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Research article Special Issues

Fast delivery of melatonin from electrospun blend polyvinyl alcohol and polyethylene oxide (PVA/PEO) fibers

  • Received: 28 March 2022 Revised: 22 May 2022 Accepted: 25 May 2022 Published: 30 May 2022
  • Water-soluble polymers possess great advantages in current drug delivery systems, such as fast delivery through polymer matrix dissolution as well as promoting solid dispersion of poorly water-soluble drugs. In this work, water-soluble polyvinyl alcohol (PVA) and polyethylene oxide (PEO) were blended (50/50) to electrospin with and without the incorporation of a model drug, melatonin (MLT), at various blend polymer concentrations. Results suggested that increasing blend PVA/PEO solution concentrations, up to 7 wt%, promoted the formation of smooth and defect-free drug-incorporating fibers with an average fiber diameter ranged from 300 to 700 nm. Mechanical properties of the blank and MLT-loaded PVA/PEO fibers showed dependence on fiber morphologies and fiber mat structures, due to polymer concentrations for electrospinning. Furthermore, the surface wettability of the blend PVA/PEO fibers were investigated and further correlated with the MLT release profile of the fibers. Results suggested that fiber mats with a more well-defined fiber structure promoted a linear release behavior within 10 minutes in vitro. These drug-incorporated fibers were compatible to human umbilical vein endothelial cells (HUVECs) up to 24 hours. In general, this work demonstrated the structure-property correlations of electrospun PVA/PEO fibers and their potential biomedical applications in fast delivery of small molecule drugs.

    Citation: Rachel Emerine, Shih-Feng Chou. Fast delivery of melatonin from electrospun blend polyvinyl alcohol and polyethylene oxide (PVA/PEO) fibers[J]. AIMS Bioengineering, 2022, 9(2): 178-196. doi: 10.3934/bioeng.2022013

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  • Water-soluble polymers possess great advantages in current drug delivery systems, such as fast delivery through polymer matrix dissolution as well as promoting solid dispersion of poorly water-soluble drugs. In this work, water-soluble polyvinyl alcohol (PVA) and polyethylene oxide (PEO) were blended (50/50) to electrospin with and without the incorporation of a model drug, melatonin (MLT), at various blend polymer concentrations. Results suggested that increasing blend PVA/PEO solution concentrations, up to 7 wt%, promoted the formation of smooth and defect-free drug-incorporating fibers with an average fiber diameter ranged from 300 to 700 nm. Mechanical properties of the blank and MLT-loaded PVA/PEO fibers showed dependence on fiber morphologies and fiber mat structures, due to polymer concentrations for electrospinning. Furthermore, the surface wettability of the blend PVA/PEO fibers were investigated and further correlated with the MLT release profile of the fibers. Results suggested that fiber mats with a more well-defined fiber structure promoted a linear release behavior within 10 minutes in vitro. These drug-incorporated fibers were compatible to human umbilical vein endothelial cells (HUVECs) up to 24 hours. In general, this work demonstrated the structure-property correlations of electrospun PVA/PEO fibers and their potential biomedical applications in fast delivery of small molecule drugs.



    Collective behaviors often appear in many classical oscillatory systems [1,4,7,17,18,25,27,29]. Recently, such classical synchronization dynamics has been extended to a quantum regime, and it is called quantum synchronization in literature. It is worthwhile mentioning from [19,20] that quantum synchronization has attracted many researchers in the quantum optics community due to its powerful applications in quantum information and quantum computing [8,14,15,16,21,28,33,34]. Among possible candidates describing quantum synchronization, we are interested in analytical studies on quantum synchronization via Wigner's formalism [30] that was first introduced by Wigner in 1932 in order to find quantum corrections to classical statistical mechanics. For the mathematical properties of the Wigner transform, we refer the reader to [32].

    To set up the stage, we begin with the Schördinger-Lohe (SL) model [19]. Let ψj=ψj(t,x):R+×RdC be the wave function of the quantum system situated at the j-th node whose dynamics is governed by the Cauchy problem to the SL model:

    {itψj=12Δψj+Vjψj+iκ2NNk=1(ψkψj,ψkψj,ψjψj),t>0,xRd,ψj(0,x)=ψ0j(x),j[N]:={1,,N}, (1)

    where Vj=Vj(x):RdR is a time independent potential at the j-th node, κ denotes a (uniform) nonnegative coupling strength between nodes, and , is the standard inner product in L2(Rd).

    Note that the Planck constant is assumed to be unity for simplicity. Like the classical Schrödinger equation, system (1) satisfies L2-conservation of the wave function ψj. We refer the reader to a recent review article [9] for the emergent dynamics of the SL model (1).

    In this paper, we study the emergent dynamics of the Cauchy problem to the WL model with identical potentials:

    {twij+pxwij+Θ[V](wij)=κ2NNk=1{(wkj+wik)(R2d(wik+wkj)dxdp)wij},t>0,(x,p)R2d,wij(0,x,p)=w0ij(x,p),i,j[N], (2)

    subject to initial constraints:

    R2dw0iidxdp=1,|R2dw0ijdxdp1|<1,ij[N]. (3)

    First, we recall the following definition of the emergent dynamics as follows.

    Definition 1.1. [3] System (2) exhibits complete aggregation if relative states tend to zero asymptotically.

    limtwijwmL2(R2d)=0,i,j,,m[N].

