Delta-shocks And Vacuum States To The Euler Equations In Gas Dynamics By The Flux Approximation

In this dissertation, we introduce the flux approximation in the Euler equations for isentropic fluids and nonisentropic fluids to study the phenomena of concentration and cavitation and the formation of delta-shocks and vacuum states in solutions. By solving the Riemann problem of the zero-pressure flow with a flux approximation, we find a family of parameterized delta-shock and constant density state, and we prove that, as the flux perturbation vanishes, they converge to the delta-shock and vacuum state to the zero-pressure flow, respectively. Then, taking different pressure laws, we construct the Riemann solutions of the Euler equations for isentropic fluids and nonisentropic fluids with the flux approximation, and study the formation of delta-shocks and vacuum states in solutions as the flux perturbation including pressure vanishes. The results show that the flux approximations of difference have their respective effect on the formation of delta-shock and vacuum state in isentropic fluids and nonisentropic fluids; both the delta shock wave and vacuum state are stable under some flux small perturbations. The results further enrich the theory of delta-shocks and vacuum states.Chapter 1 presents an introduction on the study of delta-shocks and the work of this dissertation.Chapter 2 recalls the Riemann solutions to the zero-pressure gas dynamics.Chapter 3 studies the Euler equations for isentropic fluids with the flux approx-imation. Firstly, solving the Riemann problem of the zero-pressure flow with a flux approximation, we construct parameterized delta-shock and constant density state, then we show that, as the flux perturbation vanishes, they converge to the delta-shock and vacuum state to the zero-pressure flow, respectively. Secondly, we solve the Riemann problem of the isentropic fluids with the flux approximation and prove that, as the flux perturbation including pressure vanishes, any two-shock Riemann solution tends to a delta shock wave solution to the zero-pressure flow; any two-rarefaction-wave Riemann solution tends to a two-contact-discontinuity solution to the zero-pressure flow and the nonvacuum intermediate state in between tends to a vacuum state. Thirdly, we exhibit the numerical results.Chapter 4 discusses the Euler equations for nonisentropic fluids with the flux approximation. It is rigorously proved that, as the flux perturbation including pres-sure vanishes, any Riemann solution containing two shock waves and possibly one-contact-discontinuity to the nonisentropic fluids with the flux approximation tends to a delta shock wave solution to the zero-pressure flow; any Riemann solution containing two rarefaction waves and possibly one-contact-discontinuity to the non-isentropic fluids with the flux approximation tends to a two-contact-discontinuity solution to the zero-pressure flow, and the nonvacuum intermediate state in be-tween tends to a vacuum state.. The numerical simulations are coinciding with the theoretical analysis.Chapter 5 studies the isentropic Chaplygin gas equations with a double pa-rameter flux approximation. We prove that, as the two-parameter flux perturbation including pressure vanishes, any two-shock Riemann solution and any parameterized delta shock wave solution to the perturbed isentropic Chaplygin gas equations tend to a delta shock wave solution to the zero-pressure flow; any two-rarefaction-wave Riemann solution to the perturbed isentropic Chaplygin gas equations tends to a two-contact-discontinuity solution to the zero-pressure flow and the nonvacuum in-termediate state in between tends to a vacuum state. Furthermore, we also show that, as the single parameter flux perturbation vanishes, any two-shock Riemann solution satisfying certain initial data and any parameterized delta shock wave solu-tion to the perturbed isentropic Chaplygin gas equations tend to a delta shock wave solution to the isentropic Chaplygin gas equations. The numerical simulations are consistent with the theoretical analysis.Chapter 6 considers the modified Chaplygin gas equations with the flux approx-imation. We rigorously prove that, as the triple parameter flux perturbation includ-ing pressure vanishes, any two-shock Riemann solution and any two-rarefaction-wave Riemann solution to the perturbed modified Chaplygin gas equations tend to the delta-shock and vacuum state solutions to the zero-pressure flow, respectively. Then we show that, as a double parameter flux perturbation vanishes, part of two-shock Riemann solution to the perturbed modified Chaplygin gas equations tends to a delta shock wave solution to the generalized Chaplygin gas equations. Moreover, we also show that, as a single parameter flux perturbation vanishes, part of two-shock Riemann solution and any parameterized delta shock wave solution to the perturbed generalized Chaplygin gas equations tend to a delta shock wave solution to the gen-eralized Chaplygin gas equations. Finally, we present the numerical results.Chapter 7 studies the pressureless type system with the flux approximation. By solving the Riemann problem of the pressureless type system with single parameter flux approximation, we construct parameterized delta-shock and generalized con-stant density state. We show that, as the flux perturbation vanishes, they converge to the delta-shock and vacuum state to the pressureless type system, respectively. Then, we solve the Riemann problem of the pressureless type system with a double parameter flux approximation. It is proved that, as the two-parameter flux per-turbation including pressure vanishes, any Riemann solution containing two shock waves tends to a delta shock wave solution to the pressureless type system; any Rie-mann solution containing two rarefaction waves tends to a two-contact-discontinuity solution to the pressureless type system and the nonvacuum intermediate state in between tends to a vacuum state.