The Study of Vacuum States in Nonlinear Hyperbolic Balance Laws in Sciences and Engineering(2/2)

The study of vacuum solutions to hyperbolic balance laws has been an important and challenging subject in nonlinear hyperbolic PDEs. In this project, we study the vacuum solutions of nonlinear hyperbolic systems of balance laws arising in sciences and engineering. The existence and behavior of approximate vacuum states to the compressible Euler equations are investigated. We construct the generalized solutions of the regularized Riemann problem by the method of characteristics and the modified Lax method. Using this kind of solutions as building blocks of the generalized Glimm scheme, we are able to establish the global existence of solutions near vacuum. This project includes the following three parts: 1.[Traffic Flow] We invent a new version of the multilane model near vacuum, which is equivalent to Aw—Rascle model together with discontinuous source terms. The global solutions of the generalized Riemann problem and Cauchy problem for the leading-order equations are obtained by the generalized Glimm—Temple method. The solutions are used to interpret realistic traffic phenomena. 2.[Transonic Nozzle Flow] We study the global existence and behavior of transonic solutions consisting of vacuum states in the variable area ducts. The stability of regularized Riemann problems is studied via the method of characteristics and integral factors. The global existence are established by the modified random choice method coupled with the operating-splitting method. 3.[Compressible Euler Equations] We obtain L^1 convergences and convergence rates of solutions near vacuum for both rarefaction wave and shock wave. An a priori estimate together with the iteration to the sequence of approximate solutions provides the uniform bounds of the partial derivatives of Riemann invariants. In addition, by using the convexity of the characteristic curves and the previous arguments, we are able to obtain the result. This project is closely related to the perturbation theory to hyperbolic systems of balance laws, which is still an open subject to investigate. Therefore, it is worth discussing in the future project. The research of this project will help to build up the essential results for the perturbation theory of hyperbolic PDEs.