We consider the incompressible 2D Navier-Stokes equations on the torus driven by a deterministic time quasi-periodic force and a noise that is white in time and extremely degenerate in Fourier space. We show that the asymptotic statistical behavior is characterized by a uniquely ergodic and exponentially mixing quasi-periodic invariant measure. The result is true for any value of the viscosity ν > 0. By utilizing this quasi-periodic invariant measure, we show the strong law of large numbers and central limit theorem for the continuous time inhomogeneous solution processes. Estimates of the corresponding rate of convergence are also obtained, which is the same as in the time homogeneous case for the strong law of large numbers, while the convergence rate in the central limit theorem depends on the Diophantine approximation property on the quasi-periodic frequency and the mixing rate of the quasi-periodic invariant measure. We also prove the existence of a stable quasi-periodic solution in the laminar case (when the viscosity is large). The scheme of analyzing the statistical behavior of the time inhomogeneous solution process by the quasi-periodic invariant measure could be extended to other inhomogeneous Markov processes.
College and Department
Physical and Mathematical Sciences; Mathematics
BYU ScholarsArchive Citation
Liu, Rongchang, "Statistical Properties of 2D Navier-Stokes Equations Driven by Quasi-Periodic Force and Degenerate Noise" (2022). Theses and Dissertations. 9391.
Navier-Stokes Equations, quasi-periodic invariant measure, unique ergodicity, mixing, limit theorems, rate of convergence, Diophantine condition