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
{i∂tψj=−12Δψj+Vjψj+iκ2NN∑k=1(ψk−⟨ψj,ψk⟩⟨ψj,ψj⟩ψj),t>0,x∈Rd,ψj(0,x)=ψ0j(x),j∈[N]:={1,⋯,N}, | (1) |
where
Note that the Planck constant is assumed to be unity for simplicity. Like the classical Schrödinger equation, system (1) satisfies
In this paper, we study the emergent dynamics of the Cauchy problem to the WL model with identical potentials:
{∂twij+p⋅∇xwij+Θ[V](wij)=κ2NN∑k=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,|∫R2dw0ijdxdp−1|<1,i≠j∈[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.
limt→∞‖wij−wℓm‖L2(R2d)=0,i,j,ℓ,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
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
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
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
⟨f,g⟩:=∫R2df(x,p)¯g(x,p)dxdp,‖f‖:=√⟨f,f⟩, |
where
(Fϕ)(p):=∫Rdϕ(x)e−ix⋅pdx,(F−1ϕ)(x):=1(2π)d∫Rdϕ(p)eix⋅pdp. |
For a given real-valued function
(Fy→pψ)(x,p):=∫Rdψ(x,y)e−iy⋅pdy. |
In this section, we introduce the
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
w[ψ,ϕ](x,p):=1(2π)d∫Rdψ(x+y2)¯ϕ(x−y2)eip⋅ydy,(x,p)∈R2d, | (4) |
where
2. For
Θ[V](w)(x,p):=−i(2π)d∫Rd[V(x+y2)−V(x−y2)](Fp′→yw)(x,y)eip⋅ydy=−i(2π)d∫R2d[V(x+y2)−V(x−y2)]w(x,p′)ei(p−p′)⋅ydp′dy. |
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)d∫Rdϕ(x+y2)¯ψ(x−y2)eip⋅ydy=1(2π)d∫Rd¯ψ(x+y2)ϕ(x−y2)e−ip⋅ydybyy↔−y=¯1(2π)d∫Rdψ(x+y2)¯ϕ(x−y2)eip⋅ydy=¯w[ψ,ϕ](x,p). | (5) |
2. For the case
w[ψ,ψ](x,p)=1(2π)d∫Rdψ(x+y2)¯ψ(x−y2)e−ip⋅ydy. |
Since
w[ψ,ψ]=:w[ψ]. |
Moreover, one can easily verify that
3. The
∫Rdw[ψ](x,p)dp=|ψ(x)|2. |
Moreover, the
∫R2dw[ψ,ϕ](x,p)dxdp=1(2π)d∫R3dϕ(x+y2)¯ψ(x−y2)eip⋅ydydxdp=∫Rdϕ(x)¯ψ(x)dx=⟨ϕ,ψ⟩. |
4. Since
¯Θ[V](w)(x,p)=i(2π)d∫R2d[V(x+y2)−V(x−y2)]ˉw(x,p′)e−i(p−p′)⋅ydp′dy=−i(2π)d∫R2d[V(x−y2)−V(x+y2)]ˉw(x,p′)e−i(p−p′)⋅ydp′dy=−i(2π)d∫R2d[V(x+y2)−V(x−y2)]ˉw(x,p′)ei(p−p′)⋅ydp′dybyy↔−y=Θ[V](¯w)(x,p). |
In the following lemma, we provide several properties of
Lemma 2.2. For
(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)d∫R4d[V(x+y2)−V(x−y2)]f(x,p′)g(x,p)ei(p−p′)⋅ydp′dydxdp=−i(2π)d∫R4d[V(x−y2)−V(x+y2)]f(x,p)g(x,p′)ei(p−p′)⋅ydp′dydxdp=−∫R2dΘ[V](g)⋅fdxdp. |
(ⅱ) By the definition of Definition 2.1 and Fubini's theorem, we have
∫R2dΘ[V](f)dxdp=−i(2π)d∫R4d[V(x+y2)−V(x−y2)]f(x,p′)ei(p−p′)⋅ydp′dydxdp=−i(2π)d∫R3d[V(x+y2)−V(x−y2)]eip⋅y(∫Rdf(x,p′)e−ip′⋅ydp′)dydxdp=−i(2π)d∫R3d[V(x+y2)−V(x−y2)](Fp′→yf)(x,y)eip⋅ydydxdp=−i∫Rd[V(x)−V(x)](Fp′→0f)(x,0)dx=0. |
Remark 2. If we set
∫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
For a given ensemble of wave functions
wij:=w[ψi,ψj],wi:=w[ψ]=w[ψi,ψi],i,j∈[N]. |
Then, the evolution of the WL distributions
∂twij+p⋅∇xwij=−i(2π)d∫R2d[Vi(x+y2)−Vj(x−y2)]wij(x,p′)ei(p−p′)⋅ydp′dy+κ2NN∑k=1[(wkj+wik)−(∫R2dwikdxdp∫R2dwiidxdp+∫R2dwkjdxdp∫R2dwjjdxdp)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
Proof. (ⅰ) It follows from (6) that
∂twii+∇x⋅(pwii)+Θ[Vi](wii)=κ2NN∑k=1[(wki+wik)−(∫R2dwikdxdp∫R2dwiidxdp+∫R2dwkidxdp∫R2dwiidxdp)wii]. | (7) |
Now, we integrate (7) over
ddt∫R2dwii(t,x,p)dxdp=0. |
(ⅱ) For the second assertion, we follow a similar calculation in (ⅰ).
