Global derivation of the 1D Vlasov-Poisson equation from quantum many-body dynamics with screened Coulomb potential

Xuwen Chen Department of Mathematics, University of Rochester, Rochester, NY 14627, USA xuwenmath@gmail.com , Shunlin Shen School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, Anhui Province, China slshen@ustc.edu.cn , Ping Zhang Academy of Mathematics & Systems Science and Hua Loo-Keng Center for Mathematical Sciences, Chinese Academy of Sciences, Beijing 100190, China, and School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China zp@amss.ac.cn and Zhifei Zhang School of Mathematical Sciences, Peking University, Beijing, 100871, China zfzhang@math.pku.edu.cn

Abstract.

We study the 1D quantum many-body dynamics with a screened Coulomb potential in the mean-field setting. Combining the quantum mean-field, semiclassical, and Debye length limits, we prove the global derivation of the 1D Vlasov-Poisson equation. We tackle the difficulties brought by the pure state data, whose Wigner transforms converge to Wigner measures. We find new weighted uniform estimates around which we build the proof. As a result, we obtain, globally, stronger limits, and hence the global existence of solutions to the 1D Vlasov-Poisson equation subject to such Wigner measure data, which satisfy conservation laws of mass, momentum, and energy, despite being measure solutions. This happens to solve the 1D case of an open problem regarding the conservation law of the Vlasov-Poisson equation raised in [18] by Diperna and Lions.

Key words and phrases:

Quantum Many-body Dynamics, Vlasov-Poisson Equation, Global Weak Solution, Quantum Mean-field Approximation, Semiclassical Limit

2010 Mathematics Subject Classification:

Primary 35Q55, 35Q83, 35D30; Secondary 35A01, 81V70, 82C70.

1. Introduction

Per the superposition principle, the dynamics of $N$ quantum particles interacting through a two-body interaction potential are governed by the linear $N$ -body Schrödinger equation

(1.1)

\left\{\begin{aligned} i\hbar\partial_{t}\Psi_{N,\hbar,\varepsilon}=&H_{N,\hbar,\varepsilon}\Psi_{N,\hbar,\varepsilon},\\ \Psi_{N,\hbar,\varepsilon}(0)=&\Psi_{N,\hbar}^{\mathrm{in}},\end{aligned}\right.

where $\Psi_{N,\hbar,\varepsilon}(t,x_{1},..,x_{N})\in\mathbb{C}$ is the $N$ -particle wave function at time $t$ and the Hamiltonian operator is

(1.2)

\displaystyle H_{N,\hbar,\varepsilon}=\sum_{j=1}^{N}-\frac{1}{2}\hbar^{2}\Delta_{x_{j}}+\frac{1}{N}\sum_{1\leq j<k\leq N}V_{\varepsilon}(x_{j}-x_{k}).

In many physical systems dealing with charges in which electro-magnetism is involved, an important physically observable phenomenon is the screening effect, which arises from the collective behaviors of charged particles and modifies the long-range Coulomb potential into an exponentially decaying form at a distance. The concept of a screened Coulomb potential arises in the physics of many-body systems, particularly in plasma physics, condensed matter physics, and certain areas of molecular physics. For example, for an electrically neutral system, the distribution of charges gives rise to an electric potential $V(x)$ that satisfies Poisson’s equation

\displaystyle\nabla^{2}V(x)=-\sum_{j=1}^{N}q_{j}n_{j}(x),

where $q_{j}$ is the charge and $n_{j}(x)$ is the concentration at position $x$ . Under suitable physical assumptions, one often reduces the Poisson’s equation to a simpler one

\displaystyle(\nabla^{2}-\varepsilon^{2})V(x)=\delta(x),

where the parameter $\varepsilon$ denotes the Debye length that characterizes different physical regimes. For more details on the derivation of a screened Coulomb potential, see also the standard monograph [2]. For more physical background on the screened Coulomb potential, see for instance [16, 33, 34, 40, 44, 47].

In the paper, we consider the 1D screened Coulomb potential

(1.3)

\displaystyle V_{\varepsilon}(x)=\pm\frac{1}{2}|x|e^{-\varepsilon|x|},

where the sign $\pm$ denotes defocusing/focusing. Here, the form (1.3) is a version of approximate solution to the 1D Poisson’s equation.

Totally different from the 3D Coulomb potential $\frac{1}{|x|}$ which has a slow decay at the infinity, the 1D Coulomb potential $|x|$ tends to infinity as $|x|\to\infty$ . Hence, for the 1D interacting systems, it is reasonable to consider the screened Coulomb potential model, as it seems to be counterintuitive that the interaction force grows to be infinitely large with the distance between particles increasing to infinity. From the perspective of physics, the screening effect is widely present in many physical systems. In fact, the Debye length is an experimentally observable parameter of $N$ -body systems. Some people even use that to define the experimental regimes.

Taking into account the screening effect, the effective interaction range between particles, by which different physical regimes are characterized, is quantified by the Debye length. The most interesting regime might be the Debye length limit $\varepsilon\to 0$ , as the full 1D Coulomb potential is formally recovered in the limit. Thus, not only in the theoretical physics but also in the numerical computation, it is common to take the screened model as an approximation.

Nevertheless, it is a challenge to provide a rigorous proof, as the 1D screened Coulomb potential is far from a perturbation or a regularized model for the Coulomb potential. Moreover, a key goal in mathematical physics is to understand how nonlinear equations of classical physics emerge as descriptions of quantum microscopic linear dynamics in appropriate asymptotic regimes. Staring from the quantum many-body dynamics (1.1), we are concerned with the asymptotic limit of the $N$ -body wave function as the particle number $N\to\infty$ , the Planck’s constant $\hbar\to 0$ , and the Debye length $\varepsilon\to 0$ , which leads to a kinetic equation, the Vlasov-Poisson equation

(1.4)

\left\{\begin{aligned} &\partial_{t}f+\xi\partial_{x}f+E\partial_{\xi}f=0,\\ &\partial_{x}E=\pm\int_{\mathbb{R}}fd\xi,\\ &f(0)=f_{0}.\end{aligned}\right.

The Vlasov-Poisson systems describe the evolution of the distribution function $f(t,x,\xi)$ of particle under a self-consistent electric or gravitational field. There have been many developments such as [18, 17, 39, 51] on the global well-posedness problem of weak/measure solutions to the Vlasov-Poisson equation. Moreover, as pointed out in the review [18, p.278], apart from the uniqueness and regularity, the conservation law for the weak solutions in the kinetic theory is an important open question.

The quantum many-body dynamics (1.1) and the kinetic equation (1.4) are linked by the Wigner transform, which takes the form that

(1.5)

\displaystyle f_{N,\hbar,\varepsilon}^{(1)}(t,x,\xi)=W_{\hbar}[\gamma_{N,\hbar,\varepsilon}^{(1)}](t,x,\xi)=\frac{1}{2\pi}\int_{\mathbb{R}}e^{-i\xi y}\gamma_{N,\hbar,\varepsilon}^{(1)}\left(t,x+\frac{\hbar y}{2},x-\frac{\hbar y}{2}\right)dy,

where the first marginal density is

(1.6)

\displaystyle\gamma_{N,\hbar,\varepsilon}^{(1)}(t,x,x^{\prime})=\int_{\mathbb{R}^{N-1}}\Psi_{N,\hbar,\varepsilon}(t,x,x_{2},..,x_{N})\overline{\Psi_{N,\hbar,\varepsilon}(t,x^{\prime},x_{2},..,x_{N})}dx_{2}\cdot\cdot\cdot dx_{N}.

The Wigner function $f_{N,\hbar,\varepsilon}^{(1)}(t,x,\xi)$ turns the spatial marginal density into a real-valued density on the phase space, and satisfies induced basic properties in kinetic theory such as the conservation laws of mass, momentum, and energy.

Our goal is to justify the limit process in which the Wigner function (1.5) from the quantum many-body dynamics (1.1) tends to the Vlasov-Poisson equation (1.4).

Theorem 1.1 (Main theorem).

Let $\Psi_{N,\hbar,\varepsilon}(t)$ be the solution to the $N$ -body dynamics (1.1), and $f_{N,\hbar,\varepsilon}^{(1)}(t)$ be the Wigner transform of $\gamma_{N,\hbar,\varepsilon}^{(1)}(t)$ . Assume the initial data $\Psi_{N,\hbar,\varepsilon}(0)$ is normalized and factorized in the sense that

\displaystyle\Psi_{N,\hbar}^{\mathrm{in}}=\prod_{j=1}^{N}\psi_{\hbar}^{\mathrm{in}}(x_{j}),\quad\|\psi_{\hbar}^{\mathrm{in}}\|_{L_{x}^{2}}=1,

and $\psi_{\hbar}^{\mathrm{in}}$ satisfies the uniform bounds

(1.7)

\displaystyle\||x|\psi_{\hbar,\varepsilon}^{\mathrm{in}}\|_{L_{x}^{2}}\leq C,\quad\|\hbar^{k}\partial_{x}^{k}\psi_{\hbar}^{\mathrm{in}}\|_{L_{x}^{2}}\leq C^{k}k^{k},\quad k\geq 0.

Then there exist a subsequence of $\left\{f_{N,\hbar,\varepsilon}^{(1)}\right\}$ , which we still denote by $\left\{f_{N,\hbar,\varepsilon}^{(1)}\right\}$ , and a non-negative bounded Radon measure

f(t,dx,d\xi)\in C([0,\infty);\mathcal{M}^{+}(\mathbb{R}^{2})-w^{*}),

such that

(1.8)

\displaystyle\lim_{(N,\hbar,\varepsilon)\to(\infty,0,0)}\int_{0}^{T}\iint_{\mathbb{R}^{2}}\left(f_{N,\hbar,\varepsilon}^{(1)}(t,x,\xi)-f(t,x,\xi)\right)\phi dxd\xi dt=0,\quad

for all $T>0$ and $\phi\in L_{t}^{1}([0,T];\mathcal{A})$ , where the space $\mathcal{A}$ is defined in (4.1). The Wigner measure $f(t,dx,d\xi)$ is a weak solution to the Vlasov-Poisson equation (1.4) with the initial measure datum $f(0,dx,d\xi)$ in the sense of Definition A.1. Moreover, the Wigner measure $f(t,dx,d\xi)$ satisfies the conservation laws of mass, momentum, and energy

(1.9)			$\displaystyle\iint_{\mathbb{R}^{2}}f(t,dx,d\xi)=\iint_{\mathbb{R}^{2}}f(0,dx,d\xi),$
(1.10)			$\displaystyle\iint_{\mathbb{R}^{2}}\xi f(t,dx,d\xi)=\iint_{\mathbb{R}^{2}}\xi f(0,dx,d\xi),$
(1.11)			$\displaystyle\iint_{\mathbb{R}^{2}}\xi^{2}f(t,dx,d\xi)\pm\frac{1}{2}\iint_{\mathbb{R}^{2}}\|x-y\|\rho(t,dx)\rho(t,dy)$
	$\displaystyle=$	$\displaystyle\iint_{\mathbb{R}^{2}}\xi^{2}f(0,dx,d\xi)\pm\frac{1}{2}\iint_{\mathbb{R}^{2}}\|x-y\|\rho(0,dx)\rho(0,dy),$

where $\rho(t,dx)=\int_{\mathbb{R}}f(t,dx,\xi)d\xi$ .

Remark 1.2 (Global existence and conservation laws).

One could also consider Theorem 1.1 as proof of global existence of measure solutions to (1.4) subject to such Wigner measure data with conversation of mass, momentum, and energy. Staring from the quantum many-body dynamics, we happen to solve the 1D case of an open problem regarding the conservation law of the Vlasov-Poisson equation raised in [18] by Diperna and Lions.

Remark 1.3 (Existence of initial data).

One can choose the initial data as $\psi_{\hbar}^{\mathrm{in}}*j_{\hbar}$ , where the mollifier $j_{\hbar}(x)=\hbar^{-1}j(x/\hbar)$ with $0\leq j(x)\in\mathcal{S}(\mathbb{R})$ , $\int_{\mathbb{R}}j(x)dx=1$ , and $\int_{\mathbb{R}}|\partial_{x}^{k}j|dx\leq Ck^{k}$ for all $k\geq 0$ . Then the uniform bounds (1.7) are satisfied. For example, one can choose $j(x)=\pi^{-1}e^{-x^{2}}$ .

Remark 1.4 (The torus case).

With some modifications, Theorem 1.1 can be extended to the torus case for the Coulomb potential, as the screening effect is more specialized for $\mathbb{R}$ and there is no essential difference between the unscreened and screened cases on $\mathbb{T}$ from the mathematical view.

Remark 1.5 (Fixed Debye length).

Our proof also works for any fixed Debye length. The limit equation would then be a Vlasov equation with a screened Coulomb potential characterized by the Debye length.

Currently, there have been many nice developments [6, 8, 15, 20, 28, 29, 30, 31, 37, 38, 41, 45, 48, 50] devoted to the derivation of the Vlasov-type equations from quantum systems. The semiclassical limit of the one-body Schrödinger equation leading to the Vlasov equation was first systematically studied in [38]. For the Coulomb potential case, in [38, 41], this problem was solved for a mixed state initial data

(1.12)

\displaystyle\sum_{j=1}^{\infty}\lambda_{j}^{\hbar}\psi_{j}^{\hbar}(x)\overline{\psi_{j}^{\hbar}(x^{\prime})},\quad\sum_{j=1}^{\infty}\lambda_{j}^{\hbar}=1,

under the uniform bound condition

(1.13)

\displaystyle\frac{1}{\hbar^{3}}\sum_{j=1}^{\infty}(\lambda_{j}^{\hbar})^{2}\leq C.

However, a pure state in which $j=1$ , $\lambda_{j}^{\hbar}=1$ cannot satisfy (1.13). It was then solved in [50] for the 1D case with general initial data including the pure state densities. For the higher dimensional case, apart from the local derivation for the monokinetic case such as [30, 45, 48], it remains an open problem for the global derivation of the 3D Vlasov-Poisson equation from the quantum and classical microscopic systems.

In our setting of justifying the global limit to the weak solution of the Vlasov-Poisson equation from the 1D quantum many-body dynamics, there are also several hard problems which we list below.

(1)

The problem of the pure state density. The quantum mean-field problem in the $N\to\infty$ limit is closely related to the Bose-Einstein condensate, a physical phenomenon that all particles take the same quantum state. That is, the $N$ -body wave function takes the product form that

\displaystyle\Psi_{N,\hbar,\varepsilon}(t,x_{1},..,x_{N})\sim\prod_{j=1}^{N}\psi_{\hbar,\varepsilon}(t,x_{j}),

which yields a pure state marginal density

\gamma_{N,\hbar,\varepsilon}^{(1)}(t,x,x^{\prime})\sim\psi_{\hbar,\varepsilon}(t,x)\overline{\psi_{\hbar,\varepsilon}}(t,x^{\prime}).

However, the Wigner transform of a pure state density is only known to converge to a Wigner measure as pointed by Lions and Paul in [38]. That is, to obtain the Vlasov-Poisson equation from a pure state density, we have to work in the non-smoothing setting and deal with a not only weak but also measure solution of the Vlasov-Poisson equation. Many exsiting strong-weak stability arguments such as the modulated energy method might not be valid, as the uniqueness of the limiting weak solution to the Vlasov-Poisson equation is unknown.

(2)

The non-smoothness of the potential at the origin. For the $C^{1,1}$ interaction potentials, the mean-field and semiclassical approximation to the Vlasov-type equation has been proven in [29]. The singularity at the origin hinders the application of the method in [29]. Hence, new ideas are required to deal with the singularity at the origin.

(3)

Weak convergence problem in the Debye length limit. To recover the Vlasov-Poisson equation with the full 1D Coulomb potential which is singular at both the origin and the infinity, we need to establish the limit process, which only holds in the weak sense that

\displaystyle\lim_{\varepsilon\to 0}\int_{\mathbb{R}}V_{\varepsilon}(x)\varphi(x)dx=\int_{\mathbb{R}}\frac{1}{2}|x|\varphi(x)dx,\quad\forall\varphi\in C_{c}(\mathbb{R}).

Therefore, two weak limits, the semiclassical and Debye length limits, entangle here. Especially for the convergence of the nonlinear term, it requires new uniform estimates and a cancellation structure to deal with these two weak limits at the same time.

(4)

The conservation laws for the limit measure solution. As we start from basic physics model, the linear $N$ -body dynamics, it is naturally desired that the limit solution satisfies more physical properties, such as the conservation laws. However, from the view of mathematics, like many other open problems such as the conservation of energy for the renormalized solution of the Boltzmann equation [19] and the Vlasov-Poisson equation [18, 17], it is highly non-trivial to prove these conservation laws for the limit weak measure solution.

1.1. Outline of the Proof of the Main Theorem

We divide the proof into the following five steps.

Step 1. Preliminary reduction to a one-body nonlinear Schrödinger equation.

Since the first wave of work, for example [1, 3, 21, 22, 23, 24, 25, 26, 27] and the references within on deriving the nonlinear Schrödinger equations from the quantum many-body dynamics with the delta-type and Coulomb potentials, there have been a large quantities of work on the study of the quantum mean-field limit using various methods, such as [4, 7, 5, 9, 10, 11, 12, 13, 14, 32, 35, 36, 42, 43]. One of the crucial step of the paper is to take the quantum mean-field limit and reduce the $N$ -body problem to the one-body nonlinear Schrödinger equation

\displaystyle i\hbar\partial_{t}\psi_{\hbar,\varepsilon}=

\displaystyle-\frac{1}{2}\hbar^{2}\partial_{x}^{2}\psi_{\hbar,\varepsilon}+(V_{\varepsilon}*|\psi_{\hbar,\varepsilon}|^{2})\psi_{\hbar,\varepsilon}.

We use directly the result in [4] by Ben Porat and Golse, and obtain

(1.14)

\displaystyle\|f_{N,\hbar,\varepsilon}^{(1)}(t)-f_{\hbar,\varepsilon}(t)\|_{L_{x,\xi}^{2}}\leq 4\sqrt{\frac{1}{N\hbar}}\exp\left(\sqrt{\frac{Ct}{\hbar^{3}\varepsilon}}\right),

where $f_{\hbar,\varepsilon}(t)=W_{\hbar}[\psi_{\hbar,\varepsilon}(t)]$ is the Wigner transform of the one-body wave function $\psi_{\hbar,\varepsilon}(t)$ . With this key observation, it suffices to study the limit problem for $f_{\hbar,\varepsilon}(t)$ .

Step 2. Weighted uniform energy estimates.

In Section 3, we introduce new weighted uniform estimates

(1.15)

\displaystyle\|\langle x\rangle^{\frac{1}{2}}\hbar^{k}\partial_{x}^{k}\psi_{\hbar,\varepsilon}(t)\|_{L_{x}^{2}}\leq

\displaystyle C(k,t),\quad\forall k\geq 1,

based on which we set up

(1.16)

\displaystyle\Big{\|}\langle x\rangle\hbar^{\alpha}\partial_{x}^{\alpha}\int_{\mathbb{R}}\xi^{k}f_{\hbar,\varepsilon}(t,x,\xi)d\xi\Big{\|}_{L_{x}^{1}}\leq

\displaystyle C(k,\alpha,t),

where $\langle x\rangle=\sqrt{1+x^{2}}$ . The weighted uniform estimates are new, and 1D specific for our subsequent analysis including the compactness, convergence and the conservation laws for the limit solution. The proof and usage of (1.15) and (1.16) are the key.

Step 3. Compactness and convergence. In Section 4, using the uniform estimates in Section 3, we are able to obtain higher moment difference estimates between the Wigner function and the Husimi function which is non-negative, and attain more properties. Then, we prove the compactness of the sequence $\left\{f_{\hbar,\varepsilon}(t,x,\xi)\right\}$ and justify the weak convergence (up to a subsequence) to a non-negative bounded Radon measure

\displaystyle f(t,dx,d\xi)\in C([0,\infty);\mathcal{M}^{+}(\mathbb{R}^{2})-w^{*}),

in the sense that for $\forall T>0$ , $k\geq 0$ , there hold

\displaystyle\lim_{(\hbar,\varepsilon)\to(0,0)}\int_{0}^{T}\iint_{\mathbb{R}^{2}}\left(\xi^{k}f_{\hbar,\varepsilon}(t,x,\xi)-\xi^{k}f(t,x,\xi)\right)\phi dxd\xi dt=0,\ \forall\phi\in L_{t}^{1}([0,T];\mathcal{A}),

where the test function space $\mathcal{A}$ is defined in (4.1). Our method here enables a direct proof that the limit is non-negative. Furthermore, with our method, for the convergence of the moment function $\int_{\mathbb{R}}\xi^{k}f_{\hbar,\varepsilon}(t,x,\xi)d\xi$ , we are able to prove the narrow convergence due to (1.15) and (1.16). That is,

\displaystyle\lim_{(\hbar,\varepsilon)\to(0,0)}\int_{0}^{T}\int_{\mathbb{R}}\left(\int_{\mathbb{R}}\xi^{k}f_{\hbar,\varepsilon}(t,x,\xi)d\xi-\int_{\mathbb{R}}\xi^{k}f(t,x,\xi)d\xi\right)\varphi dxdt=0,\ \forall\varphi\in L_{t}^{1}([0,T];C_{b}(\mathbb{R})).

The test functions belong to the space of the bounded continuous functions. This is the key to the conservation laws, as we can take the constant 1 as a test function now.

Step 4. Conservation laws for the limit solution.

In Section 5, we prove the conservation laws of mass, momentum, and energy for the limit measure solution as presented in (1.9)–(1.11). The mass, momentum, and kinetic energy parts follow from the narrow convergence. The difficult one is the convergence of the interaction potential energy, as the narrow convergence we obtain in Step 3 remains too weak to deal with the limit problem for the nonlinear term. To circumvent this problem, based on the weighted uniform estimates (1.15) and (1.16), we introduce a weighted transform to obtain the local strong convergence, which is the key to the convergence of the interaction potential energy.

Step 5. Convergence to the Vlasov-Poisson equation

The most intricate part is to verify the limit to the Vlasov-Poisson equation. In Sections 6–7, we follow the scheme in [50] to prove the moment convergence to the Vlasov-Poisson equation and establish the exponential decay for the limit measure, which is used to obtain the full convergence. More precisely, in Section 6, for the test function $\varphi(t,x)\xi^{k}$ , we obtain

(1.17)

\displaystyle\int_{\Omega_{T}}\int_{\mathbb{R}}\left(\partial_{t}\varphi+\xi\partial_{x}\varphi\right)\xi^{k}f(t,dx,d\xi)dt-k\int_{\Omega_{T}}\varphi\overline{E}\left(\int_{\mathbb{R}}\xi^{k-1}f(t,dx,d\xi)\right)dt=0,

where $\Omega_{T}=(0,T)\times\mathbb{R}$ and $\overline{E}$ is the Vol^′pert’s symmetric average defined in (A.1). Then we prove the exponential decay estimate that

\displaystyle\iint_{\Omega_{T}}\int_{\mathbb{R}}e^{\delta|\xi|}f(t,dx,d\xi)dt\leq C_{\delta},

based on which we prove

\displaystyle\partial_{t}f+\xi\partial_{x}f-\partial_{\xi}(\overline{E}f)=0,

in the sense of distributions in Section 7. (Notice the difference of test functions in Sections 6 and 7.) Hence, we conclude that the limit measure $f(t,dx,d\xi)$ is a weak solution to the Vlasov-Poisson equation.

During the proof of the convergence, the main difficulties lie in dealing with the vanishing problem of the remainder term

\displaystyle\mathrm{R}_{\hbar,\varepsilon}^{(k)}=i\sum_{2\leq\alpha\leq k}\binom{k}{\alpha}\frac{\hbar^{\alpha-1}}{2^{k}}(1-(-1)^{\alpha})D_{x}^{\alpha}(V_{\varepsilon}*\rho_{\hbar,\varepsilon})\int_{\mathbb{R}}\xi^{k-\alpha}f_{\hbar,\varepsilon}d\xi,

and the convergence problem of the nonlinear term

\displaystyle\left(\partial_{x}V_{\varepsilon}*\rho_{\hbar,\varepsilon}\right)\left(\int_{\mathbb{R}}\xi^{k-1}f_{\hbar,\varepsilon}d\xi\right),

both of which yield a entangled product limit problem in the double semiclassical and Debye length limit. Adding more to the difficulty, the limit point is a measure instead of a locally integrable function, which is also known to be a stubborn technical point. Nonetheless, using the new weighted uniform estimates (1.15)–(1.16) and an iteration scheme which reduces $\xi^{k+1}$ in (1.17) into $\xi^{k}$ and hence enables an induction, we can fortunately overcome these difficulties.

Finally, we put the definition of the weak solution to the Vlasov-Poisson equation in Appendix A and include some basic properties of the bounded variation functions in Appendix B. Putting together the results in the above steps $1$ – $5$ , we conclude Theorem 1.1.

2. Preliminary Reduction: Quantum Mean-field Limit

In this section, we take the quantum mean-field limit and reduce the quantum $N$ -body dynamics to the one-body nonlinear Schrödinger equation (NLS)

(2.1)

\left\{\begin{aligned} i\hbar\partial_{t}\psi_{\hbar,\varepsilon}=&-\frac{1}{2}\hbar^{2}\partial_{x}^{2}\psi_{\hbar,\varepsilon}+(V_{\varepsilon}*|\psi_{\hbar,\varepsilon}|^{2})\psi_{\hbar,\varepsilon},\\ \psi_{\hbar,\varepsilon}(0)=&\psi_{\hbar}^{\mathrm{in}}.\end{aligned}\right.

Certainly, there have been many methods developed to establish a quantitative measurement between the $N$ -body systems and the one-body NLS. Here, to make our limit problem of the quantum $N$ -body dynamics concise and clear, we use directly the result in [4], which is inspired by [42] and gives a convergence rate estimate between the $N$ -body dynamics and the one-body NLS with an explicit $\hbar$ -dependence.

Theorem 2.1 ([4, Corollary 4.2]).

Let $\Psi_{N,\hbar,\varepsilon}(t)$ be the solution to the $N$ -body dynamics (1.1) with the factorized initial data, $\gamma_{N,\hbar,\varepsilon}^{(1)}(t)$ be the first marginal density. Then it holds that

(2.2)

\displaystyle\operatorname{Tr}\Big{|}\gamma_{N,\hbar,\varepsilon}^{(1)}(t,x,x^{\prime})-\psi_{\hbar,\varepsilon}(t,x)\overline{\psi_{\hbar,\varepsilon}}(t,x^{\prime})\Big{|}\leq 4\sqrt{\frac{1}{N}}\exp\left(\frac{3}{\hbar}\int_{0}^{t}L_{\hbar,\varepsilon}(s)ds\right),

where

(2.3)

\displaystyle L_{\hbar,\varepsilon}(t):=C\|V_{\varepsilon}\|_{L_{x}^{\infty}}\|\psi_{\hbar,\varepsilon}(t)\|_{H^{2}}.

Using Theorem 2.1 and the energy estimate (3.3) in Section 3, we immediately obtain the following corollary.

Corollary 2.2.

