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 , 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
index for time direction,
The entropy condition (2.3.17) with Kruzkov entropy pairs (), reads
This defines a family of non-negative measures, ,
Differentiate (2.3.18) w.r.t. v: one finds that the indicator function, , where
satisfies the transport equation,
which corresponds to (2.2.7) with . We now apply the averaging lemma with (s = q= 1, p = 2), which tells us that as asserted.
It follows that if the non-degeneracy condition (2.3.16) holds, then the family of approximate solutions 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, §.
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
translation invariant: a bootstrap argument yields an
improved regularity, ,
In particular, if the problem is nonlinear in the sense that the non-degeneracy condition (2.2.14) holds,
then the corresponding solution operator, , has a regularization effect mapping . This could be viewed as a multidimensional generalization for Tartar's regularization result for a.e. nonlinear one-dimensional fluxes, .
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'
has a linearized symbol which fails to satisfy the non-degeneracy/non-linearity condition (2.3.16), since it vanishes 's along . This corresponds to its persistence of oscillations along , which excludes compactness.
Example #2. We consider
In this case the linearized symbol is given by
; 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 .)
Example #3. Consider
For we obtain an index of non-degeneracy/non-linearity of order .