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Degenerate parabolic equations

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 tex2html_wrap_inline6049 be a family of approximate solutions of (2.2.7) with tex2html_wrap_inline6077-compact entropy production,
Then tex2html_wrap_inline6049 is compact in tex2html_wrap_inline6105, [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, tex2html_wrap_inline6111 on a set of non-zero measure, guarantees regularization, compactness ...

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


Example. Consider the isotropic equation
Here tex2html_wrap_inline6119 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, tex2html_wrap_inline6121, (2.4.34) holds with tex2html_wrap_inline6123 and one conclude a regularizing effect of order tex2html_wrap_inline6125, i.e., tex2html_wrap_inline6127.
A particular attractive advantage of the kinetic formulation in this case, is that it applies to non-isotropic problems as well.

Eitan Tadmor
Mon Dec 8 17:34:34 PST 1997