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)