论文标题
关于$ l^2 $的收敛速率，从哈特里到vlasov $ \ unicode {x2013} $ poisson方程
On the $L^2$ Rate of Convergence in the Limit from the Hartree to the Vlasov$\unicode{x2013}$Poisson Equation
论文作者
论文摘要
使用新的稳定性估计值来差异vlasov $ \ unicode {x2013} $ poisson方程的两种解决方案的平方根，我们获得了$ l^2 $ wigner solom the Hartree方程解决方案的wigner wigner norm n Norm north Hartree方程的融合，并具有coulomb vlasov $ poissode conteriation的coulomb潜在的contression conteriation contressove。与$ \ hbar $成比例。这改善了$ \ hbar^{3/4- \ varepsilon} $收敛速率在[L.〜lafleche，c.〜saffirio中获得的$ l^2 $中，出现分析和PDE]。本文感兴趣的另一个原因是新方法，让人联想到用于证明多体schrödinger方程向Hartree $ \ Unicode {X2013} $ Fock方程的均值范围限制的方法。
Using a new stability estimate for the difference of the square roots of two solutions of the Vlasov$\unicode{x2013}$Poisson equation, we obtain the convergence in the $L^2$ norm of the Wigner transform of a solution of the Hartree equation with Coulomb potential to a solution of the Vlasov$\unicode{x2013}$Poisson equation, with a rate of convergence proportional to $\hbar$. This improves the $\hbar^{3/4-\varepsilon}$ rate of convergence in $L^2$ obtained in [L.~Lafleche, C.~Saffirio: Analysis & PDE, to appear]. Another reason of interest of this paper is the new method, reminiscent of the ones used to prove the mean-field limit from the many-body Schrödinger equation towards the Hartree$\unicode{x2013}$Fock equation for mixed states.
