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Hopf Bifurcation of Viscous Shock Waves in Compressible Gas Dynamics and MHD

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Abstract

Extending our previous results for artificial viscosity systems, we show, under suitable spectral hypotheses, that shock wave solutions of compressible Navier–Stokes and magnetohydrodynamics equations undergo Hopf bifurcation to nearby time-periodic solutions. The main new difficulty associated with physical viscosity and the corresponding absence of parabolic smoothing is the need to show that the difference between nonlinear and linearized solution operators is quadratically small in H s for data in H s. We accomplish this by a novel energy estimate carried out in Lagrangian coordinates; interestingly, this estimate is false in Eulerian coordinates. At the same time, we greatly sharpen and simplify the analysis of the previous work.

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Authors and Affiliations

  1. Institut de Mathématiques de Jussieu, Université Paris Diderot (Paris 7) et UMR CNRS 7586, Paris, France

    Benjamin Texier

  2. Indiana University, Bloomington, IN, 47405, USA

    Kevin Zumbrun

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  1. Benjamin Texier
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  2. Kevin Zumbrun
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Correspondence to Benjamin Texier.

Additional information

Communicated by A. Bressan

Research of B.T. was partially supported under NSF grant number DMS-0505780.

Research of K.Z. was partially supported under NSF grant number DMS-0300487.

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Texier, B., Zumbrun, K. Hopf Bifurcation of Viscous Shock Waves in Compressible Gas Dynamics and MHD. Arch Rational Mech Anal 190, 107–140 (2008). https://doi.org/10.1007/s00205-008-0112-x

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  • DOI: https://doi.org/10.1007/s00205-008-0112-x

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