Let $f_{N,\hbar,\varepsilon}^{(1)}(t)$ , $f_{\hbar,\varepsilon}(t)$ be the Wigner transform of $\gamma_{N,\hbar,\varepsilon}^{(1)}(t)$ , $\psi_{\hbar,\varepsilon}(t)$ respectively. We have

(2.4)

\displaystyle\|f_{N,\hbar,\varepsilon}^{(1)}(t)-f_{\hbar,\varepsilon}(t)\|_{L_{x,\xi}^{2}}\leq 4\sqrt{\frac{1}{N\hbar}}\exp\left(\sqrt{\frac{Ct}{\hbar^{3}\varepsilon}}\right).

Proof.

By Plancherel identity and the operator inequality that $\|A\|_{HS}\leq\operatorname{Tr}|A|$ , we obtain

(2.5)		$\displaystyle\\|f_{N,\hbar,\varepsilon}^{(1)}(t)-f_{\hbar,\varepsilon}(t)\\|_{L_{x,\xi}^{2}}=$	$\displaystyle\\|W_{\hbar}[\gamma_{N,\hbar,\varepsilon}^{(1)}(t)]-W_{\hbar}[\psi_{\hbar,\varepsilon}(t)]\\|_{L_{x,\xi}^{2}}$
	$\displaystyle=$	$\displaystyle\hbar^{-\frac{1}{2}}\big{\\|}\gamma_{N,\hbar,\varepsilon}^{(1)}(t,x,x^{\prime})-\psi_{\hbar,\varepsilon}(t,x)\overline{\psi_{\hbar,\varepsilon}}(t,x^{\prime})\big{\\|}_{L_{x,x^{\prime}}^{2}}$
	$\displaystyle\leq$	$\displaystyle\hbar^{-\frac{1}{2}}\operatorname{Tr}\Big{\|}\gamma_{N,\hbar,\varepsilon}^{(1)}(t,x,x^{\prime})-\psi_{\hbar,\varepsilon}(t,x)\overline{\psi_{\hbar,\varepsilon}}(t,x^{\prime})\Big{\|}.$

Using (2.2), energy estimate (3.3), and $\|V_{\varepsilon}\|_{L_{x}^{\infty}}\leq\varepsilon^{-1}$ , we arrive at (2.4). ∎

The convergence rate estimate (2.4) is enough for the limit problem up to a subsequence. Therefore, in the follow Sections 3–7, we start from the nonlinear Schrödinger equation (2.1) and justify its limit to the Vlasov-Poisson equation.

3. Weighted Uniform Higher Energy Estimates

In this section, we set up the weighted uniform higher energy estimates on the one-body wave function $\psi_{\hbar,\varepsilon}(t)$ of the nonlinear Schrödinger equation (2.1). Then using the weighted uniform estimates, we provide the higher derivative and weighted uniform estimates for the higher moments of $f_{\hbar,\varepsilon}(t,x,\xi)$ in Lemma 3.2.

Lemma 3.1 (Weighted uniform estimates).

Let $\rho_{\hbar,\varepsilon}(t)=|\psi_{\hbar,\varepsilon}(t)|^{2}$ . We have

(3.1)		$\displaystyle\\|\langle x\rangle^{2}\rho_{\hbar,\varepsilon}(t)\\|_{L_{x}^{1}}\leq$	$\displaystyle C(t),$
(3.2)		$\displaystyle\\|\langle x\rangle^{\frac{1}{2}}\hbar^{k}\partial_{x}^{k}\psi_{\hbar,\varepsilon}(t)\\|_{L_{x}^{2}}\leq$	$\displaystyle C(k,t),\quad\forall k\geq 1,$

where $\langle x\rangle=\sqrt{1+x^{2}}$ .

Proof.

Estimate (3.1) is usually called a virial estimate, while estimate (3.2) is a new weighted uniform estimate, which might not be true for the higher dimension case.

Before proving the weighted uniform estimates (3.1)–(3.2), we set up the higher energy estimates

(3.3)

\displaystyle\big{\|}\hbar^{k}\partial_{x}^{k}\psi_{\hbar,\varepsilon}(t)\big{\|}_{L_{x}^{2}}\leq C(k,t).

For $k=0$ , estimate (3.3) just follows from the mass conservation law of (2.1). For the defousing case, we can also obtain (3.3) with $k=1$ by using the energy conservation law of (2.1), as the potential energy is positive. However, such an argument is not valid for the focusing case. To provide a unified proof, we take the induction argument to deal with the general case $k\geq 1$ for both defocusing and focusing cases.

We assume that (3.3) holds for $n\leq k-1$ , and we prove it for $n=k$ . Using the nonlinear Schrödinger equation (2.1), we obtain

		$\displaystyle\frac{1}{2}\frac{d}{dt}\\|\hbar^{k}\partial_{x}^{k}\psi_{\hbar,\varepsilon}\\|_{L_{x}^{2}}^{2}$
	$\displaystyle=$	$\displaystyle\operatorname{Re}\int\overline{\partial_{t}\hbar^{k}\partial_{x}^{k}\psi_{\hbar,\varepsilon}}\hbar^{k}\partial_{x}^{k}\psi_{\hbar,\varepsilon}dx$
	$\displaystyle=$	$\displaystyle-\operatorname{Im}\int\overline{\hbar^{k-1}\partial_{x}^{k}\left[(V_{\varepsilon}*\rho_{\hbar,\varepsilon})\psi_{\hbar,\varepsilon}\right]}\hbar^{k}\partial_{x}^{k}\psi_{\hbar,\varepsilon}dx$
	$\displaystyle=$	$\displaystyle-\operatorname{Im}\int\overline{(\hbar^{k-1}\partial_{x}^{k}\left[(V_{\varepsilon}\rho_{\hbar,\varepsilon})\psi_{\hbar,\varepsilon}\right]-(V_{\varepsilon}\rho_{\hbar,\varepsilon})\hbar^{k-1}\partial_{x}^{k}\psi_{\hbar,\varepsilon})}\hbar^{k}\partial_{x}^{k}\psi_{\hbar,\varepsilon}dx.$

Then by Hölder’s inequality, Leibniz rule, and Young’s inequality, we get

(3.4)			$\displaystyle\frac{1}{2}\frac{d}{dt}\\|\hbar^{k}\partial_{x}^{k}\psi_{\hbar,\varepsilon}\\|_{L_{x}^{2}}^{2}$
	$\displaystyle\leq$	$\displaystyle\big{\\|}\hbar^{k-1}\partial_{x}^{k}\left[(V_{\varepsilon}\rho_{\hbar,\varepsilon})\psi_{\hbar,\varepsilon}\right]-(V_{\varepsilon}\rho_{\hbar,\varepsilon})\hbar^{k-1}\partial_{x}^{k}\psi_{\hbar,\varepsilon}\big{\\|}_{L_{x}^{2}}\\|\hbar^{k}\partial_{x}^{k}\psi_{\hbar,\varepsilon}\\|_{L_{x}^{2}}$
	$\displaystyle\leq$	$\displaystyle\sum_{j=1}^{k}\binom{k}{j}\\|\hbar^{j-1}\partial_{x}^{j}V_{\varepsilon}*\rho_{\hbar,\varepsilon}\\|_{L_{x}^{\infty}}\\|\hbar^{k-j}\partial_{x}^{k-j}\psi_{\hbar,\varepsilon}\\|_{L_{x}^{2}}\\|\hbar^{k}\partial_{x}^{k}\psi_{\hbar,\varepsilon}\\|_{L_{x}^{2}}$
	$\displaystyle\leq$	$\displaystyle\sum_{j=1}^{k}\binom{k}{j}\\|\partial_{x}V_{\varepsilon}\\|_{L_{x}^{\infty}}\hbar^{j-1}\\|\partial_{x}^{j-1}\rho_{\hbar,\varepsilon}\\|_{L_{x}^{1}}\\|\hbar^{k-j}\partial_{x}^{k-j}\psi_{\hbar,\varepsilon}\\|_{L_{x}^{2}}\\|\hbar^{k}\partial_{x}^{k}\psi_{\hbar,\varepsilon}\\|_{L_{x}^{2}}$
	$\displaystyle\leq$	$\displaystyle C\sum_{j=1}^{k}\binom{k}{j}\hbar^{j-1}\\|\partial_{x}^{j-1}\rho_{\hbar,\varepsilon}\\|_{L_{x}^{1}}\\|\hbar^{k-j}\partial_{x}^{k-j}\psi_{\hbar,\varepsilon}\\|_{L_{x}^{2}}\\|\hbar^{k}\partial_{x}^{k}\psi_{\hbar,\varepsilon}\\|_{L_{x}^{2}},$

where in the last inequality we have used $\|\partial_{x}V_{\varepsilon}\|_{L_{x}^{\infty}}\leq C$ . Using again Leibniz rule and Hölder’s inequality, we have

(3.5)

\displaystyle\hbar^{j-1}\|\partial_{x}^{j-1}\rho_{\hbar,\varepsilon}\|_{L_{x}^{2}}\leq

\displaystyle\sum_{j_{1}+j_{2}=j-1}\binom{j-1}{j_{1}}\|\hbar^{j_{1}}\partial_{x}^{j_{1}}\psi_{\hbar,\varepsilon}\|_{L_{x}^{2}}\|\hbar^{j_{2}}\partial_{x}^{j_{2}}\psi_{\hbar,\varepsilon}\|_{L_{x}^{2}}.

Plugging (3.5) into (3.4), we use (3.3) for the case $n<k$ to obtain

(3.6)

\displaystyle\frac{d}{dt}\|\hbar^{k}\partial_{x}^{k}\psi_{\hbar,\varepsilon}\|_{L_{x}^{2}}^{2}\leq C(k,t).

Noticing that the initial datum satisfies

\displaystyle\|\hbar^{k}\partial_{x}^{k}\psi_{\hbar,\varepsilon}(0)\|_{L_{x}^{2}}\leq C^{k}k^{k},

by (3.6) we arrive at

\displaystyle\|\hbar^{k}\partial_{x}^{k}\psi_{\hbar,\varepsilon}(t)\|_{L_{x}^{2}}\leq C(k,t),

which completes the proof of (3.3).

Now, we get into the proof of estimate (3.1). For $t\in[0,T]$ , we have

		$\displaystyle\frac{d}{dt}\int_{\mathbb{R}}e^{-2\delta\|x\|^{2}}\langle x\rangle^{2}\rho_{\hbar,\varepsilon}(t,x)dx$
	$\displaystyle=$	$\displaystyle-\int_{\mathbb{R}}e^{-2\delta\|x\|^{2}}\langle x\rangle^{2}\partial_{x}\left(\operatorname{Im}\left(\overline{\psi_{\hbar,\varepsilon}}\hbar\partial_{x}\psi_{\hbar,\varepsilon}\right)\right)dx$
	$\displaystyle=$	$\displaystyle\int_{\mathbb{R}}e^{-2\delta\|x\|^{2}}2x(1-2\delta\langle x\rangle^{2})\operatorname{Im}\left(\overline{\psi_{\hbar,\varepsilon}}\hbar\partial_{x}\psi_{\hbar,\varepsilon}\right)dx$
	$\displaystyle\lesssim$	$\displaystyle\\|e^{-\delta\|x\|^{2}}(\delta\langle x\rangle^{2}+1)\\|_{L_{x}^{\infty}}\\|e^{-\delta\|x\|^{2}}\|x\|\psi_{\hbar,\varepsilon}(t,x)\\|_{L_{x}^{2}}\\|\hbar\partial_{x}\psi_{\hbar,\varepsilon}(t,x)\\|_{L_{x}^{2}}$
	$\displaystyle\lesssim$	$\displaystyle C(T)\int_{\mathbb{R}}e^{-2\delta\|x\|^{2}}\langle x\rangle^{2}\rho_{\hbar,\varepsilon}(t,x)dx,$

where in the last inequality we used the energy estimate (3.3) with $k=1$ . By Gronwall’s inequality, we have

\displaystyle\|e^{-2\delta|x|^{2}}\langle x\rangle^{2}\rho_{\hbar,\varepsilon}(t,x)\|_{L_{x}^{1}}\leq C(t).

Letting $\delta\to 0$ and using Fatou’s lemma, we arrive at (3.1).

Next, we can prove the weighted uniform estimate (3.2). By integration by parts, Hölder’s inequality, virial estimate (3.1), and the higher energy estimates (3.3), we have

		$\displaystyle\Big{\\|}\left(\chi(\frac{x}{R})\langle x\rangle\right)^{\frac{1}{2}}\hbar^{k}\partial_{x}^{k}\psi_{\hbar,\varepsilon}\Big{\\|}_{L_{x}^{2}}^{2}$
	$\displaystyle=$	$\displaystyle\hbar^{2k}\Big{\|}\int_{\mathbb{R}}\chi(\frac{x}{R})\langle x\rangle\partial_{x}^{k}\psi_{\hbar,\varepsilon}\overline{\partial_{x}^{k}\psi_{\hbar,\varepsilon}}dx\Big{\|}$
	$\displaystyle\leq$	$\displaystyle\hbar^{2k}\sum_{\alpha=0}^{k}\binom{k}{\alpha}\Big{\|}\int_{\mathbb{R}}\psi_{\hbar,\varepsilon}\partial_{x}^{\alpha}\left(\chi(\frac{x}{R})\langle x\rangle\right)\overline{\partial_{x}^{2k-\alpha}\psi_{\hbar,\varepsilon}}dx\Big{\|}$
	$\displaystyle\leq$	$\displaystyle\\|\langle x\rangle\psi_{\hbar,\varepsilon}\\|_{L_{x}^{2}}\\|\chi(\frac{x}{R})\\|_{L_{x}^{\infty}}\\|\hbar^{2k}\partial_{x}^{2k}\psi_{\hbar,\varepsilon}\\|_{L_{x}^{2}}$
		$\displaystyle+\sum_{\alpha=1}^{k}\binom{k}{\alpha}\hbar^{\alpha}\\|\psi_{\hbar,\varepsilon}\\|_{L_{x}^{2}}\Big{\\|}\partial_{x}^{\alpha}\left(\chi(\frac{x}{R})\langle x\rangle\right)\Big{\\|}_{L_{x}^{\infty}}\\|\hbar^{2k-\alpha}\partial_{x}^{2k-\alpha}\psi_{\hbar,\varepsilon}\\|_{L_{x}^{2}}$
	$\displaystyle\leq$	$\displaystyle C(k,t),$

where in the last inequality we have used that

\displaystyle\hbar^{\alpha}\leq 1,\quad\Big{\|}\partial_{x}^{\alpha}\left(\chi(\frac{x}{R})\langle x\rangle\right)\Big{\|}_{L_{x}^{\infty}}\lesssim 1,\quad\forall\alpha\geq 1.

Sending $R\to\infty$ and using Fatou’s lemma, we arrive at (3.2). ∎

Now, we are able to provide the higher derivative and weighted uniform estimates for the higher moments of $f_{\hbar,\varepsilon}(t,x,\xi)$ . For simplicity, we define the Wigner function

(3.7)

\displaystyle W_{\hbar}[u_{1},u_{2}]:=\frac{1}{2\pi}\int_{\mathbb{R}}e^{-iz\xi}u_{1}(x+\frac{\hbar z}{2})\overline{u_{2}(x-\frac{\hbar z}{2})}dz,

and use the shorthand $W_{\hbar}[u]$ to denote $W_{\hbar}[u,u]$ . Next, we give two formulas of the Wigner function. Let $D_{x}:=\frac{1}{i}\partial_{x}$ .

•

The Wigner function with a weight function $\xi^{k}$ satisfies

(3.8)		$\displaystyle\xi^{k}W_{\hbar}[u_{1},u_{2}]=$	$\displaystyle\frac{1}{2\pi}\frac{\hbar^{k}}{2^{k}}\sum_{\alpha=0}^{k}\binom{k}{\alpha}(-1)^{k-\alpha}\int e^{-iz\xi}D_{x}^{\alpha}u_{1}(x+\frac{\hbar z}{2})\overline{D_{x}^{k-\alpha}u_{2}(x-\frac{\hbar z}{2})}dz$
	$\displaystyle=$	$\displaystyle\frac{1}{2\pi}\frac{\hbar^{k}}{2^{k}}\sum_{\alpha=0}^{k}\binom{k}{\alpha}(-1)^{k-\alpha}W_{\hbar}[D^{\alpha}u_{1},D^{k-\alpha}u_{2}].$

•

The integral of the Wigner function with a weight function $\xi^{k}$ satisfies

(3.9)

\displaystyle\int_{\mathbb{R}}\xi^{k}W_{\hbar}[u_{1},u_{2}]d\xi=\frac{\hbar^{k}}{2^{k}}\sum_{\alpha=0}^{k}\binom{k}{\alpha}(-1)^{k-\alpha}D_{x}^{\alpha}u_{1}\overline{D_{x}^{k-\alpha}u_{2}}.

Lemma 3.2.

For $t\in[0,T]$ , we have

(3.10)	$\displaystyle\Big{\\|}\langle x\rangle\hbar^{\alpha+1}\partial_{x}^{\alpha}\int_{\mathbb{R}}\xi^{k}f_{\hbar,\varepsilon}d\xi\Big{\\|}_{L_{x}^{\infty}}\leq$	$\displaystyle C(k,\alpha,t),\quad\forall k\geq 0,\ \alpha\geq 0,$
(3.11)	$\displaystyle\Big{\\|}\langle x\rangle\hbar^{\alpha}\partial_{x}^{\alpha}\int_{\mathbb{R}}\xi^{k}f_{\hbar,\varepsilon}d\xi\Big{\\|}_{L_{x}^{1}}\leq$	$\displaystyle C(k,\alpha,t),\quad\forall k\geq 0,\ \alpha\geq 0,$
(3.12)	$\displaystyle\\|\hbar^{k}\partial_{x}^{k}E_{\hbar,\varepsilon}\\|_{L_{x}^{\infty}}\leq$	$\displaystyle C(k,t),\quad\forall k\geq 0,$

where $E_{\hbar,\varepsilon}=\partial_{x}V_{\varepsilon}*\rho_{\hbar,\varepsilon}$ .

Proof.

For (3.10), by formula (3.9), Hölder’s inequality, Leibniz rule, we have

		$\displaystyle\Big{\\|}\langle x\rangle\hbar^{\alpha+1}\partial_{x}^{\alpha}\int_{\mathbb{R}}\xi^{k}f_{\hbar,\varepsilon}d\xi\Big{\\|}_{L_{x}^{\infty}}$
	$\displaystyle=$	$\displaystyle\Big{\\|}\langle x\rangle\hbar^{\alpha+1}\partial_{x}^{\alpha}\int_{\mathbb{R}}\xi^{k}W_{\hbar}[\psi_{\hbar,\varepsilon}]d\xi\Big{\\|}_{L_{x}^{\infty}}$
	$\displaystyle\leq$	$\displaystyle\sum_{k_{1}+k_{2}=k}\binom{k}{k_{1}}\Big{\\|}\langle x\rangle\hbar^{\alpha+1}\partial_{x}^{\alpha}\left(\hbar^{k_{1}}\partial_{x}^{k_{1}}\psi_{\hbar,\varepsilon}\overline{\hbar^{k_{2}}\partial_{x}^{k_{2}}\psi_{\hbar,\varepsilon}}\right)\Big{\\|}_{L_{x}^{\infty}}$
	$\displaystyle\leq$	$\displaystyle\sum_{k_{1}+k_{2}=k}\sum_{\alpha_{1}+\alpha_{2}=\alpha}\binom{k}{k_{1}}\binom{\alpha}{\alpha_{1}}\\|\langle x\rangle^{\frac{1}{2}}\hbar^{\alpha_{1}+k_{1}+\frac{1}{2}}\partial_{x}^{\alpha_{1}+k_{1}}\psi_{\hbar,\varepsilon}\\|_{L_{x}^{\infty}}\\|\langle x\rangle^{\frac{1}{2}}\hbar^{\alpha_{2}+k_{2}+\frac{1}{2}}\partial_{x}^{\alpha_{2}+k_{2}}\psi_{\hbar,\varepsilon}\\|_{L_{x}^{\infty}}$
	$\displaystyle\leq$	$\displaystyle C(k,\alpha,t),$

where in the last inequality we have used the weighted uniform estimates (3.2) and the interpolation inequality that

\displaystyle\|\langle x\rangle^{\frac{1}{2}}\hbar^{j+\frac{1}{2}}\partial_{x}^{j}\psi_{\hbar,\varepsilon}\|_{L_{x}^{\infty}}\leq\Big{\|}\hbar\partial_{x}\left(\langle x\rangle^{\frac{1}{2}}\hbar^{j}\partial_{x}^{j}\psi_{\hbar,\varepsilon}\right)\Big{\|}_{L_{x}^{2}}^{\frac{1}{2}}\|\langle x\rangle^{\frac{1}{2}}\hbar^{j}\partial_{x}^{j}\psi_{\hbar,\varepsilon}\|_{L_{x}^{2}}^{\frac{1}{2}}\leq C(j,t).

For (3.11), similarly we have

		$\displaystyle\Big{\\|}\langle x\rangle\hbar^{\alpha}\partial_{x}^{\alpha}\int_{\mathbb{R}}\xi^{k}f_{\hbar,\varepsilon}d\xi\Big{\\|}_{L_{x}^{1}}$
	$\displaystyle\leq$	$\displaystyle\sum_{k_{1}+k_{2}=k}\sum_{\alpha_{1}+\alpha_{2}=\alpha}\binom{k}{k_{1}}\binom{\alpha}{\alpha_{1}}\\|\langle x\rangle^{\frac{1}{2}}\hbar^{\alpha_{1}+k_{1}}\partial_{x}^{\alpha_{1}+k_{1}}\psi_{\hbar,\varepsilon}\\|_{L_{x}^{2}}\\|\langle x\rangle^{\frac{1}{2}}\hbar^{\alpha_{2}+k_{2}}\partial_{x}^{\alpha_{2}+k_{2}}\psi_{\hbar,\varepsilon}\\|_{L_{x}^{2}}$
	$\displaystyle\leq$	$\displaystyle C(k,\alpha,t).$

For (3.12), by Young’s inequality and (3.11), we get

\displaystyle\|\hbar^{k}\partial_{x}^{k}E_{\hbar,\varepsilon}\|_{L_{x}^{\infty}}=\big{\|}\partial_{x}V_{\varepsilon}*\left(\hbar^{k}\partial_{x}^{k}\rho_{\hbar,\varepsilon}\right)\big{\|}_{L_{x}^{\infty}}\leq\|\partial_{x}V_{\varepsilon}\|_{L_{x}^{\infty}}\|\hbar^{k}\partial_{x}^{k}\rho_{\hbar,\varepsilon}\|_{L_{x}^{1}}\leq C(k,t).

Hence, we have completed the proof of (3.10)–(3.12).

∎

4. Compactness and Narrow Convergence

In this section, with respect to the weak^∗ topology of the dual space of $\mathcal{A}$ defined in (4.1), we prove the compactness of the sequence $\left\{f_{\hbar,\varepsilon}(t,x,\xi)\right\}$ justify a weak convergence to a non-negative Radon measure $f(t,dx,d\xi)$ . In general, the Wigner transform of the wave function is only a real-valued function and may change sign. To ensure the non-negativity of the limit measure, we need to use the Husimi transform of the wave function, which fixes a non-negative sign.

In Section 4.1, we estimate the higher moment differences between the Wigner function and Husimi function, which is used to show that they have the same convergence and limit. Then in Section 4.2, it suffices to prove the convergence of the Husimi function to a non-negative measure, the proof of which relies on the weighted uniform estimates established in Section 3.

More specifically, for the convergence of the sequence $\left\{f_{\hbar,\varepsilon}(t,x,\xi)\right\}$ , we use the test function space introduced in [38]

(4.1)

\displaystyle\mathcal{A}=\left\{\phi\in C_{c}^{\infty}(\mathbb{R}^{2}):\left(\mathcal{F}_{\xi}\phi\right)(x,\eta)\in L^{1}\left(\mathbb{R}_{\eta},C_{c}\left(\mathbb{R}_{x}\right)\right)\right\},

equipped with the norm

\|\phi(x,\xi)\|_{\mathcal{A}}=\int_{\mathbb{R}}\sup_{x}\left|\left(\mathcal{F}_{\xi}\phi\right)(x,\eta)\right|d\eta,

where $\left(\mathcal{F}_{\xi}\phi\right)(x,\eta)$ is the Fourier transform of $\phi(x,\xi)$ with respect to $\xi$ .

Furthermore, for the convergence of the moment function $\int_{\mathbb{R}}\xi^{k}f_{\hbar,\varepsilon}(t,x,\xi)d\xi$ , we are able to prove the narrow convergence. That is, the test functions belong to the space of the bounded continuous functions, which we denote by $C_{b}(\mathbb{R})$ . The stronger narrow convergence is the key to the conservation laws for the limit measure in Section 5 and the moment convergence to the Vlasov-Poisson equation in Section 6.

4.1. Higher Moment Estimates between the Wigner and Husimi function

We first define the Husimi transform. For more details, see for instance [38, 49].

Definition 4.1.

Given $u\in L^{2}$ , the Husimi transform of $u$ is defined by

(4.2)

\displaystyle\widetilde{W}_{\hbar}\left[u\right]=W_{\hbar}\left[u\right]*_{(x,\xi)}G_{\hbar},

with

\displaystyle G_{\hbar}(x,\xi)=(\pi\hbar)^{-1}e^{-\frac{|x|^{2}}{\hbar}}e^{-\frac{|\xi|^{2}}{\hbar}}:=g_{\hbar}(x)g_{\hbar}(\xi).

An important property of the Husimi function is the non-negativity, that is, $\widetilde{W}_{\hbar}\left[u\right]\geq 0$ . To make use of the non-negativity of the Husimi function, we need to provide the higher moment estimates between the Wigner function and Husimi function as follows.

Lemma 4.2.

For $\phi(x,\xi)\in C_{c}^{\infty}(\mathbb{R}^{2})$ , there holds that

(4.3)

\displaystyle\Big{|}\iint_{\mathbb{R}^{2}}\xi^{k}W_{\hbar}[u_{1},u_{2}]\phi dxd\xi\Big{|}\leq\sum_{\alpha=0}^{k}\binom{k}{\alpha}\|\hbar^{\alpha}\partial_{x}^{\alpha}u_{1}\|_{L_{x}^{2}}\|\hbar^{k-\alpha}\partial_{x}^{k-\alpha}u_{2}\|_{L_{x}^{2}}\|\phi\|_{\mathcal{A}},

In particular, we obtain

(4.4)

\displaystyle\Big{|}\iint_{\mathbb{R}^{2}}\xi^{k}f_{\hbar,\varepsilon}(t,x,\xi)\phi dxd\xi\Big{|}\leq C(k,t)\|\phi\|_{\mathcal{A}}.