    In the sequel, we provide several comments on the Cauchy problem (2)–(3). First, the WL model (2) was first introduced in [3], and a priori asymptotic analysis has been studied only for the two-particle system with N=2. Second, one notices that (2) is equipped with the identical potential V for all i,j. In fact, potentials for the corresponding SL model (1) would be non-identical in general. However, for the simplicity of mathematical representation, identical potentials are considered in both [3] and our analysis of this work. Third, initial data are restricted to a suitable class. For (3)1, it is usually assumed for the classical Wigner equation so that conservation laws hold under the assumption. Precisely, it corresponds to the mass conservation for a classical Schrödinger equation (see Remark 1). Hence, it would be reasonable to employ (3)1. On the other hand, the assumption (3)2 does not appear in the study of the classical Wigner equation consisting of a single equation, whereas (2) contains N2 equations. Later, we will see from Corollary 1 that (3)2 guarantees the uniform L2-boundedness. We refer the reader to [5,6,10,11,13,22,23,26,35] for the Wigner and Wigner-type equations.

    The main results of this paper are two-fold. First, we provide the complete aggregation dynamics of (2) in a priori setting. Under the assumptions (3) on initial data, we can find an invariant set whose center plays the role of an asymptotically stable fixed point (see Lemma 3.2). Then, we obtain the uniform-boundedness of the L2-norm of a solution to the WL model and show that the L2-norms of all relative states tend to zero (see Corollary 1). For details, we refer to Theorem 3.1 in Section 3.

    Second, we provide a global existence theory of (2) combining the classical methods (fixed point theorem and semigroup theory) and exponential aggregation estimates. We highlight that this paper extends the results in [3] where the existence theory was not considered even for N=2, whereas a priori aggregation estimates were established only for N=2. For this, we first define a suitable function space X which is a subset of L2(R2d). Then, we recast the WL model as a first-order matrix-valued PDE on X and apply the fixed point theorem to show that the WL model admits a global mild solution. Furthermore, if more regularity assumptions on initial data are imposed, then we show tha a global classical solution can be obtained from the semigroup approach (see Theorem 4.1).

    The rest of this paper is organized as follows. In Section 2, we introduce generalized Wigner functions and the WL distribution matrix, and study their elementary properties. We also review previous results for the WL model. In Section 3, we provide complete aggregation estimates for the WL model in a priori setting. In Section 4, we show the global existence of mild and classical solutions depending on the regularity of initial data. Finally, Section 5 is devoted to a brief summary of this paper and some remaining issues for a future work. In Appendix A, we summarize classical results on the semigroup theory to be used for the global solvability in Section 4.

    Gallery of Notation: Throughout the paper, as long as there is no confusion, we simply use R2d instead of Rdx×Rdp or R2dx,p. Let f=f(x,p) and g=g(x,p) be two functions in L2(R2d). Then, the standard L2-inner product and the L2-norm are defined by

    f,g:=R2df(x,p)¯g(x,p)dxdp,f:=f,f,

    where ¯g(x,p) is the complex conjugate of g(x,p)C. We set the Fourier transform and its inverse transform as follows:

    (Fϕ)(p):=Rdϕ(x)eixpdx,(F1ϕ)(x):=1(2π)dRdϕ(p)eixpdp.

    For a given real-valued function ψ with two set of variables x,yRd, we define the Fourier transform in y variable as follows:

    (Fypψ)(x,p):=Rdψ(x,y)eiypdy.

    In this section, we introduce the N×N Wigner-Lohe (WL) distribution matrix associated with the SL model (1), and its governing model "the Wigner-Lohe model", and review the emergent behaviors of the 2×2 Wigner-Lohe model in [3].

    In this subsection, we show how the WL distribution matrix can be constructed from the SL model. For this, we first recall the generalized Wigner distribution and the pseudo-differential operator.

    Definition 2.1. [3]

    1. For any two complex-valued wave functions ψ,ϕL2(Rd), the generalized Wigner distribution w[ψ,ϕ] is defined by

    w[ψ,ϕ](x,p):=1(2π)dRdψ(x+y2)¯ϕ(xy2)eipydy,(x,p)R2d, (4)

    where ¯ψ is the complex conjugate of ψ.

    2. For VL(Rd) and wL2(R2d), we define the pseudo-differential operator Θ[V] as

    Θ[V](w)(x,p):=i(2π)dRd[V(x+y2)V(xy2)](Fpyw)(x,y)eipydy=i(2π)dR2d[V(x+y2)V(xy2)]w(x,p)ei(pp)ydpdy.

    Remark 1. Below, we give several comments on the generalized Wigner distribution and the pseudo-differential operator.

    1. The generalized Wigner distribution is complex conjugate symmetric in the sense that

    w[ϕ,ψ](x,p)=1(2π)dRdϕ(x+y2)¯ψ(xy2)eipydy=1(2π)dRd¯ψ(x+y2)ϕ(xy2)eipydybyyy=¯1(2π)dRdψ(x+y2)¯ϕ(xy2)eipydy=¯w[ψ,ϕ](x,p). (5)

    2. For the case ψ=ϕ, two definitions (4) and (5) yield the standard Wigner function [12]:

    w[ψ,ψ](x,p)=1(2π)dRdψ(x+y2)¯ψ(xy2)eipydy.

    Since w[ψ,ψ] coincide with the standard Wigner function, we simply denote

    w[ψ,ψ]=:w[ψ].

    Moreover, one can easily verify that w[ψ] is real-valued.

    3. The p-integral of w[ψ]=w[ψ,ψ] is the modulus square of ψ:

    Rdw[ψ](x,p)dp=|ψ(x)|2.

    Moreover, the (x,p)-integral of w[ψ,ϕ] is the inner product of ψ and ϕ:

    R2dw[ψ,ϕ](x,p)dxdp=1(2π)dR3dϕ(x+y2)¯ψ(xy2)eipydydxdp=Rdϕ(x)¯ψ(x)dx=ϕ,ψ.

