标题： 关于玻尔兹曼级数全局时间存在性和唯一性的解 显示英文标题

标题： On the global in time existence and uniqueness of solutions to the Boltzmann hierarchy

摘要： 在本文中，我们建立了玻尔兹曼层次结构的全局时间存在性和唯一性解，这是一个对于从多粒子系统严格推导玻尔兹曼方程至关重要的方程层次结构。 受可用的基于$L^{\infty}$的玻尔兹曼方程解的先验估计启发，我们为玻尔兹曼层次结构的解发展了多项式加权的$L^\infty$先验界，并通过一种称为克莱纳曼-马切多板游戏论证的组合技术重新组织求和，从而处理狄森级数中项数的阶乘增长。 本文是第一篇将这种组合技术与基于$L^{\infty}$的估计相结合，以证明玻尔兹曼层次结构温和解唯一性的作品。 我们对适定初始数据的玻尔兹曼层次结构全局时间温和解的存在性的证明是构造性的，并通过一个 Hewitt-Savage 类型定理利用已知的玻尔兹曼方程全局时间解。 显示英文摘要

摘要： In this paper we establish the global in time existence and uniqueness of solutions to the Boltzmann hierarchy, a hierarchy of equations instrumental for the rigorous derivation of the Boltzmann equation from many particles. Inspired by available $L^{\infty}$-based a-priori estimate for solutions to the Boltzmann equation, we develop the polynomially weighted $L^\infty$ a-priori bounds for solutions to the Boltzmann hierarchy and handle the factorial growth of the number of terms in the Dyson's series by reorganizing the sum through a combinatorial technique known as the Klainerman-Machedon board game argument. This paper is the first work that exploits such a combinatorial technique in conjunction with an $L^{\infty}$-based estimate to prove uniqueness of the mild solutions to the Boltzmann hierarchy. Our proof of existence of global in time mild solutions to the Boltzmann hierarchy for admissible initial data is constructive and it employs known global in time solutions to the Boltzmann equation via a Hewitt-Savage type theorem.