  We show that, for first-order systems of conservation laws with a strictly
convex entropy,in particular for the very simple so-called "inviscid" Burgers
equation,it is possible to address the Cauchy problem by a suitable convex
minimizationproblem, quite similar to some problems arising in optimal
transport or variational mean-field game theory.In the general case, we show
that smooth, shock-free, solutions can be recoveredon some sufficiently small
interval of time. In the special situation of the Burgers equation, we
furthershow that every "entropy solution" (in the sense of Kruzhkov)including
shocks, can be recovered, for arbitrarily long time intervals.