    4. Since V is real-valued, one also has

    ¯Θ[V](w)(x,p)=i(2π)dR2d[V(x+y2)V(xy2)]ˉw(x,p)ei(pp)ydpdy=i(2π)dR2d[V(xy2)V(x+y2)]ˉw(x,p)ei(pp)ydpdy=i(2π)dR2d[V(x+y2)V(xy2)]ˉw(x,p)ei(pp)ydpdybyyy=Θ[V](¯w)(x,p).

    In the following lemma, we provide several properties of Θ[V](w) in Definition 2.1.

    Lemma 2.2. For f,gL2(R2d), one has the following relations:

    (i)R2dΘ[V](f)gdxdp=R2dΘ[V](g)fdxdp.(ii)R2dΘ[V](f)dxdp=0.

    Proof. (ⅰ) We use the change of variables:

    (p,p,y)(p,p,y)

    to yield

    R2dΘ[V](f)gdxdp=i(2π)dR4d[V(x+y2)V(xy2)]f(x,p)g(x,p)ei(pp)ydpdydxdp=i(2π)dR4d[V(xy2)V(x+y2)]f(x,p)g(x,p)ei(pp)ydpdydxdp=R2dΘ[V](g)fdxdp.

    (ⅱ) By the definition of Definition 2.1 and Fubini's theorem, we have

    R2dΘ[V](f)dxdp=i(2π)dR4d[V(x+y2)V(xy2)]f(x,p)ei(pp)ydpdydxdp=i(2π)dR3d[V(x+y2)V(xy2)]eipy(Rdf(x,p)eipydp)dydxdp=i(2π)dR3d[V(x+y2)V(xy2)](Fpyf)(x,y)eipydydxdp=iRd[V(x)V(x)](Fp0f)(x,0)dx=0.

    Remark 2. If we set f=u and g=¯u in Lemma 2.2, and recall Remark 2.2(4), then one has

    R2dΘ[V](u)ˉudxdp=R2dΘ[V](ˉu)udxdp=R2d¯Θ[V](u)udxdp=¯R2dΘ[V](u)ˉudxdp.

    This yields

    Re[R2dΘ[V](u)¯udxdp]=0.

    This fact was used in [23] to show the conservation of L2-norm for the solution to the quantum Liouville equation, which can be considered as a special case of the WL model with κ=0.

    For a given ensemble of wave functions {ψj} which is a solution to the SL model (1), we set

    wij:=w[ψi,ψj],wi:=w[ψ]=w[ψi,ψi],i,j[N].

    Then, the evolution of the WL distributions {wij} is given by the following coupled system:

    twij+pxwij=i(2π)dR2d[Vi(x+y2)Vj(xy2)]wij(x,p)ei(pp)ydpdy+κ2NNk=1[(wkj+wik)(R2dwikdxdpR2dwiidxdp+R2dwkjdxdpR2dwjjdxdp)wij]. (6)

    For the detailed derivation of (6), we refer the reader to [3]. Next, we show that system (6) admits conservation laws.

    Lemma 2.3. Let {wij} be a solution to (6) which rapidly decays to zero at infinity. Then, one has

    (i)ddtR2dwii(t,x,p)dxdp=0,t>0,i[N].

    (ii)ddtR2dwij(t,x,p)dxdp=Rd(Vi(x)Vj(x))wij(t,x,0)dx,t>0,i,j[N].

    Proof. (ⅰ) It follows from (6) that

    twii+x(pwii)+Θ[Vi](wii)=κ2NNk=1[(wki+wik)(R2dwikdxdpR2dwiidxdp+R2dwkidxdpR2dwiidxdp)wii]. (7)

    Now, we integrate (7) over (x,p)R2d and then use the zero far field assumption on wij and the second estimate of Lemma 2.3 to see

    ddtR2dwii(t,x,p)dxdp=0.

    (ⅱ) For the second assertion, we follow a similar calculation in (ⅰ).

    Remark 3. (i) Consider the linear Wigner equation:

    tw+pxw+Θ[V](w)=0.

    Then by Lemma 2.3, one has

    ddtR2dw(t,x,p)dxdp=0. (8)

    However, it is worthwhile mentioning that since w(t,x,p) can take negative values, the L1-norm of w would not be conserved in general. In fact, the relation (8) corresponds to the L2-conservation of the Schrödinger equation. If we assume that

    w=w[ψ]

    for a solution ψ to the linear Schrödinger equation, then the following relation holds (see Remark 1 (3)):

    R2dw(t,x,p)dxdp=ψ(t)2L2,t>0.

    Thus, the results in Lemma 2.3 is consistent with the classical theory for the Wigner equation.

    (ii) It follows from Moyal's identity [31] that

    w(t)2=ψ(t)4.

    Hence, the linear Wigner equation enjoys L2-conservation (but not L1-conservation).

    (iii) We have shown in Lemma 2.3 that diagonal elements {wi} satisfy conservation laws. However the off-diagonal elements do not satisfy conservation laws. Hence, conservation laws hold for system (6) with identical potentials.

    From now on, we are concerned with the following special situation:

    Vi=VandR2dw0i(x,p)dxdp=1,i[N]. (9)

    In this case, the (x,p)-integrals of {wij} are constants along the dynamics (6) (see Lemma 2.3):

    R2dwi(t,x,p)dxdp=R2dw0i(x,p)dxdp=1,t0,i[N],R2dwij(t,x,p,)dxdp=R2dw0ij(x,p)dxdp,t0,i,j[N].

    Hence, the Cauchy problem for system (6) with (9) can be further simplified as follows:

    {twij+pxwij+Θ[V](wij)=κ2NNk=1[(wik+wkj)(R2d(wik+wkj)dxdp)wij],t>0,(x,p)R2d,wij(0,x,p)=w0ij(x,p),i,j[N].

