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Numerical hypocoercivity for the Kolmogorov equation


Authors: Alessio Porretta and Enrique Zuazua
Journal: Math. Comp. 86 (2017), 97-119
MSC (2010): Primary 65N06; Secondary 35L02, 35B40, 35Q84
DOI: https://doi.org/10.1090/mcom/3157
Published electronically: May 25, 2016
MathSciNet review: 3557795
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Abstract: We prove that a finite-difference centered approximation for the Kolmogorov equation in the whole space preserves the decay properties of continuous solutions as $ t \to \infty $, independently of the mesh-size parameters. This is a manifestation of the property of numerical hypo-coercivity, and it holds both for semi-discrete and fully discrete approximations. The method of proof is based on the energy methods developed by Herau and Villani, employing well-balanced Lyapunov functionals mixing different energies, suitably weighted and equilibrated by multiplicative powers in time. The decreasing character of this Lyapunov functional leads to the optimal decay of the $ L^2$-norms of solutions and partial derivatives, which are of different order because of the anisotropy of the model.


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Additional Information

Alessio Porretta
Affiliation: Dipartimento di Matematica, Università di Roma Tor Vergata, Via della Ricerca Scientifica 1, 00133 Roma, Italy
Email: porretta@mat.uniroma2.it

Enrique Zuazua
Affiliation: Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain
Email: enrique.zuazua@uam.es

DOI: https://doi.org/10.1090/mcom/3157
Received by editor(s): January 2, 2015
Published electronically: May 25, 2016
Article copyright: © Copyright 2016 American Mathematical Society