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The acoustic limit for the Boltzmann equation. (English) Zbl 0973.76075

First, the authors derive acoustic equations directly from the Boltzmann equation as a formal limit of moment equations for an appropriately scaled family of Boltzmann solutions. Then this limit is investigated for the Boltzmann equation considered over a periodic spatial domain for bounded collision kernels. Appropriately scaled families of Di Perma-Lions renormalized solutions are shown to have fluctuations that converge entropically to a unique limit described by a solution of acoustic equations for all time, provided that its initial fluctuations converge entropically to an limit associated to any given \(L^2\) initial data of acoustic equations. The associated local conservation laws are recovered in this limit.

MSC:

76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics
76A02 Foundations of fluid mechanics
45K05 Integro-partial differential equations
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