Content-Type: multipart/mixed; boundary="-------------0205220337953" This is a multi-part message in MIME format. ---------------0205220337953 Content-Type: text/plain; name="02-234.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="02-234.comments" S.Graffi, A.Martinez: Dipartimento di Matematica, Universit di Bologna, Italy (e-mail: graffi@dm.unibo.it, martinez@dm.unibo.it) M.Pulvirenti: Dipartimento di Matematica, Universit di Roma "La Sapienza", Italy (e-mail: pulvirenti@mat.uniroma.it) ---------------0205220337953 Content-Type: text/plain; name="02-234.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="02-234.keywords" Hartree equation, mean field, classical limit, Vlasov equation ---------------0205220337953 Content-Type: application/x-tex; name="GMPArc.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="GMPArc.tex" %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \documentstyle{amsppt} \magnification=1200 \vsize7.8truein \def\t#1{\text{\rm #1}} \def\Om{\Omega} \def\ov{\overline} \def\om{\omega} \def\var{\varepsilon} \def\lam{\lambda} \def\R{\Bbb R} \def\E{\Bbb E} \def\F{\Cal F} \def \N {\Bbb N} \def\pa{\partial} \def\d{\t{$\,$d}} \def\Lam{\Lambda} \def\nequ {=\hskip-3 mm/} \def\neque {=\hskip-2 mm/} \def\ds{\displaystyle} \nologo \baselineskip=18pt \TagsOnRight \NoRunningHeads \topmatter \title Mean-Field Approximation of Quantum Systems and Classical Limit \endtitle \author S. Graffi$^*$, A. Martinez $^*$ and M.Pulvirenti $^+$ \endauthor \affil $^*$ Dipartimento di Matematica, Universit\`a di Bologna, Italy \\ (e-mail: graffi\@dm.unibo.it, martinez\@dm.unibo.it) \\ $^+$ Dipartimento di Matematica, Universit\`a di Roma "La Sapienza", Italy \\ (e-mail: pulvirenti\@mat.uniroma.it) \endaffil \abstract We prove that, for a smooth two-body potentials, the quantum mean-field approximation to the nonlinear Schr\"odinger equation of the Hartree type is stable at the classical limit $h \to 0$, yielding the classical Vlasov equation. \endabstract \endtopmatter \heading Introduction \endheading Consider a system of $N$ identical classical particles of unit mass, evolving according to the dynamics generated by the following mean-field Hamiltonian $$ \Cal H =\sum_{i=1}^N \frac 12 v_i^2+\frac 1N \sum_{i0$, sufficiently small, such that for all $j\in \Bbb N$: $$ f^N_j(t)\to f_j(t)\qquad \hbox {in } \Cal D '(\Bbb R^{3j} \times \Bbb R^{3j}), \quad t\in [0,T] \tag 1.11 $$ where $f_j (t)=f (t)^{\otimes j}$ and $f(t)$ is the unique (weak) solution of the classical Vlasov equation: $$ (\pa_t+v\cdot \nabla_x+E\cdot \nabla_v) f(x,v,t)=0. \tag 1.12 $$ Here: $$ E(x,t)=-\nabla \varphi * \rho (x,t),\qquad \rho (x,t)=\int dv f(x,v,t). \tag 1.13 $$ Moreover, for $t\in [0,T]$: $$ f(x,v,t)= \rho (x,t) \delta (v-u(x,t)) \tag 1.14 $$ where the pair $(\rho, u)$ fulfills the continuity