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
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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
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References
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- 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.
- 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)
- P. Howard and M. Raoofi, Pointwise asymptotic behavior of perturbed viscous shock profiles, Adv. Differential Equations (2006) 1031–1080. MR 2263670 (2007i:35161)
- 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)
- 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)
- J. Humpherys, G. Lyng, and K. Zumbrun, Spectral stability of ideal-gas shock layers, Arch. Ration. Mech. Anal. 194 (2009), 1029–1079. MR 2563632
- 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)
- 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)
- 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.
- 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
- M. Johnson, K. Zumbrun, and P. Noble, Nonlinear stability of viscous roll waves, to appear, SIAM J. Math. Anal.
- 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)
- 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)
- 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)
- 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)
- 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)
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- 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)
- 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)
- 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)
- 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)
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- 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)
- 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)
- D. Sattinger, On the stability of waves of nonlinear parabolic systems, Adv. Math. 22 (1976) 312–355. MR 0435602 (55:8561)
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- B. Sandstede and A. Scheel, Hopf bifurcation from viscous shock waves, SIAM J. Math. Anal. 39 (2008) 2033–2052. MR 2390324 (2009f:35218)
- 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)
- B. Texier and K. Zumbrun, Galloping instability of viscous shock waves, Physica D. 237 (2008) 1553-1601. MR 2454606 (2009h:35271)
- 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)
- 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).
- K. Zumbrun, Refined wave–tracking and stability of viscous Lax shocks, Methods Appl. Anal. 7 (2000) 747–768; preprint, 1999. MR 1868555 (2002j:35211)
- 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)
- 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)
- 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)
- K. Zumbrun, Conditional stability of unstable viscous shocks, J. Differential Equations 247 (2009), no. 2, 648–671. MR 2523696
- 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)
- K. Zumbrun, Conditional stability of unstable viscous shock waves in compressible gas dynamics and MHD, to appear, Arch. for Ration. Mech. Anal.; preprint (2009).
- 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)
- 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)
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Additional Information
Kevin Zumbrun
Affiliation:
Department of Mathematics, Indiana University, Bloomington, Indiana 47405
MR Author ID:
330192
Email:
kzumbrun@indiana.edu
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.