Georgia Institute of Technology College of Sciences

Batchelor's law describes the power law spectrum of the turbulent regime of passive scalars (e.g., temperature or a dilute concentration of some tracer chemical) advected by an incompressible fluid (e.g., the Navier-Stokes equations at fixed Reynolds number), in the limit of vanishingly low molecular diffusivity. Predicted in 1959, it has been confirmed empirically in a variety of experiments, e.g. salinity concentrations among ocean currents. On the other hand, as with many turbulent regimes in physics, a true predictive theory from first principles has been missing (even a non-rigorous one), and there has been some controversy regarding the extent to which Batchelor's law is universal. 

In this talk, I will present a program of research, joint with Jacob Bedrossian (UMD) and Sam Punshon-Smith (Brown), which has rigorously proved Batchelor's law for passive scalars advected by the Navier-Stokes equations on the periodic box subjected to Sobolev regular, white-in-time body forces. The proof is a synthesis of techniques from dynamical systems and smooth ergodic theory, stochastics/probability, and fluid mechanics. To our knowledge, this work constitutes the first mathematically rigorous proof of a turbulent power law spectrum. It also establishes a template for predictive theories of passive scalar turbulence in more general settings, providing a strong argument for the universality of Batchelor's law. 

While the existence and properties of the SRB measure for the billiard map associated with a periodic Lorentz gas are well understood, there are few results regarding other types of measures for dispersing billiards. We begin by proposing a naive definition of topological entropy for the billiard map, and show that it is equivalent to several classical definitions. We then prove a variational principle for the topological entropy and proceed to construct a unique probability measure which achieves the maximum. This measure is Bernoulli and positive on open sets. An essential ingredient is a proof of the absolute continuity of the unstable foliation with respect to the measure of maximal entropy. This is joint work with Viviane Baladi.
 

We are studying the asymptotic homotopical complexity of a sequence of billiard flows on the 2D unit torus T^2 with n
circular obstacles. We get asymptotic lower and upper bounds for the radial sizes of the homotopical rotation sets and,
accordingly, asymptotic lower and upper bounds for the sequence of topological entropies. The obtained bounds are rather
close to each other, so this way we are pretty well capturing the asymptotic homotopical complexity of such systems.

Note that the sequence of topological entropies grows at the order of log(n), whereas, in sharp contrast, the order of magnitude of the sequence of metric entropies is only log(n)/n.

Also, we prove the convexity of the admissible rotation set AR, and the fact that the rotation vectors obtained from
periodic admissible trajectories form a dense subset in AR.

Considering SL(2,R) skew-product maps over circle rotations,
we prove that a renormalization transformation
associated with the golden mean alpha
has a nontrivial periodic orbit of length 3.
We also present some numerical results,
including evidence that this period 3 describes
scaling properties of the Hofstadter butterfly
near the top of the spectrum at alpha,
and scaling properties of the generalized eigenfunction
for this energy.