In 1959, Batchelor predicted that passive scalars advection in incompressible fluids at fixed Reynolds number with small diffusivity k should display a |k|^−1 power spectrum in a statistically stationary experiment, which has since been tested extensively in the physics and engineering literature. Results obtained with Alex Blumenthal and Sam Punshon-Smith provide the first mathematically rigorous proof of this law in the fixed Reynolds number case under stochastic forcing (with the caveat that in the 3d case, the Navier-Stokes equations are regularized with hyperviscosity). The origin of the spectrum is the uniform, exponential rate that all passive scalar fields mix (up to a random prefactor), which we prove using ideas from random dynamical systems such as a la Furstenberg and two-point geometric ergodicity.

Short intro: Jacob Bedrossian is a Professor of Mathematics at the University of Maryland, College Park USA. His research interests are primarily focused on the mathematical analysis of deterministic and stochastic partial differential equations arising in the study of incompressible fluid mechanics at high Reynolds number and plasma physics. For his contributions, he has been awarded a 2015 Sloan Fellowship, a 2016 NSF CAREER grant, the 2019 SIAG/APDE prize (joint with Nader Masmoudi), the 2019 IMA prize, and the 2020 Peter Lax award. He is also a 2020 Simons Fellow, a 2022 Nachdiplom lecturer at ETH Zurich, and an invited speaker at the ICM 2022 in St. Petersburg.