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. 

Ideas from physics have predicted a number of important properties of random constraint satisfaction problems such as the satisfiability threshold and the free energy (the exponential growth rate of the number of solutions).  Another prediction is the condensation regime where most of the solutions are contained in a small number of clusters and the overlap of two random solutions is concentrated on two points.  We establish this phenomena for the random regular NAESAT model.

We prove that a power series invariant of suitable ideal triangulations, defined by Frohman-Kania-Bartoszynska coincides with the power series invariant of Dimofte-Gaiotto-Gukov known as the 3D index. In partucular, we deduce that the FKB invariant is topological, and that the tetrahedron weight of the 3D index is a limit of quantum 6j symbols. Joint work with Roland van der Veen.

Given a potential function of three vector arguments, $f(x,y,z)$, which is $O(n)$-invariant, $f(Qx,Qy,Qz)=f(x,y,z)$ for all $Q$ orthogonal, we use semidefinite programming bounds to determine optimizing probability measures for interaction energies of the form $\int\int\int f(x,y,z) d\mu(x)d\mu(y)d\mu(z)$ over the sphere. This approach builds on previous use of such bounds in the discrete setting by Bachoc-Vallentin, Cohn-Woo, and Musin, and is successful for kernels which can be shown to have expansions in a particular basis, for instance certain symmetric polynomials in inner products $u=\langle x,y \rangle$, $v=\langle y,z\rangle$, and $t=\langle z, x \rangle$. For other kernels we pose conjectures on the behavior of optimizers, partially inferred through numerical studies.