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Nonlinear Stability of Time-Periodic Viscous Shocks

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Abstract

In order to understand the nonlinear stability of many types of time-periodic travelling waves on unbounded domains, one must overcome two main difficulties: the presence of embedded neutral eigenvalues and the time-dependence of the associated linear operator. This problem is studied in the context of time-periodic Lax shocks in systems of viscous conservation laws. Using spatial dynamics and a decomposition into separate Floquet eigenmodes, it is shown that the linear evolution for the time-dependent operator can be represented using a contour integral similar to that of the standard time-independent case. By decomposing the resulting Green’s distribution, the leading order behavior associated with the embedded eigenvalues is extracted. Sharp pointwise bounds are then obtained, which are used to prove that time-periodic Lax shocks are linearly and nonlinearly stable under the necessary conditions of spectral stability and minimal multiplicity of the translational eigenvalues. The latter conditions hold, for example, for small-oscillation time-periodic waves that emerge through a supercritical Hopf bifurcation from a family of time-independent Lax shocks of possibly large amplitude.

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References

  1. Texier B., Zumbrun K.: Hopf bifurcation of viscous shock waves in compressible gas- and magnetohydrodynamics. Arch. Rational Mech. Anal. 190, 107–140 (2008)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  2. Kasimov A.R., Stewart D.S.: Spinning instability of gaseous detonations. J. Fluid Mech. 466, 179–203 (2002)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  3. Sandstede B., Scheel A.: Defects in oscillatory media: toward a classification. SIAM J. Appl. Dyn. Syst. 3(1), 1–68 (2004)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  4. Texier B., Zumbrun K.: Relative Poincaré-Hopf bifurcation and galloping instability of traveling waves. Methods Appl. Anal. 12(4), 349–380 (2005)

    MATH  MathSciNet  Google Scholar 

  5. Texier B., Zumbrun K.: Galloping instability of viscous shock waves. Physica D 237, 1553–1601 (2008)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  6. Sandstede B., Scheel A.: Hopf bifurcation from viscous shock waves. SIAM J. Math. Anal. 39, 2033–2052 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  7. Howard P., Zumbrun K.: Stability of undercompressive shock profiles. J. Differ. Equ. 225(1), 308–360 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  8. Mascia C., Zumbrun K.: Pointwise Green function bounds for shock profiles of systems with real viscosity. Arch. Rational Mech. Anal. 169(3), 177–263 (2003)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  9. Peterhof D., Sandstede B., Scheel A.: Exponential dichotomies for solitary-wave solutions of semilinear elliptic equations on infinite cylinders. J. Differ. Equ. 140(2), 266–308 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  10. Sandstede B., Scheel A.: On the structure of spectra of modulated travelling waves. Math. Nachr. 232, 39–93 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  11. Zumbrun K., Howard P.: Pointwise semigroup methods and stability of viscous shock waves. Indiana Univ. Math. J. 47(3), 741–871 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  12. Raoofi M., Zumbrun K.: Stability of undercompressive viscous shock profiles of hyperbolic–parabolic systems. J. Differ. Equ. 246, 1539–1567 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  13. Liu T.P., Zumbrun K.: On nonlinear stability of general undercompressive viscous shock waves. Comm. Math. Phys. 174(2), 319–345 (1995)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  14. Mascia C., Zumbrun K.: Stability of large-amplitude viscous shock profiles of hyperbolic-parabolic systems. Arch. Rational Mech. Anal. 172(1), 93–131 (2004)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  15. Zumbrun, K.: Multidimensional stability of planar viscous shock waves. In: Advances in the Theory of Shock Waves, Birkhäuser, pp. 307–516, 2001

  16. Zumbrun, K.: Stability of large-amplitude shock waves of compressible Navier-Stokes equations. In: Handbook of Mathematical Fluid Dynamics, Vol. 3. North-Holland, Amsterdam, 311–533, 2004

  17. Zumbrun, K.: Planar stability criteria for viscous shock waves of systems with real viscosity. In: Hyperbolic Systems of Balance Laws., Lecture Notes in Mathematics, Vol. 1911. Springer, New York, 229–326, 2007

  18. Gardner R.A., Zumbrun K.: The gap lemma and geometric criteria for instability of viscous shock profiles. Comm. Pure Appl. Math. 51(7), 797–855 (1998)