Moreover, for $\phi(x,\xi)\in C_{c}^{\infty}(\mathbb{R}^{2})$ and $\varphi(x)\in C_{c}^{\infty}(\mathbb{R})$ , we have the estimates on the difference between the Wigner function and Husimi function that

(4.5)			$\displaystyle\Big{\|}\iint_{\mathbb{R}^{2}}\left(\xi^{k}\widetilde{W}_{\hbar}[u]-\xi^{k}W_{\hbar}[u]\right)\phi dxd\xi\Big{\|}$
	$\displaystyle\leq$	$\displaystyle\\|u\\|_{L_{x}^{2}}^{2}\\|\phi*_{(x,\xi)}G_{\hbar}-\phi\\|_{\mathcal{A}}+C(k)\sum_{\alpha=0}^{k-1}\sum_{\alpha_{1}+\alpha_{2}=\alpha}\hbar^{k-\alpha}\\|\hbar^{\alpha_{1}}\partial_{x}^{\alpha_{1}}u\\|_{L_{x}^{2}}\\|\hbar^{\alpha_{2}}\partial_{x}^{\alpha_{2}}u\\|_{L_{x}^{2}}\\|\phi\\|_{\mathcal{A}},$

and

(4.6)			$\displaystyle\int_{\mathbb{R}}\left(\int_{\mathbb{R}}\xi^{k}\widetilde{W}_{\hbar}[u]d\xi-\int_{\mathbb{R}}\xi^{k}W_{\hbar}[u]d\xi\right)\varphi dx$
	$\displaystyle\leq$	$\displaystyle\Big{\\|}\int_{\mathbb{R}}\xi^{k}W_{\hbar}[u]d\xi\Big{\\|}_{L_{x}^{1}}\\|\varphi*_{x}g_{\hbar}-\varphi\\|_{L_{x}^{\infty}}+C(k)\sum_{\alpha=0}^{k-1}\hbar^{k-\alpha}\Big{\\|}\int_{\mathbb{R}}\xi^{\alpha}W_{\hbar}[u]d\xi\Big{\\|}_{L_{x}^{1}}\\|\varphi\\|_{L_{x}^{\infty}}.$

Proof.

For (4.3) with $k=0$ , we have

		$\displaystyle\Big{\|}\iint_{\mathbb{R}^{2}}W_{\hbar}[u_{1},u_{2}]\phi(x,\xi)dxd\xi\Big{\|}$
	$\displaystyle\leq$	$\displaystyle\left(\int_{\mathbb{R}}\sup_{x}\|\mathcal{F}_{\xi}\phi(x,y)\|dy\right)\left(\sup_{y}\int_{\mathbb{R}}\Big{\|}u_{1}(x+\frac{\hbar y}{2})\overline{u_{2}(x-\frac{\hbar y}{2})}\Big{\|}dx\right)$
	$\displaystyle\leq$	$\displaystyle\\|u_{1}\\|_{L_{x}^{2}}\\|u_{2}\\|_{L_{x}^{2}}\\|\phi\\|_{\mathcal{A}}.$

For the $k\geq 1$ case, by formula (3.8), we obtain

\displaystyle\Big{|}\iint_{\mathbb{R}^{2}}\xi^{k}W_{\hbar}[u_{1},u_{2}]\phi dxd\xi\Big{|}\leq\sum_{\alpha=0}^{k}\binom{k}{\alpha}\|\hbar^{\alpha}\partial_{x}^{\alpha}u_{1}\|_{L_{x}^{2}}\|\hbar^{k-\alpha}\partial_{x}^{k-\alpha}u_{2}\|_{L_{x}^{2}}\|\phi\|_{\mathcal{A}},

which completes the proof of (4.3). Then (4.4) follows from (4.3) and the uniform estimate (3.3).

For (4.5), we notice that

(4.7)

\displaystyle\xi^{k}\widetilde{W}_{\hbar}[u]=\xi^{k}(W_{\hbar}[u]*_{(x,\xi)}G_{\hbar})=\sum_{\alpha=0}^{k}\binom{k}{\alpha}(\xi^{\alpha}W_{\hbar}[u])*_{(x,\xi)}(\xi^{k-\alpha}G_{\hbar}),

and hence get

		$\displaystyle\xi^{k}\widetilde{W}_{\hbar}[u]-\xi^{k}W_{\hbar}[u]$
	$\displaystyle=$	$\displaystyle(\xi^{k}W_{\hbar}[u])_{(x,\xi)}G_{\hbar}-\xi^{k}W_{\hbar}[u]+\sum_{\alpha=0}^{k-1}\binom{k}{\alpha}(\xi^{\alpha}W_{\hbar}[u])_{(x,\xi)}(\xi^{k-\alpha}G_{\hbar}).$

Using (4.3), we obtain

		$\displaystyle\Big{\|}\iint_{\mathbb{R}^{2}}\left(\xi^{k}\widetilde{W}_{\hbar}[u]-\xi^{k}W_{\hbar}[u]\right)\phi dxd\xi\Big{\|}$
	$\displaystyle\leq$	$\displaystyle\Big{\|}\iint_{\mathbb{R}^{2}}\xi^{k}W_{\hbar}[u]\left(\phi_{(x,\xi)}G_{\hbar}-\phi\right)dxd\xi\Big{\|}+\sum_{\alpha=0}^{k-1}\binom{k}{\alpha}\Big{\|}\iint_{\mathbb{R}^{2}}(\xi^{\alpha}W_{\hbar}[u])(\xi^{k-\alpha}G_{\hbar})\phi dxd\xi\Big{\|}$
	$\displaystyle\leq$	$\displaystyle\\|u\\|_{L_{x}^{2}}^{2}\\|\phi*_{(x,\xi)}G_{\hbar}-\phi\\|_{\mathcal{A}}$
		$\displaystyle+\sum_{\alpha=0}^{k-1}\sum_{\alpha_{1}+\alpha_{2}=\alpha}\binom{k}{\alpha}\binom{\alpha}{\alpha_{1}}\\|\hbar^{\alpha_{1}}\partial_{x}^{\alpha_{1}}u\\|_{L_{x}^{2}}\\|\hbar^{\alpha_{2}}\partial_{x}^{\alpha_{2}}u\\|_{L_{x}^{2}}\\|(\xi^{k-\alpha}G_{\hbar})*_{(x,\xi)}\phi\\|_{\mathcal{A}}$
	$\displaystyle\leq$	$\displaystyle\\|u\\|_{L_{x}^{2}}^{2}\\|\phi*_{(x,\xi)}G_{\hbar}-\phi\\|_{\mathcal{A}}+C(k)\sum_{\alpha=0}^{k-1}\sum_{\alpha_{1}+\alpha_{2}=\alpha}\\|\hbar^{\alpha_{1}}\partial_{x}^{\alpha_{1}}u\\|_{L_{x}^{2}}\\|\hbar^{\alpha_{2}}\partial_{x}^{\alpha_{2}}u\\|_{L_{x}^{2}}\\|\phi\\|_{\mathcal{A}}\\|\xi^{k-\alpha}G_{\hbar}\\|_{L_{x}^{1}L_{\xi}^{1}}$
	$\displaystyle\lesssim$	$\displaystyle\\|u\\|_{L_{x}^{2}}^{2}\\|\phi*_{(x,\xi)}G_{\hbar}-\phi\\|_{\mathcal{A}}+C(k)\sum_{\alpha=0}^{k-1}\sum_{\alpha_{1}+\alpha_{2}=\alpha}\\|\hbar^{\alpha_{1}}\partial_{x}^{\alpha_{1}}u\\|_{L_{x}^{2}}\\|\hbar^{\alpha_{2}}\partial_{x}^{\alpha_{2}}u\\|_{L_{x}^{2}}\hbar^{k-\alpha}\\|\phi\\|_{\mathcal{A}},$

where in the last two inequalities we have used that

	$\displaystyle\\|g*_{(x,\xi)}\phi\\|_{\mathcal{A}}\leq$	$\displaystyle\\|(\mathcal{F}_{\xi}g)*_{x}(\mathcal{F}_{\xi}\phi)\\|_{L_{\xi}^{1}L_{x}^{\infty}}\leq\\|\mathcal{F}_{\xi}g\\|_{L_{\eta}^{\infty}L_{x}^{1}}\\|\mathcal{F}_{\xi}\phi\\|_{L_{\eta}^{1}L_{x}^{\infty}}\leq\\|g\\|_{L_{x}^{1}L_{\xi}^{1}}\\|\phi\\|_{\mathcal{A}},$
	$\displaystyle\\|\xi^{k-\alpha}G_{\hbar}\\|_{L_{x}^{1}L_{\xi}^{1}}\lesssim$	$\displaystyle\hbar^{k-\alpha}.$

Therefore, we complete the proof of (4.5). Estimate (4.6) follows from a way in which we obtain (4.5).

∎

4.2. Convergence to a Non-negative Radon Measure

As we have established the difference estimates between the Wigner function and the Husimi function, which shows that they have same limit, we can use the non-negativity of the Husimi function to conclude the convergence of $\left\{f_{\hbar,\varepsilon}(t,x,\xi)\right\}$ to a non-negative Radon measure $f(t,dx,d\xi)$ .

Notation.

Here, for the convenience, we also use the notation $f(t,x,\xi)dxd\xi$ to denote the measure $f(t,dx,d\xi)$ . Hence, one should keep in mind that $f(t,x,\xi)$ is not an $L_{loc}^{1}$ function.

Lemma 4.3.

There exists a subsequence of $\left\{f_{\hbar,\varepsilon}(t,x,\xi)\right\}$ , which we still denote by $\left\{f_{\hbar,\varepsilon}(t,x,\xi)\right\}$ , and a bounded non-negative Radon measure

(4.8)

\displaystyle f(t,dx,d\xi)\in C([0,\infty);\mathcal{M}^{+}(\mathbb{R}^{2})-w^{*}),

such that for $\forall T>0$ , $k\geq 0$ , there hold

(4.9)

\displaystyle\lim_{(\hbar,\varepsilon)\to(0,0)}\int_{0}^{T}\iint_{\mathbb{R}^{2}}\left(\xi^{k}f_{\hbar,\varepsilon}(t,x,\xi)-\xi^{k}f(t,x,\xi)\right)\phi dxd\xi dt=0,\ \forall\phi\in L_{t}^{1}([0,T];\mathcal{A}),

and

(4.10)

\displaystyle\lim_{(\hbar,\varepsilon)\to(0,0)}\int_{0}^{T}\int_{\mathbb{R}}\left(\int_{\mathbb{R}}\xi^{k}f_{\hbar,\varepsilon}(t,x,\xi)d\xi-\int_{\mathbb{R}}\xi^{k}f(t,x,\xi)d\xi\right)\varphi dxdt=0,\ \forall\varphi\in L_{t}^{1}([0,T];C_{b}(\mathbb{R})),

and

(4.11)

\displaystyle\lim_{(\hbar,\varepsilon)\to(0,0)}\int_{\mathbb{R}}\left(\int_{\mathbb{R}}\xi^{k}f_{\hbar,\varepsilon}(t,x,\xi)d\xi-\int_{\mathbb{R}}\xi^{k}f(t,x,\xi)d\xi\right)\varphi dx=0,\ \forall t\geq 0,\ \varphi\in C_{b}(\mathbb{R}).

Moreover, the limit measure satisfies the weighted estimates that

(4.12)		$\displaystyle\iint_{\mathbb{R}^{2}}\langle x\rangle^{2}f(t,dx,d\xi)\leq$	$\displaystyle C(t),$
(4.13)		$\displaystyle\iint_{\mathbb{R}^{2}}\langle x\rangle\|\xi\|^{k}f(t,dx,d\xi)\leq$	$\displaystyle C(k,t).$

Proof.

As we need to prove the non-negativity of the limit point, we take the Husimi transform of the wave function $\psi_{\hbar,\varepsilon}$ , which we denote by $\widetilde{f}_{\hbar,\varepsilon}(t,x,\xi):=\widetilde{W}_{\hbar}[\psi_{\hbar,\varepsilon}]\geq 0$ .

We first prove (4.8) and (4.9). The proof can be divided into the following three steps.

Step 1. $L_{x,\xi}^{1}$ Uniform Bounds.

By estimate (4.5) on the Wigner function and the Husimi function in Lemma 4.3 and the uniform estimate (3.3), we obtain

		$\displaystyle\Big{\|}\int_{0}^{T}\iint_{\mathbb{R}^{2}}\left(\xi^{k}f_{\hbar,\varepsilon}(t,x,\xi)-\xi^{k}\widetilde{f}_{\hbar,\varepsilon}(t,x,\xi)\right)\phi dxd\xi dt\Big{\|}$
	$\displaystyle\leq$	$\displaystyle\\|\psi_{\hbar,\varepsilon}\\|_{L_{t}^{\infty}L_{x}^{2}}^{2}\\|\phi*_{(x,\xi)}G_{\hbar}-\phi\\|_{L_{t}^{1}\mathcal{A}}$
		$\displaystyle+C(k)\sum_{\alpha=0}^{k-1}\sum_{\alpha_{1}+\alpha_{2}=\alpha}\hbar^{k-\alpha}\\|\hbar^{\alpha_{1}}\partial_{x}^{\alpha_{1}}\psi_{\hbar,\varepsilon}\\|_{L_{t}^{\infty}L_{x}^{2}}\\|\hbar^{\alpha_{2}}\partial_{x}^{\alpha_{2}}\psi_{\hbar,\varepsilon}\\|_{L_{t}^{\infty}L_{x}^{2}}\\|\phi\\|_{L_{t}^{1}\mathcal{A}}$
	$\displaystyle\lesssim$	$\displaystyle\\|\phi*_{(x,\xi)}G_{\hbar}-\phi\\|_{L_{t}^{1}\mathcal{A}}+\hbar\\|\phi\\|_{L_{t}^{1}\mathcal{A}}\to 0.$

Thus, the Wigner function $f_{\hbar,\varepsilon}(t,x,\xi)$ and the Husimi function $\widetilde{f}_{\hbar,\varepsilon}(t,x,\xi)$ have the same convergence and limit. We are left to prove the convergence and the limit of the Husimi function $\widetilde{f}_{\hbar,\varepsilon}(t,x,\xi)$ .

We establish the $L_{x,\xi}^{1}$ uniform bound for $\widetilde{f}_{\hbar,\varepsilon}(t,x,\xi)$ . By formula (4.7), we use the uniform estimate (3.11) to get

	$\displaystyle\\|\xi^{2k}\widetilde{f}_{\hbar,\varepsilon}(t,x,\xi)\\|_{L_{x,\xi}^{1}}=$	$\displaystyle\iint_{\mathbb{R}^{2}}\xi^{2k}\widetilde{f}_{\hbar,\varepsilon}dxd\xi$
	$\displaystyle\leq$	$\displaystyle\sum_{\alpha=0}^{2k}\binom{2k}{\alpha}\Big{\|}\iint_{\mathbb{R}^{2}}(\xi^{\alpha}f_{\hbar,\varepsilon})*_{(x,\xi)}(\xi^{2k-\alpha}G_{\hbar})dxd\xi\Big{\|}$
	$\displaystyle=$	$\displaystyle\sum_{\alpha=0}^{2k}\binom{2k}{\alpha}\Big{\|}\left(\iint_{\mathbb{R}^{2}}\xi^{\alpha}f_{\hbar,\varepsilon}dxd\xi\right)\left(\iint_{\mathbb{R}^{2}}\xi^{2k-\alpha}G_{\hbar}dxd\xi\right)\Big{\|}$
	$\displaystyle\leq$	$\displaystyle C(2k,t).$

By Hölder’s inequality, we arrive at

(4.14)

\displaystyle\|\xi^{k}\widetilde{f}_{\hbar,\varepsilon}(t,x,\xi)\|_{L_{x,\xi}^{1}}\leq\|\xi^{2k}\widetilde{f}_{\hbar,\varepsilon}(t,x,\xi)\|_{L_{x,\xi}^{1}}^{\frac{1}{2}}\|\widetilde{f}_{\hbar,\varepsilon}(t,x,\xi)\|_{L_{x,\xi}^{1}}^{\frac{1}{2}}\leq C(k,t).

Step 2. Equicontinuity.

To obtain the equicontinuity of $\widetilde{f}_{\hbar,\varepsilon}(t,x,\xi)$ and apply the compactness argument, we prove the uniform estimates for the time-derivative of $\widetilde{f}_{\hbar,\varepsilon}$ . We take the duality argument and notice that

(4.15)

\displaystyle\iint_{\mathbb{R}^{2}}\xi^{k}\partial_{t}\widetilde{f}_{\hbar,\varepsilon}\phi dxd\xi=\iint_{\mathbb{R}^{2}}\xi^{k}\left(\partial_{t}f_{\hbar,\varepsilon}*_{(x,\xi)}G_{\hbar}\right)\phi dxd\xi.

From the nonlinear Schrödinger equation (2.1), we have

(4.16)

\displaystyle\partial_{t}f_{\hbar,\varepsilon}+\xi\partial_{x}f_{\hbar,\varepsilon}+\Theta[V_{\varepsilon},f_{\hbar,\varepsilon}]=0,

where the nonlinear term is

(4.17)

\displaystyle\Theta[V_{\varepsilon},f_{\hbar,\varepsilon}]=\frac{i}{2\pi}\iint_{\mathbb{R}^{2}}\frac{V_{\varepsilon}*\rho_{\hbar,\varepsilon}(x+\frac{\hbar y}{2})-V_{\varepsilon}*\rho_{\hbar,\varepsilon}(x-\frac{\hbar y}{2})}{\hbar}f_{\hbar,\varepsilon}(t,x,\eta)e^{-i(\xi-\eta)y}d\eta dy.

Putting (4.16) into (4.15), we obtain

	$\displaystyle\iint_{\mathbb{R}^{2}}\xi^{k}\partial_{t}\widetilde{f}_{\hbar,\varepsilon}\phi dxd\xi=$	$\displaystyle-\iint_{\mathbb{R}^{2}}\xi^{k}\left((\xi\partial_{x}f_{\hbar,\varepsilon})*_{(x,\xi)}G_{\hbar}\right)\phi dxd\xi$
		$\displaystyle-\iint_{\mathbb{R}^{2}}\xi^{k}\left(\Theta[V_{\varepsilon},f_{\hbar,\varepsilon}]*_{(x,\xi)}G_{\hbar}\right)\phi dxd\xi$
	$\displaystyle:=$	$\displaystyle I_{1}+I_{2}.$

For the linear term $I_{1}$ , we rewrite

	$\displaystyle I_{1}=$	$\displaystyle\iint_{\mathbb{R}^{2}}\xi^{k}\left((\xi f_{\hbar,\varepsilon})*_{(x,\xi)}G_{\hbar}\right)\left(\partial_{x}\phi\right)dxd\xi$
	$\displaystyle=$	$\displaystyle\sum_{\alpha=0}^{k}\binom{k}{\alpha}\iint_{\mathbb{R}^{2}}\left((\xi^{\alpha+1}f_{\hbar,\varepsilon})*_{(x,\xi)}(\xi^{k-\alpha}G_{\hbar})\right)\left(\partial_{x}\phi\right)dxd\xi$
	$\displaystyle=$	$\displaystyle\sum_{\alpha=0}^{k}\binom{k}{\alpha}\iint_{\mathbb{R}^{2}}\xi^{\alpha+1}f_{\hbar,\varepsilon}\left((\xi^{k-\alpha}G_{\hbar})*_{(x,\xi)}\left(\partial_{x}\phi\right)\right)dxd\xi.$

Using estimate (4.4), we get

(4.18)

\displaystyle|I_{1}|\lesssim\sum_{\alpha=0}^{k}\|(\xi^{k-\alpha}G_{\hbar})*_{(x,\xi)}\partial_{x}\phi\|_{\mathcal{A}}\lesssim\|\partial_{x}\phi\|_{\mathcal{A}}\lesssim\|\phi\|_{H_{x,\xi}^{2}}.

For the nonlinear term $I_{2}$ , we rewrite

(4.19)		$\displaystyle I_{2}=$	$\displaystyle-\sum_{\alpha=0}^{k}\binom{k}{\alpha}\iint_{\mathbb{R}^{2}}\left(\left(\xi^{\alpha}\Theta[V_{\varepsilon},f_{\hbar,\varepsilon}]\right)*_{(x,\xi)}\left(\xi^{k-\alpha}G_{\hbar}\right)\right)\phi dxd\xi$
	$\displaystyle=$	$\displaystyle-\sum_{\alpha=0}^{k}\binom{k}{\alpha}\iint_{\mathbb{R}^{2}}\xi^{\alpha}\Theta[V_{\varepsilon},f_{\hbar,\varepsilon}]\left(\left(\xi^{k-\alpha}G_{\hbar}\right)*_{(x,\xi)}\phi\right)dxd\xi.$

Using again estimate (4.4), we have

(4.20)			$\displaystyle\iint_{\mathbb{R}^{2}}\Theta[V_{\varepsilon},f_{\hbar,\varepsilon}]\phi dxd\xi$
	$\displaystyle=$	$\displaystyle\iint_{\mathbb{R}^{2}}f_{\hbar,\varepsilon}(t,x,\eta)\mathcal{F}_{\eta}^{-1}\left[\mathcal{F}_{y}(\phi)\frac{V_{\varepsilon}\rho_{\hbar,\varepsilon}(x+\frac{\hbar y}{2})-V_{\varepsilon}\rho_{\hbar,\varepsilon}(x-\frac{\hbar y}{2})}{\hbar}\right]dxd\eta$
	$\displaystyle\lesssim$	$\displaystyle\Big{\\|}\mathcal{F}_{\eta}^{-1}\left[\mathcal{F}_{y}(\phi)\frac{V_{\varepsilon}\rho_{\hbar,\varepsilon}(x+\frac{\hbar y}{2})-V_{\varepsilon}\rho_{\hbar,\varepsilon}(x-\frac{\hbar y}{2})}{\hbar}\right]\Big{\\|}_{\mathcal{A}}$
	$\displaystyle\leq$	$\displaystyle\Big{\\|}\mathcal{F}_{y}(\phi)\frac{V_{\varepsilon}\rho_{\hbar,\varepsilon}(x+\frac{\hbar y}{2})-V_{\varepsilon}\rho_{\hbar,\varepsilon}(x-\frac{\hbar y}{2})}{\hbar}\Big{\\|}_{L_{y}^{1}L_{x}^{\infty}}$
	$\displaystyle\leq$	$\displaystyle\\|\|y\|\mathcal{F}_{y}(\phi)\\|_{L_{y}^{1}L_{x}^{\infty}}\\|\partial_{x}V_{\varepsilon}*\rho_{\hbar,\varepsilon}\\|_{L_{x}^{\infty}}$
	$\displaystyle\lesssim$	$\displaystyle\\|\phi\\|_{H_{x,\xi}^{3}}.$

By the definition of $\Theta[V_{\varepsilon},f_{\hbar,\varepsilon}]$ in (4.17), we have

(4.21)

\displaystyle\xi^{\alpha}\Theta[V_{\varepsilon},f_{\hbar,\varepsilon}]=\sum_{\alpha_{1}+\alpha_{2}=\alpha}\binom{\alpha}{\alpha_{1}}\Theta^{(\alpha_{1})}[V_{\varepsilon},\xi^{\alpha_{2}}f_{\hbar,\varepsilon}],

where

		$\displaystyle\Theta^{(\alpha_{1})}[V_{\varepsilon},\xi^{\alpha_{2}}f_{\hbar,\varepsilon}]$
	$\displaystyle=$	$\displaystyle\frac{i}{2\pi}\iint_{\mathbb{R}^{2}}D_{y}^{\alpha_{1}}\left(\frac{V_{\varepsilon}\rho_{\hbar,\varepsilon}(x+\frac{\hbar y}{2})-V_{\varepsilon}\rho_{\hbar,\varepsilon}(x-\frac{\hbar y}{2})}{\hbar}\right)\left(\eta^{\alpha_{2}}f_{\hbar,\varepsilon}(t,x,\eta)\right)e^{-i(\xi-\eta)y}d\eta dy.$

Putting (4.21) into (4.19), in the same way as (4.20), we obtain

(4.22)		$\displaystyle\|I_{2}\|\leq$	$\displaystyle\sum_{\alpha=0}^{k}\sum_{\alpha_{1}+\alpha_{2}=\alpha}\binom{k}{\alpha}\binom{\alpha}{\alpha_{1}}\Big{\|}\iint_{\mathbb{R}^{2}}\Theta^{(\alpha_{1})}[V_{\varepsilon},\xi^{\alpha_{2}}f_{\hbar,\varepsilon}]\left(\left(\xi^{k-\alpha}G_{\hbar}\right)*_{(x,\xi)}\phi\right)dxd\xi\Big{\|}$
	$\displaystyle\lesssim$	$\displaystyle\sum_{\alpha=0}^{k}\sum_{\alpha_{1}+\alpha_{2}=\alpha}\big{\\|}\left(\xi^{k-\alpha}G_{\hbar}\right)_{(x,\xi)}\phi\big{\\|}_{H_{x,\xi}^{3}}\\|\hbar^{\alpha_{1}}\partial_{x}^{\alpha_{1}+1}V_{\varepsilon}\rho_{\hbar,\varepsilon}\\|_{L_{x}^{\infty}}$
	$\displaystyle\lesssim$	$\displaystyle C(k,t)\\|\phi\\|_{H_{x,\xi}^{3}},$

where in the last inequality we have used Young’s inequality and uniform estimate (3.12).

Combining estimates (4.18) and (4.22) on the terms $I_{1}$ and $I_{2}$ , we arrive at

\displaystyle\|\partial_{t}\xi^{k}\widetilde{f}_{\hbar,\varepsilon}\|_{H_{x,\xi}^{-3}}\leq C(k,t).

Step 3. Compactness Argument.

By Arzel $\grave{\text{a}}$ -Ascoli compactness lemma and a diagonal argument, for all $k\geq 1$ there exist a subsequence of $\left\{\widetilde{f}_{\hbar,\varepsilon}\right\}$ , which we still denote by $\left\{\widetilde{f}_{\hbar,\varepsilon}\right\}$ , and a limit point

f_{k}(t,x,\xi)\in C([0,T];H_{x,\xi}^{-3})

such that

(4.23)

\displaystyle\lim_{(\hbar,\varepsilon)\to(0,0)}\|\xi^{k}\widetilde{f}_{\hbar,\varepsilon}(t,x,\xi)-f_{k}(t,x,\xi)\|_{C([0,T];H_{x,\xi}^{-3})}=0.

Actually, we have that $f_{k}(t,x,\xi)=\xi^{k}f(t,x,\xi)$ due to that

		$\displaystyle\lim_{(\hbar,\varepsilon)\to(0,0)}\int_{0}^{T}\iint_{\mathbb{R}^{2}}\xi^{k}\widetilde{f}_{\hbar,\varepsilon}(t,x,\xi)\phi(t,x,\xi)dtdxd\xi$
	$\displaystyle=$	$\displaystyle\lim_{(\hbar,\varepsilon)\to(0,0)}\int_{0}^{T}\iint_{\mathbb{R}^{2}}\widetilde{f}_{\hbar,\varepsilon}(t,x,\xi)(\xi^{k}\phi(t,x,\xi))dtdxd\xi$
	$\displaystyle=$	$\displaystyle\int_{0}^{T}\iint_{\mathbb{R}^{2}}f(t,x,\xi)\xi^{k}\phi(t,x,\xi)dtdxd\xi.$

Moreover, by the non-negativity of the Husimi function and the $L_{x,\xi}^{1}$ uniform bound for $\widetilde{f}_{\hbar,\varepsilon}$ , we get

\displaystyle f(t,x,\xi)dxd\xi\in C([0,T];\mathcal{M}^{+}(\mathbb{R}^{2})-w^{*}).

