On the stability of nonlinear conservation laws in the Hausdorff metric.

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 genuinely multidimensional fluxes.
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https://hal.archives-ouvertes.fr/hal-00137881
Contributor : Martin Campos Pinto <>
Submitted on : Thursday, March 22, 2007 - 1:54:04 PM
Last modification on : Wednesday, May 15, 2019 - 3:43:53 AM

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Martin Campos Pinto, Albert Cohen, Wolfgang Dahmen, Ronald Devore. On the stability of nonlinear conservation laws in the Hausdorff metric.. Journal of Hyperbolic Differential Equations, World Scientific Publishing, 2005, 2, pp 25-38. ⟨hal-00137881⟩

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