Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X

   
 
 

 

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


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 [Enhancements On Off] (What's this?)

  • [AGJ] J. Alexander, R. Gardner and C.K.R.T. Jones, A topological invariant arising in the analysis of traveling waves, J. Reine Angew. Math. 410 (1990) 167-212. MR 1068805 (92d:58028)
  • [BHZ] B. Barker, J. Humpherys, and K. Zumbrun, One-dimensional stability of parallel shock layers in isentropic magnetohydrodynamics, to appear, J. Diff. Eq.
  • [BHRZ] B. Barker, J. Humpherys, , K. Rudd, and K. Zumbrun, Stability of viscous shocks in isentropic gas dynamics, Comm. Math. Phys. 281 (2008), no. 1, 231-249. MR 2403609 (2009c:35286)
  • [BDG] T.J. Bridges, G. Derks, and G. 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 (2004i:37148)
  • [CHNZ] N. Costanzino, J. Humpherys, T. Nguyen, and K. Zumbrun, Spectral stability of noncharacteristic isentropic Navier-Stokes boundary layers, Arch. Ration. Mech. Anal. 192 (2009), 537-587. MR 2505363
  • [Br] L. Q. Brin, Numerical testing of the stability of viscous shock waves. Math. Comp. 70 (2001) no. 235, 1071-1088. MR 1710652 (2001j:65118)
  • [BrZ] L. Brin and K. Zumbrun, Analytically varying eigenvectors and the stability of viscous shock waves. Seventh Workshop on Partial Differential Equations, Part I (Rio de Janeiro, 2001). Mat. Contemp. 22 (2002), 19-32. MR 1965784 (2004c:15012)
  • [BeSZ] M. Beck, B. Sandstede, and K. Zumbrun, Nonlinear stability of time-periodic viscous shocks, Arch. Ration. Mech. Anal. 196 (2010), 1011-1076. MR 2644447
  • [FS1] H. Freistühler and P. Szmolyan, Spectral stability of small shock waves, Arch. Ration. Mech. Anal. 164 (2002) 287-309. MR 1933630 (2003j:35273)
  • [FS2] H. Freistühler and P. Szmolyan. Spectral stability of small-amplitude viscous shock waves in several space dimensions, Arch. Ration. Mech. Anal. 195 (2010), 353-373. MR 2592280
  • [GZ] R. Gardner and K. 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 (99c:35152)
  • [G] J. Goodman, Remarks on the stability of viscous shock waves, in: Viscous profiles and numerical methods for shock waves (Raleigh, NC, 1990), 66-72, SIAM, Philadelphia, PA, 1991. MR 1142641
  • [GSZ] J. Goodman, A. Szepessy, and K. Zumbrun, A remark on the stability of viscous shock waves, SIAM J. Math. Anal. 25 (1994) 1463-1467. MR 1302156 (95i:35182)
  • [GMWZ] O. Guès, G. Métivier, M. Williams, and K. 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 (2005k:35273)
  • [GMWZ2] 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.
  • [HZ] P. Howard and K. Zumbrun, Stability of undercompressive viscous shock waves, J. Differential Equations 225 (2006), no. 1, 308-360; preprint 2004. MR 2228699 (2007d:35185)
  • [HR] P. Howard and M. Raoofi, Pointwise asymptotic behavior of perturbed viscous shock profiles, Adv. Differential Equations (2006) 1031-1080. MR 2263670 (2007i:35161)
  • [HRZ] P. Howard, M. Raoofi, and K. Zumbrun, Sharp pointwise bounds for perturbed viscous shock waves, J. Hyperbolic Differ. Equ. 3 (2006) 297-373; preprint 2005. MR 2229858 (2006m:35235)
  • [HLZ] J. Humpherys, O. Lafitte, and K. Zumbrun, Stability of isentropic Navier-Stokes shocks in the high-Mach number limit, Comm. Math. Phys. 293 (2010), 1-36; published online, Sept. 2009. MR 2563797 (2010i:76085)
