Thesis Defenses Seminar
Thursday, November 17, 2022, 10:00am, Physics 119
Omar Melikechi (Duke University, Mathematics)
Random Splitting of Fluid Models: Ergodicity, Convergence, and Chaos
Abstract:
In this dissertation 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 $R^D$ 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 a Markov chain, which we call a random splitting of $V$ or simply random splitting. 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 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.

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