Citation: M Saad Bin Arif, Uvais Mustafa, Shahrin bin Md. Ayob. Extensively used conventional and selected advanced maximum power point tracking techniques for solar photovoltaic applications: An overview[J]. AIMS Energy, 2020, 8(5): 935-958. doi: 10.3934/energy.2020.5.935
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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)|≲ |
where
Proof. (ⅰ) The first estimate follows from the following ODE:
\dot z_{12} = \frac{\kappa}{2}(1-z_{12}^2), \quad t > 0. |
This can be integrated explicitly:
\begin{equation} z_{12}(t) = \frac{ (1+z_{12}^0)e^{\kappa t} - (1-z_{12}^0)}{(1+z_{12}^0)e^{\kappa t} + (1-z_{12}^0)}, \quad t > 0, \end{equation} | (13) |
where
\begin{equation} z_{12}^0 \in \mathbb R, \quad z_{12}^0 < -1 \quad \text{or}\quad z_{12}^0 > 1, \end{equation} | (14) |
then the denominator of the right-hand side of (13) can be zero, and hence
\lim\limits_{t\to T_*-} z_{12}(t) = \infty, \quad T_* = \frac1 \kappa \ln \frac{ 1-z_{12}^0}{1+z_{12}^0}. |
In other words, for initial data satisfying (14),
(ⅱ) It is easy to see that
\begin{equation} \partial_t(w_1-w_2)+p\cdot\nabla_x(w_1-w_2)+\Theta[V](w_1-w_2) = -\frac{\kappa R_{12}}{2}(w_1-w_2). \end{equation} | (15) |
We multiply (15) by
\begin{equation*} \frac{ \mathrm{d}}{ \mathrm{d}t}\|w_1(t)-w_2(t)\|_{L^2}^2 = -\kappa R_{12}(t)\|w_1(t)-w_2(t)\|_{L^2}^2. \end{equation*} |
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
\begin{equation} y' \leq \alpha_1e^{-\beta_1 t}y + \alpha_2 e^{-\beta_2 t}, \quad t > 0. \end{equation} | (16) |
Then, the following assertions hold.
1. If
\alpha_1 < 0, \quad \beta_1 = 0, \quad \alpha_2 > 0, \quad \beta_2 > 0, |
there exist uniform positive constants
y(t) \leq C_0e^{-D_0 t}, \quad t \geq 0. |
2. If
\alpha_1 > 0, \quad \beta_1 > 0, \quad \alpha_2 = 0, |
there exists a uniform constant
y(t) \leq C_1 y_0, \quad t \geq 0. |
Proof. (ⅰ) By the comparison principle of ODE and method of integrating factor, we have
y(t) \leq \left( y_0 + \frac{\alpha_2}{\alpha_1 + \beta_2}\right) e^{\alpha_1 t} - \frac{\alpha_2}{\alpha_1 + \beta_2} e^{-\beta_2t}, \quad t \geq 0. |
Hence, there exist uniform positive constants
y(t) \leq C_0e^{-D_0 t}, \quad t \geq 0. |
(ⅱ) We multiply (16) with the integrating factor
\begin{equation*} \label{B-6} \exp{ \left(-\int_0^t \alpha_1 e^{-\beta_1 s} ds\right) } = \exp{ \left( -\frac{\alpha_1}{\beta_1} (1-e^{-\beta_1 t}) \right) } \end{equation*} |
to find
y(t) \leq y_0 e^{ \frac{\alpha_1}{\beta_1} (1-e^{-\beta_1t}) } \leq e^\frac{\alpha_1}{\beta_1}y_0 = : C_1y_0, \quad t \geq 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
\begin{equation} \lim\limits_{t\to\infty} \|w_{ik} - w_{jm}\| = 0, \quad i, j, k, m \in [N]. \end{equation} | (17) |
Proof. Since the proof is rather lengthy, we introduce a strategy toward the proof. We first claim:
\lim\limits_{t\to\infty} \|w_{ik} - w_{jk} \| = 0, \quad k\neq i, j \in [N]. |
For this, the key idea is to derive Grönwall's type differential inequality for
\begin{equation} \frac{ \mathrm{d}}{ \mathrm{d}t} \sum\limits_{k = 1}^N \|w_{ik}-w_{jk}\|^2 \leq - \kappa \Big ( 1- C_1 e^{- \kappa t} \Big ) \sum\limits_{k = 1}^N \|w_{ik}-w_{jk}\|^2 + C_2 e^{- \kappa t}, \quad t > 0. \end{equation} | (18) |
Then, we apply Lemma 2.5 to derive the desired zero convergence for
\|w_{ik} - w_{jm} \| \leq \|w_{ik} - w_{jk} \| + \|w_{jk} - w_{jm}\| = \|w_{ik} - w_{jk} \| + \|w_{kj} - w_{mj}\|. |
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
\begin{equation*} \label{C-2-2} z_{ij}(t) : = \int_{\mathbb{R}^{2d}}w_{ij}(t, x, p) \mathrm{d} x \mathrm{d} p, \quad i, j \in [N], \quad t > 0. \end{equation*} |
Then, it follows from Lemma 2.3 that
\begin{equation} z_i : = z_{ii} = 1, \quad i \in [N]. \end{equation} | (19) |
On the other hand, we integrate (2) with respect to
\begin{equation} \frac{dz_{ij}}{dt} = \frac{ \kappa}{2N} \sum\limits_{k = 1}^N (z_{ik} + z_{kj}) (1-z_{ij}), \quad t > 0. \end{equation} | (20) |
Due to (19), it is natural to consider the maximal diameter for the set
\mathcal{D}({\mathcal Z}(t)) : = \max\limits_{1\leq i, j\leq N} |1-z_{ij}(t)|, \quad t > 0, \quad \mathcal D(\mathcal Z^0): = \mathcal D(\mathcal Z(0)), |
which is expected to converge to zero under a suitable condition.
