| The purpose of this dissertation is to investigate the wave behavior of hyperbolic conservation laws with a moving source:{dollar}{dollar}eqalignno{lcub}{lcub}partial uoverpartial t{rcub}+{lcub}partial f(u)overpartial x{rcub}&=g(x-ct,u) &(1)cr u(x,0)&=usb0 (x) &(2)cr{rcub}{dollar}{dollar}Here g represents external effects and c is the source's speed. Using these equations, we can model several physical situations as hyperbolic conservation laws with a moving source, for example, a moving magnetic field for MHD and the nozzle flow. When the speed of the source is too close to one of the characteristic speeds, say, the i-th characteristic speed of the system, resonance occurs. The effects of the nonlinear interaction of the source term and the i-waves propagate at around the same speed. As a result, the solution is not always asymptotically stable, which makes the wave phenomena more interesting.; Our primary interest is about the time-asymptotic stability and instability of solutions constructed by Glimm's scheme and the random choice method. To determine the stability, g plays a dominant role. Without loss of generality, c is assumed to be zero. We propose the following stability criterion under this assumption:{dollar}{dollar}eqalignno{lcub}lsb{lcub}i{rcub}{lcub}partial goverpartial u{rcub}rsb{lcub}i{rcub}< &0, rm for nonlinear stability &(3)cr lsb{lcub}i{rcub} {lcub}partial goverpartial u{rcub}rsb{lcub}i{rcub}> &0, rm for nonlinear instability &(4)cr{rcub}{dollar}{dollar}Here {dollar}lsb{lcub}i{rcub}{dollar} and {dollar}rsb{lcub}i{rcub}{dollar} are the i-th normalized left and right eigenvectors of {dollar}{lcub}partial foverpartial u{rcub}{dollar} respectively. (3) and (4) are justified through the crucial estimates on the local change of the speed of the transonic shock wave. Moreover, combining with these local results, a global estimate on the nonlinear wave interactions is introduced to study the evolution of the transonic shock wave. Such an approach is absent in previous works.; We prove that if a standing shock wave {dollar}usb{lcub}*{rcub}(x){dollar} is perturbed under condition (3), it decelerates and tends to a nearby asymptotic state. However, under condition (4), it accelerates and eventually leaves the region where the source is valid. That is, there exists a solution {dollar}u(x,t){dollar} whose initial data is a perturbation of {dollar}usb{lcub}*{rcub}(x){dollar} and the total variation of {dollar}(u(x,0)-usb{lcub}*{rcub}(x)){dollar} is arbitrarily small, and yet as {dollar}ttoinfty, u(x,t){dollar} tends to one of the stable asymptotic states without any standing shock waves. |