    Article  MathSciNet  Google Scholar 

  19. Kapitula T., Sandstede B.: Stability of bright solitary-wave solutions to perturbed nonlinear Schrödinger equations. Physica D 124, 58–103 (1998). doi:10.1016/S0167-2789(98)00172-9

    Article  MATH  MathSciNet  ADS  Google Scholar 

  20. Friedman, A.: Partial differential equations of parabolic type. Prentice-Hall, Englewood cliffs, 1964

    MATH  Google Scholar 

  21. Gilbarg D., Trudinger N.S.: Elliptic partial differential equations of second order. Springer-Verlag, New York (2001)

    MATH  Google Scholar 

  22. Ladyzhenskaya O.A., Uraltseva N.N.: Linear and quasilinear elliptic equations. Academic Press, New York (1968)

    MATH  Google Scholar 

  23. Costin O., Costin R.D., Lebowitz J.L.: Time asymptotics of the Schrödinger wave function in time-periodic potentials. J. Stat. Phys. 116(1–4), 283–310 (2004)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  24. Sandstede B., Scheel A.: Relative Morse indices, Fredholm indices, and group velocities. Discr. Cont. Dyn. Syst. A 20, 139–158 (2008)

    MATH  MathSciNet  Google Scholar 

  25. Sandstede, B.: Stability of travelling waves. In: Handbook of Dynamical Systems, Vol. 2, North-Holland, Amsterdam, 983–1055, 2002

  26. Sandstede B., Scheel A.: Spectral stability of modulated travelling waves bifurcating near essential instabilities. Proc. R. Soc. Edinburgh Sect. A 130(2), 419–448 (2000). doi:10.1017/S0308210500000238

    Article  MATH  MathSciNet  Google Scholar 

  27. Zumbrun K.: Refined wave-tracking and nonlinear stability of viscous Lax shocks. Methods Appl. Anal. 7(4), 747–768 (2000)

    MATH  MathSciNet  Google Scholar 

  28. Raoofi M.: L p asymptotic behavior of perturbed viscous shock profiles. J. Hyp. Differ. Equ. 2(3), 595–644 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  29. Lord G.J., Peterhof D., Sandstede B., Scheel A.: Numerical computation of solitary waves in infinite cylindrical domains. SIAM J. Numer. Anal. 37(5), 1420–1454 (2000). doi:10.1137/S003614299833734X

    Article  MATH  MathSciNet  Google Scholar 

  30. Oh M., Sandstede B.: Evans functions for periodic waves in infinite cylindrical domains. J. Differ. Equ. 248(3), 544–555 (2009). doi:10.1016/j.jde.2009.08.003

    Article  MathSciNet  Google Scholar 

  31. Härterich J., Sandstede B., Scheel A.: Exponential dichotomies for linear non-autonomous functional differential equations of mixed type. Indiana Univ. Math. J. 51(5), 1081–1109 (2002). doi:10.1512/iumj.2002.51.2188

    Article  MATH  MathSciNet  Google Scholar 

  32. Mallet-Paret, J., Verduyn-Lunel, S.: Exponential dichotomies and Wiener Hopf factorizations for mixed-type functional differential equations. J. Differ. Equ. (to appear)

  33. Benzoni-Gavage S., Huot P., Rousset F.: Nonlinear stability of semidiscrete shock waves. SIAM J. Math. Anal. 35(3), 639–707 (2003)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Division of Applied Mathematics, Brown University, Providence, RI, 02912, USA

    Margaret Beck & Björn Sandstede

  2. Department of Mathematics, Indiana University, Bloomington, IN, 47405, USA

    Kevin Zumbrun

Authors
  1. Margaret Beck
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  2. Björn Sandstede
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  3. Kevin Zumbrun
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Correspondence to Margaret Beck.

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Communicated by C.M. Dafermos

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Beck, M., Sandstede, B. & Zumbrun, K. Nonlinear Stability of Time-Periodic Viscous Shocks. Arch Rational Mech Anal 196, 1011–1076 (2010). https://doi.org/10.1007/s00205-009-0274-1

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  • DOI: https://doi.org/10.1007/s00205-009-0274-1

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