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Multidimensional Stability of Large-Amplitude Navier–Stokes Shocks

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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 \({\mu}\), \({\mu +\eta}\), and \({\nu=\kappa/c_v}\) constant and in the physical ratios predicted by statistical mechanics, and 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 can be used in principle to decide stability for shocks in any polytropic gas, or indeed for shocks of other models, including in, particular, viscoelasticity, combustion, and magnetohydrodynamics (MHD).

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References

  1. Alexander J., Gardner R., Jones C.: A topological invariant arising in the stability analysis of travelling waves. J. Reine Angew. Math., 410, 167–212 (1990)

    MathSciNet  MATH  Google Scholar 

  2. Barker B.: Numerical proof of stability of roll waves in the small-amplitude limit for inclined thin film flow. J. Differ. Equ. 257(8), 2950–2983 (2014)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  3. Barker B., Freistühler H., Zumbrun K.: Convex entropy, Hopf bifurcation, and viscous and inviscid shock stability. Arch. Ration. Mech. Anal. 217(1), 309–372 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  4. Barker, B., Humpherys, J., Lytle, J., Zumbrun, K.: STABLAB: A MATLAB-Based Numerical Library for Evans Function Computation, 2015. Available in the github repository, nonlinear-waves/stablab

  5. Barker B., Humpherys J., Zumbrun K.: One-dimensional stability of parallel shock layers in isentropic magnetohydrodynamics. J. Differ. Equ. 249(9), 2175–2213 (2010)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  6. Barker B., Humpherys J., Rudd K., Zumbrun K.: Stability of viscous shocks in isentropic gas dynamics. Comm. Math. Phys. 281(1), 231–249 (2008)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  7. Barker B., Humpherys J., Lafitte O., Rudd K., Zumbrun K.: Stability of isentropic Navier–Stokes shocks. Appl. Math. Lett. 21(7), 742–747 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  8. Barker, B., Humpherys, J., Lyng, G., Zumbrun, K.: Practical Evans-function computation for multidimensional viscous shock waves: Eulerian vs. Lagrangian coordinates, in preparation.

  9. Barker, B., Humpherys, J., Lyng, G., Zumbrun, K.: Practical Evans-function computation for multidimensional viscous shock waves: balanced and modified balanced flux coordinates, Preprint; arxiv: 1703.02099

  10. Barker B., Humpherys J., Lyng G., Zumbrun K.: Viscous hyperstabilization of detonation waves in one space dimension. SIAM J. Appl. Math. 75(3), 885–906 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  11. Barmin A.A., Egorushkin S.A.: Stability of shock waves. Adv. Mech. 15(1–2), 3–37 (1992)

    MathSciNet  Google Scholar 

  12. Bartels R.H., Stewart G.W.: Algorithm 432: solution of the matrix equation AX + XB = C. Comm. ACM 15(9), 820–826 (1972)

    Article  MATH  Google Scholar 

  13. Batchelor, G.K.: An Introduction to Fluid Dynamics. Cambridge Mathematical Library. Cambridge University Press, Cambridge, paperback edition, 1999

  14. Bethe, H.: On the theory of shock waves for an arbitrary equation of state. Report no. 545 for the Office of Scientific Research and Development, Serial no. NDRC-B-237, 1942

  15. Boillat G., Ruggeri T.: On the shock structure problem for hyperbolic system of balance laws and convex entropy. Contin. Mech. Thermodyn. 10, 285–292 (1998)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  16. Blokhin, A.M.: Strong discontinuities in magnetohydrodynamics. Translated by A. V. Zakharov. Nova Science Publishers, Inc., Commack, NY, 1994. x+150 pp. ISBN: 1-56072-144-8.

  17. Blokhin, A., Trakhinin, Y.: Stability of strong discontinuities in fluids and MHD. in Handbook of mathematical fluid dynamics, Vol. I, 545–652, North-Holland, Amsterdam, 2002.