    In this subsection, we briefly recall the result from [3] for a two-body system. Extension to the many-body system will be discussed in the following two sections separately. We set

    w+12:=Re[w12],z12(t):=R2dw12(t,x,p)dxdp,R12(t):=Re[z12(t)]. (10)

    Then, (w1,w2,w12) satisfies the Cauchy problem:

    {tw1+pxw1+Θ[V](w1)=κ2(w+12R12w1),t>0,tw2+pxw2+Θ[V](w2)=κ2(w+12R12w2),tw12+pxw12+Θ[V](w12)=κ4(w1+w22z12w12),(w1(0),w2(0),w12(0))=(w01,w02,w012), (11)

    subject to constraints:

    R2dw01dxdp=R2dw02dxdp=1,|R2dw012dxdp|1,R2dw012dxdp1. (12)

    Theorem 2.4. [3] Let (w1,w2,w12) be a solution to (11)(12). Then, we have

    |1z12(t)|

    where is defined in (10).

    Proof. (ⅰ) The first estimate follows from the following ODE:

    This can be integrated explicitly:

    (13)

    where . If we assume that

    (14)

    then the denominator of the right-hand side of (13) can be zero, and hence can blow up in finite-time. Precisely, there exists such that

    In other words, for initial data satisfying (14), would not be bounded. Thus, should be not be small enough to prevent a finite-time blow-up, and such condition is realized as (12). Of course, condition (12) would not be optimal in the sense to guarantee the finite-time blow-up.

    (ⅱ) It is easy to see that satisfies

    (15)

    We multiply (15) by and integrate the resulting relation to obtain

    Then, Grönwall's inequality and the first estimate yield the desired second estimate.

    Before we close this section, we introduce elementary estimates to be used in the following sections.

    Lemma 2.5. Let be a -function satisfying a differential inequality:

    (16)

    Then, the following assertions hold.

    1. If and satisfy

    there exist uniform positive constants and such that

    2. If and satisfy

    there exists a uniform constant such that

    Proof. (ⅰ) By the comparison principle of ODE and method of integrating factor, we have

    Hence, there exist uniform positive constants and such that

    (ⅱ) We multiply (16) with the integrating factor

    to find

    In this section, we present complete aggregation estimates for (2)–(3) in a priori setting. Our first result can be stated as follows.

    Theorem 3.1. Let be a sufficiently smooth solution to . Then, the complete aggregation emerges asymptotically:

    (17)

    Proof. Since the proof is rather lengthy, we introduce a strategy toward the proof. We first claim:

    For this, the key idea is to derive Grönwall's type differential inequality for . To be more specific, we will show that there exist two positive constants and such that

    (18)

    Then, we apply Lemma 2.5 to derive the desired zero convergence for . Finally, the triangle inequality gives the desired result:

    The derivation of (18) will be given in Section 3.2 after some preparatory estimates in Section 3.1.

    In this subsection, we study basic estimates for (2)–(3) that will be used in the derivation of (18). We set

    Then, it follows from Lemma 2.3 that

    (19)

    On the other hand, we integrate (2) with respect to to find the finite-dimensional dynamics for :

    (20)

    Due to (19), it is natural to consider the maximal diameter for the set :

    which is expected to converge to zero under a suitable condition.

    Lemma 3.2. (Existence of a positively invariant set) Let be a solution to satisfying the relation:

    (21)

    Then, one has

    Proof. It follows from (20) that

    (22)

    Then, (22) gives

    (23)

    where . Now, we define a set and its supremum:

    By the assumption on initial data, the set is not empty. We claim:

    Suppose to the contrary, i.e.,

    Then, one has

    (24)

    On the other hand, we observe

    For , let be the extremal indices satisfying

    Hence, (23) yields

    which contradicts (24). Since does not blow up in finite time, one has

    and the set is positively invariant along the flow (20).

    Remark 4. Lemma 3.2 says that if initial data satisfy (21):

    then one has

    Thus, the -integral of is uniformly bounded in time.

    As a direct consequence of Lemma 3.2, we can also show the uniform - boundedness of . For this, we define

    Corollary 1. Let be a solution to with initial data satisfying the relation . Then, the following assertions hold.

    1. The functional decays to zero at least exponentially. In particular, one has

    2. The functional is uniformly bounded in time. Precisely, there exists a uniform positive constant such that

    Proof. (ⅰ) It follows from (20) that

    Then, we find a differential inequality for :

    Lastly, we use initial data (21) together with the above Riccati differential inequality to give the desired result.

    (ⅱ) We multiply with (2) and take real parts for the resulting relation to obtain

    (25)

    Now, we integrate (25) with respect to to find

    (26)

    Below, we present estimates for , respectively.

    Case A.1 (Estimate of ): By integration by parts, we see

    Case A.2 (Estimate of ): It directly follows from Remark 2 that

    Case A.3 (Estimate of ): We use the Cauchy-Schwarz inequality and Corollary 1 to see

    In (26), we collect all the estimates in Case A.1–Case A.3 to derive

    This yields

    (27)

    Since tends to zero exponentially, Lemma 2.5 and differential inequality (27) yield the desired estimate.

    In this subsection, we are ready to provide the proof of Theorem 3.1. First, we claim:

    (28)

    Note that if one verifies (28), then (17) follows from the triangle inequality:

    We consider the difference between and to obtain

    (29)

    Similar to the proof of Corollary 1, we multiply to (29), take real parts and integrate the resulting relation with respect to to obtain

    (30)

    Below, we present estimates of , respectively.

