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
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Additional Information
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R. Courant and K. O. Friedrichs, Supersonic flow and shock waves, Wiley-Interscience, New York, 1948
R. Courant and P. D. Lax, Nonlinear partial differential equations with two dependent variables, Comm. Pure Appl. Math. 2 (1949)
P. D. Lax, Hyperbolic systems of conservation laws II, Comm. Pure Appl. Math. 10 (1957)
P. D. Lax, Development of singularities of solutions of nonlinear partial differential equations, J. Mathematical Phys. 5 (1964)
P. D. Lax, Hyperbolic systems of conservation laws and the mathematical theory of shock waves, in Regional Conference Series in Applied Mathematics, SIAM 1973
N. J. Zabusky, Exact solution for the vibrations of a nonlinear continuous model string, J. Mathematical Phys. 3 (1962)
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)
A. Jeffrey, The evolution of discontinuities in solutions of homogeneous nonlinear hyperbolic equations having smooth initial data, J. Math. Mech. 17 (1967)
A. Jeffrey, Quasilinear hyperbolic systems and waves. Research Notes in Mathematics, Pitman, San Francisco, 1976
J. Nishida, Global smooth solutions for the second-order quasilinear wave equation with the first-order dissipation, unpublished (1975)
M. Slemrod, Instability of steady shearing flows in a nonlinear viscoelastic fluid. Arch. Rat. Mech. Anal. 68 (1979)
B. L. Rozhdestvenskii, On the discontinuity of solutions of quasilinear wave equations, Amer. Math. Soc. Transl. (2) 101 (1973)
F. John, Formation of singularities in one-dimensional nonlinear wave propagation, Comm. Pure Appl. Math. 27 (1974)
W. Kosinski, Gradient catastrophe in the solution of nonlinear hyperbolic systems, J. Math. Anal. Appl. 61 (1977)
P. H. Chang, On the breakdown phenomena of solutions of quasilinear wave equations, Mich. Math. J. 23 (1976)
V. Thomée, Difference methods for two-dimensional mixed problems for hyperbolic first-order systems, Arch. Rat. Mech. Anal. 8 (1961).
A. S. D. Wang, On free oscillations of elastic incompressible bodies in finite shear, Int. J. Engng. Sci. 7 (1969)
C. C. Wang and C. Truesdell, Introduction to rational elasticity, Nordhoff, 1973
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)
D. G. Schaeffer, A regularity theorem for conservation laws, Advances Math. 11 (1973).
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© Copyright 1980
American Mathematical Society