Quantitative error estimates for the large friction limit of Vlasov equation with nonlocal forces

José A. Carrillo; Young Pil Choi

doi:10.1016/j.anihpc.2020.02.001

Quantitative error estimates for the large friction limit of Vlasov equation with nonlocal forces

José A. Carrillo, Young Pil Choi

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

12 Citations (Scopus)

Abstract

We study an asymptotic limit of Vlasov type equation with nonlocal interaction forces where the friction terms are dominant. We provide a quantitative estimate of this large friction limit from the kinetic equation to a continuity type equation with a nonlocal velocity field, the so-called aggregation equation, by employing 2-Wasserstein distance. By introducing an intermediate system, given by the pressureless Euler equations with nonlocal forces, we can quantify the error between the spatial densities of the kinetic equation and the pressureless Euler system by means of relative entropy type arguments combined with the 2-Wasserstein distance. This together with the quantitative error estimate between the pressureless Euler system and the aggregation equation in 2-Wasserstein distance in [Commun. Math. Phys, 365, (2019), 329–361] establishes the quantitative bounds on the error between the kinetic equation and the aggregation equation.

Original language	English
Pages (from-to)	925-954
Number of pages	30
Journal	Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire
Volume	37
Issue number	4
DOIs	https://doi.org/10.1016/j.anihpc.2020.02.001
Publication status	Published - 2020 Jul 1

Bibliographical note

Funding Information:
JAC was partially supported by the EPSRC grant number EP/P031587/1 . YPC was supported by NRF grant (No. 2017R1C1B2012918 and 2017R1A4A1014735 ) and POSCO Science Fellowship of POSCO TJ Park Foundation .

Funding Information:
JAC was partially supported by the EPSRC grant number EP/P031587/1. YPC was supported by NRF grant (No. 2017R1C1B2012918 and 2017R1A4A1014735) and POSCO Science Fellowship of POSCO TJ Park Foundation.

Publisher Copyright:
© 2020 Elsevier Masson SAS

All Science Journal Classification (ASJC) codes

Analysis
Mathematical Physics
Applied Mathematics

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10.1016/j.anihpc.2020.02.001

Cite this

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title = "Quantitative error estimates for the large friction limit of Vlasov equation with nonlocal forces",

abstract = "We study an asymptotic limit of Vlasov type equation with nonlocal interaction forces where the friction terms are dominant. We provide a quantitative estimate of this large friction limit from the kinetic equation to a continuity type equation with a nonlocal velocity field, the so-called aggregation equation, by employing 2-Wasserstein distance. By introducing an intermediate system, given by the pressureless Euler equations with nonlocal forces, we can quantify the error between the spatial densities of the kinetic equation and the pressureless Euler system by means of relative entropy type arguments combined with the 2-Wasserstein distance. This together with the quantitative error estimate between the pressureless Euler system and the aggregation equation in 2-Wasserstein distance in [Commun. Math. Phys, 365, (2019), 329–361] establishes the quantitative bounds on the error between the kinetic equation and the aggregation equation.",

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note = "Funding Information: JAC was partially supported by the EPSRC grant number EP/P031587/1 . YPC was supported by NRF grant (No. 2017R1C1B2012918 and 2017R1A4A1014735 ) and POSCO Science Fellowship of POSCO TJ Park Foundation . Funding Information: JAC was partially supported by the EPSRC grant number EP/P031587/1. YPC was supported by NRF grant (No. 2017R1C1B2012918 and 2017R1A4A1014735) and POSCO Science Fellowship of POSCO TJ Park Foundation. Publisher Copyright: {\textcopyright} 2020 Elsevier Masson SAS",

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N2 - We study an asymptotic limit of Vlasov type equation with nonlocal interaction forces where the friction terms are dominant. We provide a quantitative estimate of this large friction limit from the kinetic equation to a continuity type equation with a nonlocal velocity field, the so-called aggregation equation, by employing 2-Wasserstein distance. By introducing an intermediate system, given by the pressureless Euler equations with nonlocal forces, we can quantify the error between the spatial densities of the kinetic equation and the pressureless Euler system by means of relative entropy type arguments combined with the 2-Wasserstein distance. This together with the quantitative error estimate between the pressureless Euler system and the aggregation equation in 2-Wasserstein distance in [Commun. Math. Phys, 365, (2019), 329–361] establishes the quantitative bounds on the error between the kinetic equation and the aggregation equation.

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Quantitative error estimates for the large friction limit of Vlasov equation with nonlocal forces

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