and the momentum balance equations: $$ \pa_t \rho+\hbox {div} (u\rho)=0; \quad \pa_t u+u\cdot \nabla u=-\nabla \phi*\rho. \tag 1.15a $$ More precisely, for any test function $F\in \Cal D (\Bbb R^{3j} \times \Bbb R^{3j})$, one has $$ \langle f^N_j(t)-f_j(t),F\rangle =\Cal O(h+N^{-1}) \tag 1.15b $$ for $h$ small enough and $N$ large enough. \endproclaim We refer to this particular situation as hydrodynamic for obvious reasons. \newline Theorem 1.1 is based on some regularity estimates to be established in the next section. We will write the solution of the Schr\"o\-din\-ger equation (1.7) under the form $$ \Psi_N(X_N,t)=A^N(X_N,t) e^{i\frac {S^N(X_N,t)}{h}} \tag 1.16 $$ where $S^N$ is the classical action satisfying the Hamilton-Jacobi equation: $$ \pa_t S^N +\frac 12 |\nabla S^N|^2 +W_N=0 \tag 1.17 $$ and, consequently, $A_N$ is the solution of the "transport" equation: $$ \pa_t A^N+(P^N \cdot \nabla )A^N +\frac 12 A^N \hbox {div} P^N= -h \frac i2 \Delta A^N. \tag 1.18 $$ Here $P^N:=\nabla S^N$ satisfies: $$ \pa_t P^N+(P^N \cdot \nabla )P^N = -\nabla W. \tag 1.19 $$ As we shall show later on, the above representation holds for a short time $T>0$ which may be chosen uniformly in $N$. We then consider initial data which are not of hydrodynamical type. We restrict ourselves to classical data, namely those for which the Wigner transform is positive and normalized: $$ f^N(X_N,V_N)=f^{\otimes N}(X_N,V_N) \tag 1.20 $$ where $f$ is a classical probability distribution on the one-particle phase space. Consider now a one-particle superposition of particular WKB states, given by the following density matrix: $$ \rho (x,y)=\int dw \quad e^{i\frac wh \cdot (x-y)} a(x;w)\bar a(y,w). \tag 1.21 $$ The Wigner transform of the density matrix is: $$ f_{\rho} (x,v)= (\frac 1{2\pi})^{3} \int dw \int dz e^{iz\cdot (v-w)} a(x-\frac h2 z;w)\bar a(x+\frac h2 z,w)=|a(x,v)|^2 +O(h). \tag 1.22 $$ Setting $|a(x,v)| =\sqrt{f(x,v)}$ we see that $f$ and $f_{\rho}$ are asymptotically equivalent (in $\Cal D')$) in the limit $h\to 0$. Therefore we assume as initial condition $$ \rho^N (X_N,Y_N)=\prod_{i=1}^N \rho (x_i,y_i) \tag 1.23 $$ for the density matrix (1.21) with $a (x;w)=\sqrt {f(x,w)}$. We will prove \proclaim {Theorem 1.2} Assume $\varphi \in C^{2}(\Bbb R^3)$, $\sigma \in C^{2}(\Bbb R^3)$, $\ds \partial^{\alpha}\varphi$, $\ds \partial^{\alpha}\sigma$ uniformly bounded for $|\alpha|\leq 2$, $a$ compactly supported in $w$ and $a(\cdot,w)\in C^2\cap H^2 (\Bbb R^3)$ for all $w$. Let $h=h(N) \to 0$ as $N \to \infty$. Then for all $j\in \Bbb N$ and $t\ge 0$: $$ f^N_j(t)\to f_j(t)\qquad \hbox {in } \Cal D '(\Bbb R^{3j} \times \Bbb R^{3j}), \tag 1.24 $$ where $f_j (t)=f (t)^{\otimes j}$ and $f(t)$ is the unique solution of the classical Vlasov equation: $$ (\pa_t+v\cdot \nabla_x+E\cdot \nabla_v) f(x,v,t)=0 \tag 1.25 $$ where $$ E(x,t)=-\nabla \varphi * \rho (x,t),\qquad \rho (x,t)=\int dv f(x,v,t), \tag 1.26 $$ with initial datum $f(x,v)=|a(x,v)|^2$. \endproclaim Note that here the convergence result holds globally in time. \heading 2. The classical system and its mean-field properties \endheading Consider the associated Hamiltonian system: $$ \dot X_N(t)=V_N (t);\quad \dot V_N (t)_i=-\frac 1N \sum_{j\neq i} \nabla \varphi (x_i(t)-x_j(t)) \tag 2.1 $$ where $$ X_N(t)=(x_1(t)\dots x_N(t)), \quad V_N(t)=(v_1(t)\dots v_N(t)). $$ Denote by $X_N(t,X_N,V_N)$, $V_N(t)=V_N(t,X_N,V_N)$ the solution of the Cauchy problem with initial conditions $X_N,V_N$. For a given initial datum $(X_N,V_N)$, we consider the empirical distribution, that is a one-particle time depending measure, defined by: $$ \mu^N (dx,dv,t):=\frac 1N \sum _{i=1}^N \delta (x-x_i(t))\delta (v-v_i(t)) dx dv. \tag 2.2 $$ The following facts are well known (see e.g. [1],[2],[3]). 1) $\mu^N (dx,dv,t)$ is a weak solution of the Vlasov equation (1.12). Namely, for any test function $h=h(x,v)$, setting $\langle \mu^N (t), h\rangle = \int \mu^N (dx,dv,t) h(x,v)$, we have: $$ \frac {d}{dt} \langle \mu^N (t), h\rangle= \langle \mu^N (t), v\cdot \nabla_x h\rangle - \langle \mu^N (t), \nabla_x \varphi *\mu^N (t) \cdot \nabla_v h\rangle. \tag 2.3 $$ This follows by a direct computation. \smallskip 2) Weak solutions to the Vlasov equation are continuous with respect to the initial datum in the topology of the weak convergence of the measures. \smallskip In particular 1) and 2) imply that if $\mu^N (0)\to f$ weakly, then $\mu^N (t)\to f(t)$weakly, where $f(t)$ is the unique solution to the Vlasov equation with initial datum $f$. If $f$ is sufficiently regular then $f(t)$ hinerits such a regularity and the solution is classical. \smallskip 3) Let $f^{N} (X_N,V_N,0) $ be an initial symmetric $N$-particle distribution and let $f^{N} (X_N,V_N,t)=f^{ N}(X_N(X_N,V_N,-t),V_N(X_N,V_N,-t))$ be the solution of the Liouville equation. Define the $j$-particle marginals by: $$ f^{N}_j(X_j,V_j,t):=\int dX_{N-j} \int dV_{N-j}f^{N} (X_N,V_N,t) . \tag 2.4 $$ Then, if $$ f^{N}_j \to f^{\otimes j} \tag 2.5 $$ in the limit $N \to \infty$ and in the sense of the weak convergence of the measures, where $f=f(x,v)$ is a given $1$-particle initial distribution, then $$ f^{N}_j(t) \to f^{\otimes j}(t) \tag 2.6 $$ weakly, where $f(t)$ solves the Vlasov equation with initial condition $f$. Property (2.6) is called propagation of chaos. We now specialize the above results to our hydrodynamical case. We suppose that initially: $$ f(x,v)=f(x,v,0)=\rho (x) \delta (v-u(x)) \tag 2.7 $$ (in the sequel $u=\nabla \sigma$) and denote: $$ \Phi^t(X_N):=X_N( X_N,P^N(X_N),t) \tag 2.8 $$ where $P^N(X_N):=\{ u(x_i)\}_{i=1}^N$. Then, for a given test function $F_j\in \Cal D( \Bbb