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A study on the spatial and temporal distribution of habitation sites in the Amur River Basin and its relationship with geographical environments

  • Received: 20 November 2023 Revised: 27 January 2024 Accepted: 27 February 2024 Published: 12 March 2024
  • I achieved the goal of collection and storage of 1944 habitation sites in the Amur River basin by creating the ARCGIS geo-spatial database. On this basis, by visualizing the distribution of the habitation sites in six archaeological periods and four geography plates, the spatial and temporal distribution information of these sites were interpreted quantitatively. Specifically, the quantitative indicators, such as nuclear density, location of gravity center, and cultural inheritance index, revealed the spatial distribution of the habitation sites. In addition, the elevation, slope, and distance value from the river were extracted through spatial analysis of ArcGIS, revealing the characteristics of the combination of sites and geographical conditions. The analysis of vegetation and temperature-climate conditions of the Amur River basin suggests the coupling relationship between ancient settlements and the natural environment: Arable land with an average annual temperature above 4 ℃ is the area with the highest concentration of large settlements, while other woodlands, grasslands and swamps with lower average annual temperatures are the main distribution areas of small nomadic sites.

    Citation: Wenqing Liu. A study on the spatial and temporal distribution of habitation sites in the Amur River Basin and its relationship with geographical environments[J]. AIMS Geosciences, 2024, 10(1): 172-195. doi: 10.3934/geosci.2024010

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  • I achieved the goal of collection and storage of 1944 habitation sites in the Amur River basin by creating the ARCGIS geo-spatial database. On this basis, by visualizing the distribution of the habitation sites in six archaeological periods and four geography plates, the spatial and temporal distribution information of these sites were interpreted quantitatively. Specifically, the quantitative indicators, such as nuclear density, location of gravity center, and cultural inheritance index, revealed the spatial distribution of the habitation sites. In addition, the elevation, slope, and distance value from the river were extracted through spatial analysis of ArcGIS, revealing the characteristics of the combination of sites and geographical conditions. The analysis of vegetation and temperature-climate conditions of the Amur River basin suggests the coupling relationship between ancient settlements and the natural environment: Arable land with an average annual temperature above 4 ℃ is the area with the highest concentration of large settlements, while other woodlands, grasslands and swamps with lower average annual temperatures are the main distribution areas of small nomadic sites.



    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)|eκt,w1(t)w2(t)2L2eκt,ast,

    where z12 is defined in (10).

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

    ˙z12=κ2(1z212),t>0.

    This can be integrated explicitly:

    z12(t)=(1+z012)eκt(1z012)(1+z012)eκt+(1z012),t>0, (13)

    where z012=z12(0). If we assume that

    z012R,z012<1orz012>1, (14)

    then the denominator of the right-hand side of (13) can be zero, and hence z12(t) can blow up in finite-time. Precisely, there exists 0<T< such that

    limtTz12(t)=,T=1κln1z0121+z012.

    In other words, for initial data satisfying (14), z12(t) would not be bounded. Thus, Rez012 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 w1w2 satisfies

    t(w1w2)+px(w1w2)+Θ[V](w1w2)=κR122(w1w2). (15)

    We multiply (15) by 2(w1w2) and integrate the resulting relation to obtain

    ddtw1(t)w2(t)2L2=κR12(t)w1(t)w2(t)2L2.

    Then, Grönwall's inequality and the first estimate |1z12(t)|eκt 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 y:R+R+ be a C1-function satisfying a differential inequality:

    yα1eβ1ty+α2eβ2t,t>0. (16)

    Then, the following assertions hold.

    1. If αi and βi satisfy

    α1<0,β1=0,α2>0,β2>0,

    there exist uniform positive constants C0 and D0>0 such that

    y(t)C0eD0t,t0.

    2. If αi and βi satisfy

    α1>0,β1>0,α2=0,

    there exists a uniform constant C1 such that

    y(t)C1y0,t0.

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

    y(t)(y0+α2α1+β2)eα1tα2α1+β2eβ2t,t0.

    Hence, there exist uniform positive constants C0 and D0>0 such that

    y(t)C0eD0t,t0.

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

    exp(t0α1eβ1sds)=exp(α1β1(1eβ1t))

    to find

    y(t)y0eα1β1(1eβ1t)eα1β1y0=:C1y0,t0.

    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 wij be a sufficiently smooth solution to (2)(3). Then, the complete aggregation emerges asymptotically:

    limtwikwjm=0,i,j,k,m[N]. (17)

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

    limtwikwjk=0,ki,j[N].

