The rectifiability of entropy measures in one space dimension.
This paper deals with entropy solutions to one-dimensional scalar conservation laws \(\partial_t u+\partial_x(f(u))= 0\) and for arbitrary measurable bounded data. The flux function \(f\in C^2\) and the set \(\{x: f''(x)= 0\}\) are assumed to be locally finite. The singular set (shock waves) of the entropy solutions is contained in a countable union of \(C^1\) curves, and \({\mathcal H}^1\) amost everywhere along these curves the solution \(u\) possesses left and right approximate limits \(u_{\pm}\). Here the symbol \({\mathcal H}^1\) denotes the one-dimensional Hausdorff measure. Moreover, it is proved that for any smooth entropy-entropy flux pair \((\eta,q)\) the entropy production is concentrated on the shock waves and can be computed in terms of \(u_{\pm}\). The solution \(u\) is approximately continuous \({\mathcal H}^1\) almost everywhere outside the above-mentioned union of curves. On the other hand, it is well known that for strictly convex fluxes the entropy solution is continuous outside the shocks.
scientific article; zbMATH DE number 2012740