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One-Dimensional Stability of Viscous Shock and Relaxation Profiles

In: Hyperbolic Problems: Theory, Numerics, Applications

Author

Listed:
  • Corrado Mascia

    (Università “La Sapienza”)

  • Kevin Zumbrun

    (Indiana University)

Abstract

Under the weak assumption of spectral stability, or stable point spectrum of the linearized operator about the wave, we establish sharp pointwise Green’s function bounds and consequent hnear and nonhnear stability for shock profiles of relaxation and real viscosity systems satisfying the dissipativity condition of Zeng/Kawashima. These include in particular compressible NavierStokes and MHD equations, and essentially all standard relaxation models: in particular, the discrete kinetic models of Broadwell, Jin-Xin, Natalini, Bouchut, Platkowski-Illner, and the moment closure models of Grad, Levermore, Müll er-Rugger i. A consequence is stability of small-amplitude profiles of Broadwell and Jin-Xin models and of general real viscosity systems, for each of which spectral stability has been verified in other works. These are the first complete stability results for profiles of a real viscosity system, and the first for relaxation models with nonscalar equilibrium equations1. Our results apply also in principle to large-amplitude shocks, an important direction for future investigation.

Suggested Citation

  • Corrado Mascia & Kevin Zumbrun, 2003. "One-Dimensional Stability of Viscous Shock and Relaxation Profiles," Springer Books, in: Thomas Y. Hou & Eitan Tadmor (ed.), Hyperbolic Problems: Theory, Numerics, Applications, pages 727-733, Springer.
    • Handle: RePEc:spr:sprchp:978-3-642-55711-8_68
    DOI: 10.1007/978-3-642-55711-8_68
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