Remark 3.
∂tw+p⋅∇xw+Θ[V](w)=0. |
Then by Lemma 2.3, one has
ddt∫R2dw(t,x,p)dxdp=0. | (8) |
However, it is worthwhile mentioning that since
w=w[ψ] |
for a solution
∫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.
‖w(t)‖2=‖ψ(t)‖4. |
Hence, the linear Wigner equation enjoys
From now on, we are concerned with the following special situation:
Vi=Vand∫R2dw0i(x,p)dxdp=1,i∈[N]. | (9) |
In this case, the
∫R2dwi(t,x,p)dxdp=∫R2dw0i(x,p)dxdp=1,t≥0,i∈[N],∫R2dwij(t,x,p,)dxdp=∫R2dw0ij(x,p)dxdp,t≥0,i,j∈[N]. |
Hence, the Cauchy problem for system (6) with (9) can be further simplified as follows:
{∂twij+p⋅∇xwij+Θ[V](wij)=κ2NN∑k=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,
{∂tw1+p⋅∇xw1+Θ[V](w1)=κ2(w+12−R12w1),t>0,∂tw2+p⋅∇xw2+Θ[V](w2)=κ2(w+12−R12w2),∂tw12+p⋅∇xw12+Θ[V](w12)=κ4(w1+w2−2z12w12),(w1(0),w2(0),w12(0))=(w01,w02,w012), | (11) |
subject to constraints:
∫R2dw01dxdp=∫R2dw02dxdp=1,|∫R2dw012dxdp|≤1,∫R2dw012dxdp≠−1. | (12) |
Theorem 2.4. [3] Let
|1−z12(t)|≲e−κt,‖w1(t)−w2(t)‖2L2≤e−κt,ast→∞, |
where
Proof. (ⅰ) The first estimate follows from the following ODE:
˙z12=κ2(1−z212),t>0. |
This can be integrated explicitly:
z12(t)=(1+z012)eκt−(1−z012)(1+z012)eκt+(1−z012),t>0, | (13) |
where
z012∈R,z012<−1orz012>1, | (14) |
then the denominator of the right-hand side of (13) can be zero, and hence
limt→T∗−z12(t)=∞,T∗=1κln1−z0121+z012. |
In other words, for initial data satisfying (14),
(ⅱ) It is easy to see that
∂t(w1−w2)+p⋅∇x(w1−w2)+Θ[V](w1−w2)=−κR122(w1−w2). | (15) |
We multiply (15) by
ddt‖w1(t)−w2(t)‖2L2=−κR12(t)‖w1(t)−w2(t)‖2L2. |
Then, Grönwall's inequality and the first estimate
Before we close this section, we introduce elementary estimates to be used in the following sections.
Lemma 2.5. Let
y′≤α1e−β1ty+α2e−β2t,t>0. | (16) |
Then, the following assertions hold.
