Georgia Institute of Technology College of Sciences

In this talk I will present a computer-assisted method to study solutions vanishing at infinity in differential equations on R^n. Such solutions arise naturally in various models, in the form of traveling waves or localized patterns for instance, and involve multiple challenges to address both on the numerical and on the analytical side. Using spectral techniques, I will explain how Fourier series can serve as an approximation of the solution as well as an efficient mean for the construction of a fixed-point operator for the proof. To illustrate the method, I will present applications to the constructive proof of localized patterns in the 2D Swift-Hohenberg equation and in the Gray-Scott model. The method extends to non-local equations and proofs of solitary travelling waves in the (capillary-gravity) Whitham equation will be exposed.

In 1989, Eremenko investigated the set of points that escape to infinity under iteration of a transcendental entire function, the so-called escaping set. He proved that every component of the closure of the escaping set is unbounded and conjectured that all the components of the escaping set are unbounded. Much of the recent work on the iteration of entire functions is involved in investigating properties of the escaping set, motivated by Eremenko's conjecture. We will begin by introducing many of the basic dynamical properties of iterates of an analytic function, and finally discuss constructing a transcendental entire function with a point connected component of the escaping set, providing a counterexample to Eremenko's conjecture. This is joint work with David Martí-Pete and Lasse Rempe.

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.