Formation of small scales in nonlinear PDEs

Scalar turbulence in stochastic fluid dynamics

Samuel Punshon-Smith

Brown University

I will discuss several recent results on the advection of a passive scalar by various models in stochastic fluid dynamics, including the 2D incompressible stochastic Navier-Stokes equations with non-degenerate white-in-time forcing. Specifically, I will explain how certain tools from random dynamical systems and the hypoelliptic theory of degenerate SPDE can be applied to prove that the Lagrangian flow has a positive Lyapunov exponent (Lagrangian chaos) and how this can be used to show almost sure exponential mixing of passive scalars advected by the stochastic Navier-Stokes equations (measured by decay of negative Sobolev norms). I will also present several applications to certain universal laws in passive scalar turbulence and the enhanced dissipation of the solutions to advection-diffusion equation. This talk is based on several joint works with collaborators Jacob Bedrossian and Alex Blumenthal.