1. If
α1<0,β1=0,α2>0,β2>0, |
there exist uniform positive constants
y(t)≤C0e−D0t,t≥0. |
2. If
α1>0,β1>0,α2=0, |
there exists a uniform constant
y(t)≤C1y0,t≥0. |
Proof. (ⅰ) By the comparison principle of ODE and method of integrating factor, we have
y(t)≤(y0+α2α1+β2)eα1t−α2α1+β2e−β2t,t≥0. |
Hence, there exist uniform positive constants
y(t)≤C0e−D0t,t≥0. |
(ⅱ) We multiply (16) with the integrating factor
exp(−∫t0α1e−β1sds)=exp(−α1β1(1−e−β1t)) |
to find
y(t)≤y0eα1β1(1−e−β1t)≤eα1β1y0=:C1y0,t≥0. |
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
limt→∞‖wik−wjm‖=0,i,j,k,m∈[N]. | (17) |
Proof. Since the proof is rather lengthy, we introduce a strategy toward the proof. We first claim:
limt→∞‖wik−wjk‖=0,k≠i,j∈[N]. |
For this, the key idea is to derive Grönwall's type differential inequality for
ddtN∑k=1‖wik−wjk‖2≤−κ(1−C1e−κt)N∑k=1‖wik−wjk‖2+C2e−κt,t>0. | (18) |
Then, we apply Lemma 2.5 to derive the desired zero convergence for
‖wik−wjm‖≤‖wik−wjk‖+‖wjk−wjm‖=‖wik−wjk‖+‖wkj−wmj‖. |
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
dzijdt=κ2NN∑k=1(zik+zkj)(1−zij),t>0. | (20) |
Due to (19), it is natural to consider the maximal diameter for the set
D(Z(t)):=max1≤i,j≤N|1−zij(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
D(Z0)<1. | (21) |
Then, one has
D(Z(t))<1,t>0. |
Proof. It follows from (20) that
ddt(1−zij)=−κ2NN∑k=1(zik+zkj)(1−zij),t>0. | (22) |
Then, (22) gives
|1−zij(t)|=|1−z0ij|exp[−κ2NN∑k=1∫t0(Rik+Rkj)ds],t>0, | (23) |
where
T:={T∈(0,∞):D(Z(t))<1,t∈[0,T)},T∗:=supT. |
By the assumption on initial data, the set
T∗=∞. |
Suppose to the contrary, i.e.,
T∗<∞. |
Then, one has
limt→T∗D(Z(t))=1. | (24) |
On the other hand, we observe
maxi≠j|1−zij(t)|<1⟹mini≠jRij(t)>0,t∈[0,T∗). |
For
D(Z(t))=1−zitjt. |
Hence, (23) yields
1=D(Z(T∗))=D(Z0)exp[−κ2NN∑k=1∫T∗0(Ritk+Rkjt)ds]<1, |
which contradicts (24). Since
T∗=∞, |
and the set
Remark 4. Lemma 3.2 says that if initial data satisfy (21):
|∫R2dw0ijdxdp−1|<1,i,j∈[N], |
then one has
|∫R2dwij(t,x,p)dxdp−1|<1,i,j∈[N],t>0. |
Thus, the
As a direct consequence of Lemma 3.2, we can also show the uniform
R(W(t)):=max1≤i,j≤N‖wij(t)‖,t>0. |
Corollary 1. Let
1. The functional
D(Z(t))≤D(Z0)e−κtD(Z0)e−κt+1−D(Z0)≤D(Z0)1−D(Z0)e−κt,t>0. |
2. The functional
sup0≤t<∞R(W(t))≤R∞. |
Proof. (ⅰ) It follows from (20) that
ddt(1−zij)=−κ(1−zij)+κ2NN∑k=1(1−zik+1−zkj)(1−zij). |
Then, we find a differential inequality for
