Maciej Zworski (UC Berkeley)

Optimal enhanced dissipation for geodesic flow

Abstract:

We consider geodesic flows on negatively curved compact manifolds or more generally contact Anosov flows (all these concepts will be explained). The object is to show that if X is the generator of the flow and Δ, a (negative) Laplacian, then solutions to the convection diffusion equation, ∂_t u = X u + Δu, u ≥ 0, satisfy
∥ u(t) - u ∥_L²(M) ≤ C u^-K e^-βt ∥ u(0)∥_L²(M) ,
where u is the (conserved) average of u(0) with respect to the contact volume form and K is a fixed constant. This provides many examples of very precise optimal enhanced dissipation in the sense of recent works of Bedrossian–Blumenthal–Punshon-Smith and Elgindi–Liss–Mattingly. The proof is based on results by Dyatlov and the speaker on stochastic stability of Pollicott‐Ruelle resonances, another concept which will be introduced and explained. The talk is based on joint work with Zhongkai Tao.