Propagation of Moments and Semiclassical Limit from Hartree to Vlasov Equation

Abstract

In this paper, we prove a quantitative version of the semiclassical limit from the Hartree to the Vlasov equation with singular interaction, including the Coulomb potential. To reach this objective, we also prove the propagation of velocity moments and weighted Schatten norms which implies the boundedness of the space density of particles uniformly in the Planck constant.

Appendices

Appendix A: Besov Spaces

We recall that a possible definition of Besov spaces (see e.g. [8]) can be done by defining the following norm

$$\begin{aligned} \Vert u\Vert _{B^{s}_{p,r}} = \left\| \left( 2^{sj}\Vert \Delta _ju\Vert _{L^p}\right) _{j\in \mathbb {Z}}\right\| _{\ell ^r}, \end{aligned}$$

(82)

where $\Delta _j$ is defined by

$$\begin{aligned} \Delta _j u&= 0 \qquad \qquad \qquad \qquad \qquad \qquad \, \text {when } j\le -2 \\ \Delta _{-1} u&= \hat{\chi }*u \\ \Delta _j u&= \mathcal {F}_y(\varphi (2^{-j}y))*u \qquad \qquad \text {when } j\ge 0, \end{aligned}$$

with

$$\begin{aligned}&\chi \in C^\infty _c(B(0,4/3),[0,1]) \nonumber \\&\varphi \in C^\infty _c\left( B\left( 0,8/3\right) \backslash B\left( 0,3/4\right) ,[0,1]\right) \nonumber \\&\chi + \sum _{j\ge 0} \varphi (2^{-j}\cdot ) = 1. \end{aligned}$$

(83)

We also define the space of log-Lipschitz functions by defining the norm

$$\begin{aligned} \Vert u\Vert _{LL} = \sup _{|x-y|\in (0,1)}\left( \frac{|u(x)-u(y)|}{|x-y|\left( 1+\left| \ln (|x-y|)\right| \right) }\right) , \end{aligned}$$

for measurable functions u vanishing at infinity. We have the following properties of Besov spaces

Proposition A.1

(84)

If K is the Coulomb potential such that $\Delta K = \delta _0$, then we get

$$\begin{aligned} \left| \nabla K\right| = \frac{C}{|x|^{d-1}} \in B^1_{1,\infty }. \end{aligned}$$

(85)

If $v\in L^\infty$ and $u\in B^{1}_{1,\infty }$, then

$$\begin{aligned} \Vert u*v\Vert _{B^1_{\infty ,\infty }}&\le \left\| u\right\| _{B^1_{1,\infty }} \Vert v\Vert _{L^\infty } \end{aligned}$$

(86)

$$\begin{aligned} \Vert u*v\Vert _{{\dot{H}}^1}&\le C \Vert u\Vert _{B^1_{1,\infty }} \Vert v\Vert _{L^2}. \end{aligned}$$

(87)

Proof

The proof of (84) and (85) can be found for example in [8, Chapter 2]. To prove (86), we remark that since $\Delta _j$ is a convolution by a smooth and rapidly decaying supported function, $\Delta _j(u*v) = \Delta _j(u)*v$. By Hölder’s inequality, we deduce the following inequality

$$\begin{aligned} \Vert u*v\Vert _{B^1_{\infty ,\infty }}&= \left\| \left( 2^{j}\Vert \Delta _ju*v\Vert _{L^\infty }\right) _{j\in \mathbb {Z}}\right\| _{\ell ^\infty } \\&\le \Vert v\Vert _{L^\infty } \left\| \left( 2^{j}\Vert \Delta _ju\Vert _{L^1}\right) _{j\in \mathbb {Z}}\right\| _{\ell ^\infty } = \Vert u\Vert _{B^1_{1,\infty }} \Vert v\Vert _{L^\infty }. \end{aligned}$$

To prove (87), we use the Fourier definition of $\dot{H}^1$ and the fact the Fourier transform is an isometry on $L^2$ to obtain

$$\begin{aligned} \Vert u*v\Vert _{{\dot{H}}^1}&\le C\Vert |y|\hat{u}(y)\hat{v}(y)\Vert _{L^2_y} \le C\Vert |y|\hat{u}(y)\Vert _{L^\infty _y} \Vert v\Vert _{L^2}. \end{aligned}$$

Then by using the fact that $\varphi (2^{-j} y)>0 \Leftrightarrow |y|\in 2^j[3/4,8/3]$, we obtain the existence of $j_y\ge -2$ such that $\varphi (2^{-j}y) = 0$ for any $j\notin \{j_y-1,j_y,j_y+1\}$ (If $j_y=-2$, then it means that $\chi (y)>0$). Then, by (83), we get

$$\begin{aligned} \Vert y\hat{u}(y)\Vert _{L^\infty }&= \left\| \left( \chi (y) + \sum _{j\ge 0} \varphi (2^{-j}y)\right) |y|\hat{u}(y)\right\| _{L^\infty _y} \\&\le C \left\| \sum _{k=-1}^12^{j_y+k}\mathcal {F}(\Delta _{j_y+k}u)(y)\right\| _{L^\infty } \\&\le C \sup _{j\in \mathbb {Z}}\left( 2^{j} \Vert \mathcal {F}(\Delta _ju)\Vert _{L^\infty }\right) \le C \left\| \left( 2^{j} \Vert \Delta _ju\Vert _{L^1}\right) _{j\in \mathbb {Z}}\right\| _{\ell ^\infty }. \end{aligned}$$

