### Hyperbolic Conservation Laws with a Moving Source

Wen-Ching Lien

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: $l_i \frac{\partial g}{\partial u} r_i < 0$ for nonlinear stability, $l_i \frac{\partial g}{\partial u} r_i > 0$ for nonlinear instability. Here $l_i$ and $r_i$ are the i-th normalized left and right eigenvectors of $\frac{df}{du}$ 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).

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Title:
Hyperbolic Conservation Laws with a Moving Source
Author(s):
Wen-Ching Lien, <lien@math.stanford.edu>
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