As an example one can treat convective equations
together with (possibly degenerate) diffusive terms

Assume the problem is not linearly degenerate, in the sense that

Let be a family of approximate solutions of
(2.2.7) with -compact entropy production,

Then is compact in , [22].

The case *Q* = 0 corresponds to our multidimensional discussion
in §2.3.15;
the case *A* =0 correspond possibly degenerate parabolic equations
(consult [17] and the references therein, for example).
According
to (2.4.32), satisfying the ellipticity condition,
on a set of non-zero measure,
guarantees regularization, compactness ...

Again, a second-order version of the averaging lemma enables us to quantify the gained regularity which we state as

*Example*. Consider the isotropic equation

Here and the lemma 2.4.1
applies.
The kinetic formulation of such equations was studied in [17].
In the particular case of porous media equation,
, (2.4.34) holds with
and one conclude a regularizing
effect of order , i.e.,
.

A particular attractive advantage of the kinetic formulation in this case, is
that it applies to *non-isotropic* problems as well.

Mon Dec 8 17:34:34 PST 1997