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.

**Paper:**- Available as PDF (368 Kbytes),
**Author(s):**- Manuel Portilheiro
- Athanasios E. Tzavaras, <tzavaras@math.wisc.edu>
**Publishing information:****Comments:****Submitted by:**- <tzavaras@math.wisc.edu> December 12 2004.

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