Hostname: page-component-8448b6f56d-m8qmq Total loading time: 0 Render date: 2024-04-24T12:28:32.431Z Has data issue: false hasContentIssue false
  • English
  • Français

The viscous damping prevents propagation of singularities in the system of viscoelasticity

Published online by Cambridge University Press:  14 November 2011

Piotr Rybka
Piotr Rybka
