Instantaneous shock location and one-dimensional nonlinear stability of viscous shock waves

Zumbrun, Kevin

doi:10.1090/S0033-569X-2011-01221-6

Author: Kevin Zumbrun
Journal: Quart. Appl. Math. 69 (2011), 177-202
MSC (2010): Primary 35B35
DOI: https://doi.org/10.1090/S0033-569X-2011-01221-6
Published electronically: January 19, 2011
MathSciNet review: 2807984
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We illustrate in a simple setting the instantaneous shock tracking approach to stability of viscous conservation laws introduced by Howard, Mascia, and Zumbrun. This involves a choice of the definition of instantaneous location of a viscous shock. We show that this choice is time-asymptotically equivalent both to the natural choice of least-squares fit pointed out by Goodman and to a simple phase condition used by Guès, Métivier, Williams, and Zumbrun in other contexts. More generally, we show that it is asymptotically equivalent to any location defined by a localized projection.

References

J. Alexander, R. Gardner, and C. Jones, A topological invariant arising in the stability analysis of travelling waves, J. Reine Angew. Math. 410 (1990), 167–212. MR 1068805

B. Barker, J. Humpherys, and K. Zumbrun, One-dimensional stability of parallel shock layers in isentropic magnetohydrodynamics, to appear, J. Diff. Eq.

Blake Barker, Jeffrey Humpherys, Keith Rudd, and Kevin Zumbrun, Stability of viscous shocks in isentropic gas dynamics, Comm. Math. Phys. 281 (2008), no. 1, 231–249. MR 2403609, DOI https://doi.org/10.1007/s00220-008-0487-4

Thomas J. Bridges, Gianne Derks, and Georg Gottwald, Stability and instability of solitary waves of the fifth-order KdV equation: a numerical framework, Phys. D 172 (2002), no. 1-4, 190–216. MR 1946769, DOI https://doi.org/10.1016/S0167-2789%2802%2900655-3

Nicola Costanzino, Jeffrey Humpherys, Toan Nguyen, and Kevin Zumbrun, Spectral stability of noncharacteristic isentropic Navier-Stokes boundary layers, Arch. Ration. Mech. Anal. 192 (2009), no. 3, 537–587. MR 2505363, DOI https://doi.org/10.1007/s00205-008-0153-1

Leon Q. Brin, Numerical testing of the stability of viscous shock waves, Math. Comp. 70 (2001), no. 235, 1071–1088. MR 1710652, DOI https://doi.org/10.1090/S0025-5718-00-01237-0

Leon Q. Brin and Kevin Zumbrun, Analytically varying eigenvectors and the stability of viscous shock waves, Mat. Contemp. 22 (2002), 19–32. Seventh Workshop on Partial Differential Equations, Part I (Rio de Janeiro, 2001). MR 1965784

Margaret Beck, Björn Sandstede, and Kevin Zumbrun, Nonlinear stability of time-periodic viscous shocks, Arch. Ration. Mech. Anal. 196 (2010), no. 3, 1011–1076. MR 2644447, DOI https://doi.org/10.1007/s00205-009-0274-1

H. Freistühler and P. Szmolyan, Spectral stability of small shock waves, Arch. Ration. Mech. Anal. 164 (2002), no. 4, 287–309. MR 1933630, DOI https://doi.org/10.1007/s00205-002-0215-8

Heinrich Freistühler and Peter Szmolyan, Spectral stability of small-amplitude viscous shock waves in several space dimensions, Arch. Ration. Mech. Anal. 195 (2010), no. 2, 353–373. MR 2592280, DOI https://doi.org/10.1007/s00205-009-0272-3

Robert A. Gardner and Kevin Zumbrun, The gap lemma and geometric criteria for instability of viscous shock profiles, Comm. Pure Appl. Math. 51 (1998), no. 7, 797–855. MR 1617251, DOI https://doi.org/10.1002/%28SICI%291097-0312%28199807%2951%3A7%3C797%3A%3AAID-CPA3%3E3.0.CO%3B2-1

