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

被引:0
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作者
Benjamin Texier
Kevin Zumbrun
机构
[1] Université Paris Diderot (Paris 7) et UMR CNRS 7586,Institut de Mathématiques de Jussieu
[2] Indiana University,undefined
关键词
Shock Wave; Hopf Bifurcation; Detonation Wave; Energy Estimate; Essential Spectrum;
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摘要
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 Hs for data in Hs. 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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页码:107 / 140
页数:33
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