A dynamic Laplacian for identifying Lagrangian coherent structures on weighted Riemannian manifolds. (English) Zbl 1477.37092
The mathematics of transport and mixing in nonlinear dynamical systems has received considerable attention, driven in part by applications in fluid dynamics, atmospheric and ocean dynamics, molecular dynamics, granular flow, and other biological and engineering processes. In this work the authors extend the results of the first author [Nonlinearity 28, No. 10, 3587–3622 (2015; Zbl 1352.37063)] in three ways: (1) to dynamics that is not volume-preserving, (2) to track the transport of non-uniformly distributed tracers, and (3) to dynamics on curved manifolds.
The work is very deep and contains very well structured technical developments. The paper starts with relevant background material from differential geometry. This is followed by the description of the dynamic isoperimetric problem on weighted manifolds and the Federer-Fleming theorem is stated. Finally, the authors provide details about the dynamic Laplace operator on weighted manifolds and they state the dynamic Cheeger inequality and the main convergence result. A section is then devoted to numerical experiments. The proofs are in the appendices.
The work is very deep and contains very well structured technical developments. The paper starts with relevant background material from differential geometry. This is followed by the description of the dynamic isoperimetric problem on weighted manifolds and the Federer-Fleming theorem is stated. Finally, the authors provide details about the dynamic Laplace operator on weighted manifolds and they state the dynamic Cheeger inequality and the main convergence result. A section is then devoted to numerical experiments. The proofs are in the appendices.
Reviewer: Luis Vazquez (Madrid)
MSC:
| 37M99 | Approximation methods and numerical treatment of dynamical systems |
| 37A25 | Ergodicity, mixing, rates of mixing |
| 53C21 | Methods of global Riemannian geometry, including PDE methods; curvature restrictions |
| 53A10 | Minimal surfaces in differential geometry, surfaces with prescribed mean curvature |
| 58J50 | Spectral problems; spectral geometry; scattering theory on manifolds |
| 76F25 | Turbulent transport, mixing |
| 35Q49 | Transport equations |
Keywords:
Lagrangian coherent structure; dynamic Laplace operator; weighted Riemannian manifold; transfer operator; finite-time coherent set; dynamic isoperimetric problemCitations:
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