Jonathan Goodman, Remarks on the stability of viscous shock waves, Viscous profiles and numerical methods for shock waves (Raleigh, NC, 1990) SIAM, Philadelphia, PA, 1991, pp. 66–72. MR 1142641

Jonathan Goodman, Anders Szepessy, and Kevin Zumbrun, A remark on the stability of viscous shock waves, SIAM J. Math. Anal. 25 (1994), no. 6, 1463–1467. MR 1302156, DOI https://doi.org/10.1137/S0036141092239648

Olivier Guès, Guy Métivier, Mark Williams, and Kevin Zumbrun, Existence and stability of multidimensional shock fronts in the vanishing viscosity limit, Arch. Ration. Mech. Anal. 175 (2005), no. 2, 151–244. MR 2118476, DOI https://doi.org/10.1007/s00205-004-0342-5

O. Guès, G. Métivier, M. Williams, and K. Zumbrun. Existence and stability of noncharacteristic hyperbolic-parabolic boundary-layers for the compressible Navier-Stokes and viscous MHD equations, Arch. Ration. Mech. Anal. 197 (2010), no. 1, 1–87.

Peter Howard and Kevin Zumbrun, Stability of undercompressive shock profiles, J. Differential Equations 225 (2006), no. 1, 308–360. MR 2228699, DOI https://doi.org/10.1016/j.jde.2005.09.001

Peter Howard and Mohammadreza Raoofi, Pointwise asymptotic behavior of perturbed viscous shock profiles, Adv. Differential Equations 11 (2006), no. 9, 1031–1080. MR 2263670

Peter Howard, Mohammadreza Raoofi, and Kevin Zumbrun, Sharp pointwise bounds for perturbed viscous shock waves, J. Hyperbolic Differ. Equ. 3 (2006), no. 2, 297–373. MR 2229858, DOI https://doi.org/10.1142/S021989160600080X

Jeffrey Humpherys, Olivier Lafitte, and Kevin Zumbrun, Stability of isentropic Navier-Stokes shocks in the high-Mach number limit, Comm. Math. Phys. 293 (2010), no. 1, 1–36. MR 2563797, DOI https://doi.org/10.1007/s00220-009-0885-2

Jeffrey Humpherys, Gregory Lyng, and Kevin Zumbrun, Spectral stability of ideal-gas shock layers, Arch. Ration. Mech. Anal. 194 (2009), no. 3, 1029–1079. MR 2563632, DOI https://doi.org/10.1007/s00205-008-0195-4

Jeffrey Humpherys and Kevin Zumbrun, Spectral stability of small-amplitude shock profiles for dissipative symmetric hyperbolic-parabolic systems, Z. Angew. Math. Phys. 53 (2002), no. 1, 20–34. MR 1889177, DOI https://doi.org/10.1007/s00033-002-8139-6

Jeffrey Humpherys and Kevin Zumbrun, An efficient shooting algorithm for Evans function calculations in large systems, Phys. D 220 (2006), no. 2, 116–126. MR 2253406, DOI https://doi.org/10.1016/j.physd.2006.07.003

M. Johnson and K. Zumbrun, Nonlinear stability of periodic traveling wave solutions of viscous conservation laws in dimensions one and two, to appear, SIAM J. Math. Anal.

