Communications on Applied Mathematics and Computation ›› 2024, Vol. 6 ›› Issue (1): 489-500.doi: 10.1007/s42967-023-00267-9

• ORIGINAL PAPERS • Previous Articles     Next Articles

Nonuniform Dependence on the Initial Data for Solutions of Conservation Laws

John M. Holmes, Barbara Lee Keyfitz   

  1. Department of Mathematics, The Ohio State University, Columbus, OH, 43210, USA
  • Received:2022-12-21 Revised:2022-12-21 Published:2024-04-16
  • Contact: Barbara Lee Keyfitz,E-mail:keyfitz.2@osu.edu;John M. Holmes,E-mail:holmes.782@osu.edu E-mail:keyfitz.2@osu.edu;holmes.782@osu.edu
  • Supported by:
    The first author was supported in part by the NSF Grant DMS-2247019.

Abstract: In this paper, we study systems of conservation laws in one space dimension. We prove that for classical solutions in Sobolev spaces Hs, 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.

Key words: Conservation laws, Data-to-solution map, Nonuniform dependence