  • [HLyZ] J. Humpherys, G. Lyng, and K. Zumbrun, Spectral stability of ideal-gas shock layers, Arch. Ration. Mech. Anal. 194 (2009), 1029-1079. MR 2563632
  • [HuZ] J. Humpherys and K. Zumbrun, Spectral stability of small-amplitude shock profiles for dissipative symmetric hyperbolic-parabolic systems. Z. Angew. Math. Phys. 53 (2002) 20-34. MR 1889177 (2003b:35133)
  • [HuZ2] J. Humpherys and K. Zumbrun, An efficient shooting algorithm for Evans function calculations in large systems, Phys. D 220 (2006), no. 2, 116-126. MR 2253406 (2007e:35006)
  • [JZ1] 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.
  • [JZ2] M. Johnson and K. Zumbrun, Nonlinear stability of periodic traveling wave solutions of systems of viscous conservation laws in the generic case, J. Diff. Eq. 249 (2010), 1213-1240. MR 2652171
  • [JZN] M. Johnson, K. Zumbrun, and P. Noble, Nonlinear stability of viscous roll waves, to appear, SIAM J. Math. Anal.
  • [Liu85] T.P. Liu, Nonlinear stability of shock waves for viscous conservation laws, Mem. Amer. Math. Soc. 56 (1985), no. 328, v+108 pp. MR 791863 (87a:35127)
  • [Liu97] T.P. Liu, Pointwise convergence of shock waves for viscous conservation laws, Comm. Pure Appl. Math. 50 (1997), no. 11, 1113-1182. MR 1470318 (98j:35121)
  • [LY] T.P. Liu and S.H. 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 (99b:35137)
  • [LZ2] T.P. Liu and Y. Zeng, Time-asymptotic behavior of wave propagation around a viscous shock profile, Comm. Math. Phys. 290 (2009), 23-82. MR 2520507 (2010j:35325)
  • [LRTZ] G. Lyng, M. Raoofi, B. Texier, and K. Zumbrun, Pointwise Green function bounds and stability of combustion waves, J. Differential Equations 233 (2007), no. 2, 654-698. MR 2292522 (2007m:35147)
  • [LZu] T.P. Liu and K. Zumbrun, On nonlinear stability of general undercompressive viscous shock waves, Commun. Math. Phys. 174 (1995) 319-345. MR 1362168 (97g:35110)
  • [MaZ1] C. Mascia and K. Zumbrun, Pointwise Green's function bounds and stability of relaxation shocks, Indiana Univ. Math. J. 51 (2002), no. 4, 773-904. MR 1947862 (2003k:35151)
  • [MaZ2] 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; preprint 2001. MR 2044067 (2005e:35022)
  • [MaZ3] C. Mascia and K. Zumbrun, Pointwise Green function bounds for shock profiles of systems with real viscosity. Arch. Ration. Mech. Anal. 169 (2003), no. 3, 177-263; preprint 2002. MR 2004135 (2004h:35137)
  • [MaZ4] C. Mascia and K. Zumbrun, Stability of large-amplitude viscous shock profiles of hyperbolic-parabolic systems, Arch. Ration. Mech. Anal. 172 (2004), no. 1, 93-131; preprint 2003. MR 2048568 (2005d:35166)
  • [N] K. Nishihara, A note on the stability of travelling wave solutions of Burgers' equation (English) Japan. J. Appl. Math. 2 (1985), no. 1, 27-35. MR 839318 (87j:35335b)
  • [PZ] R. Plaza and K. Zumbrun, An Evans function approach to spectral stability of small-amplitude shock profiles, J. Disc. and Cont. Dyn. Sys. 10 (2004), 885-924. MR 2073940 (2005d:35169)
  • [R] M. Raoofi, $ L^p$ asymptotic behavior of perturbed viscous shock profiles, J. Hyperbolic Differ. Equ. 2 (2005), no. 3, 595-644; preprint 2004. MR 2172698 (2006j:35163)
  • [RZ] M. Raoofi and K. Zumbrun, Stability of undercompressive viscous shock profiles of hyperbolic-parabolic systems, J. Differential Equations 246 (2009) 1539-1567. MR 2488696 (2010a:35107)
  • [S] D. Sattinger, On the stability of waves of nonlinear parabolic systems, Adv. Math. 22 (1976) 312-355. MR 0435602 (55:8561)
