Georgia Institute of Technology College of Sciences

The dynamics of a passive scalar, such as temperature or concentration, transported by an incompressible flow can be modeled by the advection-diffusion equation. Advection often results in the formation of complicated, small-scale structures and can result in solutions relaxing to equilibrium at a rate much faster than the corresponding heat equation in regimes of weak diffusion. This phenomenon is typically referred to as enhanced diffusion. In this talk, I will discuss a joint work with Tarek Elgindi and Jonathan Mattingly in which we construct an example of a divergence-free velocity field on the two-dimensional torus that results in optimal enhanced diffusion.  The flow consists of time-periodic, alternating piece-wise linear shear flows. The proof is based on the probabilistic representation formula for the advection-diffusion equation, a discrete time approximation, and ideas from hyperbolic dynamics.

Continuum theories of physics are traditionally described by local partial differential equations (PDEs). In this talk I will discuss the Sparse Physics-Informed Discovery of Empirical Relations (SPIDER) algorithm: a general algorithm combining the weak formulation, symmetry covariance, and sparse regression to discover quantitatively accurate and qualitatively simple PDEs directly from data. This method is applied to simulated 3D turbulence and experimental 2D active turbulence. A complete mathematical model is found in both cases.

In this talk, we study the long-term behaviour of Random Dynamical Systems (RDSs) conditioned upon staying in a region of the space. We use the absorbing Markov chain theory to address this problem and define relevant dynamical systems objects for the analysis of such systems. This approach aims to develop a satisfactory notion of ergodic theory for random systems with escape.

In the early 60’s J. B. Keller and D. Levy discovered a fundamental property: the instability tongues in Mathieu-type equations lose sharpness with the addition of higher-frequency harmonics in the Mathieu potentials. Twenty years later, V. Arnold discovered a similar phenomenon on the sharpness of Arnold tongues in circle maps (and rediscovered the result of Keller and Levy). In this paper we find a third class of object where a similar type of behavior takes place: area-preserving maps of the cylinder. loosely speaking, we show that periodic orbits of standard maps are extra fragile with respect to added drift (i.e. non-exactness) if the potential of the map is a trigonometric polynomial. That is, higher-frequency harmonics make periodic orbits more robust with respect to “drift". This observation was motivated by the study of traveling waves in the discretized sine-Gordon equation which in turn models a wide variety of physical systems. This is a joint work with Mark Levi.