Lemma 3.2. (Existence of a positively invariant set) Let
\begin{equation} \mathcal D(\mathcal Z^0) < 1. \end{equation} | (21) |
Then, one has
\mathcal D(\mathcal Z(t)) < 1, \quad t > 0. |
Proof. It follows from (20) that
\begin{equation} \frac{ \mathrm{d}}{ \mathrm{d}t} (1 - z_{ij}) = -\frac{ \kappa}{2N} \sum\limits_{k = 1}^N (z_{ik} + z_{kj}) (1-z_{ij}), \quad t > 0. \end{equation} | (22) |
Then, (22) gives
\begin{equation} |1-z_{ij}(t)| = |1-z_{ij}^0| \exp\left[ -\frac{ \kappa}{2N} \sum\limits_{k = 1}^N \int_0^t (R_{ik} + R_{kj}) \mathrm{d} s \right], \quad t > 0, \end{equation} | (23) |
where
\mathcal T: = \{ T \in (0, \infty) : \mathcal D(\mathcal Z(t)) < 1, \quad t\in [0, T)\}, \quad T_* : = \sup \mathcal{T}. |
By the assumption on initial data, the set
T_* = \infty. |
Suppose to the contrary, i.e.,
T_* < \infty. |
Then, one has
\begin{equation} \lim\limits_{t\to T_*} \mathcal D(\mathcal Z(t)) = 1. \end{equation} | (24) |
On the other hand, we observe
\max\limits_{i \neq j} |1-z_{ij}(t)| < 1\quad \Longrightarrow\quad \min\limits_{i \neq j} R_{ij}(t) > 0, \quad t\in [0, T_*). |
For
\mathcal D(\mathcal Z(t)) = 1-z_{i_t j_t}. |
Hence, (23) yields
1 = \mathcal D(\mathcal Z(T_*)) = \mathcal D(\mathcal Z^0) \exp \left[ {-\frac{ \kappa}{2N} \sum\limits_{k = 1}^N \int_0^{T_*} (R_{i_tk} + R_{kj_t} ) \mathrm{d} s} \right ] < 1, |
which contradicts (24). Since
T_* = \infty, |
and the set
Remark 4. Lemma 3.2 says that if initial data satisfy (21):
\left|\int_{\mathbb{R}^{2d}}w^0_{ij} \mathrm{d} x \mathrm{d} p - 1\right| < 1, \quad i, j\in[N], |
then one has
\left|\int_{\mathbb{R}^{2d}}w_{ij}(t, x, p) \mathrm{d} x \mathrm{d} p - 1\right| < 1, \quad i, j\in[N], \quad t > 0. |
Thus, the
As a direct consequence of Lemma 3.2, we can also show the uniform
\mathcal R(\mathcal W(t)) : = \max\limits_{1\leq i, j\leq N} \|w_{ij}(t) \|, \quad t > 0. |
Corollary 1. Let
1. The functional
\begin{equation*} \mathcal D( \mathcal Z(t)) \leq \frac{\mathcal D(\mathcal Z^0) e^{- \kappa t}}{\mathcal D(\mathcal Z^0) e^{- \kappa t} + 1 - \mathcal D(\mathcal Z^0)} \leq \frac{\mathcal D(\mathcal Z^0) }{1 - \mathcal D(\mathcal Z^0)}e^{- \kappa t}, \quad t > 0. \end{equation*} |
2. The functional
\sup\limits_{0\leq t < \infty} \mathcal R(\mathcal W(t)) \leq {\mathcal R}^\infty. |
Proof. (ⅰ) It follows from (20) that
\begin{align*} \frac {\mathrm{d}} {\mathrm{d}t} (1-z_{ij}) = - \kappa(1-z_{ij}) + \frac{ \kappa}{2N} \sum\limits_{k = 1}^N ( 1-z_{ik} + 1-z_{kj})(1-z_{ij}). \end{align*} |
Then, we find a differential inequality for
\frac {\mathrm{d}} {\mathrm{d}t} \mathcal D(\mathcal Z) \leq - \kappa \mathcal D(\mathcal Z) + \kappa \mathcal D(\mathcal Z)^2, \quad t > 0. |
Lastly, we use initial data (21) together with the above Riccati differential inequality to give the desired result.
