Asymptotic Behavior Of Discontinuous Solutions To Two Kind Hyperbolic-parabolic Coupled Equations

This dissertation is devoted to the study of the asymptotic behavior of discontinuous solutions to two kind hyperbolic-parabolic coupled equations—thermoelastic equations and compressible Navier-Stokes equations.The hyperbolic and parabolic operators in hyperbolic-parabolic coupled equations have important roles in determining the properties of solutions to coupled equations. It has important significance of mathematical theory to study the propagation, interaction and reflection of discontinuous waves, and it also has an important impact in mechanics and physics.In part one, we study the asymptotic behavior of discontinuous solutions to the Cauchy problem for thermoelastic equations with second sound, when the relaxation parameter tends to zero. We obtain that the propagation behavior of discontinuous solutions is mainly dominated by the hyperbolic operator in the thermoelastic equations, the discontinuities of elastic waves and the heat flux are propagated with the velocity of elastic waves. The change of the jumps about time t not only depends on the variation of physical media, but also on the growth rate of the nonlinear function. In general, these jumps decay exponentially in time t, the faster for the smaller heat conduction coefficient, while the jump of temperature will disappear immediately when t> 0. This shows the smoothing effect of the thermal conductivity in thermoelastic equations.In part two, we study the asymptotic behavior of discontinuous solutions to the initial-boundary value problem for thermoelastic equations with second sound. We obtain the reflection of the discontinuous waves at the boundary determined by the hyperbolic operator in thermoelastic equations. The propagation and reflection of the discontinuity of the heat flux are also determined mainly by elastic waves. In their propagation and reflection process, thermoelastic equations have both the hyperbole phenomenon, also effected by the parabolic operator smoothing. Moreover, both for the propagation and reflection of jumps, under certain assumption on the changes of the physical media, we observe a very interesting new phenomenon that the jumps of the elastic waves and the heat flux increase exponentially with respect to t→∞.In part three, we study the propagation of discontinuities in the linearized compress-ible Navier-Stokes equations in one space variable. By introducing the idea of relaxation factor, we get the jumps of velocity field will soon disappear when t>0, the jumps of the density and the derivative of the velocity field in the space variable will propagate along the background velocity field. In general, these jumps decay exponentially in time t, the faster for the smaller viscosity coefficient. Meanwhile, for a fixed viscosity, these jumps of the density and the derivative of the velocity field in the space variabel will increase exponentially in time t when the background velocity field decreasing very fast in the space variable x, which is a very interesting phenomenon.