Long-lasting diffusive solutions for systems of conservation laws (Q2713029)

Systems of two nonlinear conservation laws regularized by viscous or diffusive second-order terms have traveling wave solutions that are attractors for the long-time behavior of initial value problems. The associated ODE system, a planar vector field depending on parameters such as the wave speed, can exhibit complicated global structure. As part of a program to relate aspects of this structure back to the PDEs, the authors consider homoclinic cycles, 2-cycles and 3-cycles. The corresponding traveling waves, or collections of traveling waves, are combined with rarefaction waves to understand long-time behavior of solutions of initial value problems that collapse to Riemann problems in the limit of zero viscosity. Two appendices discuss the occurrence of cycles, and bifurcations that give rise to them.