    For this, the key idea is to derive Grönwall's type differential inequality for Nk=1wikwjk2. To be more specific, we will show that there exist two positive constants C1 and C2 such that

    ddtNk=1wikwjk2κ(1C1eκt)Nk=1wikwjk2+C2eκt,t>0. (18)

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

    wikwjmwikwjk+wjkwjm=wikwjk+wkjwmj.

    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

    zij(t):=R2dwij(t,x,p)dxdp,i,j[N],t>0.

    Then, it follows from Lemma 2.3 that

    zi:=zii=1,i[N]. (19)

    On the other hand, we integrate (2) with respect to (x,p)R2d to find the finite-dimensional dynamics for zij:

    dzijdt=κ2NNk=1(zik+zkj)(1zij),t>0. (20)

    Due to (19), it is natural to consider the maximal diameter for the set {1zij}:

    D(Z(t)):=max1i,jN|1zij(t)|,t>0,D(Z0):=D(Z(0)),

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

    Lemma 3.2. (Existence of a positively invariant set) Let {wij} be a solution to (2)(3) satisfying the relation:

    D(Z0)<1. (21)

    Then, one has

    D(Z(t))<1,t>0.

    Proof. It follows from (20) that

    ddt(1zij)=κ2NNk=1(zik+zkj)(1zij),t>0. (22)

    Then, (22) gives

    |1zij(t)|=|1z0ij|exp[κ2NNk=1t0(Rik+Rkj)ds],t>0, (23)

    where Rij:=Re(zij). Now, we define a set and its supremum:

    T:={T(0,):D(Z(t))<1,t[0,T)},T:=supT.

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

    T=.

    Suppose to the contrary, i.e.,

    T<.

    Then, one has

    limtTD(Z(t))=1. (24)

    On the other hand, we observe

    maxij|1zij(t)|<1minijRij(t)>0,t[0,T).

    For t>0, let (it,jt) be the extremal indices satisfying

    D(Z(t))=1zitjt.

    Hence, (23) yields

    1=D(Z(T))=D(Z0)exp[κ2NNk=1T0(Ritk+Rkjt)ds]<1,

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

    T=,

    and the set {D(Z(t))<1} is positively invariant along the flow (20).

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

    |R2dw0ijdxdp1|<1,i,j[N],

    then one has

    |R2dwij(t,x,p)dxdp1|<1,i,j[N],t>0.

    Thus, the (x,p)-integral of wij is uniformly bounded in time.

    As a direct consequence of Lemma 3.2, we can also show the uniform L2- boundedness of wij. For this, we define

    R(W(t)):=max1i,jNwij(t),t>0.

    Corollary 1. Let wij be a solution to (2)(3) with initial data satisfying the relation (21). Then, the following assertions hold.

    1. The functional D(Z(t)) decays to zero at least exponentially. In particular, one has

    D(Z(t))D(Z0)eκtD(Z0)eκt+1D(Z0)D(Z0)1D(Z0)eκt,t>0.

    2. The functional R(W(t)) is uniformly bounded in time. Precisely, there exists a uniform positive constant R such that

    sup0t<R(W(t))R.

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

    ddt(1zij)=κ(1zij)+κ2NNk=1(1zik+1zkj)(1zij).

    Then, we find a differential inequality for D(Z):

    ddtD(Z)κD(Z)+κD(Z)2,t>0.

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

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

    12t|wij|2+12px|wij|2+Re[Θ[V](wij)¯wij]=κ2NNk=1Re[(wik+wkj(zik+zkj)wij)¯wij]. (25)

    Now, we integrate (25) with respect to (x,p)R2d to find

    ddtR2d|wij|2dxdp=R2dpx|wij|2dxdpR2dRe[¯wijΘ[V](wij)]dxdp+κNNk=1R2dRe[¯wij(wik+wkj(zik+zkj)wij)]dxdp=:I11+I12+I13. (26)

    Below, we present estimates for I1k,k=1,2,3, respectively.

    Case A.1 (Estimate of I11): By integration by parts, we see

    I11=R2dpx|wij|2dxdp=0.

    Case A.2 (Estimate of I12): It directly follows from Remark 2 that

    I12=0.

    Case A.3 (Estimate of I13): We use the Cauchy-Schwarz inequality and Corollary 1 to see

    R2d(¯wijwik+¯wijwkj(zik+zkj)|wij|2)dxdp=2wij2+R2d(¯wijwik+¯wijwkj)dxdp+(1zik+1zkj)wij22wij2+2R(W)2+2D(Z)R(W)2.

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

    ddtwij22κwij2+2κR(W)2+2κD(Z)R(W)2,t>0.

