Optimal enhanced dissipation for contact Anosov flows

Abstract

We show that for a contact Anosov flow on a compact manifold $ M $, the solutions to $ \partial_t u + X u = \nu \Delta u $, $ \nu > 0 $, where $ X $ is the generator of the flow and $ \Delta $, a (negative) Laplacian for some Riemannian metric on $ M $, satisfy \[ \| u ( t ) - \underline u \|_{L^2 ( M) } \leq C \nu^{-K} e^{ - \beta t } \| u( 0 ) \|_{L^2 ( M) }, \] where $ \underline u $ is the (conserved) average of $ u (0) $ with respect to the contact volume form, and $K$, $\beta$ are fixed positive constants. Since our class of flows includes geodesic flows on manifolds of negative curvature, this provides many examples of very precise optimal enhanced dissipation in the sense of [arXiv:1911.01561] and [arXiv:2304.05374]. The proof is based on results about stochastic stability of Pollicott--Ruelle resonances [arXiv:1407.8531].