Randomly Forced Nonlinear PDEs and Statistical Hydrodynamics in 2 Space Dimensions
This booklet is based on the author's lecture-course at ETH Zürich and it addresses the incompressible isothermal Navier-Stokes equation in a two-dimensional domain with periodic boundary conditions forced randomly by the right-hand side. The topic is related to turbulence theory but heuristic considerations are suppressed only to discussion while the focus is on rigorously proved mathematical results. The author shows, for example, that statistical characteristics of a turbulent flow stabilize with time growing to characteristics independent of an initial velocity profile, that the time average of any characteristic of a turbulent flow equals the ensemble average, and that the turbulent flow is a Gaussian process at large time scales. A short introductory section on function spaces and partial differential equations (focused on the Navier-Stokes equation) is included but the reader’s probability background is assumed. The book is primarily intended for experts in probability theory and partial differential equations with a focus on theoretical fluid dynamics.