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
12∂t|wij|2+12p⋅∇x|wij|2+Re[Θ[V](wij)¯wij]=κ2NN∑k=1Re[(wik+wkj−(zik+zkj)wij)¯wij]. | (25) |
Now, we integrate (25) with respect to
ddt∫R2d|wij|2dxdp=−∫R2dp⋅∇x|wij|2dxdp−∫R2dRe[¯wijΘ[V](wij)]dxdp+κNN∑k=1∫R2dRe[¯wij(wik+wkj−(zik+zkj)wij)]dxdp=:I11+I12+I13. | (26) |
Below, we present estimates for
● Case A.1 (Estimate of
I11=−∫R2dp⋅∇x|wij|2dxdp=0. |
● Case A.2 (Estimate of
I12=0. |
● Case A.3 (Estimate of
∫R2d(¯wijwik+¯wijwkj−(zik+zkj)|wij|2)dxdp=−2‖wij‖2+∫R2d(¯wijwik+¯wijwkj)dxdp+(1−zik+1−zkj)‖wij‖2≤−2‖wij‖2+2R(W)2+2D(Z)R(W)2. |
In (26), we collect all the estimates in Case A.1–Case A.3 to derive
ddt‖wij‖2≤−2κ‖wij‖2+2κR(W)2+2κD(Z)R(W)2,t>0. |
This yields
ddtR(W)2≤2κD(Z)R(W)2,t>0. | (27) |
Since
In this subsection, we are ready to provide the proof of Theorem 3.1. First, we claim:
limt→∞‖wik−wjk‖=0,k≠i,j. | (28) |
Note that if one verifies (28), then (17) follows from the triangle inequality:
‖wik−wjm‖≤‖wik−wjk‖+‖wjk−wjm‖=‖wik−wjk‖+‖wkj−wmj‖. |
We consider the difference between
∂t(wik−wjk)+p⋅∇x(wik−wjk)+Θ[V](wik−wjk)−κ2NN∑ℓ=1[(wiℓ−wjℓ)−(ziℓwik−zjℓwjk)−zℓk(wik−wjk)]=0. | (29) |
Similar to the proof of Corollary 1, we multiply
ddt‖wik−wjk‖2=:−∫R2dp⋅∇x|wik−wjk|2dxdp−∫R2d2Re[(¯wik−¯wjk)Θ[V](wik−wjk)]dxdp+κNN∑ℓ=1∫R2dRe(Jijkℓ)dxdp=:I21+I22+I23. | (30) |
Below, we present estimates of
● Case B.1 (Estimates of
I21=I22=0. |
● Case B.2 (Estimate of
Jijkℓ=(¯wik−¯wjk)(wiℓ−wjℓ)−ziℓ|wik−wjk|2+(ziℓ−zjℓ)wjk(¯wik−¯wjk)−zℓk|wik−wjk|2=(¯wik−¯wjk)(wiℓ−wjℓ)−(ziℓ+zℓk)|wik−wjk|2+(ziℓ−zjℓ)wjk(¯wik−¯wjk). |
In (30), we combine all the estimates Case B.1–Case B.2 to find
ddt‖wik−wjk‖2≤κNN∑ℓ=1∫R2d(|wik−wjk||wiℓ−wjℓ|−Re(ziℓ+zℓk)|wik−wjk|2+|ziℓ−zjℓ||wjk||wik−wjk|)dxdp≤κNN∑ℓ=1(‖wik−wjk‖‖wiℓ−wjℓ‖−Re(ziℓ+zℓk)‖wik−wjk‖2+|ziℓ−zjℓ|‖wjk‖‖wik−wjk‖). | (31) |
If we use Corollary 1 with
ddt‖wik−wjk‖2≤−2κ(1−αe−κt)‖wik−wjk‖2+κNN∑ℓ=1‖wik−wjk‖‖wiℓ−wjℓ‖+κR∞NN∑ℓ=1|ziℓ−zjℓ|‖wik−wjk‖. | (32) |
We sum up (32) with respect to
ddtN∑k=1‖wik−wjk‖2≤−2κ(1−αe−κt)N∑k=1‖wik−wjk‖2+κNN∑k,ℓ=1‖wik−wjk‖‖wiℓ−wjℓ‖+κR∞NN∑ℓ=1|ziℓ−zjℓ|N∑k=1‖wik−wjk‖=:−2κ(1−αe−κt)N∑k=1‖wik−wjk‖2+I31+I32. | (33) |
● Case C.1 (Estimate of
I31=κNN∑k,ℓ=1‖wik−wjk‖‖wiℓ−wjℓ‖=κN(N∑k=1‖wik−wjk‖)2≤κN∑k=1‖wik−wjk‖2. | (34) |
● Case C.2 (Estimate of
I32=κR∞NN∑ℓ=1|ziℓ−1+1−zjℓ|N∑k=1‖wik−wjk‖≤4Nκ|R∞|2αe−κt. | (35) |
In (33), we combine all the estimates (34) and (35) to derive
ddtN∑k=1‖wik−wjk‖2≤−κ(1−2αe−κt)N∑k=1‖wik−wjk‖2+4Nκ|R∞|2αe−κt. |