    Case B.1 (Estimates of and ): It follows from the estimates of and in Corollary 1 that

    Case B.2 (Estimate of ): Note that

    In (30), we combine all the estimates Case B.1–Case B.2 to find

    (31)

    If we use Corollary 1 with , then (31) becomes

    (32)

    We sum up (32) with respect to to get

    (33)

    Case C.1 (Estimate of ): we use the Cauchy-Schwarz inequality to see

    (34)

    Case C.2 (Estimate of ): we use Corollary 1 to find

    (35)

    In (33), we combine all the estimates (34) and (35) to derive

    Finally, we use Lemma 2.5 to establish (28). This completes the proof of Theorem 3.1.

    In this section, we show the global existence of a unique mild solution to the Cauchy problem for the WL model (2) following the fixed point approach in [23] where a linear Wigner equation is considered. For this, we define a subset , a norm and a transport operator:

    (36)

    Then, it is easy to check that is a Banach space. In addition, since the transport operator for the linear Wigner equation maps to , it is also useful to define the domain of denoted by :

    For the WL model as a perturbation of the linear Wigner equation, it is strongly believed that is crucial for our analysis (Lemma 4.2). Now, we are ready to provide our second result on the global existence of mild and classical solutions to (2).

    Theorem 4.1. For , the following assertions hold.

    1. If initial data and the potential satisfy

    then there exists a unique mild solution to the Cauchy problem :

    2. If we impose further regularity on initial data and the potential

    then there exists a unique classical solution to the Cauchy problem :

    Proof. Since the proof is rather lengthy, we provide the proofs in Section 4.2 and Section 4.3.

    In this subsection, we follow the same strategy in [23] in which the linear Wigner equation has been treated by means of the semigroup approach. First, we begin with an elementary property of the transport operator in (36) for the WL model.

    Lemma 4.2. Let be a classical solution to . Then, the transport operator satisfies

    In other words, the transport operator for maps to .

    Proof. Since a solution to belongs to in Corollary 1, it suffices to show that

    By straightforward calculations, we observe

    (37)

    Below, we estimate one by one.

    Case C.1 (Estimate of ): we use integration by parts to find

    Hence, we see that vanishes:

    Case C.2 (Estimate of ): since and commute particularly for , we get

    By the skew-Hermitian property of in Remark 1, one has

    Case C.3 (Estimate of ): for the summand in

    In (37), we collect all the estimates in Case C.1–Case C.3 to find

    (38)

    We sum up (38) with respect to and use the Cauchy-Schwarz inequality for the last term to derive

    It follows from Corollary 1 that tends to zero exponentially fast and Lemma 2.5 gives the desired uniform boundedness.

    In this subsection, we show that the Cauchy problem for the WL model admits a unique mild solution.

    First, we rewrite (2) as a matrix form to apply the fixed point theorem. For ,

    (39)

    where is an matrix whose entries are all zero except for -component being 1, is an matrix whose elements in -th row are all one, and is an matrix whose elements in -th column are all one. Here, is understood as an matrix whose -th component is , and the integral in the right-hand side is defined in a similar way. For copies of the function space in (36), we define a set and its norm:

    Then, becomes a Banach space. For the time variable, we use the sup norm. Thus, we consider the Banach space

    equipped with the norm

    Now, we are concerned with the global solvability of (39). Let (to be determined later) and consider a map defined by the following prescription: for each , the function is a solution to the Cauchy problem:

    (40)

    We need to check well-definedness and strict contraction of .

    ● (Well-definedness of ): It suffices to show that for , its image also belongs to . As done in Corollary 1, we multiply with (40), take real parts and integrate the resulting relation with respect to to verify that .

    ● (Strict contraction): for with , our goal is to find a constant such that

    If we set and , then

    For simplicity, we set

    By straightforward calculation,

    (41)

    In addition, we observe

    (42)

    It follows from Corollary 1 that

    Then, (41) and (42) yield

    (43)

    and integrate the relation (43) to find

    Since , we obtain

    If , the map becomes a strict contraction on the closed subset of the (complete) metric space . Hence, has a unique fixed point in for each which gives a unique local solution. Then, it can be globally extended due to uniform estimate or classical way by induction. Precisely, we define by the unique solution to the main equation on with the initial data . Then, we denote

    Hence, becomes the unique global solution to the main equation with the initial data . This completes the proof.

    Next, we are concerned with a global classical solution. In order to apply Theorem A.2(2) in Appendix A for a classical solution to the Cauchy problem (2)–(3), we have to show the continuously differentiability of the coupling term containing .

    For , we introduce an matrix whose -th component is given as

    which is nonlinear with respect to the argument . Since , one can easily verify that maps from to for any . Below, we show that is indeed Lipschitz.

    Lemma 4.3. For , there exists a positive constant that may depend on time such that

    Then, the functional derivative, denoted by , is continuous. Consequently, is Lipschitz from a bounded subset of to .

    Proof. For , we define the Gâteaux derivative (or it is sometimes simply referred as the functional derivative) of at in the direction of that is denoted by :

    At each point , the Gâteaux derivative maps from to . Then by the definition of the Gâteaux derivative, we calculate for and ,

    Since , one finds

    Therefore, we verified that is a bounded linear operator on . Hence, is continuous on . Finally, we recall the Gâteaux mean value theorem in Lemma A.3:

    Here, denotes the operator norm when we regard as a linear operator which maps from to . Since we know that is a bounded linear operator, we find the desired constant:

    This shows that is Lipschitz since and belong to a bounded subset of .

    Now, we are ready to provide the second assertion of Theorem 4.1 by applying semigroup theory.

    Step A (the linear Wigner equation on ): As a first step, we begin with the linear equation in the space instead of :

    (44)

    Since (44) on has been studied in [23], we slightly modify the proof of [23] to show the existence in . In order to use Theorem A.1, we show the term is a a bounded perturbation of the transport operator in . However, when the -norm is considered, Lemma 2.2 gives

    Since is a bounded perturbation of in (see [23,Lemma 1]), we then conclude that is a linear bounded operator defined on . Hence, (44) admits a unique classical solution

    For details, we refer the reader to [23,Theorem 1].