R^3 \times \Bbb R^3)$, $$ \int f^{N}_j (t)F_j dX_j dV_j= $$ $$ \int dX_N dV_N f^{\otimes N} (X_N,V_N) F_j(X_N^j (X_N,V_N,t),V_N^j (X_N,V_N,t)) = $$ (by the Liouville theorem) $$ =\int dX_N \rho ^{\otimes N} (X_N) F_j(\Phi^t (X_N)^j,\dot \Phi^t (X_N)^j) \to $$ $$ \int dX_j dV_j f ^{\otimes j}(X_j,V_j,t) F_j(X_j,V_j) \tag 2.9 $$ in the limit $N\to \infty$. Here we are using the notation $X_N^j$ to indicate the vector $(x_1, \dots ,x_j)$ if $X_N=(x_1, \dots ,x_N)$. On the other hand, if $f(t)$ is the solution of the Vlasov equation, for a short time $t0$ independent of $N$ such that for all $N$ and all $j\in \{ 1,\dots ,N\}$, one has, $$ \| \nabla_{x_j}A^N(t) \|_{L^2} \leq C_1. \tag 2.12 $$ Moreover for $\tau \in [0,t]$, denoting by $x_i^\gamma$, $\gamma=1,2,3$ the components of $x_i$ $$ |\frac {\pa P^N_k(\Phi^{(t-\tau)}(X_N),t)}{\pa x_i^\gamma}| \leq C\left( \frac 1N +\delta_{i,k} \right) \tag 2.13 $$ and $$ |\frac {\pa \Phi^{(t-\tau)}(X_N)_k}{\pa x_i^\gamma}| \leq C\left( \frac 1N +\delta_{i,k}\right) \tag 2.14 $$ where the constant $C$ is independent of $N$, $k$, and $i$. \endproclaim We note that the estimates (2.13) and (2.14) express the weak dependence of the position and momentum of the $k$-th particle with respect to the position of the $i$-th particle at time $0$, as it is expected in a mean-field theory. \heading 3. Convergence \endheading We are now in position to prove Theorem 1.1. We first observe that, for a time interval for which estimates (2.11), (2.12), (2.13) and (2.14) hold, we have classical solutions of eq.s (1.17), (1.19) and (1.18). Therefore we can express the $j$- particle Wigner function in terms of $A^N, S^N $ and $P^N$. For $ F_j\in \Cal D (\Bbb R^{3j} \times \Bbb R^{3j})$, we have: $$ \int F_j(X_j,V_j)f^N_j(X_j,V_j,t)dX_j dV_j= $$ $$ \left(\frac 1{2\pi}\right)^ {3j} \int dX_N \int dV_j \int dY_je^{-i Y_j\cdot V_j} F_j(X_j,V_j) \Psi_N( X_N+\frac h2 Y_j,t) \bar \Psi_N( X_N-\frac h2 Y_j,t) $$ $$ =\left(\frac 1{2\pi}\right)^{\frac {3}{2} j} \int dX_N \int dY_j \tilde F_j(X_j,Y_j) A^N( X_N+\frac h2 Y_j,t) \bar A^N( X_N-\frac h2 Y_j,t) $$ $$ e^{\frac ih [S^N( X_N+\frac h2 Y_j,t)-S^N(X_N-\frac h2 Y_j,t)]}, \tag 3.1 $$ where $\tilde F_j$ is the Fourier transform of $ F_j$ in the second variable. Changing variable $X_N \to X_N-\frac h2 Y_j$ and using the fact that $$ \int dY_j \sup_{X_j} |\tilde F_j (X_j, Y_j)| |Y_j| +\int dY_j \sup_{X_j} |\nabla_{X_j}\tilde F_j(X_j, Y_j)| \leq C j, \tag 3.2 $$ we obtain that (setting $X_j=X_N^j$): $$ (3.1)=(\frac 1{2\pi})^{\frac {3j}{2}} \int dX_N \int dY_j \tilde F_j(X_j,Y_j) A^N( X_N+hY_j,t) \bar A^N( X_N,t) $$ $$ e^{\frac ih [S^N( X_N+hY_j,t)-S^N(X_N,t)]}+O(h). \tag 3.3 $$ Now Lagrange's theorem yields: $$ A^N( X_N+hY_j,t)=A^N( X_N,t)+\int_0^h d\lam \nabla_{X_j} A^N( X_N+\lam Y_j,t)\cdot Y_j $$ Moreover, since $$ \|\nabla_{X_j} A^N\|^2_{L_2}=\sum_{i=1}^j \|\nabla_{x_i} A^N\|^2_{L_2}\leq Cj; \quad \| A^N (t) \|_{L_2}=\| A^N (0) \|_{L_2}=1 $$ by (3.2) and Proposition 2.1 we get the estimate $$ \int _0^h d\lam \int dX_N \int dY_j |\tilde F_j(X_j,Y_j)| |\nabla_{X_j} A^N( X_N+\lam Y_j,t)| | A^N( X_N,t)|\leq $$ $$ h \|A^N\|_{L_2}\|\nabla_{X_j} A^N\|_{L_2} \int dY_j \sup_{X_j} |\tilde F_j (X_j, Y_j)| |Y_j| \leq Ch \sqrt {j}. \tag 3.4 $$ Hence we can conclude that: $$ (3.1)= \left(\frac1{2\pi}\right)^{\frac{3j}{2}}\int dX_N\int dY_j\tilde F_j(X_j,Y_j)|A^N( X_N,t)|^2 $$ $$ e^{\frac ih [S^N( X_N+ h Y_j,t)-S^N(X_N,t)]} +O( h). \tag 3.5 $$ Note that $O(h)$ (as well $O(\frac 1N)$ later on) depends on $j$ which however is fixed. Furthermore: $$ S^N( X_N+ h Y_j,t)-S^N(X_N,t)= \int_0^h d\lam P^N( X_N+\lam Y_j,t)^j\cdot Y_j= $$ $$ hP^N( X_N,t)^j\cdot Y_j+O(h^2) \tag 3.6 $$ again by Proposition 2.1. Here $P^N(X_N)^j$ denotes the projection on the $j$-particle subspace of the vector $P^N(X_N)$. Hence $$ (3.1)= \left(\frac 1{2\pi}\right)^{\frac {3j}{2}} \int dX_N \int dY_j \tilde F_j(X_j,Y_j) | A^N( X_N,t)|^2 e^{-iP^N( X_N,t)^j\cdot Y_j} +O( h)= $$ $$ \int dX_N F(X_j,P^N(X_N,t)^j) | A^N( X_N,t)|^2 +O( h). \tag 3.7 $$ Setting $\Gamma^N=|A^N|^2$ we have by (1.18): $$ \pa_t \Gamma^N +\hbox {div }(P^N \Gamma^N )=B^N \tag 3.8 $$ where $$ B^N=\frac i2 h (\bar A^N \Delta A^N - A^N \Delta \bar A^N). \tag 3.9 $$ The solution of eq. (3.8) has the representation: $$ \Gamma^N (X_N,t)= \rho ^{\otimes N} (\Phi^{-t}(X_N)) J_N (X_N,t) + \int_0^t ds B^N (\Phi^{-(t-s)}(X_N)) J_N (X_N,t-s), \tag 3.10 $$ where $$ J_N (X_N,t)=\det |\frac {\pa \Phi^{-t}(X_N))}{\pa X_N}|. \tag 3.11 $$ Therefore $$ \int dX_N F_j(X_j,P^N(X_N,t)^j) \Gamma^N( X_N,t) = \int dX_N \rho ^{\otimes N}(X_N) F_j(\Phi^{t}(X_N)^j,\dot \Phi^{t}(X_N)^j) $$ $$ +\int_0^t ds \int dX_N B^N (\Phi^{-(t-s)}(X_N),s) J(X_N,t-s) F_j(X_j,P^N(X_N,t)^j). \tag 3.12 $$ The last term in the r.h.s. of (3.12) can be rewritten as: $$ \int_0^t ds \int dX_N B^N (X_N,s) F_j(\Phi^{(t-s)}(X_N)^j,P^N(\Phi^{(t-s)}(X_N)^j,t)) = $$ $$ \frac {ih}{2} \int_0^t ds \int dX_N (\bar A^N \Delta A^N - A^N \Delta \bar A^N)(X_N,s) $$ $$ F_j(\Phi^{(t-s)}(X_N)^j,P^N(\Phi^{(t-s)}(X_N)^j,t)) $$ $$ =\frac {ih}{2}\int_0^t ds \int dX_N (\bar A^N \nabla A^N \cdot \nabla F_j (\dots) - A^N \nabla \bar A^N \cdot \nabla F_j (\dots)), \tag 3.13 $$ here we have integrated by parts and made use of a crucial cancellation. We now observe that $$ \sum_{i=1}^N | \nabla_{x_i} F_j (\Phi^{(t-s)}(X_N)^j,P^N(\Phi^{(t-s)}(X_N)^j,t))| \tag 3.14 $$ can be bounded by a constant dependent on $j$ but not on $N$. Indeed: $$ \pa_{x_i^\alpha}F_j (\Phi^{(t-s)}(X_N)^j,P^N(\Phi^{(t-s)}(X_N)^j,t))= $$ $$ \sum_{k=1}^j[\nabla_{y_k} F_j(Y_j,P^N(\Phi^{(t-s)}(X_N)^jj,t))|_{Y_j=\Phi^{(t-s)}(X_N)^j}\cdot \frac {\pa \Phi^{(t-s)}(X_N)_k}{\pa