Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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

   
 
 

 

Shearing motions and the formation of shocks in an elastic circular tube


Authors: R. L. Fosdick and G. P. MacSithigh
Journal: Quart. Appl. Math. 38 (1980), 191-207
MSC: Primary 35L65; Secondary 73C99
DOI: https://doi.org/10.1090/qam/580879
MathSciNet review: 580879
Full-text PDF Free Access

References | Similar Articles | Additional Information

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

  • [1] R. Courant and K. O. Friedrichs, Supersonic flow and shock waves, Wiley-Interscience, New York, 1948 MR 0029615
  • [2] R. Courant and P. D. Lax, Nonlinear partial differential equations with two dependent variables, Comm. Pure Appl. Math. 2 (1949) MR 0033443
  • [3] P. D. Lax, Hyperbolic systems of conservation laws II, Comm. Pure Appl. Math. 10 (1957) MR 0093653
  • [4] P. D. Lax, Development of singularities of solutions of nonlinear partial differential equations, J. Mathematical Phys. 5 (1964) MR 0165243
  • [5] P. D. Lax, Hyperbolic systems of conservation laws and the mathematical theory of shock waves, in Regional Conference Series in Applied Mathematics, SIAM 1973 MR 0350216
  • [6] N. J. Zabusky, Exact solution for the vibrations of a nonlinear continuous model string, J. Mathematical Phys. 3 (1962) MR 0146545
  • [7] R. C. MacCamy and V. J. Mizel, Existence and non-existence in the large of solutions of quasilinear wave equations, Arch. Rat. Mech. Anal. 25 (1967) MR 0216165
  • [8] A. Jeffrey, The evolution of discontinuities in solutions of homogeneous nonlinear hyperbolic equations having smooth initial data, J. Math. Mech. 17 (1967) MR 0236525
  • [9] A. Jeffrey, Quasilinear hyperbolic systems and waves. Research Notes in Mathematics, Pitman, San Francisco, 1976 MR 0417585
  • [10] J. Nishida, Global smooth solutions for the second-order quasilinear wave equation with the first-order dissipation, unpublished (1975)
  • [11] M. Slemrod, Instability of steady shearing flows in a nonlinear viscoelastic fluid. Arch. Rat. Mech. Anal. 68 (1979) MR 509225
  • [12] B. L. Rozhdestvenskii, On the discontinuity of solutions of quasilinear wave equations, Amer. Math. Soc. Transl. (2) 101 (1973)
  • [13] F. John, Formation of singularities in one-dimensional nonlinear wave propagation, Comm. Pure Appl. Math. 27 (1974) MR 0369934
  • [14] W. Kosinski, Gradient catastrophe in the solution of nonlinear hyperbolic systems, J. Math. Anal. Appl. 61 (1977) MR 0460912
  • [15] P. H. Chang, On the breakdown phenomena of solutions of quasilinear wave equations, Mich. Math. J. 23 (1976) MR 0460910
  • [16] V. Thomée, Difference methods for two-dimensional mixed problems for hyperbolic first-order systems, Arch. Rat. Mech. Anal. 8 (1961). MR 0129555
  • [17] A. S. D. Wang, On free oscillations of elastic incompressible bodies in finite shear, Int. J. Engng. Sci. 7 (1969)
  • [18] C. C. Wang and C. Truesdell, Introduction to rational elasticity, Nordhoff, 1973 MR 0468442
  • [19] R. J. Knops, H. A. Levine and L. E. Payne, Non-existence, instability, and growth theorems for solutions of a class of abstract nonlinear equations with applications to nonlinear elastodynamics, Arch. Rational Mech. Analysis 55 (1974) MR 0364839
  • [20] D. G. Schaeffer, A regularity theorem for conservation laws, Advances Math. 11 (1973). MR 0326178

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC: 35L65, 73C99

Retrieve articles in all journals with MSC: 35L65, 73C99


Additional Information

DOI: https://doi.org/10.1090/qam/580879
Article copyright: © Copyright 1980 American Mathematical Society

American Mathematical Society