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## Kinetic and other approximations

Theorem 2.3.1 provides an alternative route to analyze the convergence of general entropy stable multi-dimensional schemes, schemes whose convergence proof was previously accomplished by measure-valued arguments; here we refer to finite-difference, finite-volume, streamline-diffusion and spectral approximations ..., which were studied in [4, 18, 19, 15, 16, 3]. Indeed, the feature in the convergence proof of all these methods is the -compact entropy production, (2.4.33). Hence, if the underlying conservation law satisfies the non-linear degeneracy condition (2.3.22), then the corresponding family of approximate solutions, becomes compact. Moreover, if the entropy production is bounded measure, then there is actually a gain of regularity indicated in Theorem 2.3.1 and respectively, in (2.3.21) for the translation invariant case. Remark. Note that unlike the requirement for a nonpositive entropy production from measure-valued solutions (consult () in Lecture I), here we allow for an arbitrary bounded measure.

So far we have not addressed explicitly a kinetic formulation of the multidimensional conservation law (2.3.15). The study of regularizing effect for multidimensional conservation laws was originally carried out in [22] for the approximate solution constructed by the following BGK-like model, [28] (see also [2],[14]),

Here, denotes the `pseudo-Maxwellian',

which is associated with the average of ,

The key property of this kinetic approximation is the existence of a nonnegative measure, such that (The existence of such measures proved in [22] and is related to H-functions studied in [28] and Brenier's lemma [2].) Thus, we may rewrite (2.3.25) in the form

Let be an entropy pair associated with (2.3.15). Integration of (2.3.29) against implies that the corresponding macroscopic averages, , satisfy

Thus, the entropy production in this case is nonpositive and hence a bounded measure, so that Theorem 2.3.1 applies. Viewed as a measure-valued solution, convergence follows along DiPerna's theory [8]. If, moreover, the nondegeneracy condition (2.3.16) holds, then we can further quantify the -regularity (of order .)

Theorem 2.3.1 offers a further generalization beyond the original, 'kineticly' motivated discussion in [22]. Indeed, consideration of Theorem 2.3.1 reveals the intimate connection between the macroscopic assumption of bounded entropy production in (2.3.17), and an underlying kinetic formulation ({2.3.20), analogous to (2.3.29). For a recent application of the regularizing effect for a convergence study of finite-volume schemes along these lines we refer to [24].

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