Random Splitting of Fluid Models: Ergodicity, Convergence, and Chaos

In this thesis we study random splitting and apply our results to random splittings of fluid models. Random splitting is loosely defined as follows. Consider the differential equation $\dot{x}=V(x)$ where $\dot{x}$ is a time derivative and the vector field $V$ on $\mathbb{R}^D$ splits as the sum $V=\sum_{j=1}^n V_j$. In traditional operator splitting one approximates solutions of $\dot{x}=V(x)$ by composing solutions of $\dot{x}=V_j(x)$ over (typically small) deterministic time steps. Here we take these times to be independent and identically distributed random variables. This turns the aforementioned compositions into Markov chains, which we call \textit{random splittings of $V$} or simply \textit{random splittings}. We prove under relatively mild conditions that these random splittings possess a unique invariant measure (ergodicity), that their trajectories converge on average and almost surely to trajectories of the original system $\dot{x}=V(x)$ (convergence), and that, in certain cases, their top Lyapunov exponent is positive (chaos). After proving these general results, we construct random splittings of four fluid models: the conservative Lorenz-96 and Lorenz-96 equations, and Galerkin approximations of the 2d Euler and 2d Navier-Stokes equations on the torus. We prove all these random splittings are ergodic and converge to their deterministic counterparts in a certain sense, and, for conservative Lorenz-96 and 2d Euler, that their top Lyapunov exponent is positive.