APDEs Seminar

Hydrodynamical turbulence in the Navier-Stokes equations is marked by anomalous dissipation and chaos, in both the Eulerian fields and Lagrangian flow maps (called “Eulerian chaos” and “Lagrangian chaos” respectively). In this talk we discuss two recent works joint with Alex Blumenthal and Sam Punshon-Smith on progress towards understanding these two types of chaos in the stochastically-forced Navier-Stokes equations. The first line of work adapts ideas from random dynamical systems and stochastic PDEs to prove Lagrangian chaos in the stochastic Navier-Stokes equations under certain non-degeneracy conditions, further upgrades this to almost-sure proves exponential mixing of passive scalars, and uses this to provide the mathematical proof of Batchelor’s 1959 prediction regarding the power spectrum of a passive scalar advected by the fluid. The second line of work contains a new method for obtaining strictly positive lower bounds on the top Lyapunov exponent of high-dimensional, stochastic differential equations such as the weakly-damped Lorenz-96 (L96) model or Galerkin truncations of the 2d Navier-Stokes equations. The method reduces the question of (Eulerian) chaos in this class of models to a certain Lie algebraic condition on the nonlinearity. Recent work by Sam Punshon-Smith and myself shows that by using some special structure of matrix Lie algebras and ideas from computational algebraic geometry, this condition can be verified for 2d Galerkin-Navier-Stokes in all sufficiently high dimensional trunctions (for L96 it can be verified “by hand”).