Therefore, we have completed the proof of (4.8) and (4.9).

Next, we prove estimate (4.10). By (4.6) in Lemma 4.2 and the uniform estimate (3.11), for $\varphi\in L_{t}^{1}([0,T];C_{c}^{\infty}(\mathbb{R}))$ we have

		$\displaystyle\Big{\|}\int_{0}^{T}\int_{\mathbb{R}}\left(\int_{\mathbb{R}}\xi^{k}f_{\hbar,\varepsilon}(t,x,\xi)d\xi-\int_{\mathbb{R}}\xi^{k}\widetilde{f}_{\hbar,\varepsilon}(t,x,\xi)d\xi\right)\varphi dxdt\Big{\|}$
	$\displaystyle\leq$	$\displaystyle\Big{\\|}\int_{\mathbb{R}}\xi^{k}f_{\hbar,\varepsilon}d\xi\Big{\\|}_{L_{x}^{1}}\\|\varphi*_{x}g_{\hbar}-\varphi\\|_{L_{x}^{\infty}}+C(k)\sum_{\alpha=0}^{k-1}\hbar^{k-\alpha}\Big{\\|}\int_{\mathbb{R}}\xi^{\alpha}f_{\hbar,\varepsilon}d\xi\Big{\\|}_{L_{x}^{1}}\\|\varphi\\|_{L_{x}^{\infty}}$
	$\displaystyle\leq$	$\displaystyle\\|\varphi*_{x}g_{\hbar}-\varphi\\|_{L_{t}^{1}L_{x}^{\infty}}+\hbar\\|\varphi\\|_{L_{t}^{1}L_{x}^{\infty}}\to 0.$

Thus, it suffices to deal with $\int_{\mathbb{R}}\xi^{k}\widetilde{f}_{\hbar,\varepsilon}(t,x,\xi)d\xi$ . In the same way in which we obtain (4.9), we also have

\displaystyle\Big{\|}\partial_{t}\int_{\mathbb{R}}\xi^{k}\widetilde{f}_{\hbar,\varepsilon}d\xi\Big{\|}_{H_{x}^{-3}}\leq C(k,t),

which implies that there exists a limit point $F_{k}(t,x)\in C([0,T];H_{x}^{-3})$ such that

\displaystyle\lim_{(\hbar,\varepsilon)\to(0,0)}\Big{\|}\int_{\mathbb{R}}\xi^{k}\widetilde{f}_{\hbar,\varepsilon}(t,x,\xi)d\xi-F_{k}(t,x)\Big{\|}_{C([0,T];H_{x}^{-3})}=0.

We claim that $F_{k}(t,x)=\int_{\mathbb{R}}\xi^{k}f(t,x,d\xi)$ . Indeed,

	$\displaystyle\int_{0}^{T}\int_{\mathbb{R}}F_{k}\varphi dxdt=$	$\displaystyle\lim_{(\hbar,\varepsilon)\to(0,0)}\int_{0}^{T}\int_{\mathbb{R}}\left(\int_{\mathbb{R}}\xi^{k}\widetilde{f}_{\hbar,\varepsilon}(t,x,\xi)d\xi\right)\varphi dxdt$
	$\displaystyle=$	$\displaystyle\lim_{(\hbar,\varepsilon)\to(0,0)}\int_{0}^{T}\int_{\mathbb{R}^{2}}(1+\xi^{2})\xi^{k}\widetilde{f}_{\hbar,\varepsilon}(t,x,\xi)\frac{\varphi}{1+\xi^{2}}d\xi dxdt$
	$\displaystyle=$	$\displaystyle\int_{0}^{T}\int_{\mathbb{R}^{2}}(1+\xi^{2})\xi^{k}f(t,x,\xi)\frac{\varphi}{1+\xi^{2}}d\xi dxdt$
	$\displaystyle=$	$\displaystyle\int_{0}^{T}\int_{\mathbb{R}}\left(\int_{\mathbb{R}}\xi^{k}f(t,x,\xi)d\xi\right)\varphi dxdt,$

where in the second-to-last equality we have used convergence (4.23) and the fact that

\displaystyle\frac{1}{1+\xi^{2}}\varphi(x)\in H_{x,\xi}^{3}.

Hence, we conclude (4.10) for $\varphi\in L_{t}^{1}([0,T];C_{c}^{\infty}(\mathbb{R}))$ . By the weighted uniform estimate (3.11) and the fact that $C_{c}^{\infty}(\mathbb{R})$ is dense in $C_{c}(\mathbb{R})$ , we arrive at (4.10) for $\varphi\in L_{t}^{1}([0,T];C_{c}(\mathbb{R}))$ . Moreover, for $\varphi\in L_{t}^{1}([0,T];C_{b}(\mathbb{R}))$ , we write

(4.24)			$\displaystyle\int_{0}^{T}\int_{\mathbb{R}}\left(\int_{\mathbb{R}}\xi^{k}f_{\hbar,\varepsilon}(t,x,\xi)d\xi\right)\varphi dxdt$
	$\displaystyle=$	$\displaystyle\int_{0}^{T}\int_{\mathbb{R}}\left(\int_{\mathbb{R}}\xi^{k}f_{\hbar,\varepsilon}(t,x,\xi)d\xi\right)\varphi\chi(\frac{x}{R})dxdt$
		$\displaystyle+\int_{0}^{T}\int_{\mathbb{R}}\left(\int_{\mathbb{R}}\xi^{k}f_{\hbar,\varepsilon}(t,x,\xi)d\xi\right)\varphi(1-\chi(\frac{x}{R}))dxdt.$

By the weighted uniform estimate (3.11), we get

		$\displaystyle\Big{\|}\int_{0}^{T}\int_{\mathbb{R}}\left(\int_{\mathbb{R}}\xi^{k}f_{\hbar,\varepsilon}(t,x,\xi)d\xi\right)\varphi(1-\chi(\frac{x}{R}))dxdt\Big{\|}$
	$\displaystyle\leq$	$\displaystyle\frac{1}{R}\Big{\\|}\langle x\rangle\int_{\mathbb{R}}\xi^{k}f_{\hbar,\varepsilon}(t,x,\xi)d\xi\Big{\\|}_{L_{t}^{\infty}L_{x}^{1}}\\|\varphi\\|_{L_{t}^{1}L_{x}^{\infty}}\lesssim\frac{1}{R}\to 0.$

Taking $(\hbar,\varepsilon)\to(0,0)$ and then sending $R\to\infty$ , (4.24) becomes

		$\displaystyle\lim_{(\hbar,\varepsilon)\to(0,0)}\int_{0}^{T}\int_{\mathbb{R}}\left(\int_{\mathbb{R}}\xi^{k}f_{\hbar,\varepsilon}(t,x,\xi)d\xi\right)\varphi dxdt$
	$\displaystyle=$	$\displaystyle\lim_{R\to\infty}\int_{0}^{T}\int_{\mathbb{R}}\left(\int_{\mathbb{R}}\xi^{k}f(t,x,\xi)d\xi\right)\varphi\chi(\frac{x}{R})dxdt$
	$\displaystyle=$	$\displaystyle\int_{0}^{T}\int_{\mathbb{R}}\left(\int_{\mathbb{R}}\xi^{k}f(t,x,\xi)d\xi\right)\varphi dxdt,$

where in the last equality we have used the dominated convergence theorem. Hence, we complete the proof of (4.10).

Next, we handle estimate (4.11). As we have proven that

\displaystyle\lim_{(\hbar,\varepsilon)\to(0,0)}\Big{\|}\int_{\mathbb{R}}\xi^{k}\widetilde{f}_{\hbar,\varepsilon}(t,x,\xi)d\xi-\int_{\mathbb{R}}\xi^{k}f(t,x,\xi)d\xi\Big{\|}_{C([0,T];H_{x}^{-3})}=0,

by the weighted uniform bound (3.11), we can improve it to the narrow convergence in a similar way in which we obtain (4.10) and hence complete the proof of (4.11).

Finally, we deal with the estimates (4.12)–(4.13). It suffices to prove estimate (4.13), as estimate (4.12) follows similarly. By the non-negativity of $f(t,dx,d\xi)$ and Hölder’s inequality, we again only need to prove (4.13) with the weight function $\xi^{2k}$ . By the weighted uniform estimate (3.11), we have

	$\displaystyle\iint_{\mathbb{R}^{2}}\chi(\frac{x}{R})\langle x\rangle\xi^{2k}f(t,dx,d\xi)=$	$\displaystyle\lim_{(\hbar,\varepsilon)\to(0,0)}\int_{\mathbb{R}}\chi(\frac{x}{R})\langle x\rangle\left(\int_{\mathbb{R}}\xi^{2k}f_{\hbar,\varepsilon}(t,x,\xi)d\xi\right)dx$
	$\displaystyle\leq$	$\displaystyle\sup_{(\hbar,\varepsilon)}\Big{\\|}\langle x\rangle\int_{\mathbb{R}}\xi^{2k}f_{\hbar,\varepsilon}(t,x,\xi)d\xi\Big{\\|}_{L_{x}^{1}}\leq C(2k,t).$

Sending $R\to\infty$ , by Fatou’s lemma, we arrive at (4.13). ∎

5. Conservation Laws for the Limit Measure

The conservation laws of mass, momentum and energy for the Wigner function $f_{\hbar,\varepsilon}(t,x,\xi)$ are given by

(5.1)			$\displaystyle\iint_{\mathbb{R}^{2}}f_{\hbar,\varepsilon}(t,x,\xi)d\xi dx=\iint_{\mathbb{R}^{2}}f_{\hbar,\varepsilon}(0,x,\xi)d\xi dx,$
(5.2)			$\displaystyle\iint_{\mathbb{R}^{2}}\xi f_{\hbar,\varepsilon}(t,x,\xi)d\xi dx=\iint_{\mathbb{R}^{2}}\xi f_{\hbar,\varepsilon}(0,x,\xi)d\xi dx,$
(5.3)			$\displaystyle\iint_{\mathbb{R}^{2}}\xi^{2}f_{\hbar,\varepsilon}(t,x,\xi)d\xi dx+\iint_{\mathbb{R}^{2}}V_{\varepsilon}(x-y)\rho_{\hbar,\varepsilon}(t,x)\rho_{\hbar,\varepsilon}(t,y)dxdy$
	$\displaystyle=$	$\displaystyle\iint_{\mathbb{R}^{2}}\xi^{2}f_{\hbar,\varepsilon}(0,x,\xi)d\xi dx+\iint_{\mathbb{R}^{2}}V_{\varepsilon}(x-y)\rho_{\hbar,\varepsilon}(0,x)\rho_{\hbar,\varepsilon}(0,y)dxdy.$

In the section, we prove the conservation laws for the limit measure $f(t,dx,d\xi)$ .

Lemma 5.1.

The limit measure $f(t,dx,d\xi)$ satisfies the conservation laws of mass, momentum and energy

(5.4)			$\displaystyle\iint_{\mathbb{R}^{2}}f(t,dx,d\xi)=\iint_{\mathbb{R}^{2}}f(0,dx,d\xi),$
(5.5)			$\displaystyle\iint_{\mathbb{R}^{2}}\xi f(t,dx,d\xi)=\iint_{\mathbb{R}^{2}}\xi f(0,dx,d\xi),$
(5.6)			$\displaystyle\iint_{\mathbb{R}^{2}}\xi^{2}f(t,dx,d\xi)dxd\xi+\frac{1}{2}\int_{\mathbb{R}^{2}}\|x-y\|\rho(t,dx)\rho(t,dy)$
	$\displaystyle=$	$\displaystyle\iint_{\mathbb{R}^{2}}\xi^{2}f(0,dx,d\xi)dxd\xi+\frac{1}{2}\int_{\mathbb{R}^{2}}\|x-y\|\rho(0,dx)\rho(0,dy),$

where $\rho(t,dx)=\int_{\mathbb{R}}f(t,dx,\xi)d\xi$ .

Proof.

For (5.4), by the narrow convergence (4.11) in Lemma 4.3 and the conservation law of mass (5.1) for $f_{\hbar,\varepsilon}(t,x,\xi)$ , we have

	$\displaystyle\iint_{\mathbb{R}^{2}}f(t,dx,d\xi)=$	$\displaystyle\lim_{(\hbar,\varepsilon)\to(0,0)}\iint_{\mathbb{R}^{2}}f_{\hbar,\varepsilon}(t,x,\xi)dxd\xi$
	$\displaystyle=$	$\displaystyle\lim_{(\hbar,\varepsilon)\to(0,0)}\iint_{\mathbb{R}^{2}}f_{\hbar,\varepsilon}(0,x,\xi)dxd\xi$
	$\displaystyle=$	$\displaystyle\iint_{\mathbb{R}^{2}}f(0,dx,d\xi).$

In the same way, we also have the conservation law of momentum (5.5).

For the conservation law of energy (5.6), using again the narrow convergence (4.11) in Lemma 4.3, we obtain the convergence for the kinetic energy part

	$\displaystyle\iint_{\mathbb{R}^{2}}\xi^{2}f(t,dx,d\xi)=$	$\displaystyle\lim_{(\hbar,\varepsilon)\to(0,0)}\iint_{\mathbb{R}^{2}}\xi^{2}f_{\hbar,\varepsilon}(t,x,\xi)dxd\xi,$
	$\displaystyle\iint_{\mathbb{R}^{2}}\xi^{2}f(0,dx,d\xi)=$	$\displaystyle\lim_{(\hbar,\varepsilon)\to(0,0)}\iint_{\mathbb{R}^{2}}\xi^{2}f_{\hbar,\varepsilon}(0,x,\xi)dxd\xi.$

Next, we deal with the potential energy part. For simplicity, we omit the time variable and rewrite

		$\displaystyle\iint_{\mathbb{R}^{2}}\|x-y\|e^{-\varepsilon\|x-y\|}\rho_{\hbar,\varepsilon}(x)\rho_{\hbar,\varepsilon}(y)dxdy$
	$\displaystyle=$	$\displaystyle\iint_{\mathbb{R}^{2}}\|x-y\|\rho_{\hbar,\varepsilon}(x)\rho_{\hbar,\varepsilon}(y)dxdy+\iint_{\mathbb{R}^{2}}\|x-y\|(1-e^{-\varepsilon\|x-y\|})\rho_{\hbar,\varepsilon}(x)\rho_{\hbar,\varepsilon}(y)dxdy.$

By the weighted uniform estimate (3.1), we have

		$\displaystyle\iint_{\mathbb{R}^{2}}\|x-y\|(1-e^{-\varepsilon\|x-y\|})\rho_{\hbar,\varepsilon}(x)\rho_{\hbar,\varepsilon}(y)dxdy$
	$\displaystyle\leq$	$\displaystyle\varepsilon\iint_{\mathbb{R}^{2}}\|x-y\|^{2}\rho_{\hbar,\varepsilon}(x)\rho_{\hbar,\varepsilon}(y)dxdy$
	$\displaystyle\lesssim$	$\displaystyle\varepsilon\left(\int_{\mathbb{R}}\|x\|^{2}\rho_{\hbar,\varepsilon}(x)dx\right)\left(\int_{\mathbb{R}}\rho_{\hbar,\varepsilon}(y)dy\right)\lesssim\varepsilon\to 0.$

Thus, we are left to consider the convergence of the term

(5.7)

\displaystyle\iint_{\mathbb{R}^{2}}|x-y|\rho_{\hbar,\varepsilon}(x)\rho_{\hbar,\varepsilon}(y)dxdy,

To do this, we introduce a weighted transform. Set

(5.8)		$\displaystyle F_{\hbar,\varepsilon}(x)=$	$\displaystyle\int_{\mathbb{R}}\frac{\|x-y\|}{\langle x\rangle^{1+\delta}}\rho_{\hbar,\varepsilon}(y)dy,\quad F_{\hbar,\varepsilon}(\pm\infty)=0,$
(5.9)		$\displaystyle G_{\hbar,\varepsilon}(x)=$	$\displaystyle\int_{-\infty}^{x}\langle y\rangle^{1+\delta}\rho_{\hbar,\varepsilon}(y)dy,$

where $\delta\in(0,1)$ is a fixed constant. After the weighted transform, we can use the integration by parts to get

	$\displaystyle\iint_{\mathbb{R}^{2}}\|x-y\|\rho_{\hbar,\varepsilon}(x)\rho_{\hbar,\varepsilon}(y)dxdy=$	$\displaystyle\int_{\mathbb{R}}F_{\hbar,\varepsilon}(x)\partial_{x}G_{\hbar,\varepsilon}(x)dx$
	$\displaystyle=$	$\displaystyle F_{\hbar,\varepsilon}(x)G_{\hbar,\varepsilon}(x)\|^{+\infty}_{-\infty}-\int_{\mathbb{R}}G_{\hbar,\varepsilon}(x)\partial_{x}F_{\hbar,\varepsilon}(x)dx$
	$\displaystyle=$	$\displaystyle-\int_{\mathbb{R}}G_{\hbar,\varepsilon}(x)\partial_{x}F_{\hbar,\varepsilon}(x)dx.$

Next, we get into the analysis of $G_{\hbar,\varepsilon}$ and $F_{\hbar,\varepsilon}$ . Using the weighted uniform estimate (3.1), we have

\displaystyle\|G_{\hbar,\varepsilon}\|_{L_{x}^{\infty}}\leq C,\quad\|\partial_{x}G_{\hbar,\varepsilon}\|_{L_{x}^{1}}\leq C.

Together with the $L^{p}$ compactness criteria for $1\leq p<\infty$ , we conclude that there exist a subsequence of $\left\{G_{\hbar,\varepsilon}\right\}$ and an $L_{loc}^{p}$ function $G(x)$ such that

(5.10)

\displaystyle G_{\hbar,\varepsilon}\stackrel{{\scriptstyle L_{loc}^{p}}}{{\longrightarrow}}G.

In a similar way, noticing that

(5.11)

\displaystyle\partial_{x}F_{\hbar,\varepsilon}=\int_{\mathbb{R}}\frac{1}{\langle x\rangle^{1+\delta}}\frac{x-y}{|x-y|}\rho_{\hbar,\varepsilon}(y)dy-(1+\delta)\int_{\mathbb{R}}\frac{x}{\langle x\rangle^{3+\delta}}|x-y|\rho_{\hbar,\varepsilon}(y)dy,

and

\displaystyle\partial_{x}^{2}F_{\hbar,\varepsilon}=I_{\hbar,\varepsilon}^{(1)}+I_{\hbar,\varepsilon}^{(2)}+I_{\hbar,\varepsilon}^{(3)}+I_{\hbar,\varepsilon}^{(4)},

where

	$\displaystyle I_{\hbar,\varepsilon}^{(1)}=$	$\displaystyle\frac{2\rho_{\hbar,\varepsilon}(x)}{\langle x\rangle^{1+\delta}},$
	$\displaystyle I_{\hbar,\varepsilon}^{(2)}=$	$\displaystyle-(1+\delta)\int_{\mathbb{R}}\frac{x}{\langle x\rangle^{3+\delta}}\frac{x-y}{\|x-y\|}\rho_{\hbar,\varepsilon}(y)dy,$
	$\displaystyle I_{\hbar,\varepsilon}^{(3)}=$	$\displaystyle-(1+\delta)\int_{\mathbb{R}}\frac{x}{\langle x\rangle^{3+\delta}}\frac{x-y}{\|x-y\|}\rho_{\hbar,\varepsilon}(y)dy,$
	$\displaystyle I_{\hbar,\varepsilon}^{(4)}=$	$\displaystyle-(1+\delta)\int_{\mathbb{R}}\left(\frac{1}{\langle x\rangle^{3+\delta}}-(3+\delta)\frac{x^{2}}{\langle x\rangle^{5+\delta}}\right)\frac{x-y}{\|x-y\|}\rho_{\hbar,\varepsilon}(y)dy,$

we use again the weighted uniform estimate (3.1) to get

\displaystyle\|\partial_{x}F_{\hbar,\varepsilon}\|_{L_{x}^{1}\cap L_{x}^{\infty}}\leq C,\quad\|\partial_{x}^{2}F_{\hbar,\varepsilon}\|_{L_{x}^{1}}\leq C,

which implies that there exist a subsequence of $\left\{\partial_{x}F_{\hbar,\varepsilon}\right\}$ and an $L_{loc}^{p}$ function which we denote by $\partial_{x}F(x)$ such that

(5.12)

\displaystyle\partial_{x}F_{\hbar,\varepsilon}\stackrel{{\scriptstyle L_{loc}^{p}}}{{\longrightarrow}}\partial_{x}F.

In the following, we identify the limits in (5.10) and (5.12) by

(5.13)		$\displaystyle G(x)=$	$\displaystyle\int_{-\infty}^{x}\langle y\rangle^{1+\delta}\rho(dy),\quad a.e.,$
(5.14)		$\displaystyle\partial_{x}F(x)=$	$\displaystyle\partial_{x}\int_{\mathbb{R}}\frac{\|x-y\|}{\langle x\rangle^{1+\delta}}\rho(dy),\quad a.e..$

By the weighted estimate (4.12) and the fact that $\int_{\mathbb{R}}\frac{|x-y|}{\langle x\rangle^{1+\delta}}\rho(dy)$ is Lipshitz continuous, (5.13) and (5.14) are indeed well-defined.

Take a test function $\varphi(x)\in C_{c}^{\infty}(\mathbb{R})$ , we have

	$\displaystyle\int_{\mathbb{R}}\varphi(x)G_{\hbar,\varepsilon}(x)dx=$	$\displaystyle\int_{\mathbb{R}}\left(\int_{y}^{+\infty}\varphi(x)dx\right)\langle y\rangle^{1+\delta}\rho_{\hbar,\varepsilon}(y)dy$
	$\displaystyle=$	$\displaystyle A_{1}+A_{2},$

where

	$\displaystyle A_{1}=$	$\displaystyle\int_{\mathbb{R}}\chi(\frac{y}{R})\left(\int_{y}^{+\infty}\varphi(x)dx\right)\langle y\rangle^{1+\delta}\rho_{\hbar,\varepsilon}(y)dy,$
	$\displaystyle A_{2}=$	$\displaystyle\int_{\mathbb{R}}(1-\chi(\frac{y}{R}))\left(\int_{y}^{+\infty}\varphi(x)dx\right)\langle y\rangle^{1+\delta}\rho_{\hbar,\varepsilon}(y)dy.$

Using the weighted uniform estimate (3.1), we obtain

\displaystyle|A_{2}|\leq\frac{1}{R^{1-\delta}}\|\varphi\|_{L_{x}^{1}}\|\langle y\rangle^{2}\rho_{\hbar,\varepsilon}\|_{L_{y}^{1}}\to 0.

Therefore, letting first $(\hbar,\varepsilon)\to(0,0)$ and then $R\to\infty$ , by the convergence (4.11), we get

(5.15)		$\displaystyle\lim_{(\hbar,\varepsilon)\to(0,0)}\int_{\mathbb{R}}\varphi(x)G_{\hbar,\varepsilon}(x)dx=$	$\displaystyle\lim_{R\to\infty}\int_{\mathbb{R}}\chi(\frac{y}{R})\left(\int_{y}^{+\infty}\varphi(x)dx\right)\langle y\rangle^{1+\delta}\rho(dy)$
	$\displaystyle=$	$\displaystyle\int_{\mathbb{R}}\left(\int_{y}^{+\infty}\varphi(x)dx\right)\langle y\rangle^{1+\delta}\rho(dy)$
	$\displaystyle=$	$\displaystyle\int_{\mathbb{R}}\varphi(x)\left(\int_{-\infty}^{x}\langle y\rangle^{1+\delta}\rho(dy)\right)dx,$

where in the second and last equalities we have used the dominated convergence theorem and Fubini’s theorem based on the weighted estimate (4.12) that

\displaystyle\int_{\mathbb{R}}\langle y\rangle^{2}\rho(t,dy)=\iint_{\mathbb{R}^{2}}\langle y\rangle^{2}f(t,dy,d\xi)\leq C(t).

Hence, we complete the proof of (5.13) for $G(x)$ .

For (5.14), by (5.11) we have

		$\displaystyle\int_{\mathbb{R}}\varphi(x)\partial_{x}F_{\hbar,\varepsilon}(x)dx$
	$\displaystyle=$	$\displaystyle\int_{\mathbb{R}}\left(\int_{\mathbb{R}}\left(\frac{x-y}{\|x-y\|}\frac{\varphi(x)}{\langle x\rangle^{1+\delta}}-(1+\delta)\|x-y\|\frac{x\varphi(x)}{\langle x\rangle^{3+\delta}}\right)dx\right)\rho_{\hbar,\varepsilon}(y)dy.$

In a similar fashion as in (5.15), we get

		$\displaystyle\lim_{(\hbar,\varepsilon)\to(0,0)}\int_{\mathbb{R}}\varphi(x)\partial_{x}F_{\hbar,\varepsilon}(x)dx$
	$\displaystyle=$	$\displaystyle\int_{\mathbb{R}}\left(\int_{\mathbb{R}}\left(\frac{x-y}{\|x-y\|}\frac{\varphi(x)}{\langle x\rangle^{1+\delta}}-(1+\delta)\|x-y\|\frac{x\varphi(x)}{\langle x\rangle^{3+\delta}}\right)dx\right)\rho(dy)$
	$\displaystyle=$	$\displaystyle\int_{\mathbb{R}}\varphi(x)\left(\int_{\mathbb{R}}\frac{1}{\langle x\rangle^{1+\delta}}\frac{x-y}{\|x-y\|}\rho(dy)-(1+\delta)\int_{\mathbb{R}}\frac{x}{\langle x\rangle^{3+\delta}}\|x-y\|\rho(dy)\right)dx$
	$\displaystyle=$	$\displaystyle\int_{\mathbb{R}}\varphi(x)\left(\partial_{x}\int_{\mathbb{R}}\frac{\|x-y\|}{\langle x\rangle^{1+\delta}}\rho(dy)\right)dx,$

where in the last inequality we have used Leibniz rule for the Lipschitz continuous function. This completes the proof of (5.14).

Finally, we prove the convergence of the potential energy part to the desired form. By (5.11) and the weighted uniform estimate (3.1), we notice that

\displaystyle\|\langle x\rangle^{\frac{\delta}{2}}\partial_{x}F_{\hbar,\varepsilon}\|_{L_{x}^{1}}\lesssim\int_{\mathbb{R}}\frac{1}{\langle x\rangle^{1+\frac{\delta}{2}}}dx\|\langle x\rangle\rho_{\hbar,\varepsilon}\|_{L_{x}^{1}}\leq C,

and hence obtain

(5.16)

\displaystyle\Big{|}\int_{|x|\geq R}G_{\hbar,\varepsilon}(x)\partial_{x}F_{\hbar,\varepsilon}(x)dx\Big{|}\leq\frac{1}{R^{\frac{\delta}{2}}}\|G_{\hbar,\varepsilon}\|_{L_{x}^{\infty}}\|\langle x\rangle^{\frac{\delta}{2}}\partial_{x}F_{\hbar,\varepsilon}\|_{L_{x}^{1}}\leq\frac{C}{R^{\frac{\delta}{2}}}\to 0.

