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 , .
The case Q = 0 corresponds to our multidimensional discussion in §2.3.15; the case A =0 correspond possibly degenerate parabolic equations (consult  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 . 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.