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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
DOI: https://doi.org/10.1090/S0002-9947-06-04268-1
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

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.