Johns Hopkins University
Spontaneous Stochasticity, Turbulent Magnetic Dynamo, and Onsager's Conjecture on Euler Solutions
Abstract: We review the notion of "spontaneous stochasticity", which arose from the work of Richardson (1926) on turbulent 2-particle dispersion and which corresponds to a breakdown in uniqueness of solutions to ODE's with vector fields only Hoelder continuous in space. This phenomenon has been rigorously established in probabilistic models of Brownian flows (Kraichnan model) and evidence of the same phenomenon is observed in empirical data for Navier-Stokes turbulence at high Reynolds numbers. For the scalar advection-diffusion equation (Lie transported 0-forms) ``spontaneous stochasticity" implies a robustly unique class of weak solutions which dissipate energy. We discuss how "spontaneous stochasticity" influences the turbulent kinematic magnetic dynamo (Lie-transported 1-forms), both by analytical results for Brownian flows and by Lagrangian numerical studies for Navier-Stokes turbulence. There is a close analogy between the ideal magnetic induction equation for 1-forms and the incompressible Euler equation for the velocity. This analogy suggests several natural conjectures for fluid circulations in the weak solutions of the Euler equations that describe fluid turbulence, as conjectured by Onsager. These conjectures are supported also by recent work of Constantin & Iyer (2008), which implies equivalence of the incompressible Navier-Stokes equation to a stochastic Kelvin Theorem.
Friday March 1, 2013 at 3:00 PM in SEO 636