These notes originated from a series of lectures given at Waseda University in April–May
2021, supported by Top Global University Project of Waseda University. The first author …

We study a continuous data assimilation algorithm proposed by Azouani, Olson, and Titi
(AOT) in the context of an unknown viscosity. We determine the large-time error between the …

We study a nonlinear-nudging modification of the Azouani–Olson–Titi continuous data
assimilation (downscaling) algorithm for the 2D incompressible Navier–Stokes equations …

We study numerical schemes for incompressible Navier–Stokes equations using IMEX
temporal discretizations, finite element spatial discretizations, and equipped with continuous …

We study a class of discrete-time random dynamical systems with compact phase space.
Assuming that the deterministic counterpart of the system in question possesses a …

We rigorously prove the well-posedness of the formal sensitivity equations with respect to
the viscosity corresponding to the 2D incompressible Navier–Stokes equations. Moreover …

We illustrate how the notion of asymptotic coupling provides a flexible and intuitive
framework for proving the uniqueness of invariant measures for a variety of stochastic partial …

We study different approaches to implementing sparse-in-time observations into the the
Azouani-Olson-Titi data assimilation algorithm. We propose a new method which introduces …

We show that the stochastic 3D primitive equations with the Neumann boundary condition
on the top, the lateral Dirichlet boundary condition and either the Dirichlet or the Neumann …

We establish verifiable general sufficient conditions for exponential or subexponential
ergodicity of Markov processes that may lack the strong Feller property. We apply the …