×

Galloping instability of viscous shock waves. (English) Zbl 1143.76439

Summary: 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.

MSC:

76E17 Interfacial stability and instability in hydrodynamic stability
76L05 Shock waves and blast waves in fluid mechanics
35Q35 PDEs in connection with fluid mechanics
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Abouseif, G.; Toong, T. Y., Theory of unstable one-dimensional detonations, Combust. Flame, 45, 67-94 (1982)
[2] Alexander, J.; Gardner, R.; Jones, C. K.R. T., A topological invariant arising in the analysis of traveling waves, J. Reine Angew. Math., 410, 167-212 (1990) · Zbl 0705.35070
[3] Alpert, R. L.; Toong, T. Y., Periodicity in exothermic hypersonic flows about blunt projectiles, Acta Astron., 17, 538-560 (1972)
[4] Beyn, W.-J.; Lorenz, J., Stability of viscous profiles: Proofs via dichotomies, J. Dynam. Differential Equations, 18, 1, 141-195 (2006) · Zbl 1105.35060
[5] Barmin, A. A.; Egorushkin, S. A., Stability of shock waves, Adv. Mech., 15, 1-2, 3-37 (1992)
[6] B. Barker, J. Humpherys, K. Rudd, K. Zumbrun, Stability of viscous shocks in isentropic gas dynamics, 2006 (preprint); B. Barker, J. Humpherys, K. Rudd, K. Zumbrun, Stability of viscous shocks in isentropic gas dynamics, 2006 (preprint) · Zbl 1171.35071
[7] Birtea, P.; Puta, M.; Ratiu, T. S.; Tudoran, R., Symmetry breaking for toral actions in simple mechanical systems, J. Differential Equations, 216, 282-323 (2005) · Zbl 1222.70023
[8] Blokhin, A. M.; Trakhinin, Y., Stability of fast parallel and transversal MHD shock waves in plasma with pressure anisotropy, Acta Mech., 135 (1999) · Zbl 0930.76033
[9] Bourlioux, A.; Majda, A.; Roytburd, V., Theoretical and numerical structure for unstable one-dimensional detonations., SIAM J. Appl. Math., 51, 303-343 (1991) · Zbl 0731.76076
[10] Bridges, T. J.; Derks, G.; Gottwald, G., Stability and instability of solitary waves of the fifth-order KdV equation: A numerical framework, Phys. D, 172, 1-4, 190-216 (2002) · Zbl 1047.37053
[11] L. Brin, Numerical testing of the stability of viscous shock waves, Doctoral thesis, Indiana University, 1998; L. Brin, Numerical testing of the stability of viscous shock waves, Doctoral thesis, Indiana University, 1998 · Zbl 0980.65092
[12] Brin, L. Q., Numerical testing of the stability of viscous shock waves, Math. Comp., 70, 235, 1071-1088 (2001) · Zbl 0980.65092
[13] Brin, L.; Zumbrun, K., Analytically varying eigenvectors and the stability of viscous shock waves, Seventh Workshop on Partial Differential Equations, Part I (Rio de Janeiro, 2001). Seventh Workshop on Partial Differential Equations, Part I (Rio de Janeiro, 2001), Mat. Contemp., 22, 19-32 (2002) · Zbl 1044.35057
[14] Costanzino, N.; Jenssen, K.; Lyng, G.; Williams, M., Existence and stability of curved multidimensional detonation fronts, Indiana Univ. Math. J., 56, 3, 1405-1461 (2007) · Zbl 1213.76112
[15] J. Dodd, Convection stability of shock profile solutions of a modified KdV-Burgers equation, Thesis, University of Maryland, 1996; J. Dodd, Convection stability of shock profile solutions of a modified KdV-Burgers equation, Thesis, University of Maryland, 1996
[16] D’yakov, P., Zhur. Eksptl. i Teoret. Fiz., 27, 288 (1954), translation: Atomic Energy Research Establishment AERE Lib./Trans. 648 (1956)
[17] Erpenbeck, J. J., Stability of step shocks., Phys. Fluids, 5, 10, 1181-1187 (1962) · Zbl 0111.38403
[18] Erpenbeck, J. J., Nonlinear theory of unstable one-dimensional detonations, Phys. Fluids, 10, 2, 274-289 (1967) · Zbl 0158.45305
[19] Fickett, W., Stability of the square wave detonation in a model system, Physica, 16D, 358-370 (1985) · Zbl 0577.76065
