On the stability of travelling wave solutions of a one-dimensional model system for compressible viscous gas

Matsumura, Akitaka; Nishihara, Kenji

doi:10.1007/bf03167036

Cited by 255 publications

(222 citation statements)

References 6 publications

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“…rectangular) crosssection; "spinning" instabilities for a circular cross-section, with one or more "hot spots", or "combustion heads", moving spirally along the duct. Stability of shocks and detonations may be studied within a unified mathematical framework; see, e.g., [Er1,Er2,Ko1,Ko2,D,BT,FD,LS,T,K,M1,M2,M3,FM,Me,CJLW] in the inviscid case (1.1), (1.5), and [S,Go1,Go2,KM,KMN,MN,L1,L3,GX,SX,GZ,ZH,Br1,Br2,BrZ,BDG,KK,Z1,Z2,Z3,Z4,MaZ2,MaZ3,MaZ4,GMWZ1,GWMZ2,GMWZ3,HZ,BL,LyZ1,LyZ2,JLW,…”

Section: Shocks Detonations and Gallopingmentioning

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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Section: Shocks Detonations and Gallopingmentioning

confidence: 99%

Galloping instability of viscous shock waves

Texier

Zumbrun

2008

Physica D: Nonlinear Phenomena

112

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“…This method relies on the L 1 -stability properties of the approximation equations and nonlinear large time asymptotic stability of viscous shock profiles. For the initial value problems of systems of viscous conservation laws, the stability theory of viscous shock profiles was extensively studied by many authors in the past decade, see e.g., [26][27][28][29]. For the initial boundary problems, see also [30][31][32][33][34][35][36][37] for recent progress.…”

Section: Introductionmentioning

confidence: 99%

Construction of Solutions and L 1–error Estimates of Viscous Methods for Scalar Conservation Laws with Boundary

Liu

Pan

2005

Acta Math Sinica

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This paper is concerned with an initial boundary value problem for strictly convex conservation laws whose weak entropy solution is in the piecewise smooth solution class consisting of finitely many discontinuities. By the structure of the weak entropy solution of the corresponding initial value problem and the boundary entropy condition developed by Bardos-Leroux-Nedelec, we give a construction method to the weak entropy solution of the initial boundary value problem. Compared with the initial value problem, the weak entropy solution of the initial boundary value problem includes the following new interaction type: an expansion wave collides with the boundary and the boundary reflects a new shock wave which is tangent to the boundary. According to the structure and some global estimates of the weak entropy solution, we derive the global L 1 -error estimate for viscous methods to this initial boundary value problem by using the matching travelling wave solutions method. If the inviscid solution includes the interaction that an expansion wave collides with the boundary and the boundary reflects a new shock wave which is tangent to the boundary, or the inviscid solution includes some shock wave which is tangent to the boundary, then the error of the viscosity solution to the inviscid solution is bounded by O(ε 1/2 ) in L 1 -norm; otherwise, as in the initial value problem, the L 1 -error bound is O(ε| ln ε|).

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“…We also mention the earlier works [Go1,Go2,MN,KMN] in which stability under zero mass perturbations was proved for sufficiently weak (i.e., small amplitude) shocks in 1D, [SX] in which stability under general perturbations was proved for weak shocks in 1D, and [Go3,GM] in which the stability of weak planar shock solutions for viscous scalar multidimensional conservation laws was demonstrated. See also the important partial results of [L1] and [L2] for weak shocks in 1D, in which the modern picture of 1D asymptotic behavior of a perturbed shock profile (verified for strong shocks in [Ra]) was first set out, and the treatment of special nonLax shocks in [LX,FL,LZ1,LZ2] (extended to general strong shocks in [HZ, Ra]).…”

Section: Part 1 Introductionmentioning

confidence: 79%

Multidimensional viscous shocks I: Degenerate symmetrizers and long time stability

Guès¹,

Métivier

Williams

et al. 2004

J. Amer. Math. Soc.

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We use energy estimates to study the long time stability of multidimensional planar viscous shocks ψ ( x 1 ) \psi (x_1) for systems of conservation laws. Stability is proved for both zero mass and nonzero mass perturbations, and some of the results include rates of decay in time. Shocks of any strength are allowed, subject to an appropriate Evans function condition. The main tools are a conjugation argument that allows us to replace the eigenvalue equation by a problem in which the x 1 x_1 dependence of the coefficients is removed and degenerate Kreiss-type symmetrizers designed to cope with the vanishing of the Evans function for zero frequency.

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On the stability of travelling wave solutions of a one-dimensional model system for compressible viscous gas

Cited by 255 publications

References 6 publications

Galloping instability of viscous shock waves

Galloping instability of viscous shock waves

Construction of Solutions and L 1–error Estimates of Viscous Methods for Scalar Conservation Laws with Boundary

Multidimensional viscous shocks I: Degenerate symmetrizers and long time stability

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