Topological Resolution of Riemann Problems for Pairs of Conservation Laws
Frederico Palmeira | Marchesin, Dan | Azevedo, Arthur | Eschenazi, César
Conservation laws | Riemann problem | Wave curves | Wave manifold | Intermediate surface
The structure of Riemann solutions for certain systems of conservation laws can be so complicated that the classical constructions are unable to establish their global existence and stability. For systems of two conservation laws, classically the local solution is found by intersecting two wave curves specified by the Riemann data. The intersection point represents the intermediate constant state that typically appears in such solutions. In this paper, we construct the wave curves in a three dimensional manifold, which is globally foliated by shock curves, and where rarefaction and composite curves are naturally defined. The main innovation in this paper is the construction of a differentiable two-dimensional manifold of intermediate states; the local classical construction is replaced by finding the intersection of a wave curve with the intermediate manifold; a transversality argument guarantees the stability of the Riemann solution. The new construction has the potential of establishing structural stability properties globally, i.e., for all initial data. This is its main advantage over the classical construction, which is intrinsically local. The construction presented in this work can be illustrated for a family of quadratic polynomial flux functions and their perturbations.