(ⅱ) We multiply
\begin{align} \begin{aligned} & \frac12 \partial_t |w_{ij}|^2 + \frac12 p\cdot \nabla_x |w_{ij}|^2 +\text{Re}\left[ \Theta[V](w_{ij})\overline w_{ij}\right] \\ & \quad\quad\quad\quad \quad = \frac{ \kappa}{2N} \sum\limits_{k = 1}^N \mbox{Re}\left[ \Big( w_{ik} + w_{kj} - (z_{ik} + z_{kj}) w_{ij}\Big)\overline w_{ij} \right]. \end{aligned} \end{align} | (25) |
Now, we integrate (25) with respect to
\begin{align} \begin{aligned} & \frac{ \mathrm{d}}{ \mathrm{d}t} \int_{ \mathbb R^{2d}} |w_{ij}|^2 \mathrm{d} x \mathrm{d} p \\ & \quad = -\int_{ \mathbb R^{2d}}p\cdot \nabla_x |w_{ij}|^2 dx dp - \int_{ \mathbb R^{2d}} \text{Re}\left[ \overline w_{ij} \Theta[V](w_{ij}) \right] \mathrm{d} x \mathrm{d} p \\ & \quad + \frac{ \kappa}{N} \sum\limits_{k = 1}^N \int_{ \mathbb R^{2d}} \text{Re}\left[\overline w_{ij}\Big( w_{ik} + w_{kj} - (z_{ik} + z_{kj}) w_{ij}\Big)\right] \mathrm{d} x \mathrm{d} p \\ & \quad = : \mathcal I_{11} + \mathcal I_{12} + \mathcal I_{13} . \end{aligned} \end{align} | (26) |
Below, we present estimates for
● Case A.1 (Estimate of
\begin{align*} \mathcal I_{11} = - \int_{ \mathbb R^{2d}}p\cdot \nabla_x |w_{ij}|^2 \mathrm{d} x \mathrm{d} p = 0. \end{align*} |
● Case A.2 (Estimate of
\mathcal I_{12} = 0. |
● Case A.3 (Estimate of
\begin{align*} &\int_{ \mathbb R^{2d}} \Big( \overline w_{ij} w_{ik} +\overline w_{ij} w_{kj} - (z_{ik} + z_{kj} )|w_{ij}|^2 \Big) \mathrm{d} x \mathrm{d} p \\ & \quad = -2\|w_{ij}\|^2 + \int_{ \mathbb R^{2d}} ( \overline w_{ij} w_{ik} + \overline w_{ij} w_{kj}) \mathrm{d} x \mathrm{d} p + (1-z_{ik} + 1-z_{kj}) \|w_{ij}\|^2 \\ & \quad \leq -2\|w_{ij}\|^2 +2 \mathcal R(\mathcal W) ^2 + 2\mathcal D(\mathcal Z) \mathcal R(\mathcal W) ^2. \end{align*} |
In (26), we collect all the estimates in Case A.1–Case A.3 to derive
\begin{align*} \frac {\mathrm{d}} {\mathrm{d}t} \|w_{ij}\|^2 \leq -2 \kappa \|w_{ij}\|^2 + 2 \kappa\mathcal R(\mathcal W) ^2 + 2 \kappa \mathcal D(\mathcal Z) \mathcal R(\mathcal W)^2, \quad t > 0. \end{align*} |
This yields
\begin{equation} \frac {\mathrm{d}} {\mathrm{d}t} \mathcal R(\mathcal W)^2 \leq 2 \kappa \mathcal D(\mathcal Z) \mathcal R(\mathcal W)^2, \quad t > 0. \end{equation} | (27) |
Since
In this subsection, we are ready to provide the proof of Theorem 3.1. First, we claim:
\begin{equation} \lim\limits_{t\to\infty} \|w_{ik} - w_{jk} \| = 0, \quad k \neq i, j. \end{equation} | (28) |
Note that if one verifies (28), then (17) follows from the triangle inequality:
\|w_{ik} - w_{jm} \| \leq \|w_{ik} - w_{jk} \| + \|w_{jk} - w_{jm}\| = \|w_{ik} - w_{jk} \| + \|w_{kj} - w_{mj}\|. |
We consider the difference between
\begin{align} \begin{aligned} &\partial_t (w_{ik}-w_{jk}) + p \cdot \nabla_x (w_{ik}-w_{jk}) + \Theta[V](w_{ik} -w_{jk}) \\ & \quad -\frac{ \kappa}{2N} \sum\limits_{\ell = 1}^N \Big[ (w_{i\ell}-w_{j\ell}) -(z_{i\ell}w_{ik} - z_{j\ell}w_{jk}) -z_{\ell k}(w_{ik}-w_{jk}) \Big ] = 0. \end{aligned} \end{align} | (29) |
