγp,rn1. Using this in (3.24) yields

    F2(τ)Bγ1p,rn1Eω,n1(τ)Bγp,r+n1Eω,n2(τ)Bγ1p,r+R2(τ)Bγ1p,r. (3.25)

    Adding (3.23) and (3.25) gives

    F1(τ)Bγp,r+F2(τ)Bγ1p,r(M+nγ+1s)(Eω,n1(τ)Bγp,r+Eω,n2(τ)Bγ1p,r)+R1(τ)Bγp,r+R2(τ)Bγ1p,r. (3.26)

    Substituting (3.26) into (3.19), we obtain

    Eω,n(t)Bγp,r×Bγ1p,rf(t)+t0g(τ)Eω,n(τ)Bγp,r×Bγ1p,rdτ (3.27)

    where

    f(t)t0eK(tτ)/n(R1(τ)Bγp,r+R2(τ)Bγ1p,r)dτ (3.28)

    and

    g(τ)(M+nγ+1s)eK(tτ)/n(M+1)eK(tτ)/n. (3.29)

    Using Grönwall's inequality, from (3.27) we obtain

    Eω,n(t)Bγp,r×Bγ1p,rf(t)+t0g(τ)f(τ)etτg(z)dzdτ. (3.30)

    Using (3.12) along with (3.28) and (3.29), from (3.30), we obtain

    Eω,n(t)Bγp,r×Bγ1p,rnγs1, (3.31)

    which means that Eω,n(t)Bγp,r×Bγ1p,r0 as n for any max{s32,1+1p}<γ<s1.

    On the other hand, if δ(s,s+1), then noting that the solution with the given data is in Bδp,r×Bδ1p,r for any δ we have, for 0<t<T (from Theorem 2.5)

    Eω,n(t)Bδp,r×Bδ1p,r(uω,n,ρω,n)Bδp,r×Bδ1p,r+(uω,n,ρω,n)Bδp,r×Bδ1p,r2(u0ω,n,ρ0ω,n)Bδp,r×Bδ1p,r+(uω,n,ρω,n)Bδp,r×Bδ1p,r, (3.32)

    where we have used the solution size estimate in Theorem 2.5. Now, for δ<s+1, Eqs (3.5) and (3.6) imply that (u0ω,n,ρ0ω,n)Bδp,r×Bδ1p,rCδnδs and (uω,n,ρω,n)Bδp,r×Bδ1p,rCδnδs, where Cδ denotes a constant that depends only on δ, for a given p and r. So (3.32) yields

    Eω,n(t)Bδp,r×Bδ1p,rnδs. (3.33)

    We use the interpolation property, item (3) from Lemma 2.3, with θ=δsδγ, to obtain

    Eω,n(t)Bsp,r×Bs1p,rEω,n(t)θBγp,r×Bγ1p,rEω,n(t)1θBδp,r×Bδ1p,r. (3.34)

    From (3.34), using (3.31) and (3.33), we obtain

    Eω,n(t)Bsp,r×Bs1p,r(nγs1)δsδγ(nδs)sγδγ=nθ. (3.35)

    As θ(0,1), (3.35) implies that Eω,n(t)Bsp,r×Bs1p,r0 as n for any s>max{2+1p,52}. This completes the proof of Lemma 3.2.

    When p=r=2, Bs2,2 and Hs are equivalent by [2, Proposition 1.2], and so we obtain the following corollary:

    Corollary 3.3. The data-to-solution map for the two-component FW system (1.2) is not uniformly continuous from any bounded subset of Hs(T)×Hs1(T) into C([0,T];Hs(T))×C([0,T];Hs1(T)) for s>52.

    In this paper, we considered the two-component Fornberg-Whitham (FW) system (1.2) and used a sequential approach to prove that its data-to-solution map is not uniformly continuous for periodic initial data belonging to Besov spaces Bsp,r(T)×Bs1p,r(T) where s>max{2+1p,52}. As a corollary, this establishes non-uniform dependence on periodic initial data for the FW system (1.2) in Sobolev spaces Hs(T)×Hs1(T) for s>52.

    In this appendix, we provide a lower bound on cos(nx)Bsp,r for any s>0 and 1r<. By Definition 2.2,

    cos(nx)Bsp,r=(q12sqrΔqcosnxrLp)1r. (4.1)

    By Definition 2.1, Δqcos(nx)=φq(n)einx. Therefore, Δqcos(nx)Lp=φq(n), where 0<φq(n)1 for all q such that 1ln(2)ln(38n)q1ln(2)ln(43n) and φq(n)=0 otherwise, (4.1) implies that

    cos(nx)Bsp,r=(1ln(2)ln(43n)q=1ln(2)ln(38n)(2q)srφrq(n))1r.

    Since 2q38n for all terms in the summation, from the above we have

    cos(nx)Bsp,r(38)sns(1ln(2)ln(43n)q=1ln(2)ln(38n)φrq(n))1r. (4.2)

    Recall that φ0(ξ)=χ(ξ2)χ(ξ) and φq(ξ)=φ0(2qξ) for any q>1, where suppχ=[43,43] and χ=1 on [34,34]. This means that suppφq=[342q,832q] for any q1 and furthermore, φq=1 on the interval [432q,322q]. In other words, φq(n)=1 for 1ln(2)ln(23n)q1ln(2)ln(34n). Therefore, from (4.2) we have

    cos(nx)Bsp,r(38)sns(1ln(2)ln(34n)q=1ln(2)ln(23n)1)1r=(38)sns(1ln(2)[ln(34n)ln(23n)])1r=(1ln(2)ln(98))1r(38)sns=Csns,

    where Cs is a constant that depends only on s, for a given p and r. The same estimate holds for sin(nx)Bsp,r as well.

    All authors contributed equally towards conceptualization, formal analysis, investigation and methodology in this project; Writing of the original draft was done by Prerona Dutta; thereafter all authors together completed the review and editing process. All authors have read and approved the final version of the manuscript for publication.

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    We thank John Holmes, at The Ohio State University, for his valuable suggestions on this project. We would also like to thank the anonymous referees for their comments which greatly helped in improving the paper overall.

    All authors declare no conflicts of interest in this paper.



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