Abstract
We discuss the motion of nonlinear viscoelastic materials with fading memory in one space dimension. We formulate the mathematical problem, survey results for global existence of classical solutions to the initial value problem if the data are sufficiently small, and discuss in detail the development of singularities in initially smooth solutions for large data.
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References
L. Boltzmann, Zur Theorie der elastischen Nachwirkung, Ann. Phys. 7 (1876), Ergänzungsband, 624-654.
B. D. Coleman, M. E. Gurtin and I. R. Herrera, Waves in materials with memory, Arch. Rat. Mech. Anal. 19 (1965), 1–19; B. D. Coleman and M. E. Gurtin, ibid, 239-265.
C. M. Dafermos, Dissipation in materials with memory, in: J. A. Nohel, M. Renardy and A. S. Lodge (eds.), Viscoelasticity and Rheology, Academic Press, New York (1985), 221–234.
C. M. Dafermos, Development of singularities in the motion of materials with fading memory, Arch. Rat. Mech. Anal. 91 (1986), 193–205.
C.M. Dafermos and J. A. Nohel, Energy methods for nonlinear, hyperbolic Volterra integrodifferential equations, Comm. PDE 4 (1979), 219–278.
C. M. Dafermos and J. A. Nohel, A nonlinear hyperbolic Volterra equation in viscoelasticity, Amer. J. Math., Supplement (1981), 87–116.
H. Hattori, Breakdown of smooth solutions in dissipative nonlinear hyperbolic equations, Q. Appl. Math. 40 (1982/83), 113–127.
W. J. Hrusa, Global existence and asymptotic stability for a semilinear hyperbolic Volterra equation with large initial data, SIAM J. Math. Anal. 16 (1985), 110–134.
W. J. Hrusa and J. A. Nohel, Global existence and asymptotics in one-dimensional nonlinear viscoelasticity, in: P. G. Ciarlet and M. Roseau (eds.), Trends and Applications of Pure Mathematics to Mechanics, Springer Lecture Notes in Physics 195 (1984), 165–187.
W. J. Hrusa and J. A. Nohel, The Cauchy problem in one-dimensional nonlinear viscoelasticity, J. Diff. Eq. 59 (1985), 388–412.
W. J. Hrusa and M. Renardy, On a class of quasilinear partial integro-differential equations with singular kernels, J. Diff. Eq. 64 (1986), 195–220.
F. John, Formation of singularities in one-dimensional nonlinear wave propagation, Comm. Pure Appl. Math. 27 (1974), 377–405.
S. Klainerman and A. Majda, Formation of singularities for wave equations including the nonlinear vibrating string, Comm. Pure Appl. Math. 33 (1980), 241–263.
P. D. Lax, Development of singularities of solutions of nonlinear hyperbolic partial differential equations, J. Math. Phys. 5 (1964), 611–613.
R. C. MacCamy, A model for one-dimensional nonlinear viscoelasticity, Q. Appl. Math. 35 (1977), 21–33.
R. Malek-Madani and J. A. Nohel, Formation of singularities for a conservation law with memory, SIAM J. Math. Anal. 16 (1985), 530–540.
P. A. Markowich and M. Renardy, Lax-Wendroff methods for hyperbolic history value problems, SIAM J. Num. Anal. 21 (1984), 24–51; Corrigendum, SIAM J. Num. Anal. 22 (1985), 204.
M. Renardy, Recent developments and open prolems in the mathematical theory of viscoelasticity, in: J. A. Nohel, M. Renardy and A. S. Lodge (eds.), Viscoelasticity and Rheology, Academic Press, New York (1985), 345–360.
M. Slemrod, Instability of steady shearing flows in a nonlinear viscoelastic fluids, Arch. Rat. Mech. Anal. 68 (1978), 211–225.
M. Slemrod, Appendix: Breakdown of smooth shearing flow in visco-elastic fluids for two constitutive relations: the vortex sheet vs. the vortex shock, in: D. D. Joseph, Hyperbolic phenomena in the flow of viscoelastic fluids, in: J. A. Nohel, M. Renardy and A. S. Lodge (eds.), Viscoelasticity and Rheology, Academic Press, New York (1985), 309–321.
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Nohel, J.A., Renardy, M.J. (1987). Development of Singularities in Nonlinear Viscoelasticity. In: Dafermos, C., Ericksen, J.L., Kinderlehrer, D. (eds) Amorphous Polymers and Non-Newtonian Fluids. The IMA Volumes in Mathematics and Its Applications, vol 6. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-1064-1_7
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DOI: https://doi.org/10.1007/978-1-4612-1064-1_7
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