Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Suitable weak solutions and low stratification singular limit for a fluid particle interaction model

Authors: Joshua Ballew and Konstantina Trivisa
Journal: Quart. Appl. Math. 70 (2012), 469-494
MSC (2010): Primary 54C40, 14E20; Secondary 46E25, 20C20
DOI: https://doi.org/10.1090/S0033-569X-2012-01310-2
Published electronically: May 10, 2012
MathSciNet review: 2986131
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Abstract: This article deals with a fluid-particle interaction model for the evolution of particles dispersed in a viscous compressible fluid within the physical space $ \Omega \subset \mathbb{R}^3.$ The system is expressed by the continuity equation, the balance of momentum and the so-called Smoluchowski equation for the evolution of particles. The coupling between the dispersed and dense phases is obtained through the drag forces that the fluid and the particles exert mutually by the action-reaction principle. Using the relative entropy method of Dafermos and DiPerna, the global-in-time existence of suitable weak solutions is presented under reasonable physical assumptions on the initial data, the physical domain, and the external potential. In addition, the low Mach number and low stratification limits of the system are established rigorously for both bounded and unbounded domains.

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Additional Information

Joshua Ballew
Affiliation: Department of Mathematics, University of Maryland, College Park, Maryland 20742-4105
Email: jballew@math.umd.edu

Konstantina Trivisa
Affiliation: Department of Mathematics & Institute for Physical Science and Technology, University of Maryland, College Park, Maryland 20742-4105
Email: trivisa@math.umd.edu

DOI: https://doi.org/10.1090/S0033-569X-2012-01310-2
Keywords: Fluid-particle interaction, relative entropy, suitable solutions, low Mach number, low stratification singular limit, compressible and viscous fluid.
Received by editor(s): December 23, 2011
Published electronically: May 10, 2012
Additional Notes: The first author was supported in part by the John Osborn Memorial Summer Fellowship and the National Science Foundation under the grants DMS 0807815 and DMS 1109397.
The second author acknowledges the support by the National Science Foundation under the awards DMS 0807815 and DMS 1109397.
Dedicated: Dedicated to Constantine M. Dafermos on the occasion of his 70th birthday
Article copyright: © Copyright 2012 Brown University
The copyright for this article reverts to public domain 28 years after publication.

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