The Single Conservation Law

Abstract

In this chapter we shall obtain precise mathematical results on the existence and uniqueness of solutions for a single conservation law. In addition we shall also study the asymptotic behavior of our constructed solution. The existence problem will be attacked via a finite-difference method. Thus we shall replace the given differential equation by a finite-difference approximation depending on mesh parameters Ax and At. For every such pair (Δx, Δt) we shall construct a solution of the finite-difference equation, and we shall then obtain estimates which enable us to pass to the limit as the mesh parameters tend to zero in a certain definite way. The estimates which we obtain will be in the sup-norm and in the total variation-norm of the approximants, both sets of estimates being independent of the mesh parameters. It is worth noting that we are forced into obtaining bounds on the variation of the approximants, rather than (the usually encountered) bounds on derivatives, since the latter bounds would imply via the standard compactness criteria, that the limit would be continuous ; we know that this is not generally true.