Therefore, by the definition (82), we obtain (87). $\square$

Appendix B: Wasserstein distances

We recall the definition of the classical Wasserstein–(Monge–Kantorovich) distances between two probability measures $(\mu _0,\mu _1)\in \mathcal {P}(X)^2$ on a given separable Banach space X. We first define the notion of coupling by saying that $\gamma \in \mathcal {P}(X^2)$ is a coupling of $\mu _0$ and $\mu _1$ when

$$\begin{aligned} (\pi _1)_\#\gamma = \mu _0 \text { and } (\pi _2)_\#\gamma = \mu _1, \end{aligned}$$

where $\pi _1$ and $\pi _2$ are respectively the projection on the first and second variable and $\pi _\#\gamma$ denotes the pushforward of the measure $\gamma$ by the map $\pi$. In other words

$$\begin{aligned} \forall \varphi \in C_0(X), \int _{X^2} \varphi (x) \gamma (\mathrm {d}x\,\mathrm {d}y) = \int _{X} \varphi (x) \mu _0(\mathrm {d}x). \end{aligned}$$

We denote by $\Pi (\mu _0,\mu _1)$ the set of couplings of $\mu _0$ and $\mu _1$. Then we define the Wasserstein–(Monge–Kantorovich) distance in the following way

$$\begin{aligned} W_p(\mu _0,\mu _1) := \left( \inf _{\gamma \in \Pi (\mu _0,\mu _1)}\int _{X^2} \Vert x-y\Vert _X^p\gamma (\mathrm {d}x\,\mathrm {d}y)\right) ^\frac{1}{p}. \end{aligned}$$

(88)

The existence of a minimizer is well known and we refer for example to the books [57] or [53] for more properties of these distances.

The following proposition may be classical but we prove it for the sake of completeness

Proposition B.1

Let $(f_0,f_1)\in \mathcal {P}(\mathbb {R}^{2d})^2$ and for $\mathrm {i}\in \{0,1\}$, let $\rho _\mathrm {i}= (\pi _1)_\# f_\mathrm {i}$. Then

$$\begin{aligned} W_2(\rho _0,\rho _1) \le W_2(f_0,f_1). \end{aligned}$$

Proof

Let $\gamma \in \mathcal {P}(\mathbb {R}^{2d}\times \mathbb {R}^{2d})$ be the optimal transport plan from $f_0$ to $f_1$ and define $\gamma _\rho = (\pi _{1,3})_\#\gamma$ by

$$\begin{aligned} \forall \varphi \in C_0(\mathbb {R}^{2d}), \int _{\mathbb {R}^{2d}}\varphi (x,y)\gamma _\rho (\mathrm {d}x\,\mathrm {d}y) := \int _{\mathbb {R}^{4d}} \varphi (x,y)\gamma (\mathrm {d}x\,\mathrm {d}\xi \,\mathrm {d}y\,\mathrm {d}\eta ). \end{aligned}$$

Then for any $\varphi \in C_0$, since the first marginal of $\gamma$ is $f_0$,

$$\begin{aligned} \int _{\mathbb {R}^{2d}}\varphi (x)\gamma _\rho (\mathrm {d}x\,\mathrm {d}y)&= \int _{\mathbb {R}^{4d}} \varphi (x)\gamma (\mathrm {d}x\,\mathrm {d}\xi \,\mathrm {d}y\,\mathrm {d}\eta ) \\&= \int _{\mathbb {R}^{2d}}\varphi (x)f_0(\mathrm {d}x\,\mathrm {d}\xi ) \\&= \int _{\mathbb {R}^d}\varphi (x)\rho _0(\mathrm {d}x). \end{aligned}$$

Hence, the first marginal of $\gamma _\rho$ is $\rho _0$. In the same way, the second marginal of $\gamma _\rho$ is $\rho _1$, and we deduce that $\gamma _\rho \in \Pi (\rho _0,\rho _1)$. Next, let $(\varphi _n)_{n\in \mathbb {N}} \in (C_0(\mathbb {R}^{2d})\cap L^1(\gamma _\rho ))^\mathbb {N}$ be an increasing sequence of nonnegative functions converging pointwise to $(x,y)\mapsto |x-y|^2$. By definition of $\gamma _\rho$, for any $n\in \mathbb {N}$, $\varphi \in L^1(\gamma )$. Therefore, by the monotone convergence theorem,

$$\begin{aligned} \int _{\mathbb {R}^{2d}}|x-y|^2 \gamma _\rho (\mathrm {d}x\,\mathrm {d}y)&= \lim \limits _{n\rightarrow \infty } \int _{\mathbb {R}^{2d}}\varphi _n(x,y)\gamma _\rho (\mathrm {d}x\,\mathrm {d}y) \\&= \lim \limits _{n\rightarrow \infty } \int _{\mathbb {R}^{4d}} \varphi _n(x,y)\gamma (\mathrm {d}x\,\mathrm {d}\xi \,\mathrm {d}y\,\mathrm {d}\eta ) \\&= \int _{\mathbb {R}^{4d}} |x-y|^2\gamma (\mathrm {d}x\,\mathrm {d}\xi \,\mathrm {d}y\,\mathrm {d}\eta ) \\&\le \int _{\mathbb {R}^{4d}} (|x-y|^2+|\xi -\eta |^2)\gamma (\mathrm {d}x\,\mathrm {d}\xi \,\mathrm {d}y\,\mathrm {d}\eta ) = W_2(f_0,f_1)^2. \end{aligned}$$

By definition (88), we deduce

$$\begin{aligned} W_2(\rho _0,\rho _1)^2 \le \int _{\mathbb {R}^{2d}}|x-y|^2 \gamma _\rho (\mathrm {d}x\,\mathrm {d}y) \le W_2(f_0,f_1)^2, \end{aligned}$$

which proves the result. $\square$