Structural Stability Analysis Of Solutions To The Riemann Problem Of Hyperbolic Balance Laws With Discontinuous Sources

Equations of radiation hydrodynamics are important research subjects not only intheory but also in practice. In this dissertation, we will study the Riemann problemsrelated to the equations of radiation hydrodynamics. Under certain conditions, equa-tions of radiation hydrodynamics can be transformed to quasilinear balance laws withsources. Hence, we frst study the Burgers equation with discontinuous source, thenthe shock wave solutions of2×2systems with discontinuous sources, and shock wavesolutions to equations of one dimensional radiation hydrodynamics.InChapter2,weareconcernedwiththeRiemannproblemoftheBurgersequationwith a discontinuous source term, construct a global entropy solution to this Riemannproblem, and discuss the efects of the discontinuous source term on the propagationof the shock waves and rarefaction waves. The study shows that the discontinuity ofthe source term has clearly infuences on the propagation of shocks and rarefactionwaves, and produces several interesting phenomena such as the appearance of weakdiscontinuities, theproductionandabsorptionofnewshocks, artifcial”vacuums”, anddiferent types of asymptotic behavior of shocks.Next, in Chapter3, we study the behavior of shock waves in a2×2balance lawwithdiscontinuoussourceterms. Undercertainstabilityconditions,weobtaintheexis-tence of a local shock wave solution of this problem, and deduce that the discontinuoussource terms create a weak discontinuity in this shock wave solution.Finally, in Chapter4, we are concerned with the local structural stability of one-dimensionalshockwavesinradiationhydrodynamicsdescribedbytheisentropicEuler-Boltzmann coupled equations. Even though in this radiation hydrodynamics model, the radiative efects can be formally understood as source terms to the isentropic Eulerequations of hydrodynamics, however, in general the radiation feld has singularitiespropagated in an angular domain issuing from a point across which the density of fuidis discontinuous. This is the major difculty in the stability analysis of shocks. Undercertain assumptions on the radiation felds, we show there exists a local weak solu-tion to the initial value problem of the one dimensional Euler-Boltzmann equations,in which the radiation intensity is continuous, while the density and velocity of fuidare piecewise Lipschitz continuous with a strong discontinuity representing the shock-front. The existence of such a solution shows that shock waves are structurally stable,at least local in time, in the one-dimensional radiation hydrodynamics.