Quadratic systems of conservation laws with generic behavior at infinity

Abstract

We identify quadratic systems of conservation laws with “generic” behavior at infinity, where the genericity conditions derive naturally when studying weak solutions of conservation laws. Namely, we identify quadratic models for which the vector field associated with the viscosity admissibility criterion has properties at infinity that are true for an open and dense subset of the set of all planar quadratic vector fields in the metric associated with the Euclidian space of coefficients. We determine the boundaries of the regions containing, “generic” models in the parameter space of coefficients of quadratic models. We show that when crossing the boundaries of nongeneric models transversally, the Poincaré compactification of the corresponding vector field undergoes either a saddle node or a transcritical bifurcation at infinity. For quadratic models with a bounded elliptic region we calculate the loci of nongeneric models assuming the viscosity matrix to be the identity. We obtain a two-parameter normal form for such models and show that the boundaries that determine generic models in the two-dimensional parameter space correspond to the Schaeffer-Shearer classification of models with an isolated umbilic point. Since the loci of nongeneric models are invariant under the equivalence transformations that preserve weak solutions of conservation laws, understanding their behavior at infinity promises to provide an insight into a general classification of quadratic conservation laws.