  Conservation laws are usually studied in the context of sufficient regularity
conditions imposed on the flux function, usually $C^{2}$ and uniform convexity.
Some results are proven with the aid of variational methods and a unique
minimizer such as Hopf-Lax and Lax-Oleinik. We show that many of these
classical results can be extended to a flux function that is not necessarily
smooth or uniformly or strictly convex. Although uniqueness a.e. of the
minimizer will generally no longer hold, by considering the greatest (or
supremum, where applicable) of all possible minimizers, we can successfully
extend the results. One specific nonlinear case is that of a piecewise linear
flux function, for which we prove existence and uniqueness results. We also
approximate it by a smoothed, superlinearized version parameterized by
$\varepsilon$ and consider the characterization of the minimizers for the
smooth version and limiting behavior as $\varepsilon\downarrow0$ to that of the
sharp, polygonal problem. In proving a key result for the solution in terms of
the value of the initial condition, we provide a stepping stone to analyzing
the system under stochastic processes, which will be explored further in a
future paper.