x_i^\alpha}+ $$ $$ \nabla_{v_k} F_j(\Phi^{(t-s)}(X_N)_j,V_j,t))|_{V_j=P^N(\Phi^{(t-s)}(X_N)^j)}\cdot \frac {\pa P^N_k(\Phi^{(t-s)}(X_N),t)}{\pa x_i^\alpha}. $$ Then by Proposition 1.1 we have that: $$ \pa_{x^{\alpha}_i} F_j(\Phi^{(t-s)}(X_N)_j,P^N(\Phi^{(t-s)}(X_N)^j,t))=O(1) $$ if $i=1\dots j$, while $$ \pa_{x^{\alpha}_i} F (\Phi^{(t-s)}(X_N)_j,P^N(\Phi^{(t-s)}(X_N)_j,t))=O(\frac 1N) $$ if $i>j$. Hence, by (3.14): $$ |\int dX_N \bar A^N \nabla A^N \cdot \nabla F_j (\dots)|\leq \|A \|_{L_2} \sum_{i=1}^N | \nabla_{x_i} F_j (\dots)| \|\nabla_{x_i}A \|_{L_2}\leq Cj. $$ Therefore $$ (3.1)= \int dX_N \rho ^{\otimes N}(X_N) F_j(\Phi^{t}(X_N)_j,\dot \Phi^{t}(X_N)_j)+O(h)+O(\frac 1N). \tag 3.15 $$ Notice that the first term in the r.h.s. of (3.15) is purely classical so that we can apply the convergence result (2.9) to conclude the proof. The proof of Theorem 1.2 follows along the same lines. Proceeding as as above the Wigner function is in this case: $$ f^N (X_N,V_N) = (\frac 1{2\pi})^{3} \int d\Om_N \int dY_N e^{-iY_N \cdot V_N} $$ $$ A^N(X_N-\frac h2 Y_N ,\Om_N,t )\bar A^N(X_N+\frac h2 Y_N ,\Om_N,t ) $$ $$ e^{\frac ih [S^N(X_N-\frac h2 Y_N ,\Om_N,t )-S^N((X_N+\frac h2 Y_N ,\Om_N,t) ]} \tag 3.16 $$ where $A^N$ and $S^N$ are the amplitude and the action parametrized by the initial momenta $\Om_N$. $S^N$ and $A^N$ are the solution of eq.s (1.17) and (1.18) with initial conditions $ S^N (X_N,\Om_N)=\Om_N\cdot X_N$ and $A^N(X_N, \Om_N)=a^{\otimes N }(X_N, \Om_N)$. Therefore for a test function $F_j$ we have: $$ \int F_j(X_j,V_j)f^N_j(X_j,V_j,t)dX_j dV_j= $$ $$ \left(\frac 1{2\pi}\right)^{\frac {3}{2} j} \int d\Om _N \int dX_N \int dY_j \tilde F_j(X_j,Y_j) $$ $$ A^N( X_N+\frac h2 Y_j,\Om_N, t) \bar A^N( X_N-\frac h2 Y_j,\Om_N,t) e^{\frac ih [S^N( X_N+\frac h2 Y_j,\Om_N,t)-S^N(X_N-\frac h2 Y_j,\Om_N,t)]}. \tag 3.17 $$ Proceeding as in the proof of Theorem 1.1 we find: $$ (f^N_j,F)= \int d\Om_N \int dX_N f^{\otimes N}(X_N,V_N) F(X_N(t,X_N,\Om_N), \dot X_N(t,X_N,\Om_N)) $$ $$ +O( h)+O(\frac 1N). \tag 3.18 $$ Note that that $a$ is assumed compactly supported in $w$ to avoid complications with the integral in the initial momenta. However a sufficiently rapid decay of $ a(\cdot, w) $, for large $w$, would yield the same result. Of course, once more, everything holds for a small time interval. However now the smallness of the time interval depends only on the smoothness of the potential $\varphi$ (because of the particular form $\sigma (x)=w\cdot x$). Hence the convergence at time $T$ allows us to extend the argument up to $2T$, using fact 2) of Section 2 and that $f(x,v,T)$ is still compactly supported in velocity. Therefore the convergence can be extended to arbitrary times and the proof of Theorem 1.2 is now completed. \heading 4. Regularity estimates \endheading In this section we prove