A characterization of the blow-up time for the solution of a conservation law in several space variable

We derive a necessary and sufficient condition on the L^∞ Cauchy data for a conservation law in several space variables under which the solution will be locally Lipschitz continuous up to time T . The largesf such T is therefore the “blow-up” time. Roughly, our condition is that the data can be approximated by smoother functions satisfying uniformly a certain estimate. We present an example which shows that the existence of the approximations is crucial: it is not sufficient that the data itself satisfy this estimate.