We deal with solutions to transport equations
The averaging lemmas, , , , state that in the generic non-degenerate case, averaging over the velocity space, , yields a gain of spatial regularity. The prototype statement reads
Variants of the averaging lemmas were used by DiPerna and Lions to construct global weak (renormalized) solutions of Boltzmann, Vlasov-Maxwell and related kinetic systems, , ; in Bardos et. al., , averaging lemmas were used to construct solutions of the incompressible Navier-Stokes equations. We turn our attention to their use in the context of nonlinear conservation laws and related equations.
Proof.(Sketch). We shall sketch the proof in the particular case, p=q which will suffice to demonstrate the general case.
denote the set where the symbol is 'small',
and decompose the average, accordingly:
Here, represents the usual smooth partitioning relative to and its complement, . On , the symbol is 'bounded away' from zero, so we gain one derivative:
On - along the `non uniformly elliptic' rays, we have no gain of regularity, but instead, our non-degeneracy assumption implies that is a 'small' set and therefore
Both (2.2.12) and (2.2.13) are straightforward for p=2 and by estimating the corresponding multipliers, the case follows by interpolation. Finally, we consider the K-functional
The behavior of this functional, , characterize the smoothness of in the intermediate space between and : more precisely, belongs to Besov space with 'intermediate' smoothness of order .
Now set , then with appropriately scaled we find that with . This means that belongs to Besov space, and (2.2.9) (with p=q=r) follows.
In the limiting case of in (2.2.8),
one finds that if
then averaging is a compact mapping, . The case p=2 follows from Gèrard's results .