The Derivation of the Compressible Euler Equation from Quantum Many-Body Dynamics  

作　　者：Xuwen Chen Shunlin Shen Jiahao Wu Zhifei Zhang 

机构地区：[1]Department of Mathematics,University of Rochester,Rochester,NY 14627,USA [2]School of Mathematical Sciences,Peking University,Beijing 100871,China

出　　处：《Peking Mathematical Journal》2024年第1期35-90,共56页北京数学杂志（英文）

基　　金：X.Chen was supported in part by NSF grant DMS-2005469 and a Simons fellowship numbered 916862;S.Shen was supported in part by the Postdoctoral Science Foundation of China under Grant 2022M720263;Z.Zhang was supported in part by NSF of China under Grant 12171010 and 12288101.

摘　　要：We study the three-dimensional many-particle quantum dynamics in mean-field set-ting.We forge together the hierarchy method and the modulated energy method.We prove rigorously that the compressible Euler equation is the limit as the particle num-ber tends to infinity and the Planck’s constant tends to zero.We improve the previous sufficient small time hierarchy argument to any finite time via a new iteration scheme and Strichartz bounds first raised by Klainerman and Machedon in this context.We establish strong and quantitative microscopic to macroscopic convergence of mass and momentum densities up to the 1st blow up time of the limiting Euler equation.We justify that the macroscopic pressure emerges from the space-time averages of micro-scopic interactions via the Strichartz-type bounds.We have hence found a physical meaning for Strichartz-type bounds.

关 键 词：Compressible Euler equation BBGKY hierarchy Quantum many-body dynamics Klainerman-Machedon bounds Modulated energy 

分 类 号：O17[理学—数学]