Similar to the proof of Corollary 1, we multiply
\begin{align} \begin{aligned} &\frac{ \mathrm{d}}{ \mathrm{d}t} \|w_{ik}-w_{jk}\|^2 \\ & \quad = : - \int_{ \mathbb R^{2d}} p\cdot \nabla_x|w_{ik}-w_{jk}|^2 \mathrm{d} x \mathrm{d} p \\ & \quad -\int_{ \mathbb R^{2d}} 2\text{Re}\Big[ (\overline w_{ik}-\overline w_{jk})\Theta[V](w_{ik}-w_{jk}) \Big] \mathrm{d} x \mathrm{d} p \\ & \quad + \frac{ \kappa}{N} \sum\limits_{\ell = 1}^N \int_{ \mathbb R^{2d}} \text{Re}( \mathcal J_{ijk\ell}) \; \mathrm{d} x \mathrm{d} p \\ & \quad = : \mathcal I_{21} + \mathcal I_{22} +\mathcal I_{23}. \end{aligned} \end{align} | (30) |
Below, we present estimates of
● Case B.1 (Estimates of
\mathcal I_{21} = \mathcal I_{22} = 0. |
● Case B.2 (Estimate of
\begin{align*} \mathcal J_{ijk\ell} & = (\overline w_{ik} - \overline w_{jk})(w_{i\ell} -w_{j\ell}) - z_{i\ell}|w_{ik}-w_{jk}|^2 \\ & \quad \quad \quad \quad \quad \quad + (z_{i\ell}-z_{j\ell})w_{jk}(\overline w_{ik} - \overline w_{jk}) -z_{\ell k} |w_{ik}-w_{jk}|^2 \\ & = (\overline w_{ik} - \overline w_{jk})(w_{i\ell} -w_{j\ell}) -(z_{i\ell} + z_{\ell k})|w_{ik}-w_{jk}|^2\\&\qquad + (z_{i\ell}-z_{j\ell})w_{jk}(\overline w_{ik} - \overline w_{jk}). \end{align*} |
In (30), we combine all the estimates Case B.1–Case B.2 to find
\begin{align} \begin{aligned} &\frac{ \mathrm{d}}{ \mathrm{d}t} \|w_{ik}-w_{jk}\|^2 \\ & \quad \leq \frac{ \kappa}{N}\sum\limits_{\ell = 1}^N \int_{ \mathbb R^{2d}} \Big( |w_{ik}-w_{jk}||w_{i\ell}-w_{j\ell}| -\text{Re}(z_{i\ell}+z_{\ell k})|w_{ik}-w_{jk}|^2 \\ & \quad + |z_{i\ell}-z_{j\ell} ||w_{jk}| |w_{ik}-w_{jk}|\Big) \mathrm{d} x \mathrm{d} p \\ & \quad \leq \frac{ \kappa}{N}\sum\limits_{\ell = 1}^N \Big( \|w_{ik}-w_{jk}\| \|w_{i\ell}-w_{j\ell}\| -\text{Re}(z_{i\ell} + z_{\ell k})\|w_{ik}-w_{jk}\|^2 \\ & \quad + |z_{i\ell}-z_{j\ell}| \|w_{jk}\| \|w_{ik}-w_{jk}\|\Big). \end{aligned} \end{align} | (31) |
If we use Corollary 1 with
\begin{align} \begin{aligned} \frac{ \mathrm{d}}{ \mathrm{d}t} \|w_{ik}-w_{jk}\|^2 &\leq -2 \kappa ( 1- \alpha e^{- \kappa t})\|w_{ik}-w_{jk}\|^2 \\ & \quad +\frac{ \kappa}{N} \sum\limits_{\ell = 1}^N \|w_{ik}-w_{jk}\| \|w_{i\ell}-w_{j\ell}\| \\ & \quad + \frac{ \kappa\mathcal R^\infty}{N} \sum\limits_{\ell = 1}^N |z_{i\ell}- z_{j\ell} | \|w_{ik} - w_{jk}\|. \end{aligned} \end{align} | (32) |
We sum up (32) with respect to
\begin{align} \begin{aligned} & \frac{ \mathrm{d}}{ \mathrm{d}t} \sum\limits_{k = 1}^N \|w_{ik}-w_{jk}\|^2 \\ &\leq -2 \kappa ( 1- \alpha e^{- \kappa t})\sum\limits_{k = 1}^N \|w_{ik}-w_{jk}\|^2 \\ & \quad +\frac{ \kappa}{N} \sum\limits_{k, \ell = 1}^N \|w_{ik}-w_{jk}\| \|w_{i\ell}-w_{j\ell}\|\\ & \quad + \frac{ \kappa\mathcal R^\infty}{N} \sum\limits_{\ell = 1}^N |z_{i\ell}- z_{j\ell} | \sum\limits_{k = 1}^N\|w_{ik} - w_{jk}\| \\ & = : -2 \kappa ( 1- \alpha e^{- \kappa t}) \sum\limits_{k = 1}^N \|w_{ik}-w_{jk}\|^2 +\mathcal I_{31} + \mathcal I_{32}. \end{aligned} \end{align} | (33) |
