242 | RWTH Publication No: 47167 2004 IGPM242.pdf |

TITLE | On the Stability of Nonlinear Conservation Laws in the Hausdorff Metric |

AUTHORS | Martin Campos-Pinto, Albert Cohen, Wolfgang Dahmen, Ronald DeVore |

ABSTRACT | The mapping properties of the time evolution operator E(t) of nonlinear hyperbolic scalar conservation laws is investigated. It is shown that this operator is Lipschitz in the Hausdorff metric in one space dimension whenever the flux is convex and one of the initial conditions satisfies a one-sided Lipschitz condition. The Hausdorff distance between the graphs of the solutions measures the closeness in L∞ in the regions where the solutions are smooth, as well as the closeness between the locations of shocks. A similar result on Hausdorff stability is proved with respect to a perturbation of the flux function. These results complement the well known L1 contractivity of the solution operator. They are used in a subsequent paper to prove new smoothness results for solutions to such conservation laws. Negative results are proved in the case of non-convex and genuinly multidimensional fluxes. |

KEYWORDS | Conservation laws, Hausdorff metric, stability, one sided Lipschitz, perturbation of fluxes |

DOI | 10.1142/S0219891605000348 |

PUBLICATION | Journal of hyperbolic differential equations : JHDE 2(1), 14 S. (2005) |