Hydrodynamic limit for a Fokker-Planck equation with coefficients in Sobolev spaces

Ioannis Markou

doi:10.3934/nhm.2017028

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2017, Volume 12, Issue 4: 683-705. Doi: 10.3934/nhm.2017028

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Hydrodynamic limit for a Fokker-Planck equation with coefficients in Sobolev spaces

Ioannis Markou^1,2,

1.
Department of Mathematics, University of Maryland, College Park, MD 20742, USA
2.
Current address: Foundation for Research and Technology Hellas, Institute of Applied and Computational Mathematics, N. Plastira 100, Vassilika Vouton, Heraklion 70013, Greece

Received: June 30, 2016

Revised: January 31, 2017

Published: November 2017

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Abstract

In this paper we study the hydrodynamic (small mass approximation) limit of a Fokker-Planck equation. This equation arises in the kinetic description of the evolution of a particle system immersed in a viscous Stokes flow. We discuss two different methods of hydrodynamic convergence. The first method works with initial data in a weighted L² space and uses weak convergence and the extraction of convergent subsequences. The second uses entropic initial data and gives an L¹ convergence to the solution of the limit problem via the study of the relative entropy.

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Mathematics Subject Classification: Primary: 35Q70, 35Q84; Secondary: 35Q35.

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