Abstract: We introduce a new nonclassical Riemann solver for scalar conservation laws with concave-convex flux-function. This solver is based on both a kinetic relation, which determines the propagation speed of (undercompressive) nonclassical shock waves, and a nucleation criterion, which makes a choice between a classical Riemann solution and a nonclassical one. We establish the existence of (nonclassical entropy) solutions of the Cauchy problem and discuss several examples of wave interactions. We also show the existence of a class of solutions, called splitting-merging solutions, which are made of two large shocks and small BV (bounded variation) perturbations. The nucleation solvers, as we call them, are applied to (and actually motivated by) the theory of thin film flows; they help explain numerical results observed for such flows.
Conservation Laws Preprint Server <email@example.com> Last modified: Fri Dec 31 17:08:23 MET 2004