On the piecewise smoothness of entropy solutions to scalar conservation laws

The behavior and structure of entropy solutions of scalar convex conservation laws are studied. It is well known that such entropy solutions 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 high order piecewise smoothness of the entropy solution. To this end we introduce an appropriate new C_n-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. We 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 piecewise smoothness which is assumed — sometime implicitly — in many finite-dimensional computations for such discontinuous problems.