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:={f∈L2(R2d):|∫R2dfdxdp|<∞},‖f‖X:=‖f‖+|∫fdxdp|,A:=−p⋅∇x. | (36) |
Then, it is easy to check that
D(A):={f∈X:p⋅∇xf∈L2(R2d)}⊆X. |
For the WL model as a perturbation of the linear Wigner equation, it is strongly believed that
Theorem 4.1. For
1. If initial data and the potential satisfy
w0ij∈X,i,j∈[N],andV∈L∞(Rd), |
then there exists a unique mild solution to the Cauchy problem
wij∈C([0,T];X),i,j∈[N]. |
2. If we impose further regularity on initial data and the potential
w0ij∈D(A),i,j∈[N],andV∈L∞(Rd)∩L2(Rd), |
then there exists a unique classical solution to the Cauchy problem
wij∈C([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
Lemma 4.2. Let
Awij∈L2(R2d). |
In other words, the transport operator
Proof. Since a solution
sup0≤t<∞‖p⋅∇xwij‖<∞,i,j∈[N]. |
By straightforward calculations, we observe
12ddt‖p⋅∇xwij‖2=Re⟨p⋅∇x∂twij,p⋅∇xwij⟩=Re⟨p⋅∇x(−p⋅∇xwij),p⋅∇xwij⟩−⟨p⋅∇x(Θwij),p⋅∇xwij⟩+κ2NN∑k=1Re⟨p⋅∇x(wik+wkj−(zik+zkj)wij),p⋅∇xwij⟩=:I41+I42+I43. | (37) |
Below, we estimate
● Case C.1 (Estimate of
−⟨p⋅∇x(p⋅∇xwij),p⋅∇xwij⟩=−⟨N∑j=1pj∂j(p⋅∇xwij),p⋅∇xwij⟩=⟨p⋅∇xwij,N∑j=1pj∂j(p⋅∇xwij)⟩=¯⟨p⋅∇x(p⋅∇xwij),p⋅∇xwij⟩. |
Hence, we see that
I41=0. |
● Case C.2 (Estimate of
⟨p⋅∇x(Θwij),p⋅∇xwij⟩=⟨Θ[V](p⋅∇xwij),p⋅∇xwij⟩. |
By the skew-Hermitian property of
I42=0. |
● Case C.3 (Estimate of
⟨p⋅∇x(wik+wkj−(zik+zkj)wij),p⋅∇xwij⟩=⟨p⋅∇x(wik+wkj),p⋅∇xwij⟩−(zik+zkj)‖p⋅∇xwij‖2=−2‖p⋅∇xwij‖2+(1−zik+1−zkj)‖p⋅∇xwij‖2+⟨p⋅∇x(wik+wkj),p⋅∇xwij⟩≤−2‖p⋅∇xwij‖2+|1−zik+1−zkj|‖p⋅∇xwij‖2+‖p⋅∇xwij‖(‖p⋅∇xwik‖+‖p⋅∇xwkj‖). |
In (37), we collect all the estimates in Case C.1–Case C.3 to find
12ddt‖p⋅∇xwij‖2≤−κ‖p⋅∇xwij‖2+κD(Z)‖p⋅∇xwij‖2+κ2NN∑k=1‖p⋅∇xwij‖(‖p⋅∇xwik‖+‖p⋅∇xwkj‖). | (38) |
We sum up (38) with respect to
12ddtN∑i,j=1‖p⋅∇xwij‖2≤κD(Z)N∑i,j=1‖p⋅∇xwij‖2. |
It follows from Corollary 1 that
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
∂tW+p⋅∇xW+Θ[V](W)=κ2N(EijWCj+RiWEij−W∫R2d(EijWCj+RiWEij)dxdp), | (39) |
where
{X:={F=(Fij)∈(L2(R2d))⊗N2:|∫R2dFijdxdp|<∞,i,j∈[N]},‖F‖X:=‖F‖L2(R2d)⊗N2+|∫R2dFdxdp|:=maxi,j(‖Fij‖+|∫R2dFijdxdp|). |
Then,
C([0,T];X) |
equipped with the norm
|||F|||:=sup0≤t≤T‖F(t)‖X. |
Now, we are concerned with the global solvability of (39). Let
{∂tW+p⋅∇xW+Θ[V](W)=κ2N(EijWCj+RiWEij−W∫R2d(EijGCj+RiGEij)dxdp),W(0)=W0. | (40) |
We need to check well-definedness and strict contraction of
● (Well-definedness of