[20] Fickett, W., Detonation in miniature, (The Mathematics of Combustion. The Mathematics of Combustion, Frontiers in App. Math. (1985), SIAM: SIAM Philadelphia), 133-182
[21] Fickett, W.; Davis, W. C., Detonation (1979), University of California Press: University of California Press Berkeley, CA, reissued as Detonation: Theory and experiment, Dover Press, Mineola, New York, 2000
[22] Fickett; Wood, Flow calculations for pulsating one-dimensional detonations, Phys. Fluids, 9, 903-916 (1966)
[23] Francheteau, J.; Métivier, G., Existence de chocs faibles pour des systémes quasi-linéaires hyperboliques multidimensionnels, C.R.AC.Sc. Paris, Série I, 327, 725-728 (1998) · Zbl 0918.35092
[24] Freistühler, H.; Szmolyan, P., Spectral stability of small shock waves, Arch. Ration. Mech. Anal., 164, 287-309 (2002) · Zbl 1018.35010
[25] Gardner, R.; Jones, C. K.R. T., A stability index for steady state solutions of boundary value problems for parabolic systems, J. Differential Equations, 91, 2, 181-203 (1991) · Zbl 0778.35051
[26] Gardner, R.; Jones, C. K.R. T., Traveling waves of a perturbed diffusion equation arising in a phase field model, Indiana Univ. Math. J., 38, 4, 1197-1222 (1989) · Zbl 0799.35106
[27] Gardner, R.; Zumbrun, K., The Gap Lemma and geometric criteria for instability of viscous shock profiles, Commun. Pure Appl. Math., 51, 7, 797-855 (1998)
[28] Gasser, I.; Szmolyan, P., A geometric singular perturbation analysis of detonation and deflagration waves, SIAM J. Math. Anal., 24, 968-986 (1993) · Zbl 0783.76099
[29] Gilbarg, D., The existence and limit behavior of the one-dimensional shock layer, Amer. J. Math., 73, 256-274 (1951) · Zbl 0044.21504
[30] Goodman, J., Nonlinear asymptotic stability of viscous shock profiles for conservation laws, Arch. Ration. Mech. Anal., 95, 4, 325-344 (1986) · Zbl 0631.35058
[31] Goodman, J., Remarks on the stability of viscous shock waves, (Viscous Profiles and Numerical Methods for Shock Waves. Viscous Profiles and Numerical Methods for Shock Waves, (Raleigh, NC, 1990) (1991), SIAM: SIAM Philadelphia, PA), 66-72 · Zbl 0825.76399
[32] Goodman, J.; Xin, Z., Viscous limits for piecewise smooth solutions to systems of conservation laws, Arch. Ration. Mech. Anal., 121, 3, 235-265 (1992) · Zbl 0792.35115
[33] Gues, O.; Metivier, G.; Williams, M.; Zumbrun, K., Multidimensional viscous shocks I: degenerate symmetrizers and long time stability, J. Amer. Math. Soc., 18, 1, 61-120 (2005) · Zbl 1058.35163
[34] Gues, O.; Métivier, G.; Williams, M.; Zumbrun, K., Existence and stability of multidimensional shock fronts in the vanishing viscosity limit, Arch. Ration. Mech. Anal., 175, 151-244 (2004) · Zbl 1072.35122
[35] Gues, O.; Métivier, G.; Williams, M.; Zumbrun, K., Navier-Stokes regularization of multidimensional Euler shocks, Ann. Sci. cole Norm. Sup. (4), 39, 1, 75-175 (2006) · Zbl 1173.35082
[36] Hale, J.; Koçak, H., (Dynamics and Bifurcations. Dynamics and Bifurcations, Texts in Applied Mathematics, 3 (1991), Springer-Verlag: Springer-Verlag New York), xiv+568 pp · Zbl 0745.58002
[37] Henry, D., Geometric theory of semilinear parabolic equations, (Lecture Notes in Mathematics (1981), Springer-Verlag: Springer-Verlag Berlin), iv + 348 pp · Zbl 0456.35001
[38] Howard, P.; Zumbrun, K., Stability of undercompressive viscous shock waves, J. Differential Equations, 225, 1, 308-360 (2006) · Zbl 1102.35069
[39] Howard, P.; Raoofi, M., Pointwise asymptotic behavior of perturbed viscous shock profiles, Adv. Differential Equations, 11, 9, 1031-1080 (2006) · Zbl 1162.35051
[40] Howard, P.; Raoofi, M.; Zumbrun, K., Sharp pointwise bounds for perturbed shock waves, J. Hyperbolic Differ. Equ., 3, 2, 297-374 (2006) · Zbl 1103.35073