● Case C.1 (Estimate of
\begin{align} \begin{aligned} \mathcal I_{31} & = \frac{ \kappa}{N} \sum\limits_{k, \ell = 1}^N \|w_{ik}-w_{jk}\| \|w_{i\ell}-w_{j\ell}\| = \frac \kappa N \left(\sum\limits_{k = 1}^N \|w_{ik} - w_{jk}\|\right)^2\\ & \leq \kappa \sum\limits_{k = 1}^N \|w_{ik}-w_{jk}\|^2. \end{aligned} \end{align} | (34) |
● Case C.2 (Estimate of
\begin{equation} \mathcal I_{32} = \frac{ \kappa\mathcal R^\infty}{N} \sum\limits_{\ell = 1}^N |z_{i\ell}-1+1- z_{j\ell} | \sum\limits_{k = 1}^N\|w_{ik} - w_{jk}\| \leq 4 N \kappa |{\mathcal R}^\infty|^2 \alpha e^{- \kappa t}. \end{equation} | (35) |
In (33), we combine all the estimates (34) and (35) to derive
\begin{align*} \frac{ \mathrm{d}}{ \mathrm{d}t} \sum\limits_{k = 1}^N \|w_{ik}-w_{jk}\|^2 \leq - \kappa ( 1- 2\alpha e^{- \kappa t}) \sum\limits_{k = 1}^N \|w_{ik}-w_{jk}\|^2 + 4 N \kappa |{\mathcal R}^\infty|^2 \alpha e^{- \kappa t}. \end{align*} |
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
\begin{align} \begin{aligned} &\mathcal{X}: = \left\{f \in L^2( \mathbb R^{2d}) : \; \left| \int_{ \mathbb R^{2d}} f \mathrm{d} x \mathrm{d} p \right| < \infty \right\}, \\ & \; \; \|f\|_{\mathcal{X}} : = \|f\| + \left|\int f \mathrm{d} x \mathrm{d} p\right|, \; \; A: = -p \cdot \nabla_x. \end{aligned} \end{align} | (36) |
Then, it is easy to check that
D(A) : = \left \{f \in \mathcal{X} :p \cdot \nabla_x f \in L^2( \mathbb R^{2d}) \right \} \subseteq {\mathcal 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
w_{ij}^0 \in \mathcal X, \quad i, j\in [N], \quad {{and}} \quad V \in L^{\infty}(\mathbb{R}^d), |
then there exists a unique mild solution to the Cauchy problem
w_{ij} \in {C}([0, T];\mathcal X), \quad i, j\in [N]. |
2. If we impose further regularity on initial data and the potential
w_{ij}^0 \in D(A), \quad i, j\in [N], \quad {{and}} \quad V \in L^{\infty}(\mathbb{R}^d) \cap L^{2}(\mathbb{R}^d), |
then there exists a unique classical solution to the Cauchy problem
w_{ij} \in {C}([0, T];\mathcal{X}) \cap { C}^1([0, T];D(A)), \quad i, j \in [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
Aw_{ij} \in L^2( \mathbb R^{2d}). |
In other words, the transport operator
Proof. Since a solution
\sup\limits_{0\leq t < \infty} \|p\cdot \nabla_x w_{ij}\| < \infty, \quad i, j\in [N]. |
By straightforward calculations, we observe
\begin{align} \begin{aligned} &\frac{1}{2}\frac{ \mathrm{d}}{ \mathrm{d}t}\|p \cdot \nabla_x w_{ij}\|^2 \\ & \quad = \text{Re} \langle p\cdot \nabla_x \partial_t w_{ij}, p\cdot \nabla_xw_{ij} \rangle \\ & \quad = \mathrm{Re}\langle p \cdot \nabla_x (- p \cdot \nabla_x w_{ij}), p \cdot \nabla_x w_{ij} \rangle - \langle p \cdot \nabla_x (\Theta w_{ij}), p \cdot \nabla_x w_{ij} \rangle\\ & \quad + \frac{ \kappa}{2N} \sum\limits_{k = 1}^N \text{Re} \langle p\cdot \nabla_x (w_{ik} + w_{kj} - (z_{ik} + z_{kj})w_{ij} ) , p\cdot \nabla_xw_{ij} \rangle \\ & \quad = : \mathcal{I}_{41} + \mathcal{I}_{42} + \mathcal{I}_{43}. \end{aligned} \end{align} | (37) |
