标题： 关于有限温度束缚玻色气体的动力学 显示英文标题

标题： On the dynamics of finite temperature trapped Bose gases

摘要： 描述有限温度下玻色-爱因斯坦凝聚体（BEC）和热云动力学的系统由非线性薛定谔（NLS）方程和量子玻尔兹曼（QB）方程组成。 在这种有限温度的束缚玻色气体系统中，QB方程对应于热云密度分布函数的演化，而NLS是凝聚体的方程。 在这个温度范围内，量子玻尔兹曼碰撞算子是两个算子$C_{12}$和$C_{22}$的和，它们分别描述凝聚体与非凝聚原子之间的碰撞以及非凝聚原子之间的碰撞。 在BEC临界温度以上，系统简化为仅包含$C_{22}$的方程，该方程相对于$L^\infty$范数具有爆破解的正径向解（参见\cite{EscobedoVelazquez:2015:FTB}）。 另一方面，在极低温 regime 中，系统变为$C_{12}$的方程，其具有不同的（高得多的阶数）跃迁概率，该方程具有加权$L^1$范数下的唯一全局正径向解（参见）。 \cite{AlonsoGambaBinh}). 在当前的温度条件下，我们首先将QB和NLS方程解耦，然后展示空间均匀的动能系统的正径向解的整体存在性和唯一性结果。 与\cite{EscobedoVelazquez:2015:FTB}中考虑的情况不同，由于BEC的存在，碰撞积分与复杂的能量流形相关，而不是球体，因为粒子能量由Bogoliubov色散定律近似。 此外，整个系统的质量不守恒，而\cite{EscobedoVelazquez:2015:FTB}中考虑的情况质量是守恒的。 然后提供了一种新理论。 显示英文摘要

摘要： The system that describes the dynamics of a Bose-Einstein Condensate (BEC) and the thermal cloud at finite temperature consists of a nonlinear Schrodinger (NLS) and a quantum Boltzmann (QB) equations. In such a system of trapped Bose gases at finite temperature, the QB equation corresponds to the evolution of the density distribution function of the thermal cloud and the NLS is the equation of the condensate. The quantum Boltzmann collision operator in this temperature regime is the sum of two operators $C_{12}$ and $C_{22}$, which describe collisions of the condensate and the non-condenstate atoms and collisions between non-condensate atoms. Above the BEC critical temperature, the system is reduced to an equation containing only $C_{22}$, which possesses a blow-up positive radial solution with respect to the $L^\infty$ norm (cf. \cite{EscobedoVelazquez:2015:FTB}). On the other hand, at the very low temperature regime, the system becomes an equation of $C_{12}$, with a different (much higher order) transition probability, which has a unique global positive radial solution with weighted $L^1$ norm (cf. \cite{AlonsoGambaBinh}). In the current temperature regime, we first decouple the QB and NLS equations, then show a global existence and uniqueness result for positive radial solutions to the spatially homogeneous kinetic system. Different from the case considered in \cite{EscobedoVelazquez:2015:FTB}, due to the presence of the BEC, the collision integrals are associated to sophisticated energy manifolds rather than spheres, since the particle energy is approximated by the Bogoliubov dispersion law. Moreover, the mass of the full system is not conserved while it is conserved for the case considered in \cite{EscobedoVelazquez:2015:FTB}. A new theory is then supplied.