As $G_{\hbar,\varepsilon}(x)\stackrel{{\scriptstyle L_{loc}^{2}}}{{\to}}G$ , $\partial_{x}F_{\hbar,\varepsilon}\stackrel{{\scriptstyle L_{loc}^{2}}}{{\to}}\partial_{x}F$ , letting first $(\hbar,\varepsilon)\to(0,0)$ and then $R\to\infty$ , we use (5.16) and the dominated convergence theorem to get

		$\displaystyle\lim_{(\hbar,\varepsilon)\to(0,0)}\int_{\mathbb{R}}G_{\hbar,\varepsilon}(x)\partial_{x}F_{\hbar,\varepsilon}(x)dx$
	$\displaystyle=$	$\displaystyle\lim_{R\to\infty}\lim_{(\hbar,\varepsilon)\to(0,0)}\int_{\mathbb{R}}\chi(\frac{x}{R})G_{\hbar,\varepsilon}(x)\partial_{x}F_{\hbar,\varepsilon}(x)dx$
		$\displaystyle+\lim_{R\to\infty}\lim_{(\hbar,\varepsilon)\to(0,0)}\int_{\mathbb{R}}(1-\chi(\frac{x}{R}))G_{\hbar,\varepsilon}(x)\partial_{x}F_{\hbar,\varepsilon}(x)dx$
	$\displaystyle=$	$\displaystyle\lim_{R\to\infty}\int_{\mathbb{R}}\chi(\frac{x}{R})G(x)\partial_{x}F(x)dx$
	$\displaystyle=$	$\displaystyle\int_{\mathbb{R}}G(x)\partial_{x}F(x)dx.$

By Fubini’s theorem, we obtain

	$\displaystyle\int_{\mathbb{R}}G(x)\partial_{x}F(x)dx=$	$\displaystyle\int_{\mathbb{R}}\left(\int_{\mathbb{R}}1_{\left\{y\leq x\right\}}\partial_{x}F(x)dx\right)\langle y\rangle^{1+\delta}\rho(dy)$
	$\displaystyle=$	$\displaystyle-\int_{\mathbb{R}}\int_{\mathbb{R}}\frac{\|x-y\|}{\langle y\rangle^{1+\delta}}\rho(dx)\langle y\rangle^{1+\delta}\rho(dy)$
	$\displaystyle=$	$\displaystyle-\iint_{\mathbb{R}^{2}}\|x-y\|\rho(dx)\rho(dy).$

Therefore, together with the convergence of the kinetic energy part, we complete the proof of the conservation law of energy (5.6).

∎

6. Moment Convergence to the Vlasov-Poisson Equation

In the section, our goal is to prove the convergence of some subsequence of $f_{\hbar,\varepsilon}(t,x,\xi)$ to the Vlasov-Poisson equation for the test functions of the moment form

(6.1)

\displaystyle\phi(t,x,\xi)=\varphi(t,x)\xi^{k}.

That is, the limit point $f(t,x,\xi)$ satisfies the Vlasov-Poisson equation for the test functions of the form (6.1), based on which we extend it to all test function $\phi\in C_{c}^{\infty}((0,T)\times\mathbb{R}^{2})$ in Section 7.

The main result of this section is Lemma 6.1 below.

Lemma 6.1.

Let $T>0$ and $k\geq 0$ . For $\varphi\in C_{c}^{\infty}(\Omega_{T})$ , there holds that

(6.2)

\displaystyle\int_{\Omega_{T}}\int_{\mathbb{R}}\left(\partial_{t}\varphi+\xi\partial_{x}\varphi\right)\xi^{k}f(t,dx,d\xi)dt-k\int_{\Omega_{T}}\varphi\overline{E}\left(\int_{\mathbb{R}}\xi^{k-1}f(t,dx,d\xi)\right)dt=0,

where $\Omega_{T}=(0,T)\times\mathbb{R}$ and $\overline{E}$ is the Vol^′pert’s symmetric average defined in (A.1).

To motivate the proof of Lemma 6.1, from the equation (4.16) of $f_{\hbar,\varepsilon}(t,x,\xi)$ , we observe that the moment equation

(6.3)

\displaystyle\partial_{t}\int_{\mathbb{R}}\xi^{k}f_{\hbar,\varepsilon}d\xi+\partial_{x}\int_{\mathbb{R}}\xi^{k+1}f_{\hbar,\varepsilon}d\xi+kE_{\hbar,\varepsilon}\int_{\mathbb{R}}\xi^{k-1}f_{\hbar,\varepsilon}d\xi+\mathrm{R}_{\hbar,\varepsilon}^{(k)}=0,

where $E_{\hbar,\varepsilon}=\partial_{x}V_{\varepsilon}*\rho_{\hbar,\varepsilon}$ and the remainder term is

(6.4)

\mathrm{R}_{\hbar,\varepsilon}^{(k)}=\left\{\begin{aligned} &i\sum_{2\leq\alpha\leq k}\binom{k}{\alpha}\frac{\hbar^{\alpha-1}}{2^{k}}(1-(-1)^{\alpha})D_{x}^{\alpha}(V_{\varepsilon}*\rho_{\hbar,\varepsilon})\int_{\mathbb{R}}\xi^{k-\alpha}f_{\hbar,\varepsilon}d\xi,\quad&k\geq 3,\\ &0,\quad&k=0,1,2.\end{aligned}\right.

By the convergence result in Lemma 4.3, we have the convergence for the linear term

\displaystyle\lim_{(\hbar,\varepsilon)\to(0,0)}\int_{\Omega_{T}}\int_{\mathbb{R}}\left(\partial_{t}\varphi+\xi\partial_{x}\varphi\right)\xi^{k}f_{\hbar,\varepsilon}dxd\xi dt=

\displaystyle\int_{\Omega_{T}}\int_{\mathbb{R}}\left(\partial_{t}\varphi+\xi\partial_{x}\varphi\right)\xi^{k}f(t,dx,d\xi)dt.

Therefore, to conclude Lemma 6.1, we are left to prove the vanishing of the remainder term and the convergence of the nonlinear term, that is,

(6.5)		$\displaystyle\lim_{(\hbar,\varepsilon)\to(0,0)}\Big{\|}\int_{\Omega_{T}}\varphi\mathrm{R}_{\hbar,\varepsilon}^{(k)}dxdt\Big{\|}=0,$
(6.6)		$\displaystyle\lim_{(\hbar,\varepsilon)\to 0}\int_{\Omega_{T}}\varphi E_{\hbar,\varepsilon}\left(\int_{\mathbb{R}}\xi^{k-1}f_{\hbar,\varepsilon}d\xi\right)dxdt=\int_{\Omega_{T}}\varphi\overline{E}\left(\int_{\mathbb{R}}\xi^{k-1}f(t,dx,d\xi)\right)dt.$

We deal with the remainder term and prove (6.5) in Section 6.1. The convergence of the nonlinear term is usually one of the main difficulties, as it is actually a problem of convergence of the product form in the mixed limit. We prove (6.6) for the $k=1,2$ case in Section 6.2, and the general $k\geq 3$ case in Section 6.3.

6.1. Vanishing Remainder Terms via a Cancellation Structure

In the space of the strong topology, the remainder term $\mathrm{R}_{\hbar,\varepsilon}^{(m)}$ in (6.4) is only uniformly bounded in $L^{\infty}([0,T];L_{x}^{1})$ . We follow the idea of [50] to prove that the remainder term would vanish in the weak sense by an iteration scheme using a cancellation structure.

First, we provide the weighted uniform bound for the remainder term.

Lemma 6.2.

For $T>0$ and $m\geq 3$ , we have

(6.7)

\displaystyle\|\langle x\rangle\mathrm{R}_{\hbar,\varepsilon}^{(m)}\|_{L_{t}^{\infty}([0,T];L_{x}^{1})}\leq C(m,T).

Proof.

By the weighted uniform estimate (3.11) and the uniform bound (3.12), we use Hölder’s inequality to get

	$\displaystyle\\|\langle x\rangle\mathrm{R}_{\hbar,\varepsilon}^{(m)}\\|_{L_{t}^{\infty}L_{x}^{1}}\lesssim$	$\displaystyle\sum_{\alpha=0}^{m}\Big{\\|}\langle x\rangle\hbar^{\alpha-1}\partial_{x}^{\alpha}\left(V_{\varepsilon}*\rho_{\hbar,\varepsilon}\right)\int_{\mathbb{R}}\xi^{m-\alpha}f_{\hbar,\varepsilon}d\xi\Big{\\|}_{L_{t}^{\infty}L_{x}^{1}}$
	$\displaystyle\leq$	$\displaystyle\\|\hbar^{\alpha-1}\partial_{x}^{\alpha}V_{\varepsilon}*\rho_{\hbar,\varepsilon}\\|_{L_{t}^{\infty}L_{x}^{\infty}}\Big{\\|}\langle x\rangle\int_{\mathbb{R}}\xi^{m-\alpha}f_{\hbar,\varepsilon}d\xi\Big{\\|}_{L_{t}^{\infty}L_{x}^{1}}$
	$\displaystyle\leq$	$\displaystyle C(m,T).$

∎

Next, we get into the analysis of the remainder term.

Lemma 6.3.

Let $T>0$ and $k\geq 3$ . For $\varphi(t,x)\in C_{c}^{1}(\Omega_{T})$ and $\alpha=2n+1\leq k$ with $n\geq 1$ , we have

(6.8)

\displaystyle\hbar^{\alpha-1}\int_{\Omega_{T}}\varphi\partial_{x}^{\alpha}(V_{\varepsilon}*\rho_{\hbar,\varepsilon})\left(\int_{\mathbb{R}}\xi^{k-\alpha}f_{\hbar,\varepsilon}d\xi\right)dxdt=\mathcal{E}(\hbar,\varepsilon),

with

(6.9)

\displaystyle|\mathcal{E}(\hbar,\varepsilon)|\leq C(k,\alpha)\left(\hbar\|\nabla_{t,x}\varphi\|_{L_{t}^{1}L_{x}^{\infty}}+\hbar\|\varphi\|_{L_{t}^{1}L_{x}^{\infty}}+\varepsilon\|\varphi\|_{L_{t}^{1}L_{x}^{\infty}}\right).

In particular, we have the quantitative estimate that

(6.10)

\displaystyle\Big{|}\int_{\Omega_{T}}\varphi\mathrm{R}_{\hbar,\varepsilon}^{(k)}dxdt\Big{|}\leq C(k)\left(\hbar\|\nabla_{t,x}\varphi\|_{L_{t}^{1}L_{x}^{\infty}}+\hbar\|\varphi\|_{L_{t}^{1}L_{x}^{\infty}}+\varepsilon\|\varphi\|_{L_{t}^{1}L_{x}^{\infty}}\right),\quad\forall\varphi(t,x)\in C_{c}^{1}(\Omega_{T}),

and have the qualitative convergence that

(6.11)

\displaystyle\lim_{(\hbar,\varepsilon)\to(0,0)}\Big{|}\int_{\Omega_{T}}\varphi\mathrm{R}_{\hbar,\varepsilon}^{(k)}dxdt\Big{|}=0,\quad\forall\varphi(t,x)\in L_{t}^{1}([0,T];C_{b}(\mathbb{R})).

Proof.

For convenience, we take up the notation

(6.12)

\displaystyle U_{\varepsilon}(x):=

\displaystyle\frac{1}{2}|x|-V_{\varepsilon}(x)=\frac{1}{2}|x|(1-e^{-\varepsilon|x|}),

and hence rewrite

(6.13)

\displaystyle\hbar^{\alpha-1}\int_{\Omega_{T}}\varphi\partial_{x}^{\alpha}(V_{\varepsilon}*\rho_{\hbar,\varepsilon})\left(\int_{\mathbb{R}}\xi^{k-\alpha}f_{\hbar,\varepsilon}d\xi\right)dxdt=I_{1}-I_{2},

where

	$\displaystyle I_{1}=$	$\displaystyle\hbar^{\alpha-1}\int_{\Omega_{T}}\varphi\left(\partial_{x}^{\alpha}\frac{\|x\|}{2}*\rho_{\hbar,\varepsilon}\right)\left(\int_{\mathbb{R}}\xi^{k-\alpha}f_{\hbar,\varepsilon}d\xi\right)dxdt,$
	$\displaystyle I_{2}=$	$\displaystyle\hbar^{\alpha-1}\int_{\Omega_{T}}\varphi\left(\partial_{x}^{\alpha}U_{\varepsilon}*\rho_{\hbar,\varepsilon}\right)\left(\int_{\mathbb{R}}\xi^{k-\alpha}f_{\hbar,\varepsilon}d\xi\right)dxdt.$

We first deal with the term $I_{2}$ . Noting that

\displaystyle|\partial_{x}U_{\varepsilon}(x)|\leq\varepsilon|x|\quad a.e.,

we have the pointwise bound

(6.14)		$\displaystyle\hbar^{\alpha-1}\|\partial_{x}^{\alpha}(U_{\varepsilon}*\rho_{\hbar,\varepsilon})\|=$	$\displaystyle\hbar^{\alpha-1}\|\partial_{x}U_{\varepsilon}*\partial_{x}^{\alpha-1}\rho_{\hbar,\varepsilon}\|$
	$\displaystyle\leq$	$\displaystyle\hbar^{\alpha-1}\varepsilon\int_{\mathbb{R}}\|x-y\|\|\partial_{x}^{\alpha-1}\rho_{\hbar,\varepsilon}(y)\|dy$
	$\displaystyle\leq$	$\displaystyle\varepsilon\langle x\rangle\\|\langle x\rangle\hbar^{\alpha-1}\partial_{x}^{\alpha-1}\rho_{\hbar,\varepsilon}\\|_{L_{x}^{1}}$
	$\displaystyle\lesssim$	$\displaystyle\varepsilon\langle x\rangle,$

where in the last inequality we have used the uniform estimate (3.11). By (6.14), we then use Hölder’s inequality and the uniform estimate (3.11) to obtain

(6.15)

\displaystyle I_{2}\lesssim\varepsilon\|\varphi\|_{L_{t}^{1}L_{x}^{\infty}}\Big{\|}\langle x\rangle\int_{\mathbb{R}}\xi^{k-\alpha}f_{\hbar,\varepsilon}d\xi\Big{\|}_{L_{t}^{\infty}L_{x}^{1}}\lesssim\varepsilon\|\varphi\|_{L_{t}^{1}L_{x}^{\infty}}.

Next, we handle the term $I_{1}$ via an iteration scheme. For $\alpha=2n+1\leq k$ with $n\geq 1$ , we set the notation

(6.16)

\displaystyle M_{\varphi}^{(k,\alpha,j)}=\hbar^{\alpha-1}\int_{\Omega_{T}}\varphi\left(\partial_{x}^{\alpha-2}\int_{\mathbb{R}}\xi^{j}f_{\hbar,\varepsilon}d\xi\right)\left(\int_{\mathbb{R}}\xi^{k-\alpha-j}f_{\hbar,\varepsilon}d\xi\right)dxdt.

In particular, noticing that $\partial_{x}^{2}(\frac{|x|}{2})=\delta(x)$ , we have

I_{1}=\hbar^{\alpha-1}\int_{\Omega_{T}}\varphi\partial_{x}^{\alpha-2}((\partial_{x}^{2}\frac{|x|}{2})*\rho_{\hbar,\varepsilon})\left(\int_{\mathbb{R}}\xi^{k-\alpha}f_{\hbar,\varepsilon}d\xi\right)dxdt=M_{\varphi}^{(k,\alpha,0)}.

In the following, we get into the analysis of $M_{\varphi}^{(k,\alpha,j)}$ . By integration by parts, we have

	$\displaystyle M_{\varphi}^{(k,\alpha,j)}=$	$\displaystyle-\hbar^{\alpha-1}\int_{\Omega_{T}}\varphi\left(\partial_{x}^{\alpha-3}\int_{\mathbb{R}}\xi^{j}f_{\hbar,\varepsilon}d\xi\right)\left(\partial_{x}\int_{\mathbb{R}}\xi^{k-\alpha-j}f_{\hbar,\varepsilon}d\xi\right)dxdt$
		$\displaystyle-\hbar^{\alpha-1}\int_{\Omega_{T}}\partial_{x}\varphi\left(\partial_{x}^{\alpha-3}\int_{\mathbb{R}}\xi^{j}f_{\hbar,\varepsilon}d\xi\right)\left(\int_{\mathbb{R}}\xi^{k-\alpha-j}f_{\hbar,\varepsilon}d\xi\right)dxdt.$

Using the moment equation (6.3) in which we take $m=k-\alpha-j-1$ , we have

(6.17)

\displaystyle M_{\varphi}^{(k,\alpha,j)}=A_{1}+A_{2}+A_{3}+A_{4},

where

	$\displaystyle A_{1}=$	$\displaystyle\hbar^{\alpha-1}\int_{\Omega_{T}}\varphi\left(\partial_{x}^{\alpha-3}\int_{\mathbb{R}}\xi^{j}f_{\hbar,\varepsilon}d\xi\right)\left(\partial_{t}\int_{\mathbb{R}}\xi^{m}f_{\hbar,\varepsilon}d\xi\right)dxdt,$
	$\displaystyle A_{2}=$	$\displaystyle\hbar^{\alpha-1}\int_{\Omega_{T}}\varphi\left(\partial_{x}^{\alpha-3}\int_{\mathbb{R}}\xi^{j}f_{\hbar,\varepsilon}d\xi\right)\left(mE_{\hbar,\varepsilon}\int_{\mathbb{R}}\xi^{m-1}f_{\hbar,\varepsilon}d\xi\right)dxdt,$
	$\displaystyle A_{3}=$	$\displaystyle\hbar^{\alpha-1}\int_{\Omega_{T}}\varphi\left(\partial_{x}^{\alpha-3}\int_{\mathbb{R}}\xi^{j}f_{\hbar,\varepsilon}d\xi\right)\left(\mathrm{R}_{\hbar,\varepsilon}^{(m)}\right)dxdt,$
	$\displaystyle A_{4}=$	$\displaystyle-\hbar^{\alpha-1}\int_{\Omega_{T}}\partial_{x}\varphi\left(\partial_{x}^{\alpha-3}\int_{\mathbb{R}}\xi^{j}f_{\hbar,\varepsilon}d\xi\right)\left(\int_{\mathbb{R}}\xi^{k-\alpha-j}f_{\hbar,\varepsilon}d\xi\right)dxdt.$

We can directly bound the terms $A_{2}$ , $A_{3}$ and $A_{4}$ . For $A_{2}$ , by Hölder’s equality, the uniform estimates (3.10) and (3.11), we have

	$\displaystyle\|A_{2}\|\lesssim$	$\displaystyle\hbar\\|\varphi\\|_{L_{t}^{1}L_{x}^{\infty}}\Big{\\|}\hbar^{\alpha-2}\partial_{x}^{\alpha-3}\int_{\mathbb{R}}\xi^{j}f_{\hbar,\varepsilon}d\xi\Big{\\|}_{L_{t,x}^{\infty}}\\|E_{\hbar,\varepsilon}\\|_{L_{t,x}^{\infty}}\Big{\\|}\int_{\mathbb{R}}\xi^{m-1}f_{\hbar,\varepsilon}d\xi\Big{\\|}_{L_{t}^{\infty}L_{x}^{1}}$
	$\displaystyle\lesssim$	$\displaystyle\hbar\\|\varphi\\|_{L_{t}^{1}L_{x}^{\infty}}.$

In a similar way, for $A_{3}$ we use the uniform estimate (3.10) and the $L_{t}^{\infty}L_{x}^{1}$ bound for $\mathrm{R}_{\hbar,\varepsilon}^{(m)}$ to obtain

\displaystyle|A_{3}|\leq\hbar\|\varphi\|_{L_{t}^{1}L_{x}^{\infty}}\Big{\|}\hbar^{\alpha-2}\partial_{x}^{\alpha-3}\int_{\mathbb{R}}\xi^{j}f_{\hbar,\varepsilon}d\xi\Big{\|}_{L_{t,x}^{\infty}}\|\mathrm{R}_{\hbar,\varepsilon}^{(m)}\|_{L_{t}^{\infty}L_{x}^{1}}\lesssim

\displaystyle\hbar\|\varphi\|_{L_{t}^{1}L_{x}^{\infty}}.

In the same manner, we have

\displaystyle|A_{4}|\lesssim\hbar\|\partial_{x}\varphi\|_{L_{t}^{1}L_{x}^{\infty}}.

Next, we get into the analysis of the term $A_{1}$ . Using integration by parts in the time variable, we get

\displaystyle A_{1}=A_{11}+A_{12},

where

	$\displaystyle A_{11}=$	$\displaystyle-\hbar^{\alpha-1}\int_{\Omega_{T}}\varphi\left(\partial_{x}^{\alpha-3}\partial_{t}\int_{\mathbb{R}}\xi^{j}f_{\hbar,\varepsilon}d\xi\right)\left(\int_{\mathbb{R}}\xi^{m}f_{\hbar,\varepsilon}d\xi\right)dxdt,$
	$\displaystyle A_{12}=$	$\displaystyle-\hbar^{\alpha-1}\int_{\Omega_{T}}\partial_{t}\varphi\left(\partial_{x}^{\alpha-3}\int_{\mathbb{R}}\xi^{j}f_{\hbar,\varepsilon}d\xi\right)\left(\int_{\mathbb{R}}\xi^{m}f_{\hbar,\varepsilon}d\xi\right)dxdt.$

As the term $A_{12}$ can be treated in a similar way as $A_{2}$ , we have

\displaystyle|A_{12}|\lesssim\hbar\|\partial_{t}\varphi\|_{L_{t}^{1}L_{x}^{\infty}}.

For the term $A_{11}$ , we use again the moment equation (6.3) to get

\displaystyle A_{11}=A_{111}+A_{112}+A_{113},

where

	$\displaystyle A_{111}=$	$\displaystyle M_{\varphi}^{(k,\alpha,j+1)}=\hbar^{\alpha-1}\int_{\Omega_{T}}\varphi\left(\partial_{x}^{\alpha-2}\int_{\mathbb{R}}\xi^{j+1}f_{\hbar,\varepsilon}d\xi\right)\left(\int_{\mathbb{R}}\xi^{m}f_{\hbar,\varepsilon}d\xi\right)dxdt,$
	$\displaystyle A_{112}=$	$\displaystyle j\hbar^{\alpha-1}\int_{\Omega_{T}}\varphi\left(\partial_{x}^{\alpha-3}\left(E_{\hbar,\varepsilon}\int_{\mathbb{R}}\xi^{j+1}f_{\hbar,\varepsilon}d\xi\right)\right)\left(\int_{\mathbb{R}}\xi^{m}f_{\hbar,\varepsilon}d\xi\right)dxdt,$
	$\displaystyle A_{113}=$	$\displaystyle\hbar^{\alpha-1}\int_{\Omega_{T}}\varphi\left(\partial_{x}^{\alpha-3}\mathrm{R}_{\hbar,\varepsilon}^{(j)}\right)\left(\int_{\mathbb{R}}\xi^{m}f_{\hbar,\varepsilon}d\xi\right)dxdt.$

For the term $A_{112}$ , by the uniform estimates (3.10) and (3.12), we obtain

\displaystyle|A_{112}|\lesssim\hbar\|\varphi\|_{L_{t}^{1}L_{x}^{\infty}}.

For the term $A_{113}$ , using Leibniz rule, the uniform estimates (3.10) and (3.12), we have

\displaystyle\|\hbar^{\alpha-3}\partial_{x}^{\alpha-3}\mathrm{R}_{\hbar,\varepsilon}^{(j)}\|_{L_{x}^{1}}\lesssim C(j,\alpha,t),

and hence obtain

\displaystyle|A_{113}|\leq\hbar\|\varphi\|_{L_{t}^{1}L_{x}^{\infty}}\|\hbar^{\alpha-3}\partial_{x}^{\alpha-3}\mathrm{R}_{\hbar,\varepsilon}^{(j)}\|_{L_{t}^{\infty}L_{x}^{1}}\Big{\|}\hbar\int_{\mathbb{R}}\xi^{m}f_{\hbar,\varepsilon}d\xi\Big{\|}_{L_{t,x}^{\infty}}\lesssim\hbar\|\varphi\|_{L_{t}^{1}L_{x}^{\infty}}.

To sum up, we finally arrive at

(6.18)

\displaystyle M_{\varphi}^{(k,\alpha,j)}=M_{\varphi}^{(k,\alpha,j+1)}+\mathcal{O}(\hbar),

with $|\mathcal{O}(\hbar)|\lesssim\hbar\left(\|\nabla_{t,x}\varphi\|_{L_{t}^{1}L_{x}^{\infty}}+\|\varphi\|_{L_{t}^{1}L_{x}^{\infty}}\right)$ .

When $k=2l+1$ , $\alpha=2n+1$ , $l\geq n\geq 1$ , iteratively using (6.18) and integration by parts, we have

	$\displaystyle M_{\varphi}^{(k,\alpha,0)}=$	$\displaystyle M_{\varphi}^{(2l+1,2n+1,l-n)}+\mathcal{O}(\hbar)$
	$\displaystyle=$	$\displaystyle\hbar^{\alpha-1}\int_{\Omega_{T}}\varphi\left(\partial_{x}^{2n-1}\int_{\mathbb{R}}\xi^{l-n}f_{\hbar,\varepsilon}d\xi\right)\left(\int_{\mathbb{R}}\xi^{l-n}f_{\hbar,\varepsilon}d\xi\right)dxdt+\mathcal{O}(\hbar)$
	$\displaystyle=$	$\displaystyle(-1)\hbar^{\alpha-1}\int_{\Omega_{T}}\varphi\left(\partial_{x}^{2n-2}\int_{\mathbb{R}}\xi^{l-n}f_{\hbar,\varepsilon}d\xi\right)\left(\partial_{x}\int_{\mathbb{R}}\xi^{l-n}f_{\hbar,\varepsilon}d\xi\right)dxdt+\mathcal{O}(\hbar)$
	$\displaystyle=$	$\displaystyle...$
	$\displaystyle=$	$\displaystyle(-1)^{n-1}\hbar^{\alpha-1}\int_{\Omega_{T}}\varphi\left(\partial_{x}^{n}\int_{\mathbb{R}}\xi^{l-n}f_{\hbar,\varepsilon}d\xi\right)\left(\partial_{x}^{n-1}\int_{\mathbb{R}}\xi^{l-n}f_{\hbar,\varepsilon}d\xi\right)dxdt+\mathcal{O}(\hbar).$

Noticing that $\int_{\mathbb{R}}\xi^{l-n}f_{\hbar,\varepsilon}d\xi$ is real-valued, we hence have

(6.19)		$\displaystyle M_{\varphi}^{(k,\alpha,0)}=$	$\displaystyle(-1)^{n-1}\frac{\hbar^{\alpha-1}}{2}\int_{\Omega_{T}}\varphi\partial_{x}\left(\partial_{x}^{n-1}\int_{\mathbb{R}}\xi^{l-n}f_{\hbar,\varepsilon}d\xi\right)^{2}dxdt+\mathcal{O}(\hbar)$
	$\displaystyle=$	$\displaystyle(-1)^{n-1}\frac{\hbar^{\alpha-1}}{2}\int_{\Omega_{T}}\partial_{x}\varphi\left(\partial_{x}^{n-1}\int_{\mathbb{R}}\xi^{l-n}f_{\hbar,\varepsilon}d\xi\right)^{2}dxdt+\mathcal{O}(\hbar)$
	$\displaystyle\leq$	$\displaystyle\\|\partial_{x}\varphi\\|_{L_{t}^{1}L_{x}^{\infty}}\Big{\\|}\hbar^{n}\partial_{x}^{n-1}\int_{\mathbb{R}}\xi^{l-n}f_{\hbar,\varepsilon}d\xi\Big{\\|}_{L_{t}^{\infty}L_{x}^{1}}^{2}+\mathcal{O}(\hbar)$
	$\displaystyle\leq$	$\displaystyle\mathcal{O}(\hbar),$

where in the last inequality we have used the uniform estimate (3.10). Putting together the estimates (6.13), (6.15), and (6.19), we thus prove (6.8)–(6.9) for the case $k=2l+1$ .