[41] Humpherys, J.; Zumbrun, K., Spectral stability of small amplitude shock profiles for dissipative symmetric hyperbolic-parabolic systems, Z. Angew. Math. Phys., 53, 20-34 (2002) · Zbl 1006.35065
[42] Humpherys, J.; Zumbrun, K., An efficient shooting algorithm for Evans function calculations in large systems, Phys. D, 220, 2, 116-126 (2006) · Zbl 1101.65082
[43] Ikeda, T.; Ikeda, H.; Mimura, M., Hopf bifurcation of travelling pulses in some bistable reaction-diffusion systems, Methods Appl. Anal., 7, 1, 165-193 (2000) · Zbl 0982.35049
[44] Ikeda, T.; Nishiura, Y., Pattern selection for two breathers, SIAM J. Appl. Math., 54, 1, 195-230 (1994) · Zbl 0791.35063
[45] G. Iooss, Travelling water-waves, as a paradigm for bifurcations in reversible infinite-dimensional “dynamical” systems, in: Proceedings of the International Congress of Mathematicians, Vol. III (Berlin, 1998). Doc. Math. 1998, Extra Vol. III, 611-622 (electronic); G. Iooss, Travelling water-waves, as a paradigm for bifurcations in reversible infinite-dimensional “dynamical” systems, in: Proceedings of the International Congress of Mathematicians, Vol. III (Berlin, 1998). Doc. Math. 1998, Extra Vol. III, 611-622 (electronic) · Zbl 0932.37049
[46] Iooss, G., On the standing wave problem in deep water, J. Math. Fluid Mech., 4, 2, 155-185 (2002) · Zbl 1078.76014
[47] Iooss, G.; Adelmeyer, M., (Topics in Bifurcation Theory and Applications. Topics in Bifurcation Theory and Applications, Advanced Series in Nonlinear Dynamics, vol. 3 (1998), World Scientific Publishing Co., Inc: World Scientific Publishing Co., Inc River Edge, NJ), viii+186 pp · Zbl 0968.34027
[48] Iooss, G.; Kirrmann, P., Capillary gravity waves on the free surface of an inviscid fluid of infinite depth. Existence of solitary waves, Arch. Ration. Mech. Anal., 136, 1, 1-19 (1996) · Zbl 0879.76011
[49] Iooss, G.; Mielke, M., Bifurcating time-periodic solutions of Navier-Stokes equations in infinite cylinders, J. Nonlinear Sci., 1, 1, 107-146 (1991) · Zbl 0797.76010
[50] Jenssen, H. K.; Lyng, G.; Williams, M., Equivalence of low-frequency stability conditions for multidimensional detonations in three models of combustion, Indiana Univ. Math. J., 54, 1, 1-64 (2005) · Zbl 1127.35046
[51] Kato, T., Perturbation Theory for Linear Operators (1985), Springer-Verlag: Springer-Verlag Berlin Heidelberg
[52] Kasimov, A. R.; Stewart, D. S., Spinning instability of gaseous detonations, J. Fluid Mech., 466, 179-203 (2002) · Zbl 1013.76034
[53] Kawashima, S.; Matsumura, A., Asymptotic stability of traveling wave solutions of systems for one-dimensional gas motion, Commun. Math. Phys., 101, 1, 97-127 (1985) · Zbl 0624.76095
[54] Kawashima, S.; Matsumura, A.; Nishihara, K., Asymptotic behavior of solutions for the equations of a viscous heat-conductive gas, Proc. Japan Acad. Ser. A Math. Sci., 62, 7, 249-252 (1986) · Zbl 0624.76097
[55] Klainerman, S., Global existence for nonlinear wave equations, Commun. Pure Appl. Math., 33, 1, 43-101 (1980) · Zbl 0405.35056
[56] Kontorovich, V. M., Stability of shock waves in relativistic hydrodynamics, Soviet Physics JETP, 34, 7, 127-132 (1958) · Zbl 0082.40204
[57] Kontorovich, V. M., On the interaction between small disturbances and discontinuities in magnetohydrodynamics and on the stability of shock waves, Soviet Physics JETP, 35, 8, 851-858 (1959), 1216-1225 Z. Eksper. Teoret. Fiz. · Zbl 0085.21205
[58] Kreiss, H. O., Initial boundary value problems for hyperbolic systems, Commun. Pure Appl. Math., 23, 277-298 (1970) · Zbl 0193.06902
[59] Kreiss, G.; Kreiss, H. O., Stability of systems of viscous conservation laws, Commun. Pure Appl. Math., 51, 11-12, 1397-1424 (1998) · Zbl 0935.35013
