Hyperbolic Systems of Balance Laws in Astrophysics and Biology Sciences

In this project, we consider some hyperbolic systems of balance laws arise fromastrophysics and biology. The first problem we study is the hydrodynamic escapeproblem, which is one of the most popular topics in astrophysics. The model isgoverned by the 3-dimensional compressible Euler equations with source termsrelated to the gravity and the heating effect (or the radiation) from the sun. We studythe existence of the positive entropy solutions in the spherically symmetricspace-times. The global existence result is established by the generalized Glimmscheme whose building block is invented as the new version of Riemann solver.The hydrodynamic regions where is characterized by the Knudsen number in fluiddynamics, is also provided by the detailed wave interaction estimates. We show suchgeneralized Glimm method can also be applied to the Euler-Poisson equations. In thesecond problem, we study the existence and behavior of traveling waves and pulsesfor two types of Keller-Segel equations with nonlinear interaction between cells andchemical signals. The result will be accomplished by the geometric singularperturbations in dynamical systems. Numerical simulations for both problems will beprovided to support the analytical results.