Boltzmann equations with quantum effects (2): Entropy identity, existence and uniqueness of spatial decay solutions

摘要

A previous study[l] is continued by investigating the Boltzmann equation for particles with quantum effects (BQE). First, the corresponding entropy identity is proved, then if the initial data f(x, v, 0) satisfies 0 ≤ f(x, v, 0) ≤ CΦ (x, v, 0) for a constant 0 < C < ∞ and function Φ(x, v, t), we prove the existence and uniqueness of spatial decay solutions of the BQE within a given function space B(Φ) using fixed point theory. Moreover, if there is a continuous function F(x, v) which belongs to a function set, then there exists a mild solution f(x, v, t) of the BQE such that f∞(x, v) = limf(x+ vt, v, t) = F(x, v). t→=∝