[60] Kunze, M.; Schneider, G., Exchange of stability and finite-dimensional dynamics in a bifurcation problem with marginally stable continuous spectrum, Z. Angew. Math. Phys., 55, 383-399 (2004) · Zbl 1063.35029
[61] Lee, H. I.; Stewart, D. S., Calculation of linear detonation instability: one-dimensional instability of plane detonation, J. Fluid Mech., 216, 102-132 (1990) · Zbl 0698.76120
[62] Liu, T.-P., Nonlinear stability of shock waves for viscous conservation laws, Mem. Amer. Math. Soc., 56, 328 (1985), v+108 pp · Zbl 0617.35058
[63] Liu, T.-P., Interaction of nonlinear hyperbolic waves, (Liu, F.-C.; Liu, T.-P., Nonlinear Analysis (1991), World Scientific), 171-184
[64] Liu, T.-P., Pointwise convergence to shock waves for viscous conservation laws, Commun. Pure Appl. Math., 50, 11, 1113-1182 (1997) · Zbl 0902.35069
[65] Lyng, G.; Zumbrun, K., A stability index for detonation waves in Majda’s model for reacting flow, Physica D, 194, 1-2, 1-29 (2004) · Zbl 1061.35018
[66] Lyng, G.; Raoofi, M.; Texier, B.; Zumbrun, K., Pointwise Green Function Bounds and stability of combustion waves, J. Differential Equations, 233, 2, 654-698 (2007) · Zbl 1116.80017
[67] Lyng, G.; Zumbrun, K., One-dimensional stability of viscous strong detonation waves, Arch. Ration. Mech. Anal., 173, 2, 213-277 (2004) · Zbl 1067.76041
[68] Majda, A., The stability of multi-dimensional shock fronts-a new problem for linear hyperbolic equations, Mem. Amer. Math. Soc., 275 (1983)
[69] Majda, A., The existence of multi-dimensional shock fronts, Mem. Amer. Math. Soc., 281 (1983) · Zbl 0517.76068
[70] Majda, A., Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables (1984), Springer-Verlag: Springer-Verlag New York, viii+ 159 pp · Zbl 0537.76001
[71] Mascia, C.; Zumbrun, K., Pointwise Green’s function bounds and stability of relaxation shocks, Indiana Univ. Math. J., 51, 4, 773-904 (2002) · Zbl 1036.35135
[72] Mascia, C.; Zumbrun, K., Stability of small-amplitude shock profiles of symmetric hyperbolic-parabolic systems, Commun. Pure Appl. Math., 57, 7, 841-876 (2004) · Zbl 1060.35111
[73] Mascia, C.; Zumbrun, K., Pointwise Green function bounds for shock profiles of systems with real viscosity, Arch. Ration. Mech. Anal., 169, 3, 177-263 (2003) · Zbl 1035.35074
[74] Mascia, C.; Zumbrun, K., Stability of large-amplitude viscous shock profiles of hyperbolic-parabolic systems, Arch. Ration. Mech. Anal., 172, 1, 93-131 (2004) · Zbl 1058.35160
[75] Matsumura, A.; Nishihara, K., On the stability of travelling wave solutions of a one-dimensional model system for compressible viscous gas, Japan J. Appl. Math., 2, 1, 17-25 (1985) · Zbl 0602.76080
[76] McVey, U. B.; Toong, T. Y., Mechanism of instabilities in exothermic blunt-body flows, Combus. Sci. Tech., 3, 63-76 (1971)
[77] Métivier, G., Stability of multidimensional shocks, (Advances in the Theory of Shock Waves. Advances in the Theory of Shock Waves, Progr. Nonlinear Differential Equations Appl., vol. 47 (2001), Birkhäuser Boston: Birkhäuser Boston Boston, MA), 25-103 · Zbl 1017.35075
[78] Métivier-K Zumbrun, G., Symmetrizers and continuity of stable subspaces for parabolic-hyperbolic boundary value problems, Discrete Contin. Dyn. Syst., 11, 1, 205-220 (2004) · Zbl 1102.35332
[79] Nishiura, Y.; Mimura, M., Layer oscillations in reaction-diffusion systems, SIAM J. Appl. Math., 49, 2, 481-514 (1989) · Zbl 0691.35009
[80] Pego-, R. L.; Weinstein, M. I., Asymptotic stability of solitary waves, Commun. Math. Phys., 164, 2, 305-349 (1994) · Zbl 0805.35117
[81] Plaza, R.; Zumbrun, K., An Evans function approach to spectral stability of small-amplitude viscous shock profiles, J. Disc. and Cont. Dyn. Sys., 10, 885-924 (2004) · Zbl 1058.35164