  18. Blokhin A.M., Trakhinin Y.: Stability of fast parallel MHD shock waves in polytropic gas. Eur. J. Mech. B Fluids 18, 197–211 (1999)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  19. Blokhin A.M., Trakhinin Y.: Stability of fast parallel and transversal MHD shock waves in plasma with pressure anisotropy. Acta Mech. 135, 57–71 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  20. Blokhin, A.M., Trakhinin, Y.: Hyperbolic initial-boundary value problems on the stability of strong discontinuities in continuum mechanics. Hyperbolic problems: theory, numerics, applications, Vol. I (Zürich, 1998), 77–86, Internat. Ser. Numer. Math., 129, Birkhäuser, Basel, 1999

  21. Blokhin, A.M., Trakhinin, Y., Merazhov, I.Z.: On the stability of shock waves in a continuum with bulk charge. (Russian) Prikl. Mekh. Tekhn. Fiz. 39, 29–39, 1998; translation in J. Appl. Mech. Tech. Phys. 39, 184–193, 1998

  22. Blokhin, A.M., Trakhinin, Y., Merazhov, I.Z.: Investigation on stability of electrohydrodynamic shock waves. Matematiche (Catania) 52(1997), 87–114, 1998

  23. Brin, L. Q., Zumbrun, K.: Analytically varying eigenvectors and the stability of viscous shock waves. Mat. Contemp., 22, 19–32, 2002. Seventh Workshop on Partial Differential Equations, Part I (Rio de Janeiro, 2001).

  24. Erpenbeck J. J.: Stability of step shocks. Phys. Fluids, 5(10), 1181–1187 (1962)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  25. Freistühler H., Szmolyan P.: Spectral stability of small shock waves. Arch. Ration. Mech. Anal., 164(4), 287–309 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  26. Freistühler H., Szmolyan P.: Spectral stability of small-amplitude viscous shock waves in several dimensions. Arch. Ration. Mech. Anal. 195(2), 353–373 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  27. 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  MATH  Google Scholar 

  28. Gilbarg D.: The existence and limit behavior of the one-dimensional shock layer. Amer. J. Math. 73, 256–274 (1951)

    Article  MathSciNet  MATH  Google Scholar 

  29. Goodman J.: Nonlinear asymptotic stability of viscous shock profiles for conservation laws. Arch. Rational Mech. Anal., 95(4), 325–344 (1986)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  30. Goodman, J.: Remarks on the stability of viscous shock waves. In: Viscous Profiles and Numerical Methods for Shock Waves (Raleigh, NC, 1990), SIAM, Philadelphia, PA, 66–72, 1991

  31. Goodman J.: Stability of viscous scalar shock fronts in several dimensions. Trans. Am. Math. Soc. 311(2), 683–695 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  32. Goodman J., Miller J.R.: Long-time behavior of scalar viscous shock fronts in two dimensions. J. Dynam. Differential Equations 11(2), 255–277 (1999)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  33. Guès O., Métivier G., Williams M., Zumbrun K.: Navier–Stokes regularization of multidimensional Euler shocks. Ann. Sci. École Norm. Sup. 39(4), 75–175 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  34. Guès O., Métivier G., Williams M., Zumbrun K.: Existence and stability of noncharacteristic boundary layers for the compressible Navier–Stokes and MHD equations. Arch. Ration. Mech. Anal. 197(1), 1–87 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  35. Hale N., Moore, D.R.: A sixth-order extension to the MATLAB package bvp4c of J. Kierzenka and L. Shampine, Technical Report NA-08/04, Oxford University Computing Laboratory, May 2008