Below, we estimate
● Case C.1 (Estimate of
\begin{align*} & -\langle p \cdot \nabla_x ( p \cdot \nabla_x w_{ij}), p \cdot \nabla_x w_{ij} \rangle = -\left \langle \sum\limits_{j = 1}^N p_j \partial_j (p \cdot \nabla_x w_{ij}), p \cdot \nabla_x w_{ij}\right \rangle\\ & \quad = \left \langle p \cdot \nabla_x w_{ij}, \sum\limits_{j = 1}^N p_j \partial_j (p \cdot \nabla_x w_{ij})\right \rangle = \overline{\langle p \cdot \nabla_x ( p \cdot \nabla_x w_{ij}), p \cdot \nabla_x w_{ij} \rangle}. \end{align*} |
Hence, we see that
\mathcal I_{41} = 0. |
● Case C.2 (Estimate of
\begin{align*} \langle p \cdot \nabla_x (\Theta w_{ij}), p \cdot \nabla_x w_{ij} \rangle = \langle \Theta[V]( p\cdot \nabla_xw_{ij}), p\cdot \nabla_x w_{ij}\rangle. \end{align*} |
By the skew-Hermitian property of
\mathcal I_{42} = 0. |
● Case C.3 (Estimate of
\begin{align*} \begin{aligned} & \langle p\cdot \nabla_x (w_{ik} + w_{kj} - (z_{ik} + z_{kj})w_{ij} ) , p\cdot \nabla_xw_{ij} \rangle \\ & \quad = \langle p\cdot \nabla_x(w_{ik} + w_{kj}), p\cdot\nabla_x w_{ij}\rangle -(z_{ik}+z_{kj})\|p\cdot \nabla_xw_{ij}\|^2 \\ & \quad = -2\|p\cdot \nabla_xw_{ij}\|^2 + (1-z_{ik} + 1-z_{kj}) \|p\cdot \nabla_xw_{ij}\|^2\\ & \quad + \langle p\cdot \nabla_x(w_{ik} + w_{kj}), p\cdot\nabla_x w_{ij}\rangle \\ & \quad \leq -2\|p\cdot \nabla_xw_{ij}\|^2 + |1-z_{ik} + 1-z_{kj}| \|p\cdot \nabla_xw_{ij}\|^2 \\ & \quad + \| p\cdot\nabla_x w_{ij} \| \big( \|p\cdot\nabla_x w_{ik}\| + \|p\cdot\nabla_x w_{kj}\|\big). \end{aligned} \end{align*} |
In (37), we collect all the estimates in Case C.1–Case C.3 to find
\begin{align} \begin{aligned} \frac{1}{2}\frac{ \mathrm{d}}{ \mathrm{d}t}\|p \cdot \nabla_x w_{ij}\|^2 & \leq - \kappa \|p\cdot \nabla_xw_{ij}\|^2 + \kappa \mathcal D(\mathcal Z) \|p\cdot \nabla_xw_{ij}\|^2 \\ & \quad + \frac{ \kappa}{2N}\sum\limits_{k = 1}^N \| p\cdot\nabla_x w_{ij} \| \big( \|p\cdot\nabla_x w_{ik}\| + \|p\cdot\nabla_x w_{kj}\|\big). \end{aligned} \end{align} | (38) |
We sum up (38) with respect to
\frac{1}{2}\frac{ \mathrm{d}}{ \mathrm{d}t} \sum\limits_{i, j = 1}^N \|p \cdot \nabla_x w_{ij}\|^2 \leq \kappa \mathcal D(\mathcal Z) \sum\limits_{i, j = 1}^N \|p \cdot \nabla_x w_{ij}\|^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
\begin{align} \begin{aligned} & \partial_t W + p\cdot \nabla_x W + \Theta[V](W) \\ & \quad = \frac{ \kappa}{2N} \left( E_{ij} W C_j + R_i W E_{ij} - W \int_{ \mathbb R^{2d}} ( E_{ij} W C_j + R_i W E_{ij}) \mathrm{d} x \mathrm{d} p \right), \end{aligned} \end{align} | (39) |
where
\begin{cases} {\bf{X}} : = \left\{ F = (F_{ij}) \in ( L^2( \mathbb R^{2d}) )^{\otimes N^2}\; \; :\; \; \left| \int_{ \mathbb R^{2d}} F_{ij} \mathrm{d} x \mathrm{d} p \right| < \infty, \quad i, j \in [N] \right\}, \\ \|F\|_{{\bf{X}}} : = \|F\|_{L^2( \mathbb R^{2d})^{\otimes N^2}} + \left| \int_{ \mathbb R^{2d}} F \mathrm{d} x \mathrm{d} p \right|: = \max\limits_{i, j}\bigg(\|F_{ij}\| + \left| \int_{ \mathbb R^{2d}} F_{ij} \mathrm{d} x \mathrm{d} p \right|\bigg). \end{cases} |