Mathew A. Johnson and Kevin Zumbrun, Nonlinear stability of periodic traveling wave solutions of systems of viscous conservation laws in the generic case, J. Differential Equations 249 (2010), no. 5, 1213–1240. MR 2652171, DOI https://doi.org/10.1016/j.jde.2010.04.015

M. Johnson, K. Zumbrun, and P. Noble, Nonlinear stability of viscous roll waves, to appear, SIAM J. Math. Anal.

Tai-Ping Liu, Nonlinear stability of shock waves for viscous conservation laws, Mem. Amer. Math. Soc. 56 (1985), no. 328, v+108. MR 791863, DOI https://doi.org/10.1090/memo/0328

Tai-Ping Liu, Pointwise convergence to shock waves for viscous conservation laws, Comm. Pure Appl. Math. 50 (1997), no. 11, 1113–1182. MR 1470318, DOI https://doi.org/10.1002/%28SICI%291097-0312%28199711%2950%3A11%3C1113%3A%3AAID-CPA3%3E3.3.CO%3B2-8

Tai-Ping Liu and Shih-Hsien Yu, Propagation of a stationary shock layer in the presence of a boundary, Arch. Rational Mech. Anal. 139 (1997), no. 1, 57–82. MR 1475778, DOI https://doi.org/10.1007/s002050050047

Tai-Ping Liu and Yanni Zeng, Time-asymptotic behavior of wave propagation around a viscous shock profile, Comm. Math. Phys. 290 (2009), no. 1, 23–82. MR 2520507, DOI https://doi.org/10.1007/s00220-009-0820-6

Gregory Lyng, Mohammadreza Raoofi, Benjamin Texier, and Kevin Zumbrun, Pointwise Green function bounds and stability of combustion waves, J. Differential Equations 233 (2007), no. 2, 654–698. MR 2292522, DOI https://doi.org/10.1016/j.jde.2006.10.006

Tai-Ping Liu and Kevin Zumbrun, On nonlinear stability of general undercompressive viscous shock waves, Comm. Math. Phys. 174 (1995), no. 2, 319–345. MR 1362168

Corrado Mascia and Kevin Zumbrun, Pointwise Green’s function bounds and stability of relaxation shocks, Indiana Univ. Math. J. 51 (2002), no. 4, 773–904. MR 1947862, DOI https://doi.org/10.1512/iumj.2002.51.2212

C. Mascia and K. Zumbrun, Stability of small-amplitude shock profiles of symmetric hyperbolic-parabolic systems, Comm. Pure Appl. Math. 57 (2004), no. 7, 841–876. MR 2044067, DOI https://doi.org/10.1002/cpa.20023

Corrado Mascia and Kevin Zumbrun, Pointwise Green function bounds for shock profiles of systems with real viscosity, Arch. Ration. Mech. Anal. 169 (2003), no. 3, 177–263. MR 2004135, DOI https://doi.org/10.1007/s00205-003-0258-5

Corrado Mascia and Kevin Zumbrun, Stability of large-amplitude viscous shock profiles of hyperbolic-parabolic systems, Arch. Ration. Mech. Anal. 172 (2004), no. 1, 93–131. MR 2048568, DOI https://doi.org/10.1007/s00205-003-0293-2

Kenji Nishihara, A note on the stability of travelling wave solutions of Burgers’ equation, Japan J. Appl. Math. 2 (1985), no. 1, 27–35. MR 839318, DOI https://doi.org/10.1007/BF03167037

Ramon Plaza and Kevin Zumbrun, An Evans function approach to spectral stability of small-amplitude shock profiles, Discrete Contin. Dyn. Syst. 10 (2004), no. 4, 885–924. MR 2073940, DOI https://doi.org/10.3934/dcds.2004.10.885

Mohammadreza Raoofi, $L^p$ asymptotic behavior of perturbed viscous shock profiles, J. Hyperbolic Differ. Equ. 2 (2005), no. 3, 595–644. MR 2172698, DOI https://doi.org/10.1142/S0219891605000567

Mohammadreza Raoofi and Kevin Zumbrun, Stability of undercompressive viscous shock profiles of hyperbolic-parabolic systems, J. Differential Equations 246 (2009), no. 4, 1539–1567. MR 2488696, DOI https://doi.org/10.1016/j.jde.2008.10.006

D. H. Sattinger, On the stability of waves of nonlinear parabolic systems, Advances in Math. 22 (1976), no. 3, 312–355. MR 435602, DOI https://doi.org/10.1016/0001-8708%2876%2990098-0