    This yields

    ddtR(W)22κD(Z)R(W)2,t>0. (27)

    Since D(Z) 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:

    limtwikwjk=0,ki,j. (28)

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

    wikwjmwikwjk+wjkwjm=wikwjk+wkjwmj.

    We consider the difference between wik and wjk to obtain

    t(wikwjk)+px(wikwjk)+Θ[V](wikwjk)κ2NN=1[(wiwj)(ziwikzjwjk)zk(wikwjk)]=0. (29)

    Similar to the proof of Corollary 1, we multiply ¯wikwjk to (29), take real parts and integrate the resulting relation with respect to (x,p)R2d to obtain

    ddtwikwjk2=:R2dpx|wikwjk|2dxdpR2d2Re[(¯wik¯wjk)Θ[V](wikwjk)]dxdp+κNN=1R2dRe(Jijk)dxdp=:I21+I22+I23. (30)

    Below, we present estimates of I2k,k=1,2,3, respectively.

    Case B.1 (Estimates of I21 and I22): It follows from the estimates of I11 and I12 in Corollary 1 that

    I21=I22=0.

    Case B.2 (Estimate of I23): Note that

    Jijk=(¯wik¯wjk)(wiwj)zi|wikwjk|2+(zizj)wjk(¯wik¯wjk)zk|wikwjk|2=(¯wik¯wjk)(wiwj)(zi+zk)|wikwjk|2+(zizj)wjk(¯wik¯wjk).

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

    ddtwikwjk2κNN=1R2d(|wikwjk||wiwj|Re(zi+zk)|wikwjk|2+|zizj||wjk||wikwjk|)dxdpκNN=1(wikwjkwiwjRe(zi+zk)wikwjk2+|zizj|wjkwikwjk). (31)

    If we use Corollary 1 with α:=D(Z0)1D(Z0), then (31) becomes

    ddtwikwjk22κ(1αeκt)wikwjk2+κNN=1wikwjkwiwj+κRNN=1|zizj|wikwjk. (32)

    We sum up (32) with respect to k[N] to get

    ddtNk=1wikwjk22κ(1αeκt)Nk=1wikwjk2+κNNk,=1wikwjkwiwj+κRNN=1|zizj|Nk=1wikwjk=:2κ(1αeκt)Nk=1wikwjk2+I31+I32. (33)

    Case C.1 (Estimate of I31): we use the Cauchy-Schwarz inequality to see

    I31=κNNk,=1wikwjkwiwj=κN(Nk=1wikwjk)2κNk=1wikwjk2. (34)

    Case C.2 (Estimate of I32): we use Corollary 1 to find

    I32=κRNN=1|zi1+1zj|Nk=1wikwjk4Nκ|R|2αeκt. (35)

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

    ddtNk=1wikwjk2κ(12αeκt)Nk=1wikwjk2+4Nκ|R|2αeκt.

    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 X, a norm and a transport operator:

    X:={fL2(R2d):|R2dfdxdp|<},fX:=f+|fdxdp|,A:=px. (36)

    Then, it is easy to check that (X,X) is a Banach space. In addition, since the transport operator A for the linear Wigner equation maps L2(R2d) to L2(R2d), it is also useful to define the domain of A denoted by D(A):

    D(A):={fX:pxfL2(R2d)}X.

    For the WL model as a perturbation of the linear Wigner equation, it is strongly believed that D(A) 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 T(0,), the following assertions hold.

    1. If initial data and the potential satisfy

    w0ijX,i,j[N],andVL(Rd),

    then there exists a unique mild solution to the Cauchy problem (2)(3):

    wijC([0,T];X),i,j[N].

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

    w0ijD(A),i,j[N],andVL(Rd)L2(Rd),

    then there exists a unique classical solution to the Cauchy problem (2)(3):

    wijC([0,T];X)C1([0,T];D(A)),i,j[N].

    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 A in (36) for the WL model.

    Lemma 4.2. Let wij be a classical solution to (2)(3). Then, the transport operator A=px satisfies

    AwijL2(R2d).

    In other words, the transport operator A for (2) maps L2(R2d) to L2(R2d).

    Proof. Since a solution {wij} to (2)(3) belongs to L2(R2d) in Corollary 1, it suffices to show that

    sup0t<pxwij<,i,j[N].

    By straightforward calculations, we observe

    12ddtpxwij2=Repxtwij,pxwij=Repx(pxwij),pxwijpx(Θwij),pxwij+κ2NNk=1Repx(wik+wkj(zik+zkj)wij),pxwij=:I41+I42+I43. (37)

    Below, we estimate I4k,k=1,2,3 one by one.

