Flux Perturbation To The Relativistic Euler Equations

The flux-perturbation problem of two classes of equations in relativistic fluids are studied in this dissertation. The first one is the classical relativistic Euler equa-tions (CREE) modeling the conservation of energy and momentum, and the other one, the isentropic relativistic Euler equations (IREE) governing the conservation of baryon numbers and momentum. The Riemann problem of the pressureless rel-ativistic Euler equations (PREE) with a flux perturbation is firstly solved, and two kinds of interesting U-shaped pseudo-vacuum states and parameterized delta-shock solutions are discovered. Then it is shown that, as the flux perturbation vanishes, the limits of the family of parameterized delta-shock and U-shaped pseudo-vacuum solutions are exactly the delta-shock and vacuum state solutions to the PREE, re-spectively. Secondly, using the characteristic and phase plane analysis methods, by virtue of the Lorentz transformation technique, the Riemann problems of the cor-responding systems with different state equations are constructively solved case by case. We further rigorously prove that, as pressure and flux perturbation vanish, respectively, the Riemann solutions of the relativistic Euler equations tend to the delta-shock and vacuum state solutions to the PREE, which shows that both the delta shock wave and vacuum state are stable for the PREE under some flux small perturbations.Chapter 1 presents research status of relativistic equations and the work of this dissertation.Chapter 2 discusses the Riemann problem of the PREE based on the CREE and constructs the delta-shock and vacuum state solutions.Chapter 3 studies the flux-perturbation problem of the CREE. At first, we solve the Riemann problem of the PREE with a flux perturbation, and obtain two kinds of solutions involving an inverted U-shaped pseudo-vacuum state and a family of parameterized delta shock wave. Then, as the flux perturbation vanishes, we show that the limits of the family of parameterized delta-shock and inverted U-shaped pseudo-vacuum solutions are exactly the delta-shock and vacuum state solutions of the PREE, respectively. Next, the Riemann problem of the CREE with a flux per-turbation including pressure is solved. Furthermore, it is rigorously proved that, as the double parameter flux perturbation vanishes, any two-shock Riemann solution tends to a delta-shock solution to the PREE; any two-rarefaction Riemann solu-tion tends to a two-contact-discontinuity solution to the PREE and the nonvacuum intermediate state in between tends to a vacuum state.Chapter 4 studies the limiting behaviors of Riemann solutions to the CREE for modified Chaplygin gas (MCG) as pressure and flux perturbation vanish, re-spectively. We firstly solve the Riemann problem of the system, and examine the dependence of elementary waves on parameters. It is further proved that, as the dou-ble parameter pressure and triple parameter flux perturbation vanish, respectively, any Riemann solution involving two shock waves tends to a delta-shock solution to the PREE; any Riemann solution involving two rarefaction waves and a nonvacuum intermediate state tends to a vacuum state solution to the PREE.Chapter 5 solves the delta-shock and vacuum state solution to the PREE based on the IREE.Chapter 6 studies the IREE under flux perturbations. The Riemann problem of the PREE with a flux perturbation is firstly solved, and a family of U-shaped pseudo-vacuum state and parameterized delta-shock solutions are constructed. Then it is shown that, as the flux perturbation vanishes, the limits of the family of parameter-ized delta-shock and U-shaped pseudo-vacuum solutions are exactly the delta-shock and vacuum state solutions to the PREE, respectively. Secondly, we study the Rie-mann problem of the IREE with a flux perturbation for polytropic gas. We further prove that, as the pressure and two-parameter flux perturbation vanish, respective-ly, any two-shock Riemann solution tends to a delta-shock solution to the PREE, and the intermediate density between the two shocks tends to a weighted δ-measure which forms a delta shock wave; any two-rarefaction Riemann solution tends to a two-contact-discontinuity solution to the PREE, and the nonvacuum intermediate state in between tends to a vacuum.Chapter 7 studies the flux-perturbation problem of the IREE for MCG. The Riemann problem is firstly solved and Riemann solutions are constructed. Then, it is proved that, as the double parameter pressure and triple parameter flux perturbation vanish, respectively, any Riemann solution containing two shocks tends to a delta-shock solution to the PREE; any Riemann solution containing two rarefaction waves tends to a two-contact-discontinuity solution to the PREE, and the nonvacuum intermediate state in between tends to a vacuum.