On the piecewise smoothness of entropy solutions to scalar conservation laws

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Abstract
The behavior and structure of entropy solutions of scalar convex conser- vation laws are studied. It is well kn0a.n that such entropy solutioris consist of at most countable number of C1-smooth regions. We obtain new upper. bounds on the higher order derivatives of the entropy solution in any one of its C1-smoothness regions. These bounds enable us to measure the hzgh order piecewise smoothness of the entropy solution. To this end we introduce an appropriate new Cn-semi norm - localized to the smooth part of the entropy solution. and we show that the entropy solution is stable with respect to this norm. \Ye also address the question regarding the number of C1-smoothness pieces, we show that if the initial speed has a finite number of decreasing inflection points then it bounds the number of future shock discontinuities. Loosely speaking this says that in the case of such generic initial data the entropy solution consists of a finite number of smooth pieces, each of which is as smooth as the data permits. It is this type of pzecewise smoothness which is assumed - sometime implicitly - in many finite-dimensional computations for such discontinuous problems.

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