    Case C.1 (Estimate of I41): we use integration by parts to find

    px(pxwij),pxwij=Nj=1pjj(pxwij),pxwij=pxwij,Nj=1pjj(pxwij)=¯px(pxwij),pxwij.

    Hence, we see that I41 vanishes:

    I41=0.

    Case C.2 (Estimate of I42): since Θ[V] and A commute particularly for wij, we get

    px(Θwij),pxwij=Θ[V](pxwij),pxwij.

    By the skew-Hermitian property of Θ[V] in Remark 1, one has

    I42=0.

    Case C.3 (Estimate of I43): for the summand in I43

    px(wik+wkj(zik+zkj)wij),pxwij=px(wik+wkj),pxwij(zik+zkj)pxwij2=2pxwij2+(1zik+1zkj)pxwij2+px(wik+wkj),pxwij2pxwij2+|1zik+1zkj|pxwij2+pxwij(pxwik+pxwkj).

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

    12ddtpxwij2κpxwij2+κD(Z)pxwij2+κ2NNk=1pxwij(pxwik+pxwkj). (38)

    We sum up (38) with respect to i,j[N] and use the Cauchy-Schwarz inequality for the last term to derive

    12ddtNi,j=1pxwij2κD(Z)Ni,j=1pxwij2.

    It follows from Corollary 1 that D(Z) 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 W=(wij)(i,j),

    tW+pxW+Θ[V](W)=κ2N(EijWCj+RiWEijWR2d(EijWCj+RiWEij)dxdp), (39)

    where Eij is an N×N matrix whose entries are all zero except for (i,j)-component being 1, Ri is an N×N matrix whose elements in i-th row are all one, and Cj is an N×N matrix whose elements in j-th column are all one. Here, Θ[V](W) is understood as an N×N matrix whose (i,j)-th component is Θ[V](wij), and the integral in the right-hand side is defined in a similar way. For N2 copies of the function space X in (36), we define a set and its norm:

    {X:={F=(Fij)(L2(R2d))N2:|R2dFijdxdp|<,i,j[N]},FX:=FL2(R2d)N2+|R2dFdxdp|:=maxi,j(Fij+|R2dFijdxdp|).

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

    C([0,T];X)

    equipped with the norm

    |||F|||:=sup0tTF(t)X.

    Now, we are concerned with the global solvability of (39). Let T>0 (to be determined later) and consider a map T:XX defined by the following prescription: for each GX, the function W=TG is a solution to the Cauchy problem:

    {tW+pxW+Θ[V](W)=κ2N(EijWCj+RiWEijWR2d(EijGCj+RiGEij)dxdp),W(0)=W0. (40)

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

    ● (Well-definedness of T): It suffices to show that for GX, its image W=TG also belongs to X. As done in Corollary 1, we multiply ¯W with (40), take real parts and integrate the resulting relation with respect to (x,p)R2d to verify that WX.

    ● (Strict contraction): for ¯GX with ¯W=T¯G, our goal is to find a constant C(0,1) such that

    |||W¯W|||C|||G¯G|||.

    If we set ¯W=(¯wij) and ¯G=(¯gij), then

    twij+pxwij+Θ[V](ωij)=κ2NNk=1[wik+wkjwij(R2d(gik+gkj)dxdp)],t¯wij+px¯wij+Θ[V](¯ωij)=κ2NNk=1[¯wik+¯wkj¯wij(R2d(¯gik+¯gkj)dxdp)].

    For simplicity, we set

    qij(t):=R2dgijdxdp,¯qij(t):=R2d¯gijdxdp.

    By straightforward calculation,

    12ddtmax1i,jNwij¯wijCmax1i,jNwij¯wij+max1i,jNωijmax1i,jN|qij¯qij|. (41)

    In addition, we observe

    ddtmax1i,jN|zij¯zij|Cmax1i,jN|zij¯zij|+˜Cmax1i,jN|qij¯qij|. (42)

    It follows from Corollary 1 that

    max1i,jNwij(t)R,t>0.

    Then, (41) and (42) yield

    ddtW¯WXC1W¯WX+eC2tG¯GX, (43)

    and integrate the relation (43) to find

    W¯WXW0¯W0X+C1t0W¯WXds+eC2Tt0G¯GXds.

    Since W0¯W0=0, we obtain

    |||W¯W|||=sup0tTW¯WXTe(C1+C2)Tsup0tTG¯GX=Te(C1+C2)T|||G¯G|||.