Proposition 2.1. We start by considering $$ \pa_t P^N+(P^N \cdot \nabla )P^N = -\nabla W. \tag 4.1 $$ which is independent of $$ \pa_t A^N+(P^N \cdot \nabla )A^N +\frac 12 A^N \hbox {div} P^N= -h \frac i2 \Delta A^N. \tag 4.2 $$ to be considered later on. Notice that, for $s\in [0,t]$, $$ P_i^N(\phi^{(t-s)}(X_N),t)=\dot \Phi^t(Y_N(s))_i, \quad \hbox {where}\quad Y_N(s)=\Phi^{-s}(X_N),\quad i=1\dots N. \tag 4.3 $$ Using now the short-hand notation $x_i(t)=\Phi^t(X_N)_i$, $p_i(t)=\dot \Phi^t(X_N)_i$ and denoting $x_i^\alpha (t)$ and $p_i^\alpha (t)$ the $\alpha$-th components, $\alpha=1,2,3$, we have: $$ x_i(t)=x_i+\int_0^t p_i(s) ds $$ $$ p_i(t)=\nabla \sigma (x_i)-\int_0^t \frac 1N \sum_{k\neq i} \nabla \varphi (x_i(s)-x_k(s)) ds. \tag 4.4 $$ where $X_N=(x_1\dots x_N)$. Introducing the force $F^\alpha =-\pa_{x^\alpha}\varphi$, we have: $$ \frac {\pa x_i^\beta (t)}{\pa x_j^\gamma}=\delta_{i,j}\delta_{\beta,\gamma}+\int_0^t ds \frac {\pa p_i^\beta (s)}{\pa x_j^\gamma} \tag 4.5 $$ $$ \frac {\pa p_i^\beta (t)}{\pa x_j^\gamma}=\frac {\pa^2 \sigma}{\pa x_j^\gamma \pa x_i^\beta} \delta_{i,j}+\int_0^t ds \frac 1N \sum_{k\neq i} \sum_\alpha \pa_{x^\alpha} F^\beta (x_i(s)-x_k(s)) \left(\frac {\pa x_i^\alpha (s)}{\pa x_j^\gamma}- \frac {\pa x_k^\alpha (s)}{\pa x_j^\gamma}\right). \tag 4.6 $$ Hence: $$ \frac {\pa p_i^\beta (t)}{\pa x_j^\gamma}=\frac {\pa^2 \sigma}{\pa x_j^\gamma \pa x_i^\beta} \delta_{i,j}+ \int_0^t ds \frac 1N \sum_{k\neq i} \sum_\alpha \pa_{x^\alpha} F^\beta (x_i(s)-x_k(s)) (\delta_{i,j}-\delta_{k,j})\delta_{\alpha,\gamma}+ $$ $$ \int_0^t ds \int _0^s d\tau \frac 1N \sum_{k\neq i} \sum_\alpha \pa_{x^\alpha} F^\beta (x_i(s)-x_k(s)) \left(\frac {\pa p_i^\alpha (\tau)}{\pa x_j^\beta}-\frac {\pa p_k^\alpha (t)}{\pa x_j^\beta}\right). \tag 4.7 $$ We now observe that, if $i\neq j$, the first two terms in the r.h.s. of (4.7) are $O(\frac1N)$ and hence (taking $t\in [0,T]$ with $T$ small enough): $$ \left|\frac {\pa x_i^\beta (t)}{\pa x_j^\gamma}\right|+\left |\frac {\pa p_i^\beta (t)}{\pa x_j^\gamma}\right|\leq C\left(\frac 1N+\delta_{i,j}\right), \tag 4.8 $$ with $C$ independent of $N$. \remark {Remark} Higher derivatives could be handled in the same way to obtain: $$ \left|\frac {\pa^s x_i^\beta (t)}{\pa x_{j_1}^{\gamma_1}\dots \pa x_{j_s}^{\gamma_s} }\right|+ \left|\frac {\pa^s p_i^\beta (t)}{\pa x_{j_1}^{\gamma_1}\dots \pa x_{j_s}^{\gamma_s} }\right| \leq C\left(\frac 1N+\prod _{r=1}^s\delta_{i,j_r}\right), \tag 4.9 $$ assuming a stronger regularity. \endremark Furthermore, setting $Y_N(s)=(y_1(s), \dots,y_N(s))$, $$ \frac {\pa P_i^\alpha (Y_N,t)}{\pa x_{j}^{\gamma}}= \sum_k \sum_{\beta} \frac {\pa \dot \Phi^t (Y_N)^{\alpha}_i}{\pa y_{k}^{\beta}}|_{Y_N=Y_N(s)} \frac {\pa y_k^{\beta} (s)}{\pa x_j^\gamma}. \tag 4.10 $$ We note that the terms in the sum with $k\neq i$ and $k\neq