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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Linearized stability of extreme shock profiles in systems of conservation laws with viscosity
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by Robert L. Pego PDF
Trans. Amer. Math. Soc. 280 (1983), 431-461 Request permission

Abstract:

For a genuinely nonlinear hyperbolic system of conservation laws with added artificial viscosity, ${u_t} + f{(u)_x} = \varepsilon {u_{xx}}$, we prove that traveling wave profiles for small amplitude extreme shocks (the slowest and fastest) are linearly stable to perturbations in initial data chosen from certain spaces with weighted norm; i.e., we show that the spectrum of the linearized equation lies strictly in the left-half plane, except for a simple eigenvalue at the origin (due to phase translations of the profile). The weight ${e^{cx}}$ is used in components transverse to the profile, where, for an extreme shock, the linearized equation is dominated by unidirectional convection.
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Additional Information
  • © Copyright 1983 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 280 (1983), 431-461
  • MSC: Primary 35L65; Secondary 35K55
  • DOI: https://doi.org/10.1090/S0002-9947-1983-0716831-9
  • MathSciNet review: 716831