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Abstract
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We introduce a new approach to velocity-averaging lemmas in kinetic theory. This
approach — based upon the classical energy method — provides a powerful duality
principle in kinetic transport equations which allows for a natural extension of
classical averaging lemmas to previously unknown cases where the density and the
source term belong to dual spaces. More generally, this kinetic duality principle
produces regularity results where one can trade a loss of regularity or integrability
somewhere in the kinetic transport equation for a suitable opposite gain elsewhere.
Also, it looks simpler and more robust to rely on proving inequalities instead of
constructing exact parametrices.
The results in this article are introduced from a functional analytic point of view. They
are motivated by the abstract regularity theory of kinetic transport equations. However,
recall that velocity-averaging lemmas have profound implications in kinetic theory
and its related physical models. In particular, the precise formulation of such results has
the potential to lead to important applications to the regularity of renormalizations of
Boltzmann-type equations, as well as kinetic formulations of gas dynamics, for instance.
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Keywords
kinetic transport equation, averaging lemmas, energy
method, duality principle, maximal regularity,
hypoellipticity, dispersion
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Mathematical Subject Classification
Primary: 35B65, 35Q20, 35Q49, 35Q83, 82C40
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Milestones
Received: 29 June 2020
Revised: 16 January 2021
Accepted: 23 February 2021
Published: 31 July 2021
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