When $k=2l$ , $\alpha=2n+1$ , $l\geq n\geq 1$ , repeating the proof of the case $k=2l+1$ , we also have

		$\displaystyle M_{\varphi}^{(k,\alpha,0)}$
	$\displaystyle=$	$\displaystyle M_{\varphi}^{(2l,2n+1,l-n)}+\mathcal{O}(\hbar)$
	$\displaystyle=$	$\displaystyle\hbar^{\alpha-1}\int_{\Omega_{T}}\varphi\left(\partial_{x}^{2n-1}\int_{\mathbb{R}}\xi^{l-n}f_{\hbar,\varepsilon}d\xi\right)\left(\int_{\mathbb{R}}\xi^{l-n-1}f_{\hbar,\varepsilon}d\xi\right)dxdt+\mathcal{O}(\hbar)$
	$\displaystyle=$	$\displaystyle(-1)\hbar^{\alpha-1}\int_{\Omega_{T}}\varphi\left(\partial_{x}^{2n-2}\int_{\mathbb{R}}\xi^{l-n}f_{\hbar,\varepsilon}d\xi\right)\left(\partial_{x}\int_{\mathbb{R}}\xi^{l-n-1}f_{\hbar,\varepsilon}d\xi\right)dxdt+\mathcal{O}(\hbar)$
	$\displaystyle=$	$\displaystyle...$
	$\displaystyle=$	$\displaystyle(-1)^{n-1}\hbar^{\alpha-1}\int_{\Omega_{T}}\varphi\left(\partial_{x}^{n}\int_{\mathbb{R}}\xi^{l-n}f_{\hbar,\varepsilon}d\xi\right)\left(\partial_{x}^{n-1}\int_{\mathbb{R}}\xi^{l-n-1}f_{\hbar,\varepsilon}d\xi\right)dxdt+\mathcal{O}(\hbar).$

By the moment equation (6.3), in a similar way in which we obtain (6.17), we have

	$\displaystyle M_{\varphi}^{(k,\alpha,0)}=$	$\displaystyle(-1)^{n}\frac{\hbar^{\alpha-1}}{2}\int_{\Omega_{T}}\varphi\partial_{t}\left(\partial_{x}^{n-1}\int_{\mathbb{R}}\xi^{l-n-1}f_{\hbar,\varepsilon}d\xi\right)^{2}dxdt+\mathcal{O}(\hbar)$
	$\displaystyle=$	$\displaystyle(-1)^{n-1}\frac{\hbar^{\alpha-1}}{2}\int_{\Omega_{T}}\partial_{t}\varphi\left(\partial_{x}^{n-1}\int_{\mathbb{R}}\xi^{l-n-1}f_{\hbar,\varepsilon}d\xi\right)^{2}dxdt+\mathcal{O}(\hbar)$
	$\displaystyle\leq$	$\displaystyle\\|\partial_{t}\varphi\\|_{L_{t}^{1}L_{x}^{\infty}}\Big{\\|}\hbar^{n}\partial_{x}^{n-1}\int_{\mathbb{R}}\xi^{l-n-1}f_{\hbar,\varepsilon}d\xi\Big{\\|}_{L_{t}^{\infty}L_{x}^{1}}^{2}+\mathcal{O}(\hbar)$
	$\displaystyle\leq$	$\displaystyle\mathcal{O}(\hbar).$

Therefore, putting together the estimates (6.13), (6.15), and (6.19), we have completed the proof of (6.8)–(6.9) for the case $k=2l$ .

Doing the summation over $\alpha$ in (6.8), we immediately obtain (6.10). For (6.11), we apply an approximation argument and rewrite

\displaystyle\int_{\Omega_{T}}\varphi\mathrm{R}_{\hbar,\varepsilon}^{(k)}dxdt=B_{1}+B_{2}+B_{3},

where

	$\displaystyle B_{1}=$	$\displaystyle\int_{\Omega_{T}}\varphi\left(1-\chi(\frac{x}{R})\right)\mathrm{R}_{\hbar,\varepsilon}^{(k)}dxdt,$
	$\displaystyle B_{2}=$	$\displaystyle\int_{\Omega_{T}}\left(\varphi\chi(\frac{x}{R})-\varphi_{n,R}\right)\mathrm{R}_{\hbar,\varepsilon}^{(k)}dxdt,$
	$\displaystyle B_{3}=$	$\displaystyle\int_{\Omega_{T}}\varphi_{n,R}\mathrm{R}_{\hbar,\varepsilon}^{(k)}dxdt,$

and

\displaystyle\varphi_{n,R}\in C_{c}^{\infty}(\Omega_{T}),\quad\lim_{n\to\infty}\big{\|}\varphi\chi(\frac{x}{R})-\varphi_{n,R}\big{\|}_{L_{t}^{1}([0,T];L_{x}^{\infty})}=0.

By the weighted uniform estimate (6.7) on the remainder term, we have

	$\displaystyle\|B_{1}\|\leq$	$\displaystyle\frac{1}{R}\\|\varphi\\|_{L_{t}^{1}([0,T];L_{x}^{\infty})}\\|\langle x\rangle\mathrm{R}_{\hbar,\varepsilon}^{(k)}\\|_{L_{t}^{\infty}([0,T];L_{x}^{1})}\lesssim\frac{1}{R},$
	$\displaystyle\|B_{2}\|\leq$	$\displaystyle\big{\\|}\varphi\chi(\frac{x}{R})-\varphi_{n,R}\big{\\|}_{L_{t}^{1}([0,T];L_{x}^{\infty})}\\|\mathrm{R}_{\hbar,\varepsilon}^{(k)}\\|_{L_{t}^{\infty}([0,T];L_{x}^{1})}\lesssim\big{\\|}\varphi\chi(\frac{x}{R})-\varphi_{n,R}\big{\\|}_{L_{t}^{1}([0,T];L_{x}^{\infty})}.$

Hence, using (6.10) for $\varphi_{n,R}\in C_{c}^{\infty}(\Omega_{T})$ , we obtain

		$\displaystyle\lim_{(\hbar,\varepsilon)\to(0,0)}\Big{\|}\int_{\Omega_{T}}\varphi\mathrm{R}_{\hbar,\varepsilon}^{(k)}dxdt\Big{\|}$
	$\displaystyle=$	$\displaystyle\lim_{R\to\infty}\lim_{n\to\infty}\lim_{(\hbar,\varepsilon)\to(0,0)}\left(B_{1}+B_{2}+B_{3}\right)$
	$\displaystyle\lesssim$	$\displaystyle\lim_{R\to\infty}\lim_{n\to\infty}\left(\frac{1}{R}+\big{\\|}\varphi\chi(\frac{x}{R})-\varphi_{n,R}\big{\\|}_{L_{t}^{1}([0,T];L_{x}^{\infty})}\right)=0,$

which completes the proof of (6.11).

∎

6.2. Convergence of the Nonlinear Term for $k=1,2$

As a preliminary part, we prove the convergence of the nonlinear term for the $k=1,2$ case based on the weighted uniform estimates in Section 3. We first provide estimates on $E_{\hbar,\varepsilon}(t,x)$ and study its limit.

Lemma 6.4.

There holds that

(6.20)		$\displaystyle\partial_{x}E_{\hbar,\varepsilon}=$	$\displaystyle\rho_{\hbar,\varepsilon}-\partial_{x}^{2}U_{\varepsilon}*\rho_{\hbar,\varepsilon},$
(6.21)		$\displaystyle\partial_{t}E_{\hbar,\varepsilon}=$	$\displaystyle-\int_{\mathbb{R}}\xi f_{\hbar,\varepsilon}d\xi+\partial_{x}^{2}U_{\varepsilon}*\int_{\mathbb{R}}\xi f_{\hbar,\varepsilon}d\xi.$

where $U_{\varepsilon}(x)=\frac{1}{2}|x|-V_{\varepsilon}(x)$ . We have the uniform estimates on $E_{\hbar,\varepsilon}(t,x)$ that

(6.22)	$\displaystyle\\|E_{\hbar,\varepsilon}\\|_{L_{t}^{\infty}([0,T];L_{x}^{\infty})}\leq$	$\displaystyle C(T),$
(6.23)	$\displaystyle\\|\partial_{x}E_{\hbar,\varepsilon}\\|_{L_{t}^{\infty}([0,T];L_{x}^{1})}\leq$	$\displaystyle C(T),$
(6.24)	$\displaystyle\\|\partial_{t}E_{\hbar,\varepsilon}\\|_{L_{t}^{\infty}([0,T];L_{x}^{1})}\leq$	$\displaystyle C(T).$

Moreover, for $p\in[1,\infty)$ we have the strong convergence that

(6.25)

\displaystyle E_{\hbar,\varepsilon}(t,x)\to E(t,x):=\frac{1}{2}\int_{\mathbb{R}}\frac{x-y}{|x-y|}\left(\int_{\mathbb{R}}f(t,y,\xi)d\xi\right)dy,\quad L_{loc}^{p}(\Omega_{T}),

where

(6.26)

\displaystyle E(t,x)\in BV\cap L^{\infty}(\Omega_{T}).

Moreover, the limit function $E(t,x)$ satisfies

(6.27)

\displaystyle\partial_{x}E=

\displaystyle\int_{\mathbb{R}}fd\xi,\quad\partial_{t}E=\int_{\mathbb{R}}\xi fd\xi,

in the sense of measures.

Proof.

Equations (6.20) and (6.21) follow from a direct calculation using the moment equation (6.3). Estimate (6.22) follows from the uniform estimate (3.12). For (6.23), noting that

\displaystyle\partial_{x}^{2}V_{\varepsilon}=\delta(x)e^{-\varepsilon|x|}-2\varepsilon e^{-\varepsilon|x|}+\varepsilon^{2}|x|e^{-\varepsilon|x|},

we use Young’s inequality to get

	$\displaystyle\\|\partial_{x}E_{\hbar,\varepsilon}\\|_{L_{t}^{\infty}([0,T];L_{x}^{1})}=$	$\displaystyle\\|(\partial_{x}^{2}V_{\varepsilon})*\rho_{\hbar,\varepsilon}\\|_{L_{t}^{\infty}([0,T];L_{x}^{1})}$
	$\displaystyle\leq$	$\displaystyle\\|\rho_{\hbar,\varepsilon}\\|_{L_{t}^{\infty}([0,T];L_{x}^{1})}\left(1+\\|\varepsilon e^{-\varepsilon\|x\|}\\|_{L_{x}^{1}}+\\|\varepsilon^{2}\|x\|e^{-\varepsilon\|x\|}\\|_{L_{x}^{1}}\right)\leq C(T).$

For (6.24), by the moment equation (6.3) we rewrite

\displaystyle\partial_{t}E_{\hbar,\varepsilon}=V_{\varepsilon}*\partial_{t}\rho_{\hbar,\varepsilon}=-\partial_{x}^{2}V_{\varepsilon}*\int_{\mathbb{R}}\xi f_{\hbar,\varepsilon}d\xi.

Via the same way in which we obtain (6.23), we arrive at (6.24).

By the uniform estimates (6.22)–(6.24) and $L^{p}$ compactness criteria, there is a subsequence of $\left\{E_{\hbar,\varepsilon}(t,x)\right\}$ , which we still denote by $\left\{E_{\hbar,\varepsilon}(t,x)\right\}$ , and some function

E(t,x)\in BV\cap L^{\infty}(\Omega_{T}),

such that

\displaystyle E_{\hbar,\varepsilon}(t,x)\stackrel{{\scriptstyle L_{loc}^{p}(\Omega_{T})}}{{\longrightarrow}}E(t,x),\quad p\in[1,\infty).

To obtain the explicit formula of $E(t,x)$ , we consider

	$\displaystyle\int_{\Omega_{T}}E_{\hbar,\varepsilon}\varphi dxdt=$	$\displaystyle\int_{\Omega_{T}}\frac{1}{2}\left(\frac{x}{\|x\|}\rho_{\hbar,\varepsilon}\right)\varphi dxdt+\int_{\Omega_{T}}\left(\partial_{x}U_{\varepsilon}\rho_{\hbar,\varepsilon}\right)\varphi dxdt$
	$\displaystyle=$	$\displaystyle\int_{\Omega_{T}}\frac{1}{2}\rho_{\hbar,\varepsilon}\left(\frac{x}{\|x\|}\varphi\right)dxdt+\int_{\Omega_{T}}\left(\partial_{x}U_{\varepsilon}\rho_{\hbar,\varepsilon}\right)\varphi dxdt.$

On the one hand, by the pointwise estimate (6.14) that $|\partial_{x}U_{\varepsilon}*\rho_{\hbar,\varepsilon}|\lesssim\varepsilon\langle x\rangle$ and the weighted uniform estimate (3.1), we have

\displaystyle\Big{|}\int_{\Omega_{T}}\left(\partial_{x}U_{\varepsilon}*\rho_{\hbar,\varepsilon}\right)\varphi dxdt\Big{|}\lesssim\varepsilon\|\langle x\rangle\rho_{\hbar,\varepsilon}\|_{L_{t}^{\infty}L_{x}^{1}}\|\langle x\rangle\varphi\|_{L_{t}^{1}L_{x}^{1}}\to 0.

On the other hand, due to the fact that $\frac{x}{|x|}*\varphi\in L_{t}^{1}([0,T];C_{b}(\mathbb{R}))$ , we use the narrow convergence (4.10) and hence obtain

\displaystyle\lim_{(\hbar,\varepsilon)\to(0,0)}\int_{\Omega_{T}}E_{\hbar,\varepsilon}\varphi dxdt=\frac{1}{2}\int_{\Omega_{T}}\left(\int_{\mathbb{R}}f(t,x,\xi)d\xi\right)\left(\frac{x}{|x|}*\varphi\right)dxdt,

which implies formula (6.25). In the same manner, we also attain (6.27) and hence complete the proof.

∎

Now, we are able to prove the following convergence.

Lemma 6.5.

For $\varphi\in L_{t}^{1}([0,T];C_{b}(\mathbb{R}))$ , we have

(6.28)		$\displaystyle\lim_{(\hbar,\varepsilon)\to(0,0)}\int_{\Omega_{T}}\varphi E_{\hbar,\varepsilon}\left(\int_{\mathbb{R}}f_{\hbar,\varepsilon}d\xi\right)dxdt=$	$\displaystyle\int_{\Omega_{T}}\varphi\overline{E}\partial_{x}Edxdt,$
(6.29)		$\displaystyle\lim_{(\hbar,\varepsilon)\to(0,0)}\int_{\Omega_{T}}\varphi E_{\hbar,\varepsilon}\left(\int_{\mathbb{R}}\xi f_{\hbar,\varepsilon}d\xi\right)dxdt=$	$\displaystyle\int_{\Omega_{T}}\varphi\overline{E}\partial_{t}Edxdt.$

Proof.

It suffices to prove (6.28), as (6.29) follows similarly. First, we prove that (6.28) holds for $\varphi\in C_{c}^{\infty}(\Omega_{T})$ . By (6.20), we rewrite

		$\displaystyle\int_{\Omega_{T}}\varphi E_{\hbar,\varepsilon}\left(\int_{\mathbb{R}}f_{\hbar,\varepsilon}d\xi\right)dxdt$
	$\displaystyle=$	$\displaystyle\int_{\Omega_{T}}\varphi E_{\hbar,\varepsilon}\left(\partial_{x}E_{\hbar,\varepsilon}\right)dxdt+\int_{\Omega_{T}}\varphi E_{\hbar,\varepsilon}\left(\partial_{x}^{2}U_{\varepsilon}*\rho_{\hbar,\varepsilon}\right)dxdt$
	$\displaystyle:=$	$\displaystyle A_{\hbar,\varepsilon}+B_{\hbar,\varepsilon}.$

For the first term $A_{\hbar,\varepsilon}$ , by property $(1)$ of BV functions in Appendix B, we have

	$\displaystyle\lim_{(\hbar,\varepsilon)\to(0,0)}A_{\hbar,\varepsilon}=$	$\displaystyle-\lim_{(\hbar,\varepsilon)\to(0,0)}\frac{1}{2}\int_{\Omega_{T}}(\partial_{x}\varphi)\left(E_{\hbar,\varepsilon}\right)^{2}dxdt$
	$\displaystyle=$	$\displaystyle-\frac{1}{2}\int_{\Omega_{T}}(\partial_{x}\varphi)\left(E\right)^{2}dxdt$
	$\displaystyle=$	$\displaystyle\int_{\Omega_{T}}\varphi\overline{E}\left(\partial_{x}E\right)dxdt.$

For the second term $B_{\hbar,\varepsilon}$ ,

	$\displaystyle B_{\hbar,\varepsilon}=$	$\displaystyle\int_{\Omega_{T}}\varphi E_{\hbar,\varepsilon}\left(\partial_{x}^{2}U_{\varepsilon}*\rho_{\hbar,\varepsilon}\right)dxdt$
	$\displaystyle=$	$\displaystyle-\int_{\Omega_{T}}\left(\partial_{x}\varphi\right)E_{\hbar,\varepsilon}\left(\partial_{x}U_{\varepsilon}\rho_{\hbar,\varepsilon}\right)dxdt-\int_{\Omega_{T}}\varphi\left(\partial_{x}E_{\hbar,\varepsilon}\right)\left(\partial_{x}U_{\varepsilon}\rho_{\hbar,\varepsilon}\right)dxdt$
	$\displaystyle\leq$	$\displaystyle\varepsilon\\|\langle x\rangle\partial_{x}\varphi\\|_{L_{x}^{1}}\\|E_{\hbar,\varepsilon}\\|_{L_{x}^{\infty}}+\varepsilon\\|\langle x\rangle\varphi\\|_{L_{x}^{\infty}}\\|\partial_{x}E_{\hbar,\varepsilon}\\|_{L_{x}^{1}}\to 0.$

Hence, we complete the proof of (6.28) for $\varphi\in C_{c}^{\infty}(\Omega_{T})$ . Furthermore, by the uniform bound (6.22) and the weighted uniform estimate (3.11), we get the weighted estimate that

(6.30)

\displaystyle\Big{\|}\langle x\rangle E_{\hbar,\varepsilon}\left(\int_{\mathbb{R}}f_{\hbar,\varepsilon}d\xi\right)\Big{\|}_{L_{t}^{\infty}L_{x}^{1}}\leq\|E_{\hbar,\varepsilon}\|_{L_{t}^{\infty}L_{x}^{\infty}}\Big{\|}\langle x\rangle\int_{\mathbb{R}}f_{\hbar,\varepsilon}d\xi\Big{\|}_{L_{t}^{\infty}L_{x}^{1}}\leq C(T).

Via the same way in which we obtain (6.11) by an approximation argument, we arrive at (6.28) for $\varphi\in L_{t}^{1}([0,T];C_{b}(\mathbb{R}))$ and hence complete the proof.

∎

6.3. Convergence of the Nonlinear Term for $k\geq 3$

In this section, we prove the convergence of the nonlinear term for the general $k\geq 3$ case.

Lemma 6.6.

Let $T>0$ and $k\geq 3$ . For $\varphi\in L_{t}^{1}([0,T];C_{b}(\mathbb{R}))$ , there holds that

(6.31)

\displaystyle\lim_{(\hbar,\varepsilon)\to(0,0)}\int_{\Omega_{T}}\varphi E_{\hbar,\varepsilon}\left(\int_{\mathbb{R}}\xi^{k-1}f_{\hbar,\varepsilon}d\xi\right)dxdt=\int_{\Omega_{T}}\varphi\overline{E}\left(\int_{\mathbb{R}}\xi^{k-1}f(t,dx,d\xi)\right)dt.

As we have proven the base $k=1,2$ case in Lemma 6.5, we take an induction argument to prove Lemma 6.6, whose proof is postponed to the end of the section.

Induction hypothesis: For $l\leq k-1$ , $\varphi\in L_{t}^{1}([0,T];C_{b}(\mathbb{R}))$ , there holds that

(6.32)

\displaystyle\lim_{(\hbar,\varepsilon)\to(0,0)}\int_{\Omega_{T}}\varphi E_{\hbar,\varepsilon}\left(\int_{\mathbb{R}}\xi^{l-1}f_{\hbar,\varepsilon}d\xi\right)dxdt=\int_{\Omega_{T}}\varphi\overline{E}\left(\int_{\mathbb{R}}\xi^{l-1}fd\xi\right)dxdt.

Before getting into the proof, we consider the integral function of the moment function that

(6.33)

\displaystyle M_{\hbar,\varepsilon}^{(m)}(t,x):=\int_{-\infty}^{x}\int_{\mathbb{R}}\xi^{m}f_{\hbar,\varepsilon}(t,y,\xi)d\xi dy,

which plays a similar role as $E_{\hbar,\varepsilon}(t,x)$ . We set up the uniform estimates for $M_{\hbar,\varepsilon}^{(m)}(t,x)$ and study its limit function, which is important to the convergence of the nonlinear term.

Lemma 6.7.

Let $0\leq m\leq k-1$ . The function $M_{\hbar,\varepsilon}^{(m)}(t,x)$ satisfies

(6.34)

\displaystyle\partial_{t}M_{\hbar,\varepsilon}^{(m)}+\partial_{x}M_{\hbar,\varepsilon}^{(m+1)}+m\int_{-\infty}^{x}E_{\hbar,\varepsilon}\int_{\mathbb{R}}\xi^{m-1}f_{\hbar,\varepsilon}d\xi dy+\int_{-\infty}^{x}\mathrm{R}_{\hbar,\varepsilon}^{(m)}dy=0,

and enjoys the uniform estimates that

(6.35)		$\displaystyle\\|M_{\hbar,\varepsilon}^{(m)}\\|_{L_{t}^{\infty}([0,T];L_{x}^{\infty})}\leq C(T),$
(6.36)		$\displaystyle\\|\partial_{x}M_{\hbar,\varepsilon}^{(m)}\\|_{L_{t}^{\infty}([0,T];L_{x}^{1})}\leq C(m,T),$
(6.37)		$\displaystyle\\|\partial_{t}M_{\hbar,\varepsilon}^{(m)}\\|_{L_{t}^{\infty}([0,T];L_{x}^{1})}\leq C(m,T).$

Moreover, for $p\in[1,\infty)$ we have the strong convergence that

(6.38)

\displaystyle M_{\hbar,\varepsilon}^{(m)}(t,x)\to M^{(m)}(t,x):=\int_{-\infty}^{x}\int_{\mathbb{R}}\xi^{m}f(t,y,\xi)d\xi dy,\quad\text{in $L_{loc}^{p}(\Omega_{T})$}.

Finally, under the induction hypothesis (6.32), the limit function satisfies

(6.39)

\displaystyle\partial_{t}M^{(m)}+\partial_{x}M^{(m+1)}+m\int_{-\infty}^{x}\overline{E}\int_{\mathbb{R}}\xi^{m-1}fd\xi dy=0,

in the sense of measures.

Proof.

Equation (6.34) follows from the moment equation (6.3). By the weighted uniform estimates (3.11)–(3.12) and the uniform bound (6.7) on the remainder term, we have (6.35)–(6.37). In the same way in which we obtain (6.25), we get (6.38).

Next, we prove (6.39). For the linear part, we have

\displaystyle\lim_{(\hbar,\varepsilon)\to(0,0)}\int_{\Omega_{T}}\left(\partial_{t}M_{\hbar,\varepsilon}^{(m)}+\partial_{x}M_{\hbar,\varepsilon}^{(m+1)}\right)\varphi dxdt=\int_{\Omega_{T}}-M^{(m)}\partial_{t}\varphi-M^{(m+1)}\partial_{x}\varphi dxdt.

For the nonlinear part, we use the induction hypothesis to get

		$\displaystyle\lim_{(\hbar,\varepsilon)\to(0,0)}m\int_{\Omega_{T}}\left(\int_{-\infty}^{x}E_{\hbar,\varepsilon}\int_{\mathbb{R}}\xi^{m-1}f_{\hbar,\varepsilon}d\xi dy\right)\varphi dxdt$
	$\displaystyle=$	$\displaystyle\lim_{(\hbar,\varepsilon)\to(0,0)}m\int_{\Omega_{T}}\left(E_{\hbar,\varepsilon}\int_{\mathbb{R}}\xi^{m-1}f_{\hbar,\varepsilon}d\xi\right)\left(\int_{y}^{\infty}\varphi dx\right)dydt$
	$\displaystyle=$	$\displaystyle m\int_{\Omega_{T}}\left(\overline{E}\int_{\mathbb{R}}\xi^{m-1}fd\xi\right)\left(\int_{y}^{\infty}\varphi dx\right)dydt$
	$\displaystyle=$	$\displaystyle\int_{\Omega_{T}}\left(\int_{-\infty}^{x}\overline{E}\int_{\mathbb{R}}\xi^{m-1}fd\xi dy\right)\varphi dxdt.$

For the remainder term, due to the fact that $\int_{y}^{\infty}\varphi dx\in L_{t}^{1}([0,T];C_{b}(\mathbb{R}))$ , we can use (6.11) in Lemma 6.3 to get

\displaystyle\lim_{(\hbar,\varepsilon)\to(0,0)}\int_{\Omega_{T}}\left(\int_{-\infty}^{x}\mathrm{R}_{\hbar,\varepsilon}^{(m)}dy\right)\varphi dxdt=

\displaystyle\lim_{(\hbar,\varepsilon)\to(0,0)}\int_{\Omega_{T}}\mathrm{R}_{\hbar,\varepsilon}^{(m)}\left(\int_{y}^{\infty}\varphi dx\right)dydt=0.

Hence, by formula (6.34), we complete the proof of (6.39). ∎

The following lemma shows that the limit function $M^{(m)}$ satisfies an induction equation. This is the key to reduce the order of the weight function $\xi^{k}$ so that one can make use of the induction hypothesis.

Lemma 6.8.

