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On the Oberbeck–Boussinesq Approximation on Unbounded Domains

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Nonlinear Partial Differential Equations

Part of the book series: Abel Symposia ((ABEL,volume 7))

Abstract

We study the Oberbeck–Boussinesq approximation describing the motion of an incompressible, heat-conducting fluid occupying a general unbounded domain in R 3. We provide a rigorous justification of the model by means of scale analysis of the full Navier–Stokes–Fourier system in the low Mach and Froude number regime on large domains, the diameter of which is proportional to the speed of sound. Finally, we show that the total energy of any solution of the resulting Oberbeck–Boussinesq system tends to zero with growing time.

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Acknowledgements

The work of E. Feireisl was supported by Grant 201/09/0917 of GA ČR as a part of the general research programme of the Academy of Sciences of the Czech Republic, Institutional Research Plan AV0Z10190503. The work of M. Schonbek was partially supported by NSF Grant DMS-0900909.

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Authors and Affiliations

  1. Institute of Mathematics of the Academy of Sciences of the Czech Republic, Žitná 25, 115 67, Praha 1, Czech Republic

    Eduard Feireisl

  2. Department of Mathematics, University of California at Santa Cruz, Santa Cruz, CA, 95064, USA

    Maria E. Schonbek

Authors
  1. Eduard Feireisl
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  2. Maria E. Schonbek
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Correspondence to Eduard Feireisl .

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Editors and Affiliations

  1. of Science and Technology, Dept. of Mathematics, Norwegian University, Trondheim, 7491, Norway

    Helge Holden

  2. Centre of Mathematics for Applications, University of Oslo, Oslo, 0316, Norway

    Kenneth H. Karlsen

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Feireisl, E., Schonbek, M.E. (2012). On the Oberbeck–Boussinesq Approximation on Unbounded Domains. In: Holden, H., Karlsen, K. (eds) Nonlinear Partial Differential Equations. Abel Symposia, vol 7. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-25361-4_7

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