j$ are $\ds O(\frac 1{N^2})$, while the term with $k=i$ or $k=j$ are $\ds O(\frac 1{N})$. Therefore, if $i\neq j$, the full sum is $\ds O(\frac 1{N})$. If $i=j$ the sum is $O(1)$ because of the term $k=i=j$ which is indeed $O(1)$. Summarizing: $$ \left|\frac {\pa P_i^\alpha (\Phi ^{(t-s)}(X_N),t)}{\pa x_{j}^{\gamma}}\right| \leq C\left(\frac 1N+\delta_{i,j}\right). \tag 4.11 $$ \remark {Remark} A similar analysis on the higher derivatives yields: $$ \left|\frac {\pa^s P_i^\beta (\Phi ^{(t-s)}(X_N),t)}{\pa x_{j_1}^{\gamma_1}\dots \pa x_{j_s}^{\gamma_s} }\right| \leq C\left(\frac 1N+\prod _{r=1}^s\delta_{i,j_r}\right). \tag 4.12 $$ \endremark We now proceed to analyze eq. (4.2) to obtain estimate of the solution in $H^1$. Applying the operator $\nabla_{x_j}$ to the equation, we obtain: $$ \pa_t \nabla_{x_j} A^N+ ( \nabla_{x_j} P^N \cdot \nabla ) A^N +(P^N \cdot \nabla )\nabla_{x_j} A^N +\frac 12 \nabla_{x_j} A^N \hbox {div} P^N= $$ $$ - \frac 12 A^N \hbox {div} \nabla_{x_j}P^N -h \frac i2 \Delta \nabla_{x_j} A^N. \tag 4.14 $$ In computing $$ \frac {d} {dt} (\nabla_{x_j} A^N,\nabla_{x_j} A^N)= (\pa_t \nabla_{x_j} A^N,\nabla_{x_j} A^N)+(\nabla_{x_j} A^N,\pa_t \nabla_{x_j} A^N) \tag 4.15 $$ we realize that, due to the symmetry of $\Delta$, the last term does not give any contribution. Also, the sum of the terms non involving $\nabla_{x_j}P^N$ vanishes: $$ (\nabla_{x_j} A^N, P^N \cdot \nabla \nabla_{x_j} A^N)+ (P^N \cdot \nabla \nabla_{x_j} A^N, \nabla_{x_j} A^N)+ $$ $$ \frac 12 (\nabla_{x_j} A^N, \nabla_{x_j} A^N \hbox {div} P^N)+ \frac 12 (\hbox {div} P^N \nabla_{x_j} A^N, \nabla_{x_j} A^N )=0. \tag 4.16 $$ Here we use the reality of $P^N$ and the identity: $$ \int P^N \cdot \nabla | \nabla_{x_j} A^N|^2=-\int \hbox {div} P^N | \nabla_{x_j} A^N|^2 \tag 4.17 $$ We finally observe that, by eq. (4.11), $$ ( \nabla_{x_j} P^N \cdot \nabla ) A^N= (\nabla_{x_j} P^N_j \cdot \nabla_{x_j} ) A^N + \Cal O (N^{-1}\sum_{k\not= j}\| \nabla_{x_k}A^N\|_{L_2}) \tag 4.18 $$ and thus, denoting $\|A^N\|_1 := (\sum_k \| \nabla_{x_k} A^N\|_{L_2}^2)^{1/2}$ (so that $\sum_{k\not= j}\| \nabla_{x_k}A^N\|_{L_2}\leq \sqrt N \|A^N\|_1$), we arrive to the inequality: $$ \frac {d}{dt} \| \nabla_{x_j}A^N(t)\|_{L_2}^2\leq C \{\| \nabla_{x_j}A^N(t)\|_{L_2}^2 + N^{-1/2}\| A^N\|_1\|\nabla_{x_j}A^N(t)\|_{L_2}\} \tag 4.19 $$ with $C$ independent of $N$ and $j$. In particular, taking the sum over all $j$, we obtain, $$ \frac {d}{dt} \| A^N(t)\|_1^2\leq 2C \| A^N(t)\|_1^2. \tag 4.20 $$ Since at time zero $\| A^N\|_1^2$ is $\Cal O (N)$ the same conclusion holds on any time interval. Going back to Eq. (4.19), we obtain $$ \frac {d}{dt} \| \nabla_{x_j}A^N(t)\|_{L_2}^2\leq C \| \nabla_{x_j}A^N(t)\|_{L_2}^2 \tag 4.21 $$ with a new constant $C$ independent of $N$ and $j$. 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