Ergodic theory for energetically open compressible fluid flows

0542582 - MÚ 2022 RIV NL eng J - Journal Article
Fanelli, F. - Feireisl, Eduard - Hofmanová, M.
Ergodic theory for energetically open compressible fluid flows.
Physica. D. Roč. 423, September (2021), č. článku 132914. ISSN 0167-2789. E-ISSN 1872-8022
R&D Projects: GA ČR(CZ) GA18-05974S
Institutional support: RVO:67985840
Keywords : barotropic Navier-Stokes system * ergodic theory * inflow/outflow boundary conditions * stationary statistical solution
OECD category: Pure mathematics
Impact factor: 3.751, year: 2021
Method of publishing: Limited access
https://doi.org/10.1016/j.physd.2021.132914

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∈R). In particular, the ergodic averages converge for any trajectory provided its ω-limit set in the trajectory space supports a unique (in law) stationary solution.
Permanent Link: http://hdl.handle.net/11104/0319969