Then,
C([0, T]; {\bf{X}}) |
equipped with the norm
{ \vert\kern-0.25ex \vert\kern-0.25ex \vert {F} \vert\kern-0.25ex \vert\kern-0.25ex \vert}: = \sup\limits_{0 \leq t \leq T} \|F(t) \|_{{\bf{X}}}. |
Now, we are concerned with the global solvability of (39). Let
\begin{equation} \begin{cases} \partial_t W + p\cdot \nabla_x W + \Theta[V](W) \\ \quad = \frac{ \kappa}{2N} \left( E_{ij} W C_j + R_i W E_{ij} - W \int_{ \mathbb R^{2d}} ( E_{ij} G C_j + R_i G E_{ij}) \mathrm{d} x \mathrm{d} p \right), \\ W(0) = W^0. \end{cases} \end{equation} | (40) |
We need to check well-definedness and strict contraction of
● (Well-definedness of
● (Strict contraction): for
{ \vert\kern-0.25ex \vert\kern-0.25ex \vert {W- \overline W } \vert\kern-0.25ex \vert\kern-0.25ex \vert} \leq C { \vert\kern-0.25ex \vert\kern-0.25ex \vert {G -\overline G} \vert\kern-0.25ex \vert\kern-0.25ex \vert} . |
If we set
\begin{align*} &\partial_t w_{ij} + p \cdot \nabla_x w_{ij} + \Theta[V](\omega_{ij})\\& = \frac{ \kappa}{2N} \sum\limits_{k = 1}^N \left[ w_{ik} + w_{kj} - w_{ij} \left( \int_{ \mathbb R^{2d}} (g_{ik} + g_{kj}) \mathrm{d} x \mathrm{d} p \right)\right], \\ &\partial_t\overline w_{ij} + p \cdot \nabla_x \overline w_{ij} + \Theta[V](\overline \omega_{ij})\\& = \frac{ \kappa}{2N} \sum\limits_{k = 1}^N \left[ \overline w_{ik} + \overline w_{kj} - \overline w_{ij} \left( \int_{ \mathbb R^{2d}} (\overline g_{ik} + \overline g_{kj}) \mathrm{d} x \mathrm{d} p \right)\right]. \end{align*} |
For simplicity, we set
q_{ij}(t) : = \int_{ \mathbb R^{2d}} g_{ij} \mathrm{d} x \mathrm{d} p, \quad \overline q_{ij} (t) : = \int_{ \mathbb R^{2d}} \overline g_{ij} \mathrm{d} x \mathrm{d} p. |
By straightforward calculation,
\begin{equation} \frac12\frac {\mathrm{d}} {\mathrm{d}t} \max\limits_{1\leq i, j\leq N} \|w_{ij} - \overline w_{ij}\| \leq C \max\limits_{1\leq i, j\leq N} \|w_{ij} - \overline w_{ij}\| + \max\limits_{1\leq i, j\leq N} \|\omega_{ij}\| \max\limits_{1\leq i, j \leq N} \left| q_{ij} - \overline q_{ij} \right| . \end{equation} | (41) |
In addition, we observe
\begin{equation} \frac {\mathrm{d}} {\mathrm{d}t} \max\limits_{1\leq i, j\leq N} |z_{ij} - \overline z_{ij}| \leq C \max\limits_{1\leq i, j\leq N} |z_{ij} - \overline z_{ij}| + \tilde C \max\limits_{1\leq i, j\leq N} |q_{ij} - \overline q_{ij}|. \end{equation} | (42) |
It follows from Corollary 1 that
\max\limits_{1\leq i, j\leq N } \|w_{ij}(t) \| \leq \mathcal R^\infty, \quad t > 0. |
Then, (41) and (42) yield
\begin{equation} \frac {\mathrm{d}} {\mathrm{d}t} \|W- \overline W\|_{{\bf{X}}} \leq C_1 \|W- \overline W\|_{{\bf{X}}} + e^{C_2t} \|G- \overline G\|_{{\bf{X}}}, \end{equation} | (43) |
and integrate the relation (43) to find
\begin{align*} \|W- \overline W\|_{\bf{X}} \leq \|W^0- \overline W^0\|_{\bf{X}} + C_1 \int_0^t \|W- \overline W\|_{\bf{X}} \mathrm{d} s + e^{C_2T} \int_0^t \|G- \overline G \|_{\bf{X}} \mathrm{d} s. \end{align*} |
