The ergodic hypothesis is examined for energetically open fluid systems represented by the barotropic Navier–Stokes equations with general inflow/outflow boundary conditions. We show that any globally bounded trajectory generates a stationary statistical solution, which is interpreted as a stochastic process with continuous trajectories supported by the family of weak solutions of the problem. The abstract Birkhoff–Khinchin theorem is applied to obtain convergence (in expectation and a.s.) of ergodic averages for any bounded Borel measurable function of state variables associated to any stationary solution. Finally, we show that validity of the ergodic hypothesis is determined by the behavior of entire solutions (i.e. a solution defined for any $t\in R$). In particular, the ergodic averages converge for any trajectory provided its $\omega$-limit set in the trajectory space supports a unique (in law) stationary solution.

对遍历性假设进行了研究，考察了以正压Navier-Stokes方程为代表的，具有一般流入/流出边界条件的能量开放流体系统。我们表明，任何全局有界的轨迹都会生成平稳的统计解，这被解释为具有连续的轨迹的随机过程，该轨迹由问题的弱解族支持。应用抽象的Birkhoff-Khinchin定理来获得与任何平稳解相关的状态变量的任何有界Borel可测函数的遍历平均值的收敛性（预期和作为）。最后，我们证明了遍历假说的有效性取决于整个解决方案的行为（即为任何解决方案定义的解决方案$Ť\in \mathrm{[R}$）。特别是，遍历平均值会收敛于任何轨迹，只要其轨迹$\omega$轨迹空间中的-limit集支持唯一的（在法律上）固定的解决方案。