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Positive Lyapunov exponents for the Galerkin-Navier-Stokes equations with stochastic forcing

PDE and Applied Math Seminar

Speaker: Sam Punshon-Smith, IAS
Location: Zoom and MSB2112
Start time: Thu, Oct 21 2021, 4:10PM

In this talk I will discuss a recently introduced 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 (joint with Jacob Bedrossian and Alex Blumenthal). 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. The method we proposed combines (A) a new identity connecting the Lyapunov exponents to a Fisher information of the stationary measure of the Markov process tracking tangent directions (the so-called "projective process"); and (B) an L1-based hypoelliptic regularity 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 that the projective process satisfies Hörmander's condition. I will also discuss the recent work of Jacob Bedrossian 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.