English
Modal based hypocoercivity methods on the torus and the real line with application to Goldstein-Taylor models
Virtualconference
Auteurs : Arnold, Anton (Auteur de la Conférence)
CIRM (Editeur )
35B40 - Asymptotic behavior of solutions of PDE
35S05 - General theory of $\Psi$DO
82C40 - Kinetic theory of gases
35Q82 - PDEs in connection with statistical mechanics
Ressources complémentaires :
https://www.cirm-math.fr/RepOrga/2355/Slides/slide_Anton_ARNOLD.pdf
CIRM (Editeur )
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Résumé :
We are concerned with deriving sharp exponential decay estimates (i.e. with maximum rate and minimum multiplicative constant) for linear, hypocoercive evolution equations. Using a modal decomposition of the model allows to assemble a Lyapunov functional using Lyapunov matrix inequalities for each Fourier mode.
We shall illustrate the approach on the 1D Goldstein-Taylor model, a2-velocity transport-relaxation equation. On the torus the lowest Fourier modes determine the spectral gap of the whole equation in $L^{2}$. By contrast, on the whole real line the Goldstein-Taylor model does not have a spectral gap, since the decay rate of the Fourier modes approaches zero in the small mode limit. Hence, the decay is reduced to algebraic.
In the final part of the talk we consider the Goldstein-Taylor model with non-constant relaxation rate, which is hence not amenable to a modal decomposition. In this case we construct a Lyapunov functional of pseudodifferential nature, one that is motivated by the modal analysis in the constant case.The robustness of this approach is illustrated on a multi-velocity GoldsteinTaylor model, yielding explicit rates of convergence to the equilibrium.
This is joint work with J. Dolbeault, A. Einav, C. Schmeiser, B. Signorello, and T. Wöhrer.
Keywords : BGK equation; hypocoercivity; large time behaviour; exponential decay; Lyapunov functional
Codes MSC : We shall illustrate the approach on the 1D Goldstein-Taylor model, a2-velocity transport-relaxation equation. On the torus the lowest Fourier modes determine the spectral gap of the whole equation in $L^{2}$. By contrast, on the whole real line the Goldstein-Taylor model does not have a spectral gap, since the decay rate of the Fourier modes approaches zero in the small mode limit. Hence, the decay is reduced to algebraic.
In the final part of the talk we consider the Goldstein-Taylor model with non-constant relaxation rate, which is hence not amenable to a modal decomposition. In this case we construct a Lyapunov functional of pseudodifferential nature, one that is motivated by the modal analysis in the constant case.The robustness of this approach is illustrated on a multi-velocity GoldsteinTaylor model, yielding explicit rates of convergence to the equilibrium.
This is joint work with J. Dolbeault, A. Einav, C. Schmeiser, B. Signorello, and T. Wöhrer.
Keywords : BGK equation; hypocoercivity; large time behaviour; exponential decay; Lyapunov functional
35B40 - Asymptotic behavior of solutions of PDE
35S05 - General theory of $\Psi$DO
82C40 - Kinetic theory of gases
35Q82 - PDEs in connection with statistical mechanics
Ressources complémentaires :
https://www.cirm-math.fr/RepOrga/2355/Slides/slide_Anton_ARNOLD.pdf
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Informations sur la Vidéo
Réalisateur : Hennenfent, GuillaumeLangue : Anglais Date de publication : 09/04/2021 Date de captation : 22/03/2021 Sous collection : Research talks arXiv category : Analysis of PDEs Domaine : Analysis and its Applications ; PDE Format : MP4 (.mp4) - HD Durée : 00:40:31 Audience : Researchers Download : https://videos.cirm-math.fr/2021-03-22_Arnold.mp4 
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Informations sur la Rencontre
Nom de la rencontre : Jean Morlet Chair 2021- Conference: Kinetic Equations: From Modeling Computation to Analysis / Chaire Jean-Morlet 2021 - Conférence : Equations cinétiques : Modélisation, Simulation et AnalyseOrganisateurs de la rencontre : Bostan, Mihaï ; Jin, Shi ; Mehrenberger, Michel ; Montibeller, Celine Dates : 22/03/2021 - 26/03/2021 Année de la rencontre : 2021 URL Congrès : https://www.chairejeanmorlet.com/2355.html
Données de citation
DOI : 10.24350/CIRM.V.19733203Citer cette vidéo: Arnold, Anton (2021). Modal based hypocoercivity methods on the torus and the real line with application to Goldstein-Taylor models. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19733203 URI : http://dx.doi.org/10.24350/CIRM.V.19733203 |
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Bibliographie
- ARNOLD, Anton, EINAV, Amit, SIGNORELLO, Beatrice, et al. Large time convergence of the non-homogeneous Goldstein-Taylor Equation. Journal of Statistical Physics, 2021, vol. 182, no 2, p. 1-35. - https://doi.org/10.1007/s10955-021-02702-8
- ARNOLD, Anton, DOLBEAULT, Jean, SCHMEISER, Christian, et al. Sharpening of decay rates in Fourier based hypocoercivity methods. arXiv preprint arXiv:2012.09103, 2020. - https://arxiv.org/abs/2012.09103
