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

Ioannis Markou; Ioannis Markou

doi:10.3934/nhm.2017028

Networks and Heterogeneous Media

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: 01 July 2016 Revised: 01 February 2017
Primary: 35Q70, 35Q84; Secondary: 35Q35

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.
- Hydrodynamic limit,
- Fokker-Planck equation,
- weak compactness,
- relative entropy
Citation: Ioannis Markou. Hydrodynamic limit for a Fokker-Planck equation with coefficients in Sobolev spaces[J]. Networks and Heterogeneous Media, 2017, 12(4): 683-705. doi: 10.3934/nhm.2017028

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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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