Transport, Fluids and Mixing

Positive Lyapunov exponents for 2d Galerkin-Navier-Stokes with stochastic forcing

speaker: Jacob Bedrossian (University of Maryland)

abstract: In this talk we discuss our recently introduced methods 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 (joint with Alex Blumenthal and Sam Punshon-Smith). This hallmark of chaos has long been observed in these models, however, no mathematical proof had previously been made for any type of deterministic or stochastic forcing. We propose a new method which has the ability to obtain quantitative estimates on the top Lyapunov exponents of high-dimensional, weakly dissipative SDEs. It combines (A) a new identity connecting the Lyapunov exponents to a Fisher information of the Markov process tracking tangent directions; and (B) a quantitative hypoelliptic estimate to show that this (degenerate) Fisher information is an upper bound on some fractional regularity. For L96 and GNSE, we then further reduce the lower bound of the top Lyapunov exponent to proving a certain Lie algebraic condition on the nonlinearity. I will also discuss the recent work of Sam Punshon-Smith and I on verifying this condition for the 2d Galerkin-Navier-Stokes equations in a rectangular, periodic box of any aspect ratio using some special structure of matrix Lie algebras and ideas from computational algebraic geometry (for L96 it can be verified "by hand").

timetable:
Thu 27 Jan, 15:00 - 15:50,
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