● (Strict contraction): for
|||W−¯W|||≤C|||G−¯G|||. |
If we set
∂twij+p⋅∇xwij+Θ[V](ωij)=κ2NN∑k=1[wik+wkj−wij(∫R2d(gik+gkj)dxdp)],∂t¯wij+p⋅∇x¯wij+Θ[V](¯ωij)=κ2NN∑k=1[¯wik+¯wkj−¯wij(∫R2d(¯gik+¯gkj)dxdp)]. |
For simplicity, we set
qij(t):=∫R2dgijdxdp,¯qij(t):=∫R2d¯gijdxdp. |
By straightforward calculation,
12ddtmax1≤i,j≤N‖wij−¯wij‖≤Cmax1≤i,j≤N‖wij−¯wij‖+max1≤i,j≤N‖ωij‖max1≤i,j≤N|qij−¯qij|. | (41) |
In addition, we observe
ddtmax1≤i,j≤N|zij−¯zij|≤Cmax1≤i,j≤N|zij−¯zij|+˜Cmax1≤i,j≤N|qij−¯qij|. | (42) |
It follows from Corollary 1 that
max1≤i,j≤N‖wij(t)‖≤R∞,t>0. |
Then, (41) and (42) yield
ddt‖W−¯W‖X≤C1‖W−¯W‖X+eC2t‖G−¯G‖X, | (43) |
and integrate the relation (43) to find
‖W−¯W‖X≤‖W0−¯W0‖X+C1∫t0‖W−¯W‖Xds+eC2T∫t0‖G−¯G‖Xds. |
Since
|||W−¯W|||=sup0≤t≤T‖W−¯W‖X≤Te(C1+C2)Tsup0≤t≤T‖G−¯G‖X=Te(C1+C2)T|||G−¯G|||. |
If
W(t)=Wn(t−nT),t∈[nT,(n+1)T],n≥0. |
Hence,
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
(F(W))(i,j):=κ2NN∑k=1(zik+zkj)wij=κ2NN∑k=1(∫R2d(wik+wkj)dxdp)wij, |
which is nonlinear with respect to the argument
Lemma 4.3. For
‖F(U)−F(V)‖X≤C‖U−V‖X. |
Then, the functional derivative, denoted by
Proof. For
DF(U)(V):=limτ→0F(U+τV)−F(U)τ=ddτF(U+τV)|τ=0. |
At each point
(DF(U)(V))(i,j)=κ2NN∑k=1(uij∫R2d(vik+vkj)dxdp+vij∫R2d(uik+ukj)dxdp). |
Since
‖DF(U)(V)‖X≤2κ‖U‖X⋅‖V‖X. |
Therefore, we verified that
‖F(U)−F(V)‖X≤‖U−V‖X⋅supt∈[0,1]‖DF(tU+(1−t)V)‖op. |
Here,
‖F(U)−F(V)‖X≤2κsupt∈[0,1]‖tU+(1−t)V‖X⋅‖U−V‖X. |
This shows that
Now, we are ready to provide the second assertion of Theorem 4.1 by applying semigroup theory.
● Step A (the linear Wigner equation on
∂twij+p⋅∇xwij+Θ[V](wij)=0. | (44) |
Since (44) on
‖Θ[V]w‖X=‖Θ[V]w‖. |
Since
wij∈C([0,T];X)∩C1([0,T];D(A)). |
For details, we refer the reader to [23,Theorem 1].
● Step B (the WL model on
wij∈C([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
(i) A is the infinitestimal generator of a
‖T(t)‖≤Meωt. |
(ii)
Then,
‖S(t)‖≤Me(ω+M‖B‖)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
2. Let
Finally, we recall Gâteaux's mean value theorem. We denote the directional derivative of
Lemma A.3. [2,Proposition A.2] Let
‖f(y)−f(x)‖Y≤‖x−y‖Xsup0≤θ≤1‖Df(θx+(1−θ)y‖L(X,Y), |
where
δvf(x):=limt→∞f(x+tv)−f(x)t. |
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1. | Seung-Yeal Ha, Gyuyoung Hwang, Dohyun Kim, On the semiclassical limit of the Schrödinger-Lohe model and concentration estimates, 2024, 65, 0022-2488, 10.1063/5.0194571 |