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Application: Oscillatory Profiles in Systems of Hyperbolic Balance Laws

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Part of the Lecture Notes in Mathematics book series (LNM,volume 2117)

Abstract

Conservation laws arise in various physical models including fluid dynamics (Joseph, Appl. Math. Sci., 84, 1990), magneto-hydro-dynamics (Freistühler and Szmolyan, SIAM J. Math. Anal. 26(1), 112–128, 1995), elasticity (Keyfitz and Kranzer, Arch. Ration. Mech. Anal. 72, 219–241, 1980), multiphase flow in oil recovery (Marchesin et al., SIAM J. Appl. Math. 57, 1189–1215, 1997), cosmology (Smoller and Temple, Phys. Rev. D 51, 2733–2743, 1995), and many more.Here we combine a strictly hyperbolic conservation law with a (stiff) source term. Both parts, alone, are “tame”: the conservation law may form shocks, but in general stays piecewise smooth. The source term, alone, will describe a simple, stable kinetic behavior: all trajectories eventually converge monotonically to some equilibrium. The balance law constructed of these two parts, however, can support profiles with oscillatory tails. They emerge from Poincaré–Andronov–Hopf bifurcations without parameters in the associated traveling-wave system.

Keywords

  • Source Term
  • Multiphase Flow
  • Heteroclinic Orbit
  • Turing Instability
  • Numerical Artifact

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Fig. 7.1

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Liebscher, S. (2015). Application: Oscillatory Profiles in Systems of Hyperbolic Balance Laws. In: Bifurcation without Parameters. Lecture Notes in Mathematics, vol 2117. Springer, Cham. https://doi.org/10.1007/978-3-319-10777-6_7

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