    Step B (the WL model on ): Next, we recall from Lemma 4.2 that is continuously differentiable and then apply Theorem A.2(2) to guarantee that a mild solution obtained from the first assertion of Theorem 4.1 indeed becomes a classical solution to (2)–(3):

    This completes the proof.

    In this paper, we have studied the complete aggregation estimate and the global existence of the Wigner-Lohe(WL) model which describes quantum synchronization in the Wigner picture. By taking the Wigner transform on the Schrödinger-Lohe model with identical potentials, we formally derived the WL model which is an integro-differential equation. Compared to the linear Wigner equation, one of the main difficulty to deal with the WL model lies in the lack of conservation laws. However, fortunately, we can overcome the loss of several conserved quantities via collective dynamics. For the WL model, we first establish complete aggregation estimates that can be achieved with an exponential convergence rate in a priori setting. Next, we show that the WL model admits a unique global mild solution by the standard fixed point theorem and if we impose further regularity on initial data, a unique global classical solution can be obtained by using the semigroup theory. Of course, there are still lots of untouched issues. For instance, we focused on the identical WL model where external one-body potentials are assumed to be the same. Thus, the extension of collective dynamics and global solvability of the WL model with non-identical potentials are left for a future work.

    In this appendix, we briefly summarize several results in [24] on the semigroup theory to show the existence of evolution equations. The first theorem deals with the bounded perturbation of a linear equation.

    Theorem A.1. [24] Let be a Banach space, and let and be operators on such that

    (i) A is the infinitestimal generator of a -semigroup on satisfying

    (ii) is a bounded linear operator on .

    Then, is the infinitestimal generator of a -semigroup on satisfying

    Consider the following abstract Cauchy problem:

    (45)

    In next theorem, we recall the result on the mild and classical solutions of (45).

    Theorem A.2. [24] The following assertions hold.

    1. Let be continuous in on and uniformly Lipschitz continuous (with a Lipschitz constant ) on . If is the infinitestimal generator of a semigroup for on , then for every , the initial value problem has a unique mild solution . Moreover, the mapping is Lipschitz continuous from into .

    2. Let be the infinitestimal generator of a semigroup for on . If is continuously differentiable from into , then the mild solution of with is a classical solution of the initial value problem.

    Finally, we recall Gâteaux's mean value theorem. We denote the directional derivative of at in direction by :

    Lemma A.3. [2,Proposition A.2] Let be a function between Banach spaces and . If is Gâteaux differentiable, then for ,

    where is a bounded linear operator . Here, is the directional derivative of at in direction :


    Acknowledgments



    The authors thank the Department of Mechanical Engineering, College of Engineering, at the University of Texas at Tyler for supporting the investigation of blend PVA/PEO fibers for oral drug delivery.