Let $0\leq j\leq k-1$ , $0\leq m\leq k-1$ . Under the induction hypothesis (6.32), for $\varphi\in C_{c}^{\infty}(\Omega_{T})$ , we have

(6.40)			$\displaystyle\int_{\Omega_{T}}\varphi\overline{M}^{(j)}\left(\partial_{x}M^{(m+1)}\right)dxdt$
	$\displaystyle=$	$\displaystyle\int_{\Omega_{T}}\varphi\overline{M}^{(j+1)}\left(\partial_{x}M^{(m)}\right)dxdt+I_{1}^{(j,m)}+I_{2}^{(j,m)}+I_{3}^{(j,m)}+I_{4}^{(j,m)},$

where $\overline{M}^{(j)}$ is the Vol^′pert’s symmetric average defined in (A.1) and

	$\displaystyle I_{1}^{(j,m)}=$	$\displaystyle\int_{\Omega_{T}}(\partial_{t}\varphi)M^{(j)}M^{(m)}dxdt,$
	$\displaystyle I_{2}^{(j,m)}=$	$\displaystyle-m\int_{\Omega_{T}}\varphi\left(\int_{-\infty}^{x}\overline{E}\int_{\mathbb{R}}\xi^{j-1}fd\xi dy\right)M^{(m)}dxdt,$
	$\displaystyle I_{3}^{(j,m)}=$	$\displaystyle\int_{\Omega_{T}}\left(\partial_{x}\varphi\right)M^{(j+1)}M^{(m)}dxdt,$
	$\displaystyle I_{4}^{(j,m)}=$	$\displaystyle-j\int_{\Omega_{T}}\varphi M^{(j)}\left(\int_{-\infty}^{x}\overline{E}\int_{\mathbb{R}}\xi^{m-1}fd\xi dy\right)dxdt.$

Proof.

We consider the test function of the form

\left(\varphi M^{(j)}\right)*\eta_{\sigma}\in C_{c}^{\infty}(\Omega_{T}),

where $\varphi\in C_{c}^{\infty}(\Omega_{T})$ and $\eta_{\sigma}(t,x)=\sigma^{-2}\eta(t/\sigma,x/\sigma)$ is a smooth mollifier and approximation of the identity. Putting the test function into the limit equation (6.39), we obtain

		$\displaystyle\int_{\Omega_{T}}\left(\left(\varphi M^{(j)}\right)*\eta_{\sigma}\right)\left(\partial_{x}M^{(m+1)}\right)dxdt$
	$\displaystyle=$	$\displaystyle\int_{\Omega_{T}}\left(\partial_{t}\left(\varphi M^{(j)}\right)*\eta_{\sigma}\right)M^{(m)}dxdt$
		$\displaystyle-m\int_{\Omega_{T}}\left(\left(\varphi M^{(j)}\right)*\eta_{\sigma}\right)\left(\int_{-\infty}^{x}\overline{E}\int_{\mathbb{R}}\xi^{m-1}fd\xi dy\right)dxdt$
	$\displaystyle=$	$\displaystyle\int_{\Omega_{T}}\varphi\left(\partial_{t}M^{(j)}\right)(M^{(m)}\eta_{\sigma})dxdt+\int_{\Omega_{T}}\left(\partial_{t}\varphi\right)M^{(j)}(M^{(m)}\eta_{\sigma})dxdt$
		$\displaystyle-m\int_{\Omega_{T}}\left(\left(\varphi M^{(j)}\right)*\eta_{\sigma}\right)\left(\int_{-\infty}^{x}\overline{E}\int_{\mathbb{R}}\xi^{m-1}fd\xi dy\right)dxdt$
	$\displaystyle:=$	$\displaystyle\int_{\Omega_{T}}\varphi\left(\partial_{t}M^{(j)}\right)(M^{(m)}*\eta_{\sigma})dxdt+I_{1,\sigma}^{(j,m)}+I_{2,\sigma}^{(j,m)},$

where

	$\displaystyle I_{1,\sigma}^{(j,m)}=$	$\displaystyle\int_{\Omega_{T}}\left(\partial_{t}\varphi\right)M^{(j)}(M^{(m)}*\eta_{\sigma})dxdt,$
	$\displaystyle I_{2,\sigma}^{(j,m)}=$	$\displaystyle-m\int_{\Omega_{T}}\left(\left(\varphi M^{(j)}\right)*\eta_{\sigma}\right)\left(\int_{-\infty}^{x}\overline{E}\int_{\mathbb{R}}\xi^{m-1}fd\xi dy\right)dxdt.$

Using again (6.39) for $\partial_{t}M^{(j)}$ , we expand

		$\displaystyle\int_{\Omega_{T}}\varphi\left(\partial_{t}M^{(j)}\right)(M^{(m)}*\eta_{\sigma})dxdt$
	$\displaystyle=$	$\displaystyle-\int_{\Omega_{T}}\varphi\left(\partial_{x}M^{(j+1)}\right)(M^{(m)}*\eta_{\sigma})dxdt$
		$\displaystyle-j\int_{\Omega_{T}}\varphi\left(\int_{-\infty}^{x}\overline{E}\int_{\mathbb{R}}\xi^{m-1}fd\xi dy\right)(M^{(m)}*\eta_{\sigma})dxdt$
	$\displaystyle=$	$\displaystyle\int_{\Omega_{T}}\varphi M^{(j+1)}(\partial_{x}M^{(m)}\eta_{\sigma})dxdt+\int_{\Omega_{T}}\left(\partial_{x}\varphi\right)M^{(j+1)}(M^{(m)}\eta_{\sigma})dxdt$
		$\displaystyle-j\int_{\Omega_{T}}\varphi\left(\int_{-\infty}^{x}\overline{E}\int_{\mathbb{R}}\xi^{m-1}fd\xi dy\right)(M^{(m)}*\eta_{\sigma})dxdt$
	$\displaystyle:=$	$\displaystyle\int_{\Omega_{T}}\left(\left(\varphi M^{(j+1)}\right)*\eta_{\sigma}\right)\partial_{x}M^{(m)}dxdt+I_{3,\sigma}^{(j,m)}+I_{4,\sigma}^{(j,m)},$

where

	$\displaystyle I_{3,\sigma}^{(j,m)}=$	$\displaystyle\int_{\Omega_{T}}\left(\partial_{x}\varphi\right)M^{(j+1)}(M^{(m)}*\eta_{\sigma})dxdt,$
	$\displaystyle I_{4,\sigma}^{(j,m)}=$	$\displaystyle-j\int_{\Omega_{T}}\varphi\left(\int_{-\infty}^{x}\overline{E}\int_{\mathbb{R}}\xi^{m-1}fd\xi dy\right)(M^{(m)}*\eta_{\sigma})dxdt.$

Therefore, we arrive at

(6.41)			$\displaystyle\int_{\Omega_{T}}\left(\left(\varphi M^{(j)}\right)*\eta_{\sigma}\right)\left(\partial_{x}M^{(m+1)}\right)dxdt$
	$\displaystyle=$	$\displaystyle\int_{\Omega_{T}}\left(\left(\varphi M^{(j+1)}\right)*\eta_{\sigma}\right)\partial_{x}M^{(m)}dxdt+I_{1,\sigma}^{(j,m)}+I_{2,\sigma}^{(j,m)}+I_{3,\sigma}^{(j,m)}+I_{4,\sigma}^{(j,m)}.$

By the dominated convergence theorem, we have

\displaystyle\lim_{\sigma\to 0}I_{i,\sigma}^{(j,m)}=I_{i}^{(j,m)},\quad i=1,2,3,4.

By the properties (5)-(6) of BV functions at the Appendix B, we have

	$\displaystyle\lim_{\sigma\to 0}\int_{\Omega_{T}}\left(\left(\varphi M^{(j)}\right)*\eta_{\sigma}\right)\left(\partial_{x}M^{(m+1)}\right)dxdt=$	$\displaystyle\int_{\Omega_{T}}\varphi\overline{M}^{(j)}\left(\partial_{x}M^{(m+1)}\right)dxdt,$
	$\displaystyle\lim_{\sigma\to 0}\int_{\Omega_{T}}\left(\left(\varphi M^{(j+1)}\right)*\eta_{\sigma}\right)\left(\partial_{x}M^{(m)}\right)dxdt=$	$\displaystyle\int_{\Omega_{T}}\varphi\overline{M}^{(j+1)}\left(\partial_{x}M^{(m)}\right)dxdt.$

Sending $\sigma\to 0$ in (6.41), we complete the proof of (6.40).

∎

Next, we prove that the function $M_{\hbar,\varepsilon}^{(m)}(t,x)$ , which is similar to its limit function $M^{(m)}(t,x)$ , also has an induction structure.

Lemma 6.9.

Let $0\leq j\leq k-1$ , $0\leq m\leq k-1$ . Under the induction hypothesis (6.32), we have

(6.42)			$\displaystyle\lim_{(\hbar,\varepsilon)\to(0,0)}\int_{\Omega_{T}}\varphi M_{\hbar,\varepsilon}^{(j)}\left(\partial_{x}M_{\hbar,\varepsilon}^{(m+1)}\right)dxdt$
	$\displaystyle=$	$\displaystyle\lim_{(\hbar,\varepsilon)\to(0,0)}\int_{\Omega_{T}}\varphi M_{\hbar,\varepsilon}^{(j+1)}\left(\partial_{x}M_{\hbar,\varepsilon}^{(m)}\right)dxdt+I_{1}^{(j,m)}+I_{2}^{(j,m)}+I_{3}^{(j,m)}+I_{4}^{(j,m)}.$

Proof.

Using equation (6.34) for $\partial_{x}M_{\hbar,\varepsilon}^{(m+1)}$ , we get

\displaystyle\int_{\Omega_{T}}\varphi M_{\hbar,\varepsilon}^{(j)}\left(\partial_{x}M_{\hbar,\varepsilon}^{(m+1)}\right)dxdt=

\displaystyle A_{\hbar,\varepsilon,1}^{(j,m)}+A_{\hbar,\varepsilon,2}^{(j,m)}+A_{\hbar,\varepsilon,3}^{(j,m)},

where

	$\displaystyle A_{\hbar,\varepsilon,1}^{(j,m)}=$	$\displaystyle-\int_{\Omega_{T}}\varphi M_{\hbar,\varepsilon}^{(j)}\left(\partial_{t}M_{\hbar,\varepsilon}^{(m)}\right),$
	$\displaystyle A_{\hbar,\varepsilon,2}^{(j,m)}=$	$\displaystyle-m\int_{\Omega_{T}}\varphi M_{\hbar,\varepsilon}^{(j)}\left(\int_{-\infty}^{x}E_{\hbar,\varepsilon}\int_{\mathbb{R}}\xi^{m-1}f_{\hbar,\varepsilon}d\xi dy\right)dxdt,$
	$\displaystyle A_{\hbar,\varepsilon,3}^{(j,m)}=$	$\displaystyle-\int_{\Omega_{T}}\varphi M_{\hbar,\varepsilon}^{(j)}\left(\int_{-\infty}^{x}\mathrm{R}_{\hbar,\varepsilon}^{(m)}(y)dy\right)dxdt.$

Using again equation (6.34) for $\partial_{t}M_{\hbar,\varepsilon}^{(j)}$ , we expand

	$\displaystyle A_{\hbar,\varepsilon,1}^{(j,m)}=$	$\displaystyle\int_{\Omega_{T}}(\partial_{t}\varphi)M_{\hbar,\varepsilon}^{(j)}M_{\hbar,\varepsilon}^{(m)}dxdt+\int_{\Omega_{T}}\varphi\left(\partial_{t}M_{\hbar,\varepsilon}^{(j)}\right)M_{\hbar,\varepsilon}^{(m)}dxdt$
	$\displaystyle=$	$\displaystyle A_{\hbar,\varepsilon,10}^{(j,m)}+A_{\hbar,\varepsilon,11}^{(j,m)}+A_{\hbar,\varepsilon,12}^{(j,m)}+A_{\hbar,\varepsilon,13}^{(j,m)}+A_{\hbar,\varepsilon,14}^{(j,m)},$

where

	$\displaystyle A_{\hbar,\varepsilon,10}^{(j,m)}=$	$\displaystyle\int_{\Omega_{T}}(\partial_{t}\varphi)M_{\hbar,\varepsilon}^{(j)}M_{\hbar,\varepsilon}^{(m)}dxdt,$
	$\displaystyle A_{\hbar,\varepsilon,11}^{(j,m)}=$	$\displaystyle\int_{\Omega_{T}}\varphi M_{\hbar,\varepsilon}^{(j+1)}\left(\partial_{x}M_{\hbar,\varepsilon}^{(m)}\right)dxdt,$
	$\displaystyle A_{\hbar,\varepsilon,12}^{(j,m)}=$	$\displaystyle\int_{\Omega_{T}}\left(\partial_{x}\varphi\right)M_{\hbar,\varepsilon}^{(j+1)}M_{\hbar,\varepsilon}^{(m)}dxdt,$
	$\displaystyle A_{\hbar,\varepsilon,13}^{(j,m)}=$	$\displaystyle-j\int_{\Omega_{T}}\varphi\left(\int_{-\infty}^{x}E_{\hbar,\varepsilon}\int_{\mathbb{R}}\xi^{j-1}f_{\hbar,\varepsilon}d\xi dy\right)M_{\hbar,\varepsilon}^{(m)}dxdt,$
	$\displaystyle A_{\hbar,\varepsilon,14}^{(j,m)}=$	$\displaystyle-\int_{\Omega_{T}}\varphi\left(\int_{-\infty}^{x}\mathrm{R}_{\hbar,\varepsilon}^{(m)}dy\right)M_{\hbar,\varepsilon}^{(m)}dxdt.$

Therefore, we arrive at

		$\displaystyle\int_{\Omega_{T}}\varphi M_{\hbar,\varepsilon}^{(j)}\left(\partial_{x}M_{\hbar,\varepsilon}^{(m+1)}\right)dxdt$
	$\displaystyle=$	$\displaystyle\int_{\Omega_{T}}\varphi M_{\hbar,\varepsilon}^{(j+1)}\left(\partial_{x}M_{\hbar,\varepsilon}^{(m)}\right)dxdt+A_{\hbar,\varepsilon,2}^{(j,m)}+A_{\hbar,\varepsilon,3}^{(j,m)}+A_{\hbar,\varepsilon,10}^{(j,m)}+A_{\hbar,\varepsilon,12}^{(j,m)}+A_{\hbar,\varepsilon,13}^{(j,m)}+A_{\hbar,\varepsilon,14}^{(j,m)}.$

We are left to prove that

	$\displaystyle\lim_{(\hbar,\varepsilon)\to(0,0)}A_{\hbar,\varepsilon,2}^{(j,m)}=$	$\displaystyle-m\int_{\Omega_{T}}\varphi M^{(j)}\left(\int_{-\infty}^{x}\overline{E}\int_{\mathbb{R}}\xi^{m-1}fd\xi dy\right)dxdt=I_{2}^{(j,m)},$
	$\displaystyle\lim_{(\hbar,\varepsilon)\to(0,0)}A_{\hbar,\varepsilon,3}^{(j,m)}=$	$\displaystyle 0,$
	$\displaystyle\lim_{(\hbar,\varepsilon)\to(0,0)}A_{\hbar,\varepsilon,10}^{(j,m)}=$	$\displaystyle\int_{\Omega_{T}}(\partial_{t}\varphi)M^{(j)}M^{(m)}dxdt=I_{1}^{(j,m)},$
	$\displaystyle\lim_{(\hbar,\varepsilon)\to(0,0)}A_{\hbar,\varepsilon,12}^{(j,m)}=$	$\displaystyle\int_{\Omega_{T}}\left(\partial_{x}\varphi\right)M^{(j+1)}M^{(m)}dxdt=I_{3}^{(j,m)},$
	$\displaystyle\lim_{(\hbar,\varepsilon)\to(0,0)}A_{\hbar,\varepsilon,13}^{(j,m)}=$	$\displaystyle-j\int_{\Omega_{T}}\varphi M^{(j)}\left(\int_{-\infty}^{x}\overline{E}\int_{\mathbb{R}}\xi^{m-1}fd\xi dy\right)dxdt=I_{4}^{(j,m)},$
	$\displaystyle\lim_{(\hbar,\varepsilon)\to(0,0)}A_{\hbar,\varepsilon,14}^{(j,m)}=$	$\displaystyle 0.$

It suffices to prove the limits for $A_{\hbar,\varepsilon,2}^{(j,m)}$ , $A_{\hbar,\varepsilon,3}^{(j,m)}$ , and $A_{\hbar,\varepsilon,10}^{(j,m)}$ , as the others can be dealt with in a similar way.

For $A_{\hbar,\varepsilon,2}^{(j,m)}$ , we rewrite

\displaystyle A_{\hbar,\varepsilon,2}^{(j,m)}=

\displaystyle-m\int_{\Omega_{T}}\left(\int_{y}^{\infty}\varphi M_{\hbar,\varepsilon}^{(j)}dx\right)\left(E_{\hbar,\varepsilon}\int_{\mathbb{R}}\xi^{m-1}f_{\hbar,\varepsilon}d\xi\right)dydt.

On the one hand, we have the $L_{x}^{\infty}$ convergence that

\displaystyle\Big{\|}\int_{y}^{\infty}\varphi M_{\hbar,\varepsilon}^{(j)}dx-\int_{y}^{\infty}\varphi M^{(j)}dx\Big{\|}_{L_{x}^{\infty}}\leq\|\varphi\|_{L_{x}^{2}}\|M_{\hbar,\varepsilon}^{(j)}-M^{(j)}\|_{L_{x,loc}^{2}}\to 0.

On the other hand, by the induction hypothesis for $l\leq k-1$ , we have

\displaystyle\lim_{(\hbar,\varepsilon)\to(0,0)}\int_{\Omega_{T}}\varphi E_{\hbar,\varepsilon}\left(\int_{\mathbb{R}}\xi^{m-1}f_{\hbar,\varepsilon}d\xi\right)dxdt=\int_{\Omega_{T}}\varphi\overline{E}\left(\int_{\mathbb{R}}\xi^{m-1}fd\xi\right)dxdt,

for $\varphi\in L_{t}^{1}([0,T];C_{b}(\mathbb{R}))$ . Therefore, we obtain

	$\displaystyle\lim_{(\hbar,\varepsilon)\to(0,0)}A_{\hbar,\varepsilon,2}^{(j,m)}=$	$\displaystyle-m\int_{\Omega_{T}}\left(\int_{y}^{\infty}\varphi M^{(j)}dx\right)\left(\overline{E}\int_{\mathbb{R}}\xi^{m-1}fd\xi\right)dydt$
	$\displaystyle=$	$\displaystyle-m\int_{\Omega_{T}}\varphi M^{(j)}\left(\int_{-\infty}^{x}\overline{E}\int_{\mathbb{R}}\xi^{m-1}fd\xi dy\right)dxdt$
	$\displaystyle=$	$\displaystyle I_{2}^{(j,m)}.$

For $A_{\hbar,\varepsilon,3}^{(j,m)}$ , we rewrite

	$\displaystyle A_{\hbar,\varepsilon,3}^{(j,m)}=$	$\displaystyle-\int_{\Omega_{T}}\left(\int_{y}^{+\infty}\varphi M_{\hbar,\varepsilon}^{(j)}dx\right)\mathrm{R}_{\hbar,\varepsilon}^{(m)}dydt$
	$\displaystyle=$	$\displaystyle-\int_{\Omega_{T}}\left(1-\chi(\frac{y}{R})+\chi(\frac{y}{R})\right)\left(\int_{y}^{+\infty}\varphi M_{\hbar,\varepsilon}^{(j)}dx\right)\mathrm{R}_{\hbar,\varepsilon}^{(m)}dydt.$

By the quantitative estimate (6.10) in Lemma 6.3 and the weighted uniform bound (6.7) on the remainder term, we have

	$\displaystyle\big{\|}A_{\hbar,\varepsilon,3}^{(j,m)}\big{\|}\lesssim$	$\displaystyle\hbar\Big{\\|}\nabla_{t,y}\left(\chi(\frac{y}{R})\int_{y}^{+\infty}\varphi M_{\hbar,\varepsilon}^{(j)}dx\right)\Big{\\|}_{L_{t}^{1}L_{x}^{\infty}}+(\hbar+\varepsilon)\Big{\\|}\chi(\frac{y}{R})\int_{y}^{+\infty}\varphi M_{\hbar,\varepsilon}^{(j)}dx\Big{\\|}_{L_{t}^{1}L_{x}^{\infty}}$
		$\displaystyle+\frac{1}{R}\Big{\\|}\int_{y}^{+\infty}\varphi M_{\hbar,\varepsilon}^{(j)}dx\Big{\\|}_{L_{t}^{1}L_{x}^{\infty}}\\|\langle x\rangle\mathrm{R}_{\hbar,\varepsilon}^{(m)}\\|_{L_{t}^{\infty}L_{x}^{1}}$
	$\displaystyle\leq$	$\displaystyle\hbar\left(\\|\varphi\\|_{L_{t}^{1}L_{x}^{\infty}}+\\|\partial_{t}\varphi\\|_{L_{t}^{1}L_{x}^{1}}\right)\left(\\|M_{\hbar,\varepsilon}^{(j)}\\|_{L_{t}^{\infty}L_{x}^{\infty}}+\\|\partial_{t}M_{\hbar,\varepsilon}^{(j)}\\|_{L_{t}^{\infty}L_{x}^{1}}\right)$
		$\displaystyle+\left(\frac{\hbar}{R}+\hbar+\varepsilon+\frac{1}{R}\right)\\|\varphi\\|_{L_{t}^{1}L_{x}^{\infty}}\\|M_{\hbar,\varepsilon}^{(j)}\\|_{L_{t}^{\infty}L_{x}^{\infty}}$
	$\displaystyle\lesssim$	$\displaystyle\hbar+\varepsilon+\frac{1}{R}\to 0,$

where in the last inequality we have used the uniform bounds (6.35)–(6.37) on $M_{\hbar,\varepsilon}^{(j)}$ .

For $A_{\hbar,\varepsilon,10}^{(j,m)}$ , noting that

\displaystyle M_{\hbar,\varepsilon}^{(j)}\stackrel{{\scriptstyle L_{loc}^{2}}}{{\longrightarrow}}M^{(j)},

we immediately get

	$\displaystyle\lim_{(\hbar,\varepsilon)\to(0,0)}A_{\hbar,\varepsilon,10}^{(j,m)}=$	$\displaystyle\lim_{(\hbar,\varepsilon)\to(0,0)}\int_{\Omega_{T}}(\partial_{t}\varphi)M_{\hbar,\varepsilon}^{(j)}M_{\hbar,\varepsilon}^{(m)}dxdt$
	$\displaystyle=$	$\displaystyle\int_{\Omega_{T}}(\partial_{t}\varphi)M^{(j)}M^{(m)}dxdt=I_{1}^{(j,m)}.$

Hence, we complete the proof of Lemma 6.9.

∎

Now, we are able to prove the moment convergence of the nonlinear term, which is Lemma 6.6, the last remaining part of the proof of Lemma 6.1.

Proof of Lemma 6.6.

By the uniform bound (6.22) and the weighted uniform estimate (3.11), we have

\displaystyle\Big{\|}\langle x\rangle E_{\hbar,\varepsilon}\left(\int_{\mathbb{R}}\xi^{k-1}f_{\hbar,\varepsilon}d\xi\right)dxdt\Big{\|}_{L_{t}^{\infty}L_{x}^{1})}\leq\|E_{\hbar,\varepsilon}\|_{L_{t}^{\infty}L_{x}^{1}}\Big{\|}\langle x\rangle\int_{\mathbb{R}}\xi^{k-1}f_{\hbar,\varepsilon}d\xi\Big{\|}_{L_{t}^{\infty}L_{x}^{1}}\leq C(k).

Following the same process as the $k=1,2$ case in Lemma 6.5, it suffices to prove

(6.43)

\displaystyle\lim_{(\hbar,\varepsilon)\to(0,0)}\int_{\Omega_{T}}\varphi E_{\hbar,\varepsilon}\left(\int_{\mathbb{R}}\xi^{k-1}f_{\hbar,\varepsilon}d\xi\right)dxdt=\int_{\Omega_{T}}\varphi\overline{E}\left(\int_{\mathbb{R}}\xi^{k-1}f(t,dx,d\xi)\right)dt,

for $\varphi\in C_{c}^{\infty}(\Omega_{T})$ .

First, we get into the analysis of the term on the left hand side of (6.43). Noting that

	$\displaystyle E_{\hbar,\varepsilon}=$	$\displaystyle\partial_{x}V_{\varepsilon}\rho_{\hbar,\varepsilon}=\partial_{x}\left(\frac{\|x\|}{2}+U_{\varepsilon}(x)\right)\rho_{\hbar,\varepsilon},$
	$\displaystyle\partial_{x}\left(\frac{\|x\|}{2}*\rho_{\hbar,\varepsilon}\right)=$	$\displaystyle\left(\frac{x}{2\|x\|}*\rho_{\hbar,\varepsilon}\right)=\int_{-\infty}^{x}\rho_{\hbar,\varepsilon}(y)dy-\frac{1}{2}=M_{\hbar,\varepsilon}^{(0)}-\frac{1}{2},$

we rewrite

\displaystyle\int_{\Omega_{T}}\varphi E_{\hbar,\varepsilon}\left(\int_{\mathbb{R}}\xi^{k-1}f_{\hbar,\varepsilon}d\xi\right)dxdt=A_{1}+A_{2}+A_{3},

where

	$\displaystyle A_{1}=$	$\displaystyle\int_{\Omega_{T}}\varphi M_{\hbar,\varepsilon}^{(0)}\left(\partial_{x}M_{\hbar,\varepsilon}^{(k-1)}\right)dxdt,$
	$\displaystyle A_{2}=$	$\displaystyle-\frac{1}{2}\int_{\Omega_{T}}\varphi\left(\int_{\mathbb{R}}\xi^{k-1}f_{\hbar,\varepsilon}d\xi\right)dxdt,$
	$\displaystyle A_{3}=$	$\displaystyle\int_{\Omega_{T}}\varphi\left(\partial_{x}U_{\varepsilon}*\rho_{\hbar,\varepsilon}\right)\left(\int_{\mathbb{R}}\xi^{k-1}f_{\hbar,\varepsilon}d\xi\right)dxdt.$

For term $A_{3}$ , using pointwise estimate (6.14) that $|\partial_{x}U_{\varepsilon}*\rho_{\hbar,\varepsilon}|\lesssim\varepsilon\langle x\rangle$ and the weighted estimate (3.11), we get

	$\displaystyle\|A_{3}\|=$	$\displaystyle\Big{\|}\int_{\Omega_{T}}\varphi\left(\partial_{x}U_{\varepsilon}*\rho_{\hbar,\varepsilon}\right)\left(\int_{\mathbb{R}}\xi^{k-1}f_{\hbar,\varepsilon}d\xi\right)dxdt\Big{\|}$
	$\displaystyle\leq$	$\displaystyle\varepsilon\\|\varphi\\|_{L_{t}^{1}L_{x}^{\infty}}\Big{\\|}\langle x\rangle\int_{\mathbb{R}}\xi^{k-1}f_{\hbar,\varepsilon}d\xi\Big{\\|}_{L_{t}^{\infty}L_{x}^{1}}\to 0.$

Therefore, we obtain

(6.44)			$\displaystyle\lim_{(\hbar,\varepsilon)\to(0,0)}\int_{\Omega_{T}}\varphi E_{\hbar,\varepsilon}\left(\int_{\mathbb{R}}\xi^{k-1}f_{\hbar,\varepsilon}d\xi\right)dxdt$
	$\displaystyle=$	$\displaystyle\lim_{(\hbar,\varepsilon)\to(0,0)}\int_{\Omega_{T}}\varphi M_{\hbar,\varepsilon}^{(0)}\left(\partial_{x}M_{\hbar,\varepsilon}^{(k-1)}\right)dxdt-\frac{1}{2}\int_{\Omega_{T}}\varphi\left(\int_{\mathbb{R}}\xi^{k-1}fd\xi\right)dxdt.$

On the other hand, by (6.25) and conservation of mass (5.4) in Lemma 5.1, we have

	$\displaystyle E(t,x)=$	$\displaystyle\int_{\mathbb{R}}\frac{x-y}{2\|x-y\|}\left(\int_{\mathbb{R}}f(t,y,\xi)d\xi\right)dy$
	$\displaystyle=$	$\displaystyle\int_{-\infty}^{x}\int_{\mathbb{R}}f(t,y,\xi)d\xi dy-\frac{1}{2}=M^{(0)}(t,x)-\frac{1}{2},$

and hence obtain

(6.45)			$\displaystyle\int_{\Omega_{T}}\varphi\overline{E}\left(\int_{\mathbb{R}}\xi^{k-1}f(t,dx,d\xi)\right)dt$
	$\displaystyle=$	$\displaystyle\int_{\Omega_{T}}\varphi\overline{M}^{(0)}\left(\partial_{x}M^{(k-1)}\right)dxdt-\frac{1}{2}\int_{\Omega_{T}}\varphi\left(\int_{\mathbb{R}}\xi^{k-1}fd\xi\right)dxdt.$

Comparing (6.44) with (6.45), to conclude (6.43), we are left to prove

(6.46)

\displaystyle\lim_{(\hbar,\varepsilon)\to(0,0)}\int_{\Omega_{T}}\varphi M_{\hbar,\varepsilon}^{(0)}\left(\partial_{x}M_{\hbar,\varepsilon}^{(k-1)}\right)dxdt=\int_{\Omega_{T}}\varphi\overline{M}^{(0)}\left(\partial_{x}M^{(k-1)}\right)dxdt.

