Premium
Hyperbolic conservation laws with a moving source
Author(s) -
Lien WenChing
Publication year - 1999
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/(sici)1097-0312(199909)52:9<1075::aid-cpa2>3.0.co;2-4
Subject(s) - transonic , conservation law , shock wave , mathematics , shock (circulatory) , nonlinear system , monotone polygon , eigenvalues and eigenvectors , instability , stability (learning theory) , mathematical analysis , physics , geometry , mechanics , computer science , medicine , quantum mechanics , machine learning , aerodynamics
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 criteria:$$\matrix{l_i {\partial g\over \partial u} r_i < 0 &\quad \hbox{for nonlinear stability,}\hfill\cr l_i {\partial g\over \partial u} r_i > 0 &\quad \hbox{for nonlinear instability.}\hfill\cr }$$Here l i and r i are the i th normalized left and right eigenvectors of ${df \over du}$ , respectively. Through the local analysis on the evolution of the speed and strength of the transonic shock wave, the above criterion can be justified. It turns out that the speed of the transonic shock wave is monotone increasing (decreasing) most of the time in the unstable (stable) case. This is shown by introducing a global functional on nonlinear wave interactions, based on the Glimm scheme. In particular, together with the local analysis, we can study the shock speed globally. Such a global approach is absent in the previous works. Using this strategy, we prove the existence of solutions and verify the asymptotic stability (or instability). © 1999 John Wiley & Sons, Inc.