    Conflict of interest



    The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

    [1] Szunerits S, Boukherroub R (2018) Heat: A highly efficient skin enhancer for transdermal drug delivery. Front Bioeng Biotechnol 6: 15. https://doi.org/10.3389/fbioe.2018.00015
    [2] Alshehri S, Imam SS, Hussain A, et al. (2020) Potential of solid dispersions to enhance solubility, bioavailability, and therapeutic efficacy of poorly water-soluble drugs: newer formulation techniques, current marketed scenario and patents. Drug Deliv 27: 1625-1643. https://doi.org/10.1080/10717544.2020.1846638
    [3] Prajapati SK, Jain A, Jain A, et al. (2019) Biodegradable polymers and constructs: A novel approach in drug delivery. Eur Polym J 120: 109191. https://doi.org/10.1016/j.eurpolymj.2019.08.018
    [4] Waly AL, Abdelghany AM, Tarabiah AE (2021) Study the structure of selenium modified polyethylene oxide/polyvinyl alcohol (PEO/PVA) polymer blend. J Mater Res Technol 14: 2962-2969. https://doi.org/10.1016/j.jmrt.2021.08.078
    [5] Bala R, Khanna S, Pawar P (2014) Design optimization and in vitro - in vivo evaluation of orally dissolving strips of clobazam. J Drug Deliv 2014: 1-15. https://doi.org/10.1155/2014/392783
    [6] Yun YH, Lee BK, Park K (2015) Controlled drug delivery: Historical perspective for the next generation. J Controlled Release 219: 2-7. https://doi.org/10.1016/j.jconrel.2015.10.005
    [7] Chou S-F, Carson D, Woodrow KA (2015) Current strategies for sustaining drug release from electrospun nanofibers. J Controlled Release 220: 584-591. https://doi.org/10.1016/j.jconrel.2015.09.008
    [8] Gizaw M, Thompson J, Faglie A, et al. (2018) Electrospun fibers as a dressing material for drug and biological agent delivery in wound healing applications. Bioengineering 5: 9. https://doi.org/10.3390/bioengineering5010009
    [9] Chou S-F, Woodrow KA (2017) Relationships between mechanical properties and drug release from electrospun fibers of PCL and PLGA blends. J Mech Behav Biomed Mater 65: 724-733. https://doi.org/10.1016/j.jmbbm.2016.09.004
    [10] Ball C, Chou S-F, Jiang Y, et al. (2016) Coaxially electrospun fiber-based microbicides facilitate broadly tunable release of maraviroc. Mater Sci Eng C 63: 117-124. https://doi.org/10.1016/j.msec.2016.02.018
    [11] Bagheri M, Validi M, Gholipour A, et al. (2022) Chitosan nanofiber biocomposites for potential wound healing applications: Antioxidant activity with synergic antibacterial effect. Bioeng Transl Med 7: e10254. https://doi.org/10.1002/btm2.10254
    [12] Moroni I, Garcia-Bennett AE (2021) Effects of absorption kinetics on the catabolism of melatonin released from CAP-coated mesoporous silica drug delivery vehicles. Pharmaceutics 13: 1436. https://doi.org/10.3390/pharmaceutics13091436
    [13] Al-Zaqri N, Pooventhiran T, Alsalme A, et al. (2020) Structural and physico-chemical evaluation of melatonin and its solution-state excited properties, with emphasis on its binding with novel coronavirus proteins. J Mol Liq 318: 114082. https://doi.org/10.1016/j.molliq.2020.114082
    [14] Hawkins BC, Burnett E, Chou S-F (2022) Physicomechanical properties and in vitro release behaviors of electrospun ibuprofen-loaded blend PEO/EC fibers. Mater Today Commun 30: 103205. https://doi.org/10.1016/j.mtcomm.2022.103205
    [15] ASTM D1708-18Standard test method for tensile properties of plastics by use of microtensile specimens, West Conshohocken, PA, ASTM International (2018).
    [16] ASTM D5034-21Standard test method for breaking strength and elongation of textile fabrics (grab test), West Conshohocken, PA, ASTM International (2021).
    [17] Hekmati AH, Khenoussi N, Nouali H, et al. (2014) Effect of nanofiber diameter on water absorption properties and pore size of polyamide-6 electrospun nanoweb. Text Res J 84: 2045-2055. https://doi.org/10.1177/0040517514532160
    [18] Daescu M, Toulbe N, Baibarac M, et al. (2020) Photoluminescence as a complementary tool for UV-VIS spectroscopy to highlight the photodegradation of drugs: A case study on melatonin. Molecules 25: 3820. https://doi.org/10.3390/molecules25173820
    [19] Haro Durand L, Vargas G, Vera-Mesones R, et al. (2017) In vitro human umbilical vein endothelial cells response to ionic dissolution products from lithium-containing 45S5 bioactive glass. Materials 10: 740. https://doi.org/10.3390/ma10070740
    [20] Amariei N, Manea LR, Bertea AP, et al. (2017) The influence of polymer solution on the properties of electrospun 3D nanostructures. IOP Conf Ser Mater Sci Eng 209: 012092. https://doi.org/10.1088/1757-899X/209/1/012092
    [21] Datta R, Yelash L, Schmid F, et al. (2021) Shear-thinning in oligomer melts—Molecular origins and applications. Polymers 13: 2806. https://doi.org/10.3390/polym13162806
    [22] Mirtič J, Balažic H, Zupančič Š, et al. (2019) Effect of solution composition variables on electrospun alginate nanofibers: Response surface analysis. Polymers 11: 692. https://doi.org/10.3390/polym11040692
    [23] Briscoe B, Luckham P, Zhu S (2000) The effects of hydrogen bonding upon the viscosity of aqueous poly(vinyl alcohol) solutions. Polymer 41: 3851-3860. https://doi.org/10.1016/S0032-3861(99)00550-9
    [24] Vlachou M, Siamidi A, Anagnostopoulou D, et al. (2022) Modified release of the pineal hormone melatonin from matrix tablets containing poly(L-lactic acid) and its PLA-co-PEAd and PLA-co-PBAd copolymers. Polymers 14: 1504. https://doi.org/10.3390/polym14081504
    [25] Angel N, Li S, Yan F, et al. (2022) Recent advances in electrospinning of nanofibers from bio-based carbohydrate polymers and their applications. Trends Food Sci Technol 120: 308-324. https://doi.org/10.1016/j.tifs.2022.01.003
    [26] Son WK, Youk JH, Lee TS, et al. (2004) The effects of solution properties and polyelectrolyte on electrospinning of ultrafine poly(ethylene oxide) fibers. Polymer 45: 2959-2966. https://doi.org/10.1016/j.polymer.2004.03.006
    [27] Filip P, Peer P (2019) Characterization of poly(ethylene oxide) nanofibers—Mutual relations between mean diameter of electrospun nanofibers and solution characteristics. Processes 7: 948. https://doi.org/10.3390/pr7120948