[82] Raoofi, M., \(L^p\) asymptotic behavior of perturbed viscous shock profiles, J. Hyperbolic Differ. Equ., 2, 3, 595-644 (2005) · Zbl 1085.35103
[83] Sandstede, B.; Scheel, A., Essential instability of pulses and bifurcations to modulated travelling waves, Proc. R. Soc. Edinburgh Sect. A, 129, 6, 1263-1290 (1999) · Zbl 0956.35065
[84] Sandstede, B.; Scheel, A., Essential instabilities of fronts: bifurcation and bifurcation failure, Dyn. Sys., 16, 1-28 (2001) · Zbl 1055.37069
[85] B. Sandstede, A. Scheel, Hopf bifurcation from viscous shock waves, SIAM J. Math. Anal. (in press); B. Sandstede, A. Scheel, Hopf bifurcation from viscous shock waves, SIAM J. Math. Anal. (in press) · Zbl 1195.35043
[86] Sattinger, D., On the stability of waves of nonlinear parabolic systems, Adv. Math., 22, 312-355 (1976) · Zbl 0344.35051
[87] Szepessy, A.; Xin, Z., Nonlinear stability of viscous shock waves, Arch. Ration. Mech. Anal., 122, 53-103 (1993) · Zbl 0803.35097
[88] Texier, B.; Zumbrun, K., Relative Poincaré-Hopf bifurcation and galloping instability of traveling waves, Methods Anal. and Appl., 12, 4, 349-380 (2005) · Zbl 1370.37136
[89] B. Texier, K. Zumbrun, Hopf bifurcation of viscous shock waves in compressible gas dynamics and MHD, Arch. Ration. Mech. Anal. (in press); B. Texier, K. Zumbrun, Hopf bifurcation of viscous shock waves in compressible gas dynamics and MHD, Arch. Ration. Mech. Anal. (in press) · Zbl 1155.76037
[90] B. Texier, K. Zumbrun, Transition to longitudinal instability of detonation waves is generically associated with Hopf bifurcation to time-periodic galloping solutions (in preparation); B. Texier, K. Zumbrun, Transition to longitudinal instability of detonation waves is generically associated with Hopf bifurcation to time-periodic galloping solutions (in preparation) · Zbl 1217.35138
[91] Trakhinin, Y., A complete 2D stability analysis of fast MHD shocks in an ideal gas, Commun. Math. Phys., 236, 1, 65-92 (2003) · Zbl 1088.76014
[92] C. Wulff, Theory of Meandering and Drifting Spiral Waves in Reaction-Diffusion Systems, Dissertation, Freie Universität Berlin, 1996; C. Wulff, Theory of Meandering and Drifting Spiral Waves in Reaction-Diffusion Systems, Dissertation, Freie Universität Berlin, 1996
[93] Zumbrun, K., Multidimensional stability of planar viscous shock waves, (Advances in the Theory of Shock Waves. Advances in the Theory of Shock Waves, Progr. Nonlinear Differential Equations Appl., vol. 47 (2001), Birkhäuser Boston: Birkhäuser Boston Boston, MA), 307-516 · Zbl 0989.35089
[94] Zumbrun, K., Stability of large-amplitude shock waves of compressible Navier-Stokes equations, (Handbook of Mathematical Fluid Dynamics, vol. III (2004), North-Holland: North-Holland Amsterdam), 311-533 · Zbl 1222.35156
[95] Zumbrun, K., Planar stability criteria for viscous shock waves of systems with real viscosity, (Hyperbolic Systems of Balance Laws. Hyperbolic Systems of Balance Laws, Lecture Notes in Math., vol. 1911 (2007), Springer: Springer Berlin), 229-326 · Zbl 1138.35061
[96] Zumbrun, K., Refined wave-tracking and nonlinear stability of viscous lax shocks, Methods Appl. Anal., 7, 747-768 (2000) · Zbl 1007.35048
[97] Zumbrun, K.; Howard, P., Pointwise semigroup methods and stability of viscous shock waves, Indiana Mathematics Journal. Indiana Mathematics Journal, Indiana Univ. Math. J., 51, 4, 1017-1021 (2002)
[98] Zumbrun, K.; Serre, D., Viscous and inviscid stability of multidimensional planar shock fronts, Indiana Univ. Math. J., 48, 937-992 (1999) · Zbl 0944.76027
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.