By Lemma 6.8 and Lemma 6.9, the equality (6.46) is equivalent to

(6.47)

\displaystyle\lim_{(\hbar,\varepsilon)\to(0,0)}\int_{\Omega_{T}}\varphi M_{\hbar,\varepsilon}^{(1)}\left(\partial_{x}M_{\hbar,\varepsilon}^{(k-2)}\right)dxdt=\int_{\Omega_{T}}\varphi\overline{M}^{(1)}\left(\partial_{x}M^{(k-2)}\right)dxdt.

Iteratively using Lemma 6.8 and Lemma 6.9, we are left to prove

(6.48)		$\displaystyle\lim_{(\hbar,\varepsilon)\to(0,0)}\int_{\Omega_{T}}\varphi M_{\hbar,\varepsilon}^{(n)}\left(\partial_{x}M_{\hbar,\varepsilon}^{(n)}\right)dxdt=$	$\displaystyle\int_{\Omega_{T}}\varphi\overline{M}^{(n)}\left(\partial_{x}M^{(n)}\right)dxdt,\quad k=2n+1,$
(6.49)		$\displaystyle\lim_{(\hbar,\varepsilon)\to(0,0)}\int_{\Omega_{T}}\varphi M_{\hbar,\varepsilon}^{(n)}\left(\partial_{x}M_{\hbar,\varepsilon}^{(n+1)}\right)dxdt=$	$\displaystyle\int_{\Omega_{T}}\varphi\overline{M}^{(n)}\left(\partial_{x}M^{(n+1)}\right)dxdt,\quad k=2n.$

For (6.48), by integration by parts we get

	$\displaystyle\lim_{(\hbar,\varepsilon)\to(0,0)}\int_{\Omega_{T}}\varphi M_{\hbar,\varepsilon}^{(n)}\left(\partial_{x}M_{\hbar,\varepsilon}^{(n)}\right)dxdt=$	$\displaystyle-\frac{1}{2}\lim_{(\hbar,\varepsilon)\to(0,0)}\int_{\Omega_{T}}\left(\partial_{x}\varphi\right)\left(M_{\hbar,\varepsilon}^{(n)}\right)^{2}dxdt$
	$\displaystyle=$	$\displaystyle-\frac{1}{2}\int_{\Omega_{T}}\left(\partial_{x}\varphi\right)\left(M^{(n)}\right)^{2}dxdt$
	$\displaystyle=$	$\displaystyle\int_{\Omega_{T}}\varphi\overline{M}^{(n)}\left(\partial_{x}M^{(n)}\right)dxdt,$

where in the last equality we have used the fact that $\partial_{x}(M^{(n)})^{2}=2\overline{M}^{(n)}\partial_{x}M^{(n)}$ .

For (6.49), by the equations (6.34) and (6.39), it suffices to prove

\displaystyle\lim_{(\hbar,\varepsilon)\to(0,0)}\int_{\Omega_{T}}\varphi M_{\hbar,\varepsilon}^{(n)}\left(\partial_{t}M_{\hbar,\varepsilon}^{(n)}\right)dxdt=

\displaystyle\int_{\Omega_{T}}\varphi\overline{M}^{(n)}\left(\partial_{t}M^{(n)}\right)dxdt.

This can be done in the same way in which we obtain (6.48). Hence, we complete the proof of Lemma 6.6. ∎

7. Full Convergence to the Vlasov-Poisson Equation

In the section, we prove the limit measure $f(t,dx,d\xi)$ satisfies the Vlasov-Poisson equation in the weak sense. Let

(7.1)

\displaystyle\mu:=\partial_{t}f+\xi\partial_{x}f-\partial_{\xi}(\overline{E}f).

We use the following lemma to conclude the full convergence to the Vlasov-Poisson equation, that is, $\mu(t,x,\xi)=0$ in the sense of distributions.

Lemma 7.1 ([50, $p.620$ ]).

Let $\Omega_{T}=(0,T)\times\mathbb{R}$ , $\delta$ be an arbitrary positive constant. Assume $f(t,dx,d\xi)$ satisfies the following conditions.

$(1)$

Exponential decay:

(7.2) $\displaystyle\iint_{\Omega_{T}}\int_{\mathbb{R}}e^{\delta|\xi|}f(t,dx,d\xi)dt\leq C_{\delta}.$
$(2)$

For all test functions of the form $\phi(t,x,\xi)=\varphi(t,x)\xi^{m}$ , $\varphi(t,x)\in C_{c}^{\infty}(\Omega_{T})$ , there holds that

(7.3) $\displaystyle\iint_{\Omega_{T}}\int_{\mathbb{R}}\phi d\mu(t,x,\xi)=0.$

Then $\mu(t,x,\xi)=0$ in the sense of distributions.

By the moment convergence in Lemma 6.1, we have verified the condition $(2)$ in Lemma 7.1. Therefore, we are left to prove the exponential decay condition.

Lemma 7.2.

Let $T>0$ . There holds that

(7.4)

\displaystyle\iint_{\mathbb{R}^{2}}\xi^{2k}f(t,dx,d\xi)\leq C^{2k}(2k)^{2k}e^{t},\quad\forall t\in[0,T].

In particular, there exists a positive constant $\delta$ such that

(7.5)

\displaystyle\iint_{\mathbb{R}^{2}}e^{\delta|\xi|}f(t,dx,d\xi)\leq C_{\delta}e^{t},\quad\forall t\in[0,T].

Proof.

Recalling the moment equation (6.3) that

\displaystyle\partial_{t}\int_{\mathbb{R}}\xi^{m}f_{\hbar,\varepsilon}d\xi+\partial_{x}\int_{\mathbb{R}}\xi^{m+1}f_{\hbar,\varepsilon}d\xi+mE_{\hbar,\varepsilon}\int_{\mathbb{R}}\xi^{m-1}f_{\hbar,\varepsilon}d\xi+\mathrm{R}_{\hbar,\varepsilon}^{(m)}=0,

we have

		$\displaystyle\iint_{\mathbb{R}^{2}}\xi^{2k}f_{\hbar,\varepsilon}(t,x,\xi)d\xi dx$
	$\displaystyle=$	$\displaystyle\iint_{\mathbb{R}^{2}}\xi^{2k}f_{\hbar,\varepsilon}(0,x,\xi)d\xi dx+2k\int_{\Omega_{t}}E_{\hbar,\varepsilon}\left(\int_{\mathbb{R}}\xi^{2k-1}f_{\hbar,\varepsilon}d\xi\right)dxd\tau+\int_{\Omega_{t}}\mathrm{R}_{\hbar,\varepsilon}^{(2k)}dxd\tau.$

By the narrow convergence in Lemma 4.3, Lemma 6.3, and Lemma 6.6, taking $\varphi(\tau,x)=1$ and letting $(\hbar,\varepsilon)\to(0,0)$ , we obtain

\displaystyle\iint_{\mathbb{R}^{2}}\xi^{2k}f(t,dx,d\xi)=

\displaystyle\iint_{\mathbb{R}^{2}}\xi^{2k}f(0,dx,d\xi)+2k\int_{\Omega_{t}}\overline{E}\int_{\mathbb{R}}\xi^{2k-1}f(\tau,dx,d\xi)d\tau.

For the initial data, we have

	$\displaystyle\iint_{\mathbb{R}^{2}}\xi^{2k}f(0,dx,d\xi)=$	$\displaystyle\lim_{\hbar\to 0}\iint_{\mathbb{R}^{2}}\xi^{2k}f_{\hbar}(0,x,\xi)d\xi dx$
	$\displaystyle=$	$\displaystyle\lim_{\hbar\to 0}\frac{\hbar^{2k}}{2^{2k}}\sum_{\alpha=0}^{2k}\binom{2k}{\alpha}(-1)^{2k-\alpha}\int_{\mathbb{R}}D_{x}^{\alpha}\psi_{\hbar}^{\mathrm{in}}\overline{D_{x}^{2k-\alpha}\psi_{\hbar}^{\mathrm{in}}}dx$
	$\displaystyle\leq$	$\displaystyle\sup_{\hbar}\frac{1}{2^{2k}}\sum_{\alpha=0}^{2k}\binom{2k}{\alpha}\\|\hbar^{\alpha}\partial_{x}^{\alpha}\psi_{\hbar}^{\mathrm{in}}\\|_{L_{x}^{2}}\\|\hbar^{2k-\alpha}\partial_{x}^{2k-\alpha}\psi_{\hbar}^{\mathrm{in}}\\|_{L_{x}^{2}}$
	$\displaystyle\leq$	$\displaystyle C^{2k}(2k)^{2k},$

where in the last inequality we have used the initial condition (1.7) that

\|\hbar^{\alpha}\partial_{x}^{\alpha}\psi_{\hbar}^{\mathrm{in}}\|_{L_{x}^{2}}\leq C^{\alpha}\alpha^{\alpha}.

For the nonlinear part, using that $2k|\xi|^{2k-1}\leq(2k)^{2k}+\xi^{2k}$ , we get

		$\displaystyle 2k\Big{\|}\int_{\Omega_{t}}\overline{E}\int_{\mathbb{R}}\xi^{2k-1}f(\tau,dx,d\xi)d\tau\Big{\|}$
	$\displaystyle\leq$	$\displaystyle\\|\overline{E}\\|_{L_{t,x}^{\infty}}\int_{0}^{t}\iint_{\mathbb{R}^{2}}((2k)^{2k}+\xi^{2k})f(\tau,dx,d\xi)d\tau$
	$\displaystyle\leq$	$\displaystyle(2k)^{2k}T+\int_{0}^{t}\iint_{\mathbb{R}^{2}}\xi^{2k}f(\tau,dx,d\xi)d\tau,$

where in the last inequality we have used that $\|\overline{E}\|_{L_{t,x}^{\infty}}\leq 1$ in (6.26). Thus, we arrive at

\displaystyle\iint_{\mathbb{R}^{2}}\xi^{2k}f(t,dx,d\xi)\leq C^{2k}(2k)^{2k}+T(2k)^{2k}+\int_{0}^{t}\iint_{\mathbb{R}^{2}}\xi^{2k}f(\tau,dx,d\xi)d\tau.

Then by Gronwall’s inequality, we get

(7.6)

\displaystyle\iint_{\mathbb{R}^{2}}\xi^{2k}f(t,dx,d\xi)\leq(C^{2k}+T)(2k)^{2k}e^{t}.

For the exponential decay (7.5), provided that $C\delta<1$ , we have

	$\displaystyle\iint_{\mathbb{R}^{2}}e^{\delta\|\xi\|}f(t,dx,d\xi)\leq$	$\displaystyle\iint_{\mathbb{R}^{2}}\left(e^{\delta\|\xi\|}+e^{-\delta\|\xi\|}\right)f(t,dx,d\xi)$
	$\displaystyle=$	$\displaystyle 2\sum_{k=0}^{\infty}\frac{\delta^{2k}}{2k!}\iint_{\mathbb{R}^{2}}\xi^{2k}f(t,dx,d\xi)$
	$\displaystyle\leq$	$\displaystyle 2e^{t}\sum_{k=0}^{\infty}\frac{\delta^{2k}(C^{2k}+T)(2k)^{2k}}{2k!}<\infty.$

∎

Appendix A Measure Solutions to the Vlasov-Poisson Equation

Let us recall the definition of weak measure solutions from [51] by Zheng and Majda.

Definition A.1.

A pair $(E(t,x),f(t,x,\xi))$ of a function and a bounded non-negative Radon measure is called a weak solution to the Vlasov-Poisson equation (1.4) if for any $T>0$ there hold

(1)

$E(t,x)\in(BV\cap L^{\infty})(\Omega_{T})$ , where $\Omega_{T}=(0,T)\times\mathbb{R}$ ;
(2)

$f(t,x,\xi)\in L^{\infty}(0,\infty;\mathcal{M}^{+}(\mathbb{R}^{2}))$ ;
(3)

$E(t,x)=\frac{x}{|x|}*\int_{\mathbb{R}}f(t,x,\xi)d\xi$ a.e.;

(4)

$\forall\phi\in C_{c}^{\infty}((0,T)\times\mathbb{R}^{2})$ ,

\displaystyle\int_{0}^{T}\iint_{\mathbb{R}^{2}}\left(\partial_{t}\phi\right)f+\left(\partial_{x}\phi\right)\xi fdxd\xi dt-\int_{0}^{T}\int_{\mathbb{R}}\overline{E}\int_{\mathbb{R}}\left(\partial_{\xi}\phi\right)f(d\xi)dxdt=0.

(5)

$f\in C^{0,1}([0,T);H^{-L}(\mathbb{R}^{2}))$ for some $L>0$ .

The term $\overline{E}(t,x)$ in the above definition is the Vol^′pert’s symmetric average:

(A.1)

\displaystyle\overline{E}(t,x)=\left\{\begin{aligned} &E(t,x)&\quad\text{if $E(t,x)$ is approximately continuous at $(t,x)$,}\\ &\frac{E_{l}(t,x)+E_{r}(t,x)}{2}&\quad\text{if $E(t,x)$ has a jump at $(t,x)$.}\end{aligned}\right.

where $E_{l}(t,x)$ and $E_{r}(t,x)$ denote, respectively, the left and right limits of $E(t,x)$ at a discontinuity line at $(t,x)$ .

Appendix B Basic Properties of Bounded Variation Functions

We provide some basic properties of BV functions which are used in the paper. For more details, see for instance [46], or [50, 51].

Let $\Omega$ be a Borel measurable subset of $\mathbb{R}^{2}$ .

(1)

If $E\in BV(\Omega)\cap L^{\infty}(\Omega)$ , then

$\displaystyle E^{2}\in BV(\Omega),\quad\nabla E^{2}=2\overline{E}\nabla E$

in the sense of measures.
(2)

If $u,v\in BV(\Omega)$ , then $\overline{u}$ is almost everywhere defined and measurable with respect to $\nabla v$ . Furthermore, $\overline{u}$ is integrable with respect to $\nabla v$ if $u$ is bounded.
(3)

If $u,v\in BV(\Omega)$ , $\overline{u}$ is locally integrable with respect to $\nabla v$ and $\overline{v}$ is locally integrable with respect to $\nabla u$ . Then $uv\in BV(\Omega)$ and

$\displaystyle\nabla(uv)=\overline{u}\nabla v+\overline{v}\nabla u.$
(4)

$\overline{\varphi E}=\varphi\overline{E}$ if $\varphi\in C^{1}(\Omega)$ .
(5)

Let $u\in BV(\Omega)\cap L^{\infty}(\Omega)$ , and $\eta_{\sigma}$ be an approximation of the identity. Then

(B.1) $\displaystyle u*\eta_{\sigma}\to\overline{u}\quad\mathcal{H}^{1}-a.e.$

as $\sigma\to 0$ . Here, $\mathcal{H}^{1}$ denotes the one-dimensional Hausdorff measure.
(6)

$\nabla u$ is absolutely continuous with respect to $\mathcal{H}^{1}$ for any $u\in BV(\Omega)$ .
(7)

$\overline{E}$ is $\mathcal{H}^{1}$ -a.e. defined for any $E\in BV$ .

Acknowledgements X. Chen was supported in part by U.S. NSF grant DMS-2406620. P. Zhang was supported in part by National Key R $\&$ D Program of China under Grant 2021YFA1000800 and NSF of China under Grants 12288201, 12031006. Z. Zhang was supported in part by National Key R&D Program of China under Grant 2023YFA1008801 and NSF of China under Grant 12288101.

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(2.5)		$\displaystyle\\|f_{N,\hbar,\varepsilon}^{(1)}(t)-f_{\hbar,\varepsilon}(t)\\|_{L_{x,\xi}^{2}}=$	$\displaystyle\\|W_{\hbar}[\gamma_{N,\hbar,\varepsilon}^{(1)}(t)]-W_{\hbar}[\psi_{\hbar,\varepsilon}(t)]\\|_{L_{x,\xi}^{2}}$
	$\displaystyle=$	$\displaystyle\hbar^{-\frac{1}{2}}\big{\\|}\gamma_{N,\hbar,\varepsilon}^{(1)}(t,x,x^{\prime})-\psi_{\hbar,\varepsilon}(t,x)\overline{\psi_{\hbar,\varepsilon}}(t,x^{\prime})\big{\\|}_{L_{x,x^{\prime}}^{2}}$
	$\displaystyle\leq$	$\displaystyle\hbar^{-\frac{1}{2}}\operatorname{Tr}\Big{\|}\gamma_{N,\hbar,\varepsilon}^{(1)}(t,x,x^{\prime})-\psi_{\hbar,\varepsilon}(t,x)\overline{\psi_{\hbar,\varepsilon}}(t,x^{\prime})\Big{\|}.$

		$\displaystyle\frac{d}{dt}\int_{\mathbb{R}}e^{-2\delta\|x\|^{2}}\langle x\rangle^{2}\rho_{\hbar,\varepsilon}(t,x)dx$
	$\displaystyle=$	$\displaystyle-\int_{\mathbb{R}}e^{-2\delta\|x\|^{2}}\langle x\rangle^{2}\partial_{x}\left(\operatorname{Im}\left(\overline{\psi_{\hbar,\varepsilon}}\hbar\partial_{x}\psi_{\hbar,\varepsilon}\right)\right)dx$
	$\displaystyle=$	$\displaystyle\int_{\mathbb{R}}e^{-2\delta\|x\|^{2}}2x(1-2\delta\langle x\rangle^{2})\operatorname{Im}\left(\overline{\psi_{\hbar,\varepsilon}}\hbar\partial_{x}\psi_{\hbar,\varepsilon}\right)dx$
	$\displaystyle\lesssim$	$\displaystyle\\|e^{-\delta\|x\|^{2}}(\delta\langle x\rangle^{2}+1)\\|_{L_{x}^{\infty}}\\|e^{-\delta\|x\|^{2}}\|x\|\psi_{\hbar,\varepsilon}(t,x)\\|_{L_{x}^{2}}\\|\hbar\partial_{x}\psi_{\hbar,\varepsilon}(t,x)\\|_{L_{x}^{2}}$
	$\displaystyle\lesssim$	$\displaystyle C(T)\int_{\mathbb{R}}e^{-2\delta\|x\|^{2}}\langle x\rangle^{2}\rho_{\hbar,\varepsilon}(t,x)dx,$

		$\displaystyle\Big{\|}\iint_{\mathbb{R}^{2}}W_{\hbar}[u_{1},u_{2}]\phi(x,\xi)dxd\xi\Big{\|}$
	$\displaystyle\leq$	$\displaystyle\left(\int_{\mathbb{R}}\sup_{x}\|\mathcal{F}_{\xi}\phi(x,y)\|dy\right)\left(\sup_{y}\int_{\mathbb{R}}\Big{\|}u_{1}(x+\frac{\hbar y}{2})\overline{u_{2}(x-\frac{\hbar y}{2})}\Big{\|}dx\right)$
	$\displaystyle\leq$	$\displaystyle\\|u_{1}\\|_{L_{x}^{2}}\\|u_{2}\\|_{L_{x}^{2}}\\|\phi\\|_{\mathcal{A}}.$

		$\displaystyle\Big{\|}\iint_{\mathbb{R}^{2}}\left(\xi^{k}\widetilde{W}_{\hbar}[u]-\xi^{k}W_{\hbar}[u]\right)\phi dxd\xi\Big{\|}$
	$\displaystyle\leq$	$\displaystyle\Big{\|}\iint_{\mathbb{R}^{2}}\xi^{k}W_{\hbar}[u]\left(\phi_{(x,\xi)}G_{\hbar}-\phi\right)dxd\xi\Big{\|}+\sum_{\alpha=0}^{k-1}\binom{k}{\alpha}\Big{\|}\iint_{\mathbb{R}^{2}}(\xi^{\alpha}W_{\hbar}[u])(\xi^{k-\alpha}G_{\hbar})\phi dxd\xi\Big{\|}$
	$\displaystyle\leq$	$\displaystyle\\|u\\|_{L_{x}^{2}}^{2}\\|\phi*_{(x,\xi)}G_{\hbar}-\phi\\|_{\mathcal{A}}$
		$\displaystyle+\sum_{\alpha=0}^{k-1}\sum_{\alpha_{1}+\alpha_{2}=\alpha}\binom{k}{\alpha}\binom{\alpha}{\alpha_{1}}\\|\hbar^{\alpha_{1}}\partial_{x}^{\alpha_{1}}u\\|_{L_{x}^{2}}\\|\hbar^{\alpha_{2}}\partial_{x}^{\alpha_{2}}u\\|_{L_{x}^{2}}\\|(\xi^{k-\alpha}G_{\hbar})*_{(x,\xi)}\phi\\|_{\mathcal{A}}$
	$\displaystyle\leq$	$\displaystyle\\|u\\|_{L_{x}^{2}}^{2}\\|\phi*_{(x,\xi)}G_{\hbar}-\phi\\|_{\mathcal{A}}+C(k)\sum_{\alpha=0}^{k-1}\sum_{\alpha_{1}+\alpha_{2}=\alpha}\\|\hbar^{\alpha_{1}}\partial_{x}^{\alpha_{1}}u\\|_{L_{x}^{2}}\\|\hbar^{\alpha_{2}}\partial_{x}^{\alpha_{2}}u\\|_{L_{x}^{2}}\\|\phi\\|_{\mathcal{A}}\\|\xi^{k-\alpha}G_{\hbar}\\|_{L_{x}^{1}L_{\xi}^{1}}$
	$\displaystyle\lesssim$	$\displaystyle\\|u\\|_{L_{x}^{2}}^{2}\\|\phi*_{(x,\xi)}G_{\hbar}-\phi\\|_{\mathcal{A}}+C(k)\sum_{\alpha=0}^{k-1}\sum_{\alpha_{1}+\alpha_{2}=\alpha}\\|\hbar^{\alpha_{1}}\partial_{x}^{\alpha_{1}}u\\|_{L_{x}^{2}}\\|\hbar^{\alpha_{2}}\partial_{x}^{\alpha_{2}}u\\|_{L_{x}^{2}}\hbar^{k-\alpha}\\|\phi\\|_{\mathcal{A}},$

	$\displaystyle\\|g*_{(x,\xi)}\phi\\|_{\mathcal{A}}\leq$	$\displaystyle\\|(\mathcal{F}_{\xi}g)*_{x}(\mathcal{F}_{\xi}\phi)\\|_{L_{\xi}^{1}L_{x}^{\infty}}\leq\\|\mathcal{F}_{\xi}g\\|_{L_{\eta}^{\infty}L_{x}^{1}}\\|\mathcal{F}_{\xi}\phi\\|_{L_{\eta}^{1}L_{x}^{\infty}}\leq\\|g\\|_{L_{x}^{1}L_{\xi}^{1}}\\|\phi\\|_{\mathcal{A}},$
	$\displaystyle\\|\xi^{k-\alpha}G_{\hbar}\\|_{L_{x}^{1}L_{\xi}^{1}}\lesssim$	$\displaystyle\hbar^{k-\alpha}.$

Global derivation of the 1D Vlasov-Poisson equation from quantum many-body dynamics with screened Coulomb potential

Abstract.

Key words and phrases:

2010 Mathematics Subject Classification:

1. Introduction

Theorem 1.1 (Main theorem).

Remark 1.2 (Global existence and conservation laws).

Remark 1.3 (Existence of initial data).

Remark 1.4 (The torus case).

Remark 1.5 (Fixed Debye length).

1.1. Outline of the Proof of the Main Theorem

2. Preliminary Reduction: Quantum Mean-field Limit

Theorem 2.1 ([4, Corollary 4.2]).

Corollary 2.2.

Proof.

3. Weighted Uniform Higher Energy Estimates

Lemma 3.1 (Weighted uniform estimates).

Proof.

Lemma 3.2.

Proof.

4. Compactness and Narrow Convergence

4.1. Higher Moment Estimates between the Wigner and Husimi function

Definition 4.1.

Lemma 4.2.

Proof.

4.2. Convergence to a Non-negative Radon Measure

Notation.

Lemma 4.3.

Proof.

5. Conservation Laws for the Limit Measure

Lemma 5.1.

Proof.

6. Moment Convergence to the Vlasov-Poisson Equation

Lemma 6.1.

6.1. Vanishing Remainder Terms via a Cancellation Structure

Lemma 6.2.

Proof.

Lemma 6.3.

Proof.

6.2. Convergence of the Nonlinear Term for k=1,2k=1,2

Lemma 6.4.

Proof.

Lemma 6.5.

Proof.

6.3. Convergence of the Nonlinear Term for k≥3k\geq 3

Lemma 6.6.

Lemma 6.7.

Proof.

Lemma 6.8.

Proof.

Lemma 6.9.

Proof.

Proof of Lemma 6.6.

7. Full Convergence to the Vlasov-Poisson Equation

Lemma 7.1 ([50, p​.620p.620]).

Lemma 7.2.

Proof.

Appendix A Measure Solutions to the Vlasov-Poisson Equation

Definition A.1.

Appendix B Basic Properties of Bounded Variation Functions

References

6.2. Convergence of the Nonlinear Term for $k=1,2$

6.3. Convergence of the Nonlinear Term for $k\geq 3$

Lemma 7.1 ([50, $p.620$ ]).