Nonlinear Hyperbolic Resonant Systems of Balance Laws(2/2)

In this project we deal with the following two topics. The first one is to study the global in time existence and behavior of composite waves of resonant hyperbolic systems of balance laws. The entirely resonant hyperbolic systems have the property that all the eigenvalues of Jacobian matrix of a flux are coincided in the whole phase domain. We give an example of a weak solution with vacuum for the classical Riemann problem of some entirely resonant system to indicate that the self-similar Riemann solutions are not an appropriate building block of Glimm scheme to the Cauchy problem of such systems. Instead, we invent a generalized Riemann problem with regularized Riemann data. The weak solutions of such generalized Riemann problem consists of constant states separated by the composite hyperbolic wave, which are the combination of nonlinear hyperbolic waves and contact discontinuities. Such composite waves have reasonable values of total variations so that the generalized Glimm scheme can be applied to establish the global existence of weak solutions for the entirely resonant systems. For the generalized Glimm scheme to the Cauchy problem of our system, we impose a justified C-F-L condition. Modified wave interaction estimates are provided for the stability of scheme. The results of this paper indicate that the resonance of solutions provides an effect of singularity which cannot be coupled with the singularity of Riemann data. It means that the generalized Riemann solutions with composite hyperbolic waves provide a more appropriate building block for the generalized Glimm scheme to entirely resonant system.Secondly, we consider a multilane model of traffic flow, which is governed by a hyperbolic system of balance laws. The system of balance laws is given as a 2 by 2 nonlinear hyperbolic system with a discontinuous source term. The global existence of entropy solutions to the Cauchy problem of this multi-lanes model is established by a new version of the generalized Glimm method. The generalized solutions of the Riemann problem, which is the building block of the generalized Glimm scheme, are constructed by Lax's method and an invention of perturbations solving linearized hyperbolic equations with modified source terms. The residuals are estimated for the consistency of the generalized Glimm scheme. The wave interaction estimates are provided for the decay of Glimm functionals and the result of the asymptotic behavior of solutions.