Multidimensional viscous shocks I: Degenerate symmetrizers and long time stability

Guès, Olivier; Métivier, Guy; Williams, Mark L.; Zumbrun, Kevin

doi:10.1090/s0894-0347-04-00470-9

Cited by 53 publications

(36 citation statements)

References 49 publications

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“…In this paper, we continue the program begun in [10][11][12], extending the results of [12] to a general class of hyperbolic-parabolic systems containing in particular the Navier-Stokes equations of compressible gas dynamics. Viscous regularizations are shown to exist assuming the uniform Evans condition; the condition is known to hold for weak shocks [34].…”

Section: Introductionmentioning

confidence: 99%

Navier–Stokes regularization of multidimensional Euler shocks

Guès

Métivier

Williams

et al. 2006

Annales Scientifiques de l’École Normale Supérieure

138

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We establish existence and stability of multidimensional shock fronts in the vanishing viscosity limit for a general class of conservation laws with "real", or partially parabolic, viscosity including the Navier-Stokes equations of compressible gas dynamics with standard or van der Waals-type equation of state. More precisely, given a curved Lax shock solution u 0 of the corresponding inviscid equations for which (i) each of the associated planar shocks tangent to the shock front possesses a smooth viscous profile and (ii) each of these viscous profiles satisfies a uniform spectral stability condition expressed in terms of an Evans function, we construct nearby smooth viscous shock solutions u ε of the viscous equations converging to u 0 as viscosity ε → 0, and establish for these sharp linearized stability estimates generalizing those of Majda in the inviscid case. Conditions (i)-(ii) hold always for shock waves of sufficiently small amplitude, but in general may fail for large amplitudes. We treat the viscous shock problem considered here as a representative of a larger class of multidimensional boundary problems arising in the study of viscous fluids, characterized by sharp spectral conditions rather than symmetry hypotheses, which can be analyzed by Kreiss-type symmetrizers. Compared to the strictly parabolic (artificial viscosity) case, the main new features of the analysis appear in the high frequency estimates for the linearized problem. In that regime we use frequency-dependent conjugators to decouple parabolic components that are smoothed from hyperbolic components (like density in Navier-Stokes) that are not. The construction of the conjugators and the subsequent estimates depend on a careful spectral analysis of the linearized operator.

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Section: Introductionmentioning

confidence: 99%

Navier–Stokes regularization of multidimensional Euler shocks

Guès

Métivier

Williams

et al. 2006

Annales Scientifiques de l’École Normale Supérieure

138

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“…We point out further, as observed by Serre [8] in the one-dimensional case, that the result of Theorem 1.1 is completely analogous to the corresponding relation established in [12] for the linearized dispersion relation associated with a perturbed viscous traveling front u = ū(x • ν − st), lim z→±∞ = u ± , in which the WKB expansion corresponds to matched asymptotics joining an outer, hyperbolic solution and an inner, viscous profile. See [11,Section 1.3] for a formal derivation in the shock case, starting with the same rescaling (x, t) → ( x, t); see also the related discussion of long-time vs. small-viscosity problems in [2]. Rigorous verification may be found in [2,11].…”

Section: Introductionmentioning

confidence: 99%

Low-Frequency Stability Analysis of Periodic Traveling-Wave Solutions of Viscous Conservation Laws in Several Dimensions

Oh¹,

Zumbrun²

2006

Z. Anal. Anwend.

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We generalize work of Oh & Zumbrun and Serre on spectral stability of spatially periodic traveling waves of systems of viscous conservation laws from the one-dimensional to the multi-dimensional setting. Specifically, we extend to multi-dimensions the connection observed by Serre between the linearized dispersion relation near zero frequency of the linearized equations about the wave and the homogenized system obtained by slow modulation (WKB) approximation. This may be regarded as partial justification of the WKB expansion; an immediate consequence is that hyperbolicity of the multi-dimensional homogenized system is a necessary condition for stability of the wave. As pointed out by Oh & Zumbrun in one dimension, description of the low-frequency dispersion relation is also a first step in the determination of time-asymptotic behavior.

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“…The idea of associating a viscous profile with an inviscid discontinuity seems to date back to Rankine [105] and is widely known for when the singularity is a shock, as for instance in compressible fluid mechanics (see the recent achievements by Guès, Métivier, Williams and Zumbrun in [75,74,72,73,70]). However since they are characteristic and conservative, the vortex patches are very different from the shocks of the compressible fluid mechanics (which are non-characteristic and dissipative; cf.…”

Section: The Viscous Casementioning

confidence: 99%

Viscous profiles of vortex patches

Sueur¹

2013

J. Inst. Math. Jussieu

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We deal with the incompressible Navier-Stokes equations with vortex patches as initial data. Such data describe an initial configuration for which the vorticity is discontinuous across a hypersurface. We give an asymptotic expansion of the solutions in the vanishing viscosity limit which exhibits an internal layer where the fluid vorticity has a sharp variation. This layer moves with the flow of the Euler equations.

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Multidimensional viscous shocks I: Degenerate symmetrizers and long time stability

Cited by 53 publications

References 49 publications

Navier–Stokes regularization of multidimensional Euler shocks

Navier–Stokes regularization of multidimensional Euler shocks

Low-Frequency Stability Analysis of Periodic Traveling-Wave Solutions of Viscous Conservation Laws in Several Dimensions

Viscous profiles of vortex patches

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