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 = νΔu $, $ ν> 0 $, where $ X $ is the generator of the flow and $ Δ$, a (negative) Laplacian for some Riemannian metric on $ M $, satisfy \[ \| u ( t ) - \underline u \|_{L^2 ( M) } \leq C ν^{-K} e^{ - βt } \| u( 0 ) \|_{L^2 ( M) }, \] where $ \underline u $ is the (conserved) average of $ u (0) $ with respect to the contact volume form, and $K$, $β$ 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].

Figures (2)

• Figure 1: An illustration of an exponentially mixing contact flow: the geodesic flow on $S^* \Sigma$ where $\Sigma = \Gamma \backslash D ( 0, 1 )$ is the Bolza surface, a genus two surface of constant negative curvature; $z \in D ( 0 , 1 )$ (the Poincaré disc) is the variable in the fundamental domain of $\Gamma$ and $e^{ i \theta }$ is the direction of the geodesic. The figures show the evolution of a neighbourhood of $z = 0$, $\theta = \pi/5$ (approximated by $10^5$ uniformly distributed points) under the flow at times $t = 5, 8$. The flow is periodic ${}\! \! \! \! \mod \! \Gamma$ in $z$ and ${} \!\!\ \!\! \! \! \mod \!2 \pi$ in $\theta$ -- see https://math.berkeley.edu/ zworski/bolzamix.mp4 for an animation. For the flow in the base see https://rb.gy/xfssmg. We are grateful to Semyon Dyatlov for help in producing these figures and movies.
• Figure 2: The contour of deformation used in the proof of the main theorem.

Theorems & Definitions (2)

• proof : Proof of \ref{['e:gap_res']}
• proof : Proof of \ref{['e:resolventbound']}