Stability of small-amplitude shock profiles of symmetric hyperbolic-parabolic systems

Communications on Pure and Applied Mathematics

Volumn 57, Issue 7, 2004, Pages 841-876

  Mascia, C ^a   Zumbrun, K ^b  

^a SAPIENZA UNIVERSITY OF ROME   (Italy)

^b INDIANA UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 2942554868     PISSN: 00103640     EISSN: None     Source Type: Journal    
DOI: 10.1002/cpa.20023     Document Type: Article

References (52)

1. Benzoni-Gavage, S.; Serre, D.; Zumbrun K. Altemate Evans functions and viscous shock waves. SIAM J. Math. Anal. 32 (2001), no. 5, 929-962.
2. Conley, C.; Smoller, J. On the structure of magnetohydrodynamic shock waves. Comm. Pure Appl. Math. 27 (1974), 367-375.
3. Erpenbeck, J. J. Stability of step shocks. Phys. Fluids 5 (1962), no. 10, 1181-1187.
4. Freistühler, H. Small amplitude intermediate magnetohydrodynamic shock waves. Phys. Scripta T74 (1998), 26-29.
5. Freistühler, H. Profiles for small Laxian shock waves in Kawashima type systems of conservation laws. Preprint, 2000.
6. Freistühler, H.; Zumbrun, K. Examples of unstable viscous shock waves. Unpublished notes, Institut für Mathematik, RWTH Aachen, Germany, February 1998.
7. Fries, C. Nonlinear asymptotic stability of general small-amplitude viscous Laxian shock waves. J. Differential Equations 146 (1998), no. 1, 185-202.
8. Fries, C. Stability of viscous shock waves associated with non-convex modes. Arch. Ration. Mech. Anal. 152 (2000), no. 2, 141-186.
9. Gardner, R. A.; Zumbrun, K. The gap lemma and geometric criteria for instability of viscous shock profiles. Comm. Pure Appl. Math. 51 (1998), no. 7, 797-855.
10. Gilbarg, D. The existence and limit behavior of the one-dimensional shock layer. Amer. J. Math. 73 (1951), 256-274.
11. Goodman, J. Nonlinear asymptotic stability of viscous shock profiles for conservation laws. Arch. Rational Mech. Anal. 95 (1986), no. 4, 325-344.
12. Goodman, J. Remarks on the stability of viscous shock waves. Viscous profiles and numerical methods for shock waves (Raleigh, NC, 1990), 66-72. SIAM, Philadelphia, 1991.
13. Hoff, D.; Zumbrun, K. Multi-dimensional diffusion waves for the Navier-Stokes equations of compressible flow. Indiana Univ. Math. J. 44 (1995), no. 2, 603-676.
14. Howard, P.; Zumbrun, K. A tracking mechanism for one-dimensional viscous shock waves. Preprint, 1999.
15. Howard, P.; Zumbrun, K. Pointwise estimates for dispersive-diffusive shock waves. Arch. Rational Mech. Anal. 155 (2000), no. 2, 85-169.
16. Humpherys, J.; Zumbrun, K. Spectral stability of small-amplitude shock profiles for dissipative symmetric hyperbolic-parabolic systems. Z. Angew. Math. Phys. 53 (2002), no. 1, 20-34.
17. Kanel', Ja. I. A model system of equations for the one-dimensional motion of a gas. Differencial'nye Uravnenija 4 (1968), 721-734.
18. Kapitula, T. Stability of weak shocks in λ.-ω systems. Indiana Univ. Math. J. 40 (1991), no. 4, 1193-1219.
19. ∞ spaces. J. Differential Equations 112 (1994), no. 1, 179-215.
20. Kawashima, S. Systems of a hyperbolic-parabolic composite type, with applications to the equations of magnetohydrodynamics. Doctoral thesis, Kyoto University, 1983.
21. Kawashima, S.; Matsumura, A. Asymptotic stability of traveling wave solutions of systems for one-dimensional gas motion. Comm. Math. Phys. 101 (1985), no. 1, 97-127.
22. 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 (1986), no. 7, 249-252.
23. 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, 11. SIAM, Philadelphia, 1973.
24. Liu, T.-P. The Riemann problem for general systems of conservation laws. J. Differential Equations 18 (1975), 218-234.
25. Liu, T.-P. Shock waves for compressible Navier-Stokes equations are stable. Comm. Pure Appl. Math. 39 (1986), no. 5, 565-594.
