Nonuniform Dependence on the Initial Data for Solutions of Conservation Laws

John M. Holmes; Barbara Lee Keyfitz

doi:10.1007/s42967-023-00267-9

Communications on Applied Mathematics and Computation ›› 2023, Vol. 6 ›› Issue (1) : 489-500. DOI: 10.1007/s42967-023-00267-9

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Nonuniform Dependence on the Initial Data for Solutions of Conservation Laws

John M. Holmes¹ ,
Barbara Lee Keyfitz¹^,b

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Abstract

In this paper, we study systems of conservation laws in one space dimension. We prove that for classical solutions in Sobolev spaces $H^{s}$ , with s > 3/2, the data-to-solution map is not uniformly continuous. Our results apply to all nonlinear scalar conservation laws and to nonlinear hyperbolic systems of two equations.

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John M. Holmes, Barbara Lee Keyfitz. Nonuniform Dependence on the Initial Data for Solutions of Conservation Laws. Communications on Applied Mathematics and Computation, 2023, 6(1): 489‒500 https://doi.org/10.1007/s42967-023-00267-9