Hopf Bifurcation From Viscous Shock Waves

Sandstede, Björn; Scheel, Arnd

doi:10.1137/060675587

Cited by 24 publications

(31 citation statements)

References 20 publications

(24 reference statements)

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“…does not have any purely imaginary eigenvalues or Floquet exponents, then (14) admits a hyperbolic splitting: Choose η > 0 so that all eigenvalues or Floquet exponents of (14) have distance strictly larger than η from the imaginary axis, then there exist a constant C and a ξ-…”

Section: Relative Morse Indicesmentioning

confidence: 99%

“…Similarly, for each U ∈ E u ± (ξ * ), there exists a solution U (ξ) of the above equation (14) which is defined for ξ < ξ * so that…”

Section: Relative Morse Indicesmentioning

confidence: 99%

“…This may give existence proofs that are simpler than those given in [14,17,18] though we do not know whether the stability proof given in [14] can be simplified using this approach.…”

mentioning

confidence: 99%

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Relative Morse indices, Fredholm indices, and group velocities

Sandstede¹,

Scheel²

2008

Discrete &Amp; Continuous Dynamical Systems - A

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Abstract. We discuss Fredholm properties of the linearization of partial differential equations on cylindrical domains about travelling and modulated waves. We show that the Fredholm index of each such linearization is given by a relative Morse index which depends only on the asymptotic coefficients. Several strategies are outlined that help to compute relative Morse indices using crossing numbers of spatial eigenvalues, and the results are applied to prove the existence of small viscous shock waves in hyperbolic conservation laws upon adding small localized time-periodic source terms.

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Section: Relative Morse Indicesmentioning

confidence: 99%

“…Similarly, for each U ∈ E u ± (ξ * ), there exists a solution U (ξ) of the above equation (14) which is defined for ξ < ξ * so that…”

Section: Relative Morse Indicesmentioning

confidence: 99%

See 1 more Smart Citation

Relative Morse indices, Fredholm indices, and group velocities

Sandstede¹,

Scheel²

2008

Discrete &Amp; Continuous Dynamical Systems - A

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“…Since the completion of this work, there have been several further developments. In [SS3], Sandstede and Scheel recover and somewhat sharpen our results using spatial dynamics techniques, answering the question posed in Section 1.7 of what results may be obtained by these methods, obtaining the additional information of exponential localization of solutions and exchange of spectral stability. In [TZ3,TZ3], we extend our results to shock and detonation waves of systems with physical viscosity, at the same time greatly sharpening and simplifiying the basic cancellation estimate.…”

Section: Acknowledgement Thanks To Claude Bardos For Pointing Out Thmentioning

confidence: 99%

Galloping instability of viscous shock waves

Texier

Zumbrun

2008

Physica D: Nonlinear Phenomena

112

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Motivated by physical and numerical observations of time oscillatory "galloping", "spinning", and "cellular" instabilities of detonation waves, we study Poincaré-Hopf bifurcation of traveling-wave solutions of viscous conservation laws. The main difficulty is the absence of a spectral gap between oscillatory modes and essential spectrum, preventing standard reduction to a finite-dimensional center manifold. We overcome this by direct Lyapunov-Schmidt reduction, using detailed pointwise bounds on the linearized solution operator to carry out a nonstandard implicit function construction in the absence of a spectral gap. The key computation is a space-time stability estimate on the transverse linearized solution operator reminiscent of Duhamel estimates carried out on the full solution operator in the study of nonlinear stability of spectrally stable traveling waves.

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“…These innovations not only make possible the uniform treatment of large-amplitude NavierStokes shocks, but also for the first time practical multidimensional stability computations in a variety of different settings. Two particularly interesting directions for further investigation are viscous MHD shocks and detonations, for both of which instabilities are known to occur, with interesting associated bifurcations, and for which the effects of viscosity are as yet unclear [60,66,74,75,69,81,13]. A related direction for study is on the possible relation between types of instabilities and an associated convex entropy; see [4,49,76] for related one-dimensional investigations.…”

Section: 2mentioning

confidence: 99%

Multidimensional Stability of Large-Amplitude Navier–Stokes Shocks

Humpherys

Lyng

Zumbrun

2017

Arch Rational Mech Anal

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Abstract. Extending results of Humpherys-Lyng-Zumbrun in the one-dimensional case, we use a combination of asymptotic ODE estimates and numerical Evans-function computations to examine the multidimensional stability of planar Navier-Stokes shocks across the full range of shock amplitudes, including the infinite-amplitude limit, for monatomic or diatomic ideal gas equations of state and viscosity and heat conduction coefficients µ, µ + η, and ν = κ/c v in the physical ratios predicted by statistical mechanics, with Mach number M > 1.035. Our results indicate unconditional stability within the parameter range considered; this agrees with the results of Erpenbeck and Majda for the corresponding inviscid case of Euler shocks. Notably, this study includes the first successful numerical computation of an Evans function associated with the multidimensional stability of a viscous shock wave. The methods introduced may be used in principle to decide stability for any γ-law gas, γ > 1, and arbitrary µ > |η| ≥ 0, ν > 0 or, indeed, for shocks of much more general models and equations, including in particular viscoelasticity, combustion, and magnetohydrodynamics (MHD).

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Hopf Bifurcation From Viscous Shock Waves

Cited by 24 publications

References 20 publications

Relative Morse indices, Fredholm indices, and group velocities

Relative Morse indices, Fredholm indices, and group velocities

Galloping instability of viscous shock waves

Multidimensional Stability of Large-Amplitude Navier–Stokes Shocks

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