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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Hydrodynamic limits for kinetic equations and the diffusive approximation of radiative transport for acoustic waves
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by Manuel Portilheiro and Athanasios E. Tzavaras PDF
Trans. Amer. Math. Soc. 359 (2007), 529-565 Request permission

Abstract:

We consider a class of kinetic equations, equipped with a single conservation law, which generate $L^{1}$-contractions. We discuss the hydrodynamic limit to a scalar conservation law and the diffusive limit to a (possibly) degenerate parabolic equation. The limits are obtained in the “dissipative” sense, equivalent to the notion of entropy solutions for conservation laws, which permits the use of the perturbed test function method and allows for simple proofs. A general compactness framework is obtained for the diffusive scaling in $L^{1}$. The radiative transport equations, satisfied by the Wigner function for random acoustic waves, present such a kinetic model that is endowed with conservation of energy. The general theory is used to validate the diffusive approximation of the radiative transport equation.
References
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Additional Information
  • Manuel Portilheiro
  • Affiliation: Institute of Applied and Computational Mathematics, FORTH, 71 110 Heraklion, Crete, Greece
  • Address at time of publication: Departamento de Matemática, Faculdade de Ciências e Tecnologia da Universidade de Coimbra, Apartado 3008, 3001-454 Coimbra, Portugal
  • Email: portilhe@tem.uoc.gr, portilhe@mat.uc.pt
  • Athanasios E. Tzavaras
  • Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706 – and – Institute of Applied and Computational Mathematics, FORTH, 71110 Heraklion, Crete, Greece
  • Address at time of publication: Department of Mathematics, University of Maryland, College Park, Maryland 20742
  • Email: tzavaras@math.wisc.edu, tzavaras@math.umd.edu
  • Received by editor(s): September 30, 2004
  • Published electronically: September 19, 2006
  • © Copyright 2006 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 359 (2007), 529-565
  • MSC (2000): Primary 35L65, 78A40, 82C40
  • DOI: https://doi.org/10.1090/S0002-9947-06-04268-1
  • MathSciNet review: 2255185