| Summary: | 博士 === 國立中央大學 === 數學研究所 === 100 === In this dissertation we consider the asymptotic behavior of solutions for regularized equations to some nonlinear hyperbolic balance laws arising from the following topics: the viscous gas flow through discontinuous nozzle, viscous traffic flow model, and the atmosphere hydrodynamic escape model. Through the dynamical system theory approach, we can transfer our steady-state problem into a singularly perturbed problem. By analyzing the system in different scales, we are able to construct the singular stationary wave solutions. By using the technique of geometric singular perturbations, we can show there exist true stationary solutions for our problems shadowing the singular stationary wave solutions. For some special degenerate singular solutions, we apply more advanced theory from geometric singular perturbation to prove the persistence of these solutions under the perturbation. Moreover, in the first topics, we introduce a new entropy condition to ensure the uniqueness of the stationary solutions, and in the second topics, we also analyze the stability of stationary wave solutions.
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