## Hyperbolic Conservation Laws with a Moving Source (1997)

Citations: | 1 - 0 self |

### BibTeX

@MISC{Lien97hyperbolicconservation,

author = {Wen-ching Lien},

title = {Hyperbolic Conservation Laws with a Moving Source},

year = {1997}

}

### OpenURL

### Abstract

The purpose of this paper is to investigate the wave behavior of hyperbolic conservation laws with a moving source. When the speed of the source is close to one of the characteristic speeds of the system, nonlinear resonance occurs and instability may result. We will study solutions with a single transonic shock wave for a general system u t + f(u) x = g(x; u). Suppose that the i-th characteristic speed is close to zero. We propose the following stability criterion: l i @g @u r i ! 0 for nonlinear stability, l i @g @u r i ? 0 for nonlinear instability Here l i and r i are the i-th normalized left and right eigenvectors of df du respectively. By using a variation of the Glimm scheme and studying the evolution of the single transonic shock wave, we prove the existence of solutions and verify the asymptotic stability (or instability). 1 Introduction In this paper, we study the time-asymptotic stability and instability of solutions to systems of conservation laws with a moving sourc...

### Citations

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Citation Context ...eometry of a duct ( i.e. the sign of a 0 (x) ) completely determines the stability of the flow: flows through an expanding duct are stable; transonic flows through a contracting duct are unstable(Liu =-=[16]-=-). A moving magnetic field for magnetohydrodynamics (MHD) can also be expressed as a moving source (Hoffman [9]). Motivated by the studies of physical models, we assume that the source has finite stre... |

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