Balanced flux formulations for multidimensional Evans-function computations for viscous shocks
Authors:
Blake Barker, Jeffrey Humpherys, Gregory Lyng and Kevin Zumbrun
Journal:
Quart. Appl. Math. 76 (2018), 531-545
MSC (2010):
Primary 35Q35, 76L05, 35Pxx
DOI:
https://doi.org/10.1090/qam/1492
Published electronically:
October 25, 2017
MathSciNet review:
3805041
Full-text PDF
Abstract |
References |
Similar Articles |
Additional Information
Abstract: The Evans function is a powerful tool for the stability analysis of viscous shock profiles; zeros of this function carry stability information. In the one-dimensional case, it is typical to compute the Evans function using Goodman’s integrated coordinates (1986); this device facilitates the search for zeros of the Evans function by winding number arguments. Although integrated coordinates are not available in the multidimensional case, we show here that there is a choice of coordinates which gives similar advantages.
References
- B. Barker, J. Humpherys, G. Lyng, and K. Zumbrun, Euler vs. Lagrange: The role of coordinates in practical Evans-function computations, ArXiv e-prints (2017), 1–20.
- Blake Barker, Jeffrey Humpherys, Joshua Lytle, and Kevin Zumbrun, STABLAB: a MATLAB-based numerical library for evans function computation, 2015, Available in the github repository, nonlinear-waves/stablab.
- 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
- Blake Barker, Rafael Monteiro, and Kevin Zumbrun, Transverse bifurcation of viscous slow mhd shocks, 2017, in preparation.
- 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
- 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, Nonlinear asymptotic stability of viscous shock profiles for conservation laws, Arch. Rational Mech. Anal. 95 (1986), no. 4, 325–344. MR 853782, DOI https://doi.org/10.1007/BF00276840
- 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
- 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, Gregory Lyng, and Kevin Zumbrun, Multidimensional stability of large-amplitude navier–stokes shocks, Archive for Rational Mechanics and Analysis (2017), 1–51.
- 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
- Andrew Majda, The existence and stability of multidimensional shock fronts, Bull. Amer. Math. Soc. (N.S.) 4 (1981), no. 3, 342–344. MR 609047, DOI https://doi.org/10.1090/S0273-0979-1981-14908-9
- 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
- Guy Métivier and Kevin Zumbrun, Large viscous boundary layers for noncharacteristic nonlinear hyperbolic problems, Mem. Amer. Math. Soc. 175 (2005), no. 826, vi+107. MR 2130346, DOI https://doi.org/10.1090/memo/0826
- 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
- 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
- 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 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
References
- B. Barker, J. Humpherys, G. Lyng, and K. Zumbrun, Euler vs. Lagrange: The role of coordinates in practical Evans-function computations, ArXiv e-prints (2017), 1–20.
- Blake Barker, Jeffrey Humpherys, Joshua Lytle, and Kevin Zumbrun, STABLAB: a MATLAB-based numerical library for evans function computation, 2015, Available in the github repository, nonlinear-waves/stablab.
- 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
- Blake Barker, Rafael Monteiro, and Kevin Zumbrun, Transverse bifurcation of viscous slow mhd shocks, 2017, in preparation.
- 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
- 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%24%5Clangle%24797%3A%3AAID-CPA3%24%5Crangle%243.0.CO%3B2-1
- Jonathan Goodman, Nonlinear asymptotic stability of viscous shock profiles for conservation laws, Arch. Rational Mech. Anal. 95 (1986), no. 4, 325–344. MR 853782, DOI https://doi.org/10.1007/BF00276840
- 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
- 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, Gregory Lyng, and Kevin Zumbrun, Multidimensional stability of large-amplitude navier–stokes shocks, Archive for Rational Mechanics and Analysis (2017), 1–51.
- 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
- Andrew Majda, The existence and stability of multidimensional shock fronts, Bull. Amer. Math. Soc. (N.S.) 4 (1981), no. 3, 342–344. MR 609047, DOI https://doi.org/10.1090/S0273-0979-1981-14908-9
- 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
- Guy Métivier and Kevin Zumbrun, Large viscous boundary layers for noncharacteristic nonlinear hyperbolic problems, Mem. Amer. Math. Soc. 175 (2005), no. 826, vi+107. MR 2130346, DOI https://doi.org/10.1090/memo/0826
- 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
- 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
- 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 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
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC (2010):
35Q35,
76L05,
35Pxx
Retrieve articles in all journals
with MSC (2010):
35Q35,
76L05,
35Pxx
Additional Information
Blake Barker
Affiliation:
Department of Mathematics, Brigham Young University, Provo, Utah 84603
Email:
blake@math.byu.edu
Jeffrey Humpherys
Affiliation:
Department of Mathematics, Brigham Young University, Provo, Utah 84603
MR Author ID:
358503
Email:
jeffh@math.byu.edu
Gregory Lyng
Affiliation:
Department of Mathematics, University of Wyoming, Laramie, Wyoming 82071
MR Author ID:
739709
Email:
glyng@uwyo.edu
Kevin Zumbrun
Affiliation:
Department of Mathematics, Indiana University, Bloomington, Indiana 47405
MR Author ID:
330192
Email:
kzumbrun@indiana.edu
Received by editor(s):
March 6, 2017
Received by editor(s) in revised form:
September 21, 2017
Published electronically:
October 25, 2017
Additional Notes:
The first author was partially supported by NSF grant DMS-0801745.
The second author was partially supported by NSF grant DMS-0847074 (CAREER)
The third author was partially supported by NSF grants DMS-0845127 (CAREER) and DMS-1413273
The last author was partially supported by NSF grant DMS-0801745
Article copyright:
© Copyright 2017
Brown University
