Travelling waves in nonlinearly viscoelastic media and shock structure in elastic media
Authors:
Stuart S. Antman and Reza Malek-Madani
Journal:
Quart. Appl. Math. 46 (1988), 77-93
MSC:
Primary 73D15; Secondary 35L67, 73D40, 73F99
DOI:
https://doi.org/10.1090/qam/934683
MathSciNet review:
934683
Full-text PDF Free Access
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Additional Information
S. S. Antman, Material constraints in continuum mechanics, Atti Acc. Naz. Lincei, Rendiconti, Cl. Sci. fis. mat. nat. Ser. VII, 70, 256–264 (1982)
- Stuart S. Antman and Zhong Heng Guo, Large shearing oscillations of incompressible nonlinearly elastic bodies, J. Elasticity 14 (1984), no. 3, 249–262. MR 760032, DOI https://doi.org/10.1007/BF00041137
- Stuart S. Antman, J. L. Ericksen, David Kinderlehrer, and Ingo Müller (eds.), Metastability and incompletely posed problems, The IMA Volumes in Mathematics and its Applications, vol. 3, Springer-Verlag, New York, 1987. MR 870005
- Joel A. Smoller and Charles C. Conley, Shock waves as limits of progressive wave solutions of higher order equations. II, Comm. Pure Appl. Math. 25 (1972), 133–146. MR 306721, DOI https://doi.org/10.1002/cpa.3160250203
- Charles C. Conley and Joel A. Smoller, Topological methods in the theory of shock waves, Partial differential equations (Proc. Sympos. Pure Math., Vol. XXIII, Univ. California, Berkeley, Calif., 1971) Amer. Math. Soc., Providence, R.I., 1973, pp. 293–302. MR 0340836
- Constantine M. Dafermos, The mixed initial-boundary value problem for the equations of nonlinear one-dimensional viscoelasticity, J. Differential Equations 6 (1969), 71–86. MR 241831, DOI https://doi.org/10.1016/0022-0396%2869%2990118-1
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Ya. I. Kanel’, On a model system of equations of one-dimensional gas motion, (in Russian), Diff. Urav. 4, 721–734. Engl. Trans., Diff. Eqs. 4, 374–380 (1968)
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S. S. Antman, Material constraints in continuum mechanics, Atti Acc. Naz. Lincei, Rendiconti, Cl. Sci. fis. mat. nat. Ser. VII, 70, 256–264 (1982)
S. S. Antman and Zhong Heng Guo, Large shearing oscillations of incompressible nonlinearly elastic bodies, J. Elasticity 14, 249–262 (1984)
S. S. Antman and R. Malek-Madani, Dissipative mechanisms, in Metastability and incompletely posed problems, edited by S. S. Antman, J. L. Ericksen, D. Kinderlehrer, I. Müller, IMA volumes in Mathematics and its Applications, Vol. 3, Springer-Verlag, 1–16, 1987
C. C. Conley and J. A. Smoller, Shock waves as limits of progressive wave solutions of higher-order equations, II, Comm. Pure Appl. Math. 25, 131–146 (1972)
C. C. Conley and J. A. Smoller, Topological methods in the theory of shock waves, Proc. Symp. Pure Math. 23, 293–302 (1973)
C. Dafermos, The mixed initial-boundary value problem for the equations of nonlinear one-dimensional viscoelasticity, J. Diff. Eqs. 6, 71–86 (1969)
C. M. Dafermos, Global smooth solutions to the initial-boundary value problem for the equations of one-dimensional nonlinear thermoviscoelasticity, SIAM J. Math. Anal. 13, 397–408 (1982)
R. DiPerna, Uniqueness of solutions to hyperbolic conservation laws, Indiana Univ. Math. Journal 28, 137–188 (1979)
A. Friedman, Partial differential equations of parabolic type, Prentice-Hall, 1964
J. Goodman, Nonlinear asymptotic stability of viscous shock profiles for conservation laws, Arch. Rat. Mech. Anal. 95, 325–344 (1986)
R. Hagan and M. Slemrod, The viscosity-capillarity admissibility criterion for shocks and phase transitions, Arch. Rat. Mech. Anal. 83, 333–361 (1983)
Ya. I. Kanel’, On a model system of equations of one-dimensional gas motion, (in Russian), Diff. Urav. 4, 721–734. Engl. Trans., Diff. Eqs. 4, 374–380 (1968)
B. Keyfitz and H. Kranzer, A system of non-strictly hyperbolic conservation laws arising in elasticity theory, Arch. Rat. Mech. Anal. 72, 219–241 (1980)
O. A. Ladyženskaja, V. A. Solonnikov, and N. N. Ural’ceva, Linear and quasilinear equations of parabolic type, Translations of Mathematical Monographs 23, Amer. Math. Soc., Providence, R.I., 1968
P. D. Lax, Hyperbolic systems of conservation laws, II, Comm. Pure Appl. Math. 10, 537–566 (1957)
T.-P. Liu, The Riemann problem for general systems of conservation laws, J. Diff. Eqs. 18 218–234 (1975)
T.-P. Liu, Nonlinear stability of shock waves for viscous conservation laws, Memoirs of the AMS 328, 1985
T.-P. Liu, Shock waves for compressible Navier-Stokes equations are stable, Comm. Pure Appl. Math. 39, 565–594 (1986)
R. C. MacCamy, Existence, uniqueness and stability of ${u_{tt}} = \frac {\partial }{{\partial x}}\left ( {\sigma \left ( {{u_x}} \right ) + \lambda \left ( {{u_x}} \right ){u_{xt}}} \right )$, Indiana Univ. Math. J. 20, 231–238 (1970)
A. Majda and R. Pego, Stable viscosity matrices for systems of conservation laws, J. Diff. Eqs. 56, 229–262 (1985)
R. S. Rivlin and J. L. Ericksen, Stress-deformation relations for isotropic materials, J. Rat. Mech. Anal. 4, 323–425 (1955)
M. Shearer, The Riemann problem for the planar motion of an elastic string, J. Diff. Eqs. 61, 149–163 (1986)
J. A. Smoller, Shock waves and reaction-diffusion equations, Springer-Verlag, 1983
C. Truesdell and W. Noll, The non-linear field theories of mechanics, Handbuch der Physik III/3, Springer-Verlag, 1965
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© Copyright 1988
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