    [28] Ziyadi H, Baghali M, Bagherianfar M, et al. (2021) An investigation of factors affecting the electrospinning of poly (vinyl alcohol)/kefiran composite nanofibers. Adv Compos Hybrid Mater 4: 768-779. https://doi.org/10.1007/s42114-021-00230-3
    [29] Mwiiri FK, Daniels R (2020) Influence of PVA molecular weight and concentration on electrospinnability of birch bark extract-loaded nanofibrous scaffolds intended for enhanced wound healing. Molecules 25: 4799. https://doi.org/10.3390/molecules25204799
    [30] Cho D, Netravali AN, Joo YL (2012) Mechanical properties and biodegradability of electrospun soy protein isolate/PVA hybrid nanofibers. Polym Degrad Stab 97: 747-754. https://doi.org/10.1016/j.polymdegradstab.2012.02.007
    [31] Amjadi M, Fatemi A (2020) Tensile behavior of high-density polyethylene including the effects of processing technique, thickness, temperature, and strain rate. Polymers 12: 1857. https://doi.org/10.3390/polym12091857
    [32] Séguéla R (2007) Plasticity of semi-crystalline polymers: crystal slip versus melting-recrystallization. E-Polym 7: 32. https://doi.org/10.1515/epoly.2007.7.1.382
    [33] Devangamath SS, Lobo B, Masti SP, et al. (2020) Thermal, mechanical, and AC electrical studies of PVA–PEG–Ag2S polymer hybrid material. J Mater Sci Mater Electron 31: 2904-2917. https://doi.org/10.1007/s10854-019-02835-3
    [34] Rashid TU, Gorga RE, Krause WE (2021) Mechanical properties of electrospun fibers—A critical review. Adv Eng Mater 23: 2100153. https://doi.org/10.1002/adem.202100153
    [35] Morel A, Domaschke S, Urundolil Kumaran V, et al. (2018) Correlating diameter, mechanical and structural properties of poly(l-lactide) fibres from needleless electrospinning. Acta Biomater 81: 169-183. https://doi.org/10.1016/j.actbio.2018.09.055
    [36] Parab RS, Rao GK (2019) Understanding the mechanical properties of polymer blends in the presence of plasticizers and other additives. Int J Pharm Sci Rev Res 54: 84-91.
    [37] Croisier F, Duwez A-S, Jérôme C, et al. (2012) Mechanical testing of electrospun PCL fibers. Acta Biomater 8: 218-224. https://doi.org/10.1016/j.actbio.2011.08.015
    [38] Ghasemi M, Singapati AY, Tsianou M, et al. (2017) Dissolution of semicrystalline polymer fibers: Numerical modeling and parametric analysis. AIChE J 63: 1368-1383. https://doi.org/10.1002/aic.15615
    [39] Hirsch E, Pantea E, Vass P, et al. (2021) Probiotic bacteria stabilized in orally dissolving nanofibers prepared by high-speed electrospinning. Food Bioprod Process 128: 84-94. https://doi.org/10.1016/j.fbp.2021.04.016
    [40] Zhang J, Yan X, Tian Y, et al. (2020) Synthesis of a new water-soluble melatonin derivative with low toxicity and a strong effect on sleep aid. ACS Omega 5: 6494-6499. https://doi.org/10.1021/acsomega.9b04120
    [41] Khan MQ, Kharaghani D, Nishat N, et al. (2019) Preparation and characterizations of multifunctional PVA/ZnO nanofibers composite membranes for surgical gown application. J Mater Res Technol 8: 1328-1334. https://doi.org/10.1016/j.jmrt.2018.08.013
    [42] Song X, Gao Z, Ling F, et al. (2012) Controlled release of drug via tuning electrospun polymer carrier. J Polym Sci Part B Polym Phys 50: 221-227. https://doi.org/10.1002/polb.23005
    [43] Mihailiasa M, Caldera F, Li J, et al. (2016) Preparation of functionalized cotton fabrics by means of melatonin loaded β-cyclodextrin nanosponges. Carbohydr Polym 142: 24-30. https://doi.org/10.1016/j.carbpol.2016.01.024
    [44] Gulino EF, Citarrella MC, Maio A, et al. (2022) An innovative route to prepare in situ graded crosslinked PVA graphene electrospun mats for drug release. Compos Part Appl Sci Manuf 155: 106827. https://doi.org/10.1016/j.compositesa.2022.106827
    [45] Torres-Martínez EJ, Vera-Graziano R, Cervantes-Uc JM, et al. (2020) Preparation and characterization of electrospun fibrous scaffolds of either PVA or PVP for fast release of sildenafil citrate. E-Polym 20: 746-758. https://doi.org/10.1515/epoly-2020-0070
    [46] Carvalho LD de, Peres BU, Maezono H, et al. (2019) Doxycycline release and antibacterial activity from PMMA/PEO electrospun fiber mats. J Appl Oral Sci 27: e20180663. https://doi.org/10.1590/1678-7757-2018-0663
    [47] Eskitoros-Togay ŞM, Bulbul YE, Tort S, et al. (2019) Fabrication of doxycycline-loaded electrospun PCL/PEO membranes for a potential drug delivery system. Int J Pharm 565: 83-94. https://doi.org/10.1016/j.ijpharm.2019.04.073
    [48] Yu D-G, Shen X-X, Branford-White C, et al. (2009) Oral fast-dissolving drug delivery membranes prepared from electrospun polyvinylpyrrolidone ultrafine fibers. Nanotechnology 20: 055104. https://doi.org/10.1088/0957-4484/20/5/055104
    [49] Samprasit W, Akkaramongkolporn P, Ngawhirunpat T, et al. (2015) Fast releasing oral electrospun PVP/CD nanofiber mats of taste-masked meloxicam. Int J Pharm 487: 213-222. https://doi.org/10.1016/j.ijpharm.2015.04.044
    [50] Huang C-Y, Hu K-H, Wei Z-H (2016) Comparison of cell behavior on pva/pva-gelatin electrospun nanofibers with random and aligned configuration. Sci Rep 6: 37960. https://doi.org/10.1038/srep37960
    [51] Yang JM, Yang JH, Tsou SC, et al. (2016) Cell proliferation on PVA/sodium alginate and PVA/poly(γ-glutamic acid) electrospun fiber. Mater Sci Eng C 66: 170-177. https://doi.org/10.1016/j.msec.2016.04.068
    [52] Carrasco-Torres G, Valdés-Madrigal M, Vásquez-Garzón V, et al. (2019) Effect of silk fibroin on cell viability in electrospun scaffolds of polyethylene oxide. Polymers 11: 451. https://doi.org/10.3390/polym11030451
    [53] Cerqueira A, Romero-Gavilán F, Araújo-Gomes N, et al. (2020) A possible use of melatonin in the dental field: Protein adsorption and in vitro cell response on coated titanium. Mater Sci Eng C 116: 111262. https://doi.org/10.1016/j.msec.2020.111262
    [54] Cheng J, Yang H, Gu C, et al. (2018) Melatonin restricts the viability and angiogenesis of vascular endothelial cells by suppressing HIF-1α/ROS/VEGF. Int J Mol Med 43: 945-955. https://doi.org/10.3892/ijmm.2018.4021
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