  36. Hopf E.: The partial differential equation \({u_t + uu_x=\mu u^2}\). Comm. Pure Appl. Math. 3, 201–230 (1950)

    Article  MathSciNet  Google Scholar 

  37. Hoff D., Zumbrun K.: Pointwise Green’s function bounds for multi-dimensional scalar viscous shock fronts. J. Differential Equations 183, 368–408 (2002)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  38. Hoff D., Zumbrun K.: Asymptotic behavior of multi-dimensional scalar viscous shock fronts. Indiana Univ. Math. J. 49, 427–474 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  39. Howard P., Raoofi M., Zumbrun K.: Sharp pointwise bounds for perturbed viscous shock waves. J. Hyperbolic Differ. Equ. 3(2), 297–373 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  40. Humpherys, J., Lyng, G., Zumbrun. K.: Spectral stability of ideal-gas shock layers. Arch. Ration. Mech. Anal. 194(3), 1029–1079, 2009

  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)

    Article  MathSciNet  MATH  Google Scholar 

  42. Humpherys J., Zumbrun K.: An efficient shooting algorithm for evans function calculations in large systems. Physica D, 220(2), 116–126 (2006)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  43. Humpherys J., Lafitte O., Zumbrun K.: Stability of viscous shock profiles in the high Mach number limit. Comm. Math. Phys. 293(1), 1–36 (2010)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  44. Il’in A.M., Oleinik O.A.: Behavior of the solution of the Cauchy problem for certain quasilinear equations for unbounded increase of time. AMS Translations 42(2), 19–23 (1964)

    Google Scholar 

  45. Kato, T.: Perturbation theory for linear operators. Classics in Mathematics. Springer-Verlag, Berlin, 1995. Reprint of the 1980 edition.

  46. Kawashima, S.: Systems of a hyperbolic–parabolic composite type, with applications to the equations of magnetohydrodynamics. thesis, Kyoto University, 1983

  47. Kawashima S., Shizuta Y.: On the normal form of the symmetric hyperbolic-parabolic systems associated with the conservation laws. Tohoku Math. J., 40, 449–464 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  48. Kawashima S., Matsumura A.: Asymptotic stability of traveling wave solutions of systems for one-dimensional gas motion. Comm. Math. Phys. 101(1), 97–127 (1985)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  49. 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)

    Article  MathSciNet  MATH  Google Scholar 

  50. Lax, P.D.: Hyperbolic systems of conservation laws and the mathematical theory of shock waves. Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics No. 11. Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1973. v+48 pp.

  51. Leger N., Vasseur A.: Relative entropy and the stability of shocks and contact discontinuities for systems of conservation laws with non-BV perturbations. Arch. Ration. Mech. Anal. 201(1), 271–302 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  52. Liu, T.-P.: Nonlinear stability of shock waves for viscous conservation laws, Mem. Amer. Math. Soc. 56(328), v+108 pp., 1985

  53. Liu T.-P.: Pointwise convergence to shock waves for viscous conservation laws. Comm. Pure Appl. Math. 50(11), 1113–1182 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  54. Majda, A.: The stability of multi-dimensional shock fronts. Mem. Amer. Math. Soc., 43(275), iv+95 pp, 1983

  55. Majda, A.: The existence of multi-dimensional shock fronts. Mem. Amer. Math. Soc., 41(281):v+93 pp, 1983

  56. Majda A.: The existence and stability of multidimensional shock fronts. Bull. Amer. Math. Soc. (N.S.), 4(3), 342–344 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  57. Majda, A.: Compressible fluid flow and systems of conservation laws in several space variables. Springer-Verlag, New York, viii+ 159 pp., 1984

  58. Métivier, G.: Stability of multidimensional shocks. Advances in the theory of shock waves, Progr. Nonlinear Differential Equations Appl., Birkhäuser Boston, Boston, MA, 47, 25–103, 2001

  59. Métivier, G., Zumbrun, K.: Large viscous boundary layers for noncharacteristic nonlinear hyperbolic problems. Mem. Amer. Math. Soc., 175(826), vi+107, 2005