  • [SX] A. Szepessy and Z. Xin, Nonlinear stability of viscous shock waves, Arch. Ration. Mech. Anal. 122 (1993) 53-103. MR 1207241 (93m:35125)
  • [SS] B. Sandstede and A. Scheel, Hopf bifurcation from viscous shock waves, SIAM J. Math. Anal. 39 (2008) 2033-2052. MR 2390324 (2009f:35218)
  • [TZ1] B. Texier and K. Zumbrun, Relative Poincaré-Hopf bifurcation and galloping instability of traveling waves, Methods Appl. Anal. 12 (2005), no. 4, 349-380. MR 2258314 (2007h:37118)
  • [TZ2] B. Texier and K. Zumbrun, Galloping instability of viscous shock waves, Physica D. 237 (2008) 1553-1601. MR 2454606 (2009h:35271)
  • [TZ3] B. Texier and K. Zumbrun, Hopf bifurcation of viscous shock waves in gas dynamics and MHD, Arch. Ration. Mech. Anal. 190 (2008) 107-140. MR 2434902 (2009g:35239)
  • [TZ4] 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).
  • [Z1] K. Zumbrun, Refined wave-tracking and stability of viscous Lax shocks, Methods Appl. Anal. 7 (2000) 747-768; preprint, 1999. MR 1868555 (2002j:35211)
  • [Z2] K. Zumbrun, Planar stability criteria for multidimensional viscous shock waves, Hyperbolic systems of balance laws, 229-326, Lecture Notes in Math., 1911, Springer, Berlin, 2007. (Lectures given at the C.I.M.E. Summer School held in Cetraro, July 14-21, 2003; preprint, 2003.) MR 2348937 (2008k:35315)
  • [Z3] K. Zumbrun, Formation of diffusion waves in a scalar conservation law with convection, Trans. Amer. Math. Soc. 347 (1995), no. 3, 1023-1032. MR 1283568 (95e:35124)
  • [Z4] K. Zumbrun, Stability of large-amplitude shock waves of compressible Navier-Stokes equations, with an appendix by Helge Kristian Jenssen and Gregory Lyng, in Handbook of mathematical fluid dynamics. Vol. III, 311-533, North-Holland, Amsterdam, 2004. MR 2099037 (2006f:35229)
  • [Z5] K. Zumbrun, Conditional stability of unstable viscous shocks, J. Differential Equations 247 (2009), no. 2, 648-671. MR 2523696
  • [Z6] K. Zumbrun, Multidimensional stability of planar viscous shock waves, Advances in the theory of shock waves, 307-516, Progr. Nonlinear Differential Equations Appl., 47, Birkhäuser Boston, Boston, MA, 2001. MR 1842778 (2002k:35200)
  • [Z7] K. Zumbrun, Conditional stability of unstable viscous shock waves in compressible gas dynamics and MHD, to appear, Arch. for Ration. Mech. Anal.; preprint (2009).
  • [ZH] K. Zumbrun and P. Howard, Pointwise semigroup methods and stability of viscous shock waves. Indiana Mathematics Journal 47 (1998), 741-871; Errata, Indiana Univ. Math. J. 51 (2002), no. 4, 1017-1021. MR 1665788 (99m:35157); MR 1947866 (2004a:35155)
  • [ZS] K. Zumbrun and D. Serre, Viscous and inviscid stability of multidimensional planar shock fronts, Indiana Univ. Math. J. 48 (1999) 937-992. MR 1736972 (2001h:35122)

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC (2010): 35B35

Retrieve articles in all journals with MSC (2010): 35B35


Additional Information

Kevin Zumbrun
Affiliation: Department of Mathematics, Indiana University, Bloomington, Indiana 47405
Email: kzumbrun@indiana.edu

DOI: https://doi.org/10.1090/S0033-569X-2011-01221-6
Keywords: Viscous shock waves; nonlinear stability; pointwise Green function bounds
Received by editor(s): September 28, 2009
Published electronically: January 19, 2011
Additional Notes: The author’s research was partially supported under NSF grants no. DMS-0300487 and DMS-0801745
Article copyright: © Copyright 2011 Brown University
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society