    If Te(C1+C2)T<1, the map T becomes a strict contraction on the closed subset of the (complete) metric space X. Hence, T has a unique fixed point in X for each W0X which gives a unique local solution. Then, it can be globally extended due to uniform L2 estimate or classical way by induction. Precisely, we define Wn by the unique solution to the main equation on [0,T] with the initial data Wn+1(0)=Wn(T). Then, we denote

    W(t)=Wn(tnT),t[nT,(n+1)T],n0.

    Hence, W becomes the unique global solution to the main equation with the initial data W0. 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 κ.

    For W=(wij)(i,j), we introduce an N×N matrix F(W) whose (i,j)-th component is given as

    (F(W))(i,j):=κ2NNk=1(zik+zkj)wij=κ2NNk=1(R2d(wik+wkj)dxdp)wij,

    which is nonlinear with respect to the argument W. Since WX, one can easily verify that F maps from [0,T]×X to X for any T>0. Below, we show that F is indeed Lipschitz.

    Lemma 4.3. For U,VX, there exists a positive constant C>0 that may depend on time T such that

    F(U)F(V)XCUVX.

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

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

    DF(U)(V):=limτ0F(U+τV)F(U)τ=ddτF(U+τV)|τ=0.

    At each point UX, the Gâteaux derivative maps from X to X. Then by the definition of the Gâteaux derivative, we calculate for U=(uij) and V=(vij),

    (DF(U)(V))(i,j)=κ2NNk=1(uijR2d(vik+vkj)dxdp+vijR2d(uik+ukj)dxdp).

    Since U,VX, one finds

    DF(U)(V)X2κUXVX.

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

    F(U)F(V)XUVXsupt[0,1]DF(tU+(1t)V)op.

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

    F(U)F(V)X2κsupt[0,1]tU+(1t)VXUVX.

    This shows that F is Lipschitz since U and V belong to a bounded subset of X.

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

    Step A (the linear Wigner equation on X): As a first step, we begin with the linear equation in the space X instead of L2(R2d):

    twij+pxwij+Θ[V](wij)=0. (44)

    Since (44) on L2(R2d) has been studied in [23], we slightly modify the proof of [23] to show the existence in X. In order to use Theorem A.1, we show the term Θ[V] is a a bounded perturbation of the transport operator A=px in X. However, when the X-norm is considered, Lemma 2.2 gives

    Θ[V]wX=Θ[V]w.

    Since Θ[V] is a bounded perturbation of A in L2(R2d) (see [23,Lemma 1]), we then conclude that Θ[V] is a linear bounded operator defined on X. Hence, (44) admits a unique classical solution

    wijC([0,T];X)C1([0,T];D(A)).

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

    Step B (the WL model on D(A)): Next, we recall from Lemma 4.2 that F 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):

    wijC([0,T];X)C1([0,T];D(A)),i,j[N].

    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 X be a Banach space, and let A and B be operators on X such that

    (i) A is the infinitestimal generator of a C0-semigroup T(t) on X satisfying

    T(t)Meωt.

    (ii) B is a bounded linear operator on X.

    Then, A+B is the infinitestimal generator of a C0-semigroup S(t) on X satisfying

    S(t)Me(ω+MB)t.

    Consider the following abstract Cauchy problem:

    {du(t)dt+Au(t)=f(t,u(t)),t>t0,u(t0)=u0. (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 f:[t0,T]×XX be continuous in t on [t0,T] and uniformly Lipschitz continuous (with a Lipschitz constant L) on X. If A is the infinitestimal generator of a C0 semigroup T(t) for t0 on X, then for every u0X, the initial value problem (45) has a unique mild solution uC([t0,T];X). Moreover, the mapping u0u is Lipschitz continuous from X into C([t0,T];X).

    2. Let A be the infinitestimal generator of a C0 semigroup T(t) for t0 on X. If f:[t0,T]×XX is continuously differentiable from [t0,T]×X into X, then the mild solution of (45) with u0D(A) is a classical solution of the initial value problem.

    Finally, we recall Gâteaux's mean value theorem. We denote the directional derivative of f at x in direction v by δvf(x):

    Lemma A.3. [2,Proposition A.2] Let f:XY be a function between Banach spaces X and Y. If f is Gâteaux differentiable, then for x,yX,

    f(y)f(x)YxyXsup0θ1Df(θx+(1θ)yL(X,Y),

    where Df(x) is a bounded linear operator Df(x):vδvf(x). Here, δvf(x) is the directional derivative of f at x in direction v:

    δvf(x):=limtf(x+tv)f(x)t.


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