  60. Métivier G., Zumbrun K.: Symmetrizers and continuity of stable subspaces for parabolic–hyperbolic boundary value problems. Discrete Contin. Dyn. Syst. 11(1), 205–220 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  61. Monteiro R.: Transverse steady bifurcation of viscous shock solutions of a system of parabolic conservation laws in a strip. J. Differential Equations 257(6), 2035–2077 (2014)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  62. 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)

    Article  MathSciNet  MATH  Google Scholar 

  63. Mascia C., Zumbrun K.: Stability of small-amplitude shock profiles of symmetric hyperbolic-parabolic systems. Comm. Pure Appl. Math., 57(7), 841–876 (2004)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  65. Nguyen T.: Stability of multi-dimensional viscous shocks for symmetric systems with variable multiplicities. Duke Math. J. 150(3), 577–614 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  66. Plaza R., Zumbrun K.: An Evans function approach to spectral stability of small-amplitude shock profiles. J. Disc. and Cont. Dyn. Sys., 10, 885–924 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  67. Pogan, A., Yao, J., Zumbrun, K.: O(2) Hopf bifurcation of viscous shock waves in a channel. In preparation

  68. Raoofi R., Zumbrun K.: Stability of undercompressive viscous shock profiles of hyperbolic-parabolic systems. J. Differential Equations 246(4), 1539–1567 (2009)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  69. Rosenhead, L.: A discussion on the first and second viscosities of fluids. Proc. Roy. Soc. London Ser. A, 226(1164), 1–6, 1954

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

    Article  MathSciNet  MATH  Google Scholar 

  71. Szepessy A., Xin Z.P.: Nonlinear stability of viscous shock waves. Arch. Rational Mech. Anal., 122(1), 53–103 (1993)

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  73. Texier B., Zumbrun K.: Hopf bifurcation of viscous shock waves in gas dynamics and MHD. Arch. Ration. Mech. Anal. 190, 107–140 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  74. Texier B., Zumbrun K.: Entropy criteria and stability of extreme shocks: a remark on a paper of Leger and Vasseur. Proc. Amer. Math. Soc. 143(2), 749–754 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  75. Weyl H.: Shock waves in arbitrary fluids. Comm. Pure Appl. Math, 2(2–3), 103–122 (1949)

    Article  MathSciNet  MATH  Google Scholar 

  76. Zumbrun, K.: Multidimensional stability of planar viscous shock waves. In Advances in the theory of shock waves, Progr. Nonlinear Differential Equations Appl., Vol. 47, pages 307–516. Birkhäuser Boston, Boston, MA, 2001

  77. Zumbrun, K.: Stability of large-amplitude shock waves of compressible Navier-Stokes equations. In: Handbook of mathematical fluid dynamics. Vol. III, pages 311–533. North-Holland, Amsterdam, 2004. With an appendix by Helge Kristian Jenssen and Gregory Lyng

  78. Zumbrun, K.: Planar stability criteria for multidimensional viscous shock waves. Hyperbolic systems of balance laws, 229–326, Lecture Notes in Math., 1911, Springer, Berlin, 2007

  79. Zumbrun K.: The refined stability condition and cellular instabilities of viscous shock waves. Phys. D 239(13), 1180–1187 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  81. Zumbrun K., Serre D.: Viscous and inviscid stability of planar shock fronts. Indiana Univ. Math. J., 48(3), 937–992 (1999)

    Article  MathSciNet  MATH  Google Scholar 

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Correspondence to Kevin Zumbrun.

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Communicated by A. Bressan

J.H. was partially supported by NSF grant DMS-CAREER-0847074. G.L. was partially supported by NSF grants DMS-CAREER-0845127 and DMS-1413273. K.Z. was partially supported by NSF grants DMS-0300487 and DMS-0801745.

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Humpherys, J., Lyng, G. & Zumbrun, K. Multidimensional Stability of Large-Amplitude Navier–Stokes Shocks. Arch Rational Mech Anal 226, 923–973 (2017). https://doi.org/10.1007/s00205-017-1147-7

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