Since
\begin{align*} { \vert\kern-0.25ex \vert\kern-0.25ex \vert {W- \overline W} \vert\kern-0.25ex \vert\kern-0.25ex \vert} & = \sup\limits_{0\leq t \leq T} \|W-\overline W\|_{\bf{X}} \leq Te^{(C_1+C_2)T} \sup\limits_{0\leq t \leq T} \|G- \overline G \|_{\bf{X}}\\ & = Te^{(C_1+C_2)T} { \vert\kern-0.25ex \vert\kern-0.25ex \vert {G- \overline G} \vert\kern-0.25ex \vert\kern-0.25ex \vert}. \end{align*} |
If
W(t) = W_n(t-nT), \quad t \in [nT, (n+1)T], \quad n\geq0. |
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
\begin{equation*} \label{E-12} (F(W))_{(i, j)}: = \frac{\kappa}{2N}\sum\limits_{k = 1}^N(z_{ik} + z_{kj})w_{ij} = \frac{ \kappa}{2N} \sum\limits_{k = 1}^N \left( \int_{ \mathbb R^{2d}} (w_{ik} + w_{kj}) \mathrm{d} x \mathrm{d} p \right)w_{ij}, \end{equation*} |
which is nonlinear with respect to the argument
Lemma 4.3. For
\begin{align*} \|F(U) - F(V)\|_{\bf{X}} \leq C\|U -V \|_{{\bf{X}}}. \end{align*} |
Then, the functional derivative, denoted by
Proof. For
\mathrm{D}F(U)(V) : = \lim\limits_{\tau \to 0} \frac{ F(U + \tau V) - F(U) }{\tau} = \frac{ \mathrm{d}}{ \mathrm{d} \tau} F(U+ \tau V) \biggl|_{\tau = 0}. |
At each point
\begin{align*} (\mathrm{D}F(U)(V))_{(i, j)} = \frac{\kappa}{2N}\sum\limits_{k = 1}^N \left(u_{ij} \int_{ \mathbb R^{2d}}(v_{ik} + v_{kj}) \mathrm{d} x \mathrm{d} p + v_{ij}\int_{ \mathbb R^{2d}}(u_{ik} + u_{kj}) \mathrm{d} x \mathrm{d} p\right). \end{align*} |
Since
\|\mathrm{D}F(U)(V) \|_{\bf{X}}\leq 2 \kappa \|U\|_{{\bf{X}}} \cdot \|V\|_{{\bf{X}}}. |
Therefore, we verified that
\|F(U) - F(V)\|_{\bf{X}} \leq \|U- V \|_{\bf{X}} \cdot \sup\limits_{t \in [0, 1]}\|\mathrm{D}F(tU + (1-t)V)\|_{ \text{op}}. |
Here,
\|F(U) - F(V)\|_{\bf{X}} \leq 2\kappa \sup\limits_{t \in [0, 1]}\|tU + (1-t)V\|_{\bf{X}} \cdot \|U - V \|_{\bf{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
\begin{equation} \partial_t w_{ij} + p \cdot \nabla_x w_{ij} + \Theta[V](w_{ij}) = 0. \end{equation} | (44) |
Since (44) on
\|\Theta[V]w\|_{\mathcal{X}} = \|\Theta[V]w\|. |
Since
w_{ij} \in C([0, T];\mathcal{X}) \cap C^1([0, T];D(A)). |
For details, we refer the reader to [23,Theorem 1].
● Step B (the WL model on
w_{ij}\in C([0, T];\mathcal{X}) \cap C^1([0, T];D(A)), \quad i, j \in [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)\| \leq Me^{\omega t}. |
(ii)
Then,
\|S(t)\| \leq Me^{(\omega + M\|B\|)t}. |
Consider the following abstract Cauchy problem:
\begin{equation} \begin{cases} \frac{ \mathrm{d} u(t)}{ \mathrm{d}t} + Au(t) = f(t, u(t)), \quad t > t_0, \\ u(t_0) = u_0. \end{cases} \end{equation} | (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 \leq \|x-y\|_X \sup\limits_{0 \leq \theta \leq 1} \|Df(\theta x + (1-\theta)y\|_{\mathcal{L}(X, Y)}, |
where
\delta_v f(x) : = \lim\limits_{t \rightarrow \infty} \frac{f(x+tv) - f(x)}{t}. |
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