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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Hydrodynamic limits for kinetic equations and the diffusive approximation of radiative transport for acoustic waves


Authors: Manuel Portilheiro and Athanasios E. Tzavaras
Journal: Trans. Amer. Math. Soc. 359 (2007), 529-565
MSC (2000): Primary 35L65, 78A40, 82C40
Published electronically: September 19, 2006
MathSciNet review: 2255185
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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.


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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

DOI: http://dx.doi.org/10.1090/S0002-9947-06-04268-1
PII: S 0002-9947(06)04268-1
Keywords: Diffusive limit, radiative transport
Received by editor(s): September 30, 2004
Published electronically: September 19, 2006
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.