Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Stability of viscous scalar shock fronts in several dimensions


Author: Jonathan Goodman
Journal: Trans. Amer. Math. Soc. 311 (1989), 683-695
MSC: Primary 35K30; Secondary 35B35, 35L67
DOI: https://doi.org/10.1090/S0002-9947-1989-0978372-9
MathSciNet review: 978372
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We prove nonlinear stability of planar shock front solutions for viscous scalar conservation laws in two or more space dimensions. The proof uses the "integrated equation" and an effective equation for the motion of the front itself. We derive energy estimates that balance terms from the integrated equation with terms from the front motion equation.


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

  • [DS] N. Dunford and J. T. Schwartz, Linear operators, Wiley, New York, 1958.
  • [F] Gerald B. Folland, Introduction to partial differential equations, Princeton University Press, Princeton, N.J., 1976. Preliminary informal notes of university courses and seminars in mathematics; Mathematical Notes. MR 0599578
  • [Ge] I. M. Gel′fand, Some problems in the theory of quasi-linear equations, Uspehi Mat. Nauk 14 (1959), no. 2 (86), 87–158 (Russian). MR 0110868
  • [Go] Jonathan Goodman, Nonlinear asymptotic stability of viscous shock profiles for conservation laws, Arch. Rational Mech. Anal. 95 (1986), no. 4, 325–344. MR 853782, https://doi.org/10.1007/BF00276840
  • [IO] A. M. Il' in and O. A. Oleinik, Behavior or the solution of the Cauchy problem for certain quasilinear equations for unbounded increase of the time, Amer. Math. Soc. Transl. (2) 42 (1964), 19-23.
  • [KM] Shuichi Kawashima and Akitaka Matsumura, Asymptotic stability of traveling wave solutions of systems for one-dimensional gas motion, Comm. Math. Phys. 101 (1985), no. 1, 97–127. MR 814544
  • [Li] Tai-Ping Liu, Nonlinear stability of shock waves for viscous conservation laws, Mem. Amer. Math. Soc. 56 (1985), no. 328, v+108. MR 791863, https://doi.org/10.1090/memo/0328
  • [Lu] G. Ludford (Ed.), Reacting flows: Combustion and chemical reators, Proc. '85 AMS/SIAM Summer Seminar in Appl. Math., Cornell Univ., Amer. Math. Soc., Providence, R.I., 1986.
  • [S] D. H. Sattinger, On the stability of waves of nonlinear parabolic systems, Advances in Math. 22 (1976), no. 3, 312–355. MR 0435602, https://doi.org/10.1016/0001-8708(76)90098-0

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 35K30, 35B35, 35L67

Retrieve articles in all journals with MSC: 35K30, 35B35, 35L67


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1989-0978372-9
Article copyright: © Copyright 1989 American Mathematical Society