Anders Szepessy and Zhou Ping Xin, Nonlinear stability of viscous shock waves, Arch. Rational Mech. Anal. 122 (1993), no. 1, 53–103. MR 1207241, DOI https://doi.org/10.1007/BF01816555

Björn Sandstede and Arnd Scheel, Hopf bifurcation from viscous shock waves, SIAM J. Math. Anal. 39 (2008), no. 6, 2033–2052. MR 2390324, DOI https://doi.org/10.1137/060675587

Benjamin Texier and Kevin Zumbrun, Relative Poincaré-Hopf bifurcation and galloping instability of traveling waves, Methods Appl. Anal. 12 (2005), no. 4, 349–380. MR 2258314, DOI https://doi.org/10.4310/MAA.2005.v12.n4.a1

Benjamin Texier and Kevin Zumbrun, Galloping instability of viscous shock waves, Phys. D 237 (2008), no. 10-12, 1553–1601. MR 2454606, DOI https://doi.org/10.1016/j.physd.2008.03.008

Benjamin Texier and Kevin Zumbrun, Hopf bifurcation of viscous shock waves in compressible gas dynamics and MHD, Arch. Ration. Mech. Anal. 190 (2008), no. 1, 107–140. MR 2434902, DOI https://doi.org/10.1007/s00205-008-0112-x

B. Texier and K. Zumbrun, Transition to longitudinal instability of detonation waves is generically associated with Hopf bifurcation to time-periodic galloping solutions, to appear, Comm. Math. Phys.; preprint (2008).

Kevin Zumbrun, Refined wave-tracking and nonlinear stability of viscous Lax shocks, Methods Appl. Anal. 7 (2000), no. 4, 747–768. MR 1868555, DOI https://doi.org/10.4310/MAA.2000.v7.n4.a8

Kevin Zumbrun, Planar stability criteria for viscous shock waves of systems with real viscosity, Hyperbolic systems of balance laws, Lecture Notes in Math., vol. 1911, Springer, Berlin, 2007, pp. 229–326. MR 2348937, DOI https://doi.org/10.1007/978-3-540-72187-1_4

Kevin R. Zumbrun, Formation of diffusion waves in a scalar conservation law with convection, Trans. Amer. Math. Soc. 347 (1995), no. 3, 1023–1032. MR 1283568, DOI https://doi.org/10.1090/S0002-9947-1995-1283568-8

Kevin Zumbrun, Stability of large-amplitude shock waves of compressible Navier-Stokes equations, Handbook of mathematical fluid dynamics. Vol. III, North-Holland, Amsterdam, 2004, pp. 311–533. With an appendix by Helge Kristian Jenssen and Gregory Lyng. MR 2099037

Kevin Zumbrun, Conditional stability of unstable viscous shocks, J. Differential Equations 247 (2009), no. 2, 648–671. MR 2523696, DOI https://doi.org/10.1016/j.jde.2009.02.017

Kevin Zumbrun, Multidimensional stability of planar viscous shock waves, Advances in the theory of shock waves, Progr. Nonlinear Differential Equations Appl., vol. 47, Birkhäuser Boston, Boston, MA, 2001, pp. 307–516. MR 1842778

K. Zumbrun, Conditional stability of unstable viscous shock waves in compressible gas dynamics and MHD, to appear, Arch. for Ration. Mech. Anal.; preprint (2009).

Kevin Zumbrun and Peter Howard, Pointwise semigroup methods and stability of viscous shock waves, Indiana Univ. Math. J. 47 (1998), no. 3, 741–871. MR 1665788, DOI https://doi.org/10.1512/iumj.1998.47.1604

K. Zumbrun and D. Serre, Viscous and inviscid stability of multidimensional planar shock fronts, Indiana Univ. Math. J. 48 (1999), no. 3, 937–992. MR 1736972, DOI https://doi.org/10.1512/iumj.1999.48.1765