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Pointwise Green's function approach to stability for scalar conservation laws
Author(s) -
Howard Peter
Publication year - 1999
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/(sici)1097-0312(199910)52:10<1295::aid-cpa6>3.0.co;2-m
Subject(s) - pointwise , conservation law , mathematics , perturbation (astronomy) , norm (philosophy) , mathematical analysis , scalar (mathematics) , law , physics , geometry , quantum mechanics , political science
We study the pointwise behavior of perturbations from a viscous shock solution to a scalar conservation law, obtaining an estimate independent of shock strength. We find that for a perturbation with initial data decaying algebraically or slower, the perturbation decays in time at the rate of decay of the integrated initial data in any L p norm, p ≥ 1. Stability in any L p norm is a direct consequence. The approach taken is that of obtaining pointwise estimates on the perturbation through a Duhamel's principle argument that employs recently developed pointwise estimates on the Green's function for the linearized equation. © 1999 John Wiley & Sons, Inc.