Optimal enhanced dissipation for contact Anosov flows
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866911775999393792 |
|---|---|
| author | Tao, Zhongkai Zworski, Maciej |
| author_facet | Tao, Zhongkai Zworski, Maciej |
| contents | 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]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_01000 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Optimal enhanced dissipation for contact Anosov flows Tao, Zhongkai Zworski, Maciej Analysis of PDEs Dynamical Systems Spectral Theory 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]. |
| title | Optimal enhanced dissipation for contact Anosov flows |
| topic | Analysis of PDEs Dynamical Systems Spectral Theory |
| url | https://arxiv.org/abs/2311.01000 |