26. Liu, T.-P. Pointwise convergence to shock waves for viscous conservation laws. Comm. Pure Appl. Math. 50(1997), no. 11, 1113-1182.
27. Liu, T.-P.; Zeng, Y. Large time behavior of solutions for general quasilinear hyperbolic-parabolic systems of conservation laws. Mem. Amer. Math. Soc. 125 (1997), no. 599.
28. Liu, T.-P.; Zeng, Y. Private communication, September 2001 (manuscript in preparation).
29. Liu, T.-P.; Zumbrun, K. Nonlinear stability of an undercompressive shock for complex Burgers equation. Comm. Math. Phys. 168 (1995), no. 1, 163-186.
30. Liu, T.-P.; Zumbrun, K. On nonlinear stability of general undercompressive viscous shock waves. Comm. Math. Phys. 174 (1995), no. 2, 319-345.
31. Majda, A. The existence of multidimensional shock fronts. Mem. Amer. Math. Soc. 43 (1983), no. 281.
32. Majda, A. The stability of multidimensional shock fronts. Mem. Amer. Math. Soc. 41 (1983), no. 275.
33. Majda, A. Compressible fluid flow and systems of conservation laws in several space variables. Springer, New York, 1984.
34. Majda, A.; Pego, R. Stable viscosity matrices for systems of conservation laws. J. Differential Equations 56 (1985), no. 2, 229-262.
35. Mascia, C.; Zumbrun, K. Pointwise Green's function bounds and stability of relaxation shocks. Indiana Univ. Math. J. 51 (2002), no. 4, 773-904.
36. Mascia, C.; Zumbrun, K. Pointwise Green function bounds for shock profiles of systems with real viscosity. Arch. Ration. Mech. Anal. 169 (2003), no. 3, 177-263.
37. Mascia, C.; Zumbrun, K. Stability of large amplitude viscous shock profiles of hyperbolic-parabolic systems. Arch. Ration. Mech. Anal., to appear.
38. Matsumura, A.; Nishihara, K. On the stability of travelling wave solutions of a one-dimensional model system for compressible viscous gas. Japan J. Appl. Math. 2 (1985), no. 1, 17-25.
39. Pego, R. L. Stable viscosities and shock profiles for systems of conservation laws. Trans. Amer. Math. Soc. 282 (1984), no. 2, 749-763.
40. Sattinger, D. On the stability of waves of nonlinear parabolic systems. Advances in Math. 22 (1976), no. 3, 312-355.
41. Serre, D.; Zumbrun, K. Boundary layer stability in real-vanishing viscosity limit. Comm. Math. Phys. 221 (2001), no. 2, 267-292.
42. Shizuta, Y.; Kawashima, S. Systems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation. Hokkaido Math. J. 14 (1985), no. 2, 249-275.
43. Smoller, J. Shock waves and reaction-diffusion equations. 2nd ed. Grundlehren der Mathematischen Wissenschaften, 258. Springer, New York, 1994.
44. 1 asymptotic behavior of compressible, isentropic, viscous 1-d flow. Comm. Pure Appl. Math. 47 (1994), no. 8, 1053-1082.
45. Zeng, Y. Gas dynamics in thermal nonequilibrium and general hyperbolic systems with relaxation. Arch. Ration. Mech. Anal. 150 (1999), no. 3, 225-279.
46. Zumbrun, K. Stability of viscous shock waves. Lecture notes, Indiana University, Bloomington, 1998.
47. Zumbrun, K. Refined wave-tracking and nonlinear stability of viscous Lax shocks. Methods Appl. Anal. 7 (2000), no. 4, 747-768.
48. Zumbrun, K. Multidimensional stability of planar viscous shock waves. Advances in the theory of shock waves, 307-516. Progress in Nonlinear Differential Equations and Their Applications, 47. Birkhäuser, Boston, 2001.
49. Zumbrun, K. Stability of large-amplitude shock waves of compressible Navier-Stokes equations. Preprint, 2003. Handbook of fluid dynamics, to appear.
50. Zumbrun, K. Planar stability criteria for viscous shock waves of systems with real viscosity. CIME summer school notes. Preprint, 2004.
51. Zumbrun, K.; Howard, P. Pointwise semigroup methods and stability of viscous shock waves. Indiana Univ. Math. J. 47 (1998), no. 3, 741-871.
52. Zumbrun, K.; Serre, D. Viscous and inviscid stability of multidimensional planar shock fronts. Indiana Univ. Math. J. 48 (1999), no. 3, 937-992.