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Regularizing effect revisited (tex2html_wrap_inline6027)

In this section we resume our discussion on the regularization effect of nonlinear conservation laws. The averaging lemma enables us to identify the proper notion of 'nonlinearity' in the multivariate case, which guarantee compactness.

The following result, adapted from [22], is in the heart of matter.


Remark. Note that the bounded measure of entropy production in (2.3.17) need not be negative; general bounded measures will do.

Proof.To simplify notations, we use the customary tex2html_wrap_inline6033 index for time direction,
The entropy condition (2.3.17) with Kruzkov entropy pairs (gif), reads
This defines a family of non-negative measures, tex2html_wrap_inline6035,
Differentiate (2.3.18) w.r.t. v: one finds that the indicator function, tex2html_wrap_inline6039, where
satisfies the transport equation,
which corresponds to (2.2.7) with tex2html_wrap_inline6041 gif. We now apply the averaging lemma with (s = q= 1, p = 2), which tells us that tex2html_wrap_inline6045 as asserted. tex2html_wrap_inline6019

It follows that if the non-degeneracy condition (2.3.16) holds, then the family of approximate solutions tex2html_wrap_inline6049 is compact and strong convergence follows. In this context we refer to the convergence statement for measure-valued solutions for general multidimensional scalar conservation laws - approximate solutions measured by their nonpositive entropy production outlined in Lecture I, §gif.

Here, Theorem 2.3.1 yields even more, by quantifying the regularity of approximate solutions with bounded entropy productions in terms of the non-degeneracy condition (2.3.16). In fact, more can be said if the solution operator associated with tex2html_wrap_inline6049 is translation invariant: a bootstrap argument yields an improved regularity, [22],
In particular, if the problem is nonlinear in the sense that the non-degeneracy condition (2.2.14) holds,
then the corresponding solution operator, tex2html_wrap_inline6053, has a regularization effect mapping tex2html_wrap_inline6055. This could be viewed as a multidimensional generalization for Tartar's regularization result for a.e. nonlinear one-dimensional fluxes, tex2html_wrap_inline5941.

We continue with few multidimensional examples which illustrate the relation between the non-degeneracy condition, (2.3.16) and regularity.

Example #1. The 'two-dimensional Burgers' equation' (2.1.6),
has a linearized symbol tex2html_wrap_inline6059 which fails to satisfy the non-degeneracy/non-linearity condition (2.3.16), since it vanishes tex2html_wrap_inline6061's along tex2html_wrap_inline6063. This corresponds to its persistence of oscillations along tex2html_wrap_inline6065, which excludes compactness.

Example #2. We consider

In this case the linearized symbol is given by tex2html_wrap_inline6067; Here we have
(just consider the second-order touch-point at v=1). Hence, the solution operator associated with (2.3.23) is compact ( - in fact, mapping tex2html_wrap_inline6071.)

Example #3. Consider
For tex2html_wrap_inline6073 we obtain an index of non-degeneracy/non-linearity of order tex2html_wrap_inline6075.

next up previous contents
Next: Kinetic and other approximations Up: Kinetic Formulations and Regularity Previous: Velocity averaging lemmas ()

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