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1 Kinetic Equations: From Newton to Boltzmann
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stillmissing.Forexample,theexistenceofsolutions,globallydefinedforpositive
time, for all nonnegative arbitrarily large initial data with finite mass, energy,
moment of inertia and entropy has been proven only in 1989 in an important
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paper of Ron DiPerna and Pierre-Louis Lions [9].
These solutions are obtained by a so-called renormalisation procedure of
the Boltzmann equation dealing effectively with the quadratic nature of the
collision integral. They are rather weak, so that neither their uniqueness nor
their conservation of energy is ascertained! The lack of smoothness of the
renormalized solutions of the Boltzmann equation is also a major obstacle
in the quest of the mathematically rigorous justification of its small Knudsen
number limits, although in recent years a lot of progress has been made in this
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area, see [1] and consecutive papers by the same authors .
Clearly, the gas dynamics Boltzmann equation is still in the core of mod-
ern kinetic theory. However, other applications of Boltzmann-type equations

Fig. 1.6. Stratocumulus stratiformis translucidus (background) and cumulus (foreground)

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http://www-groups.dcs.st-and.ac.uk/∼history/Mathematicians/Lions.html
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check http://www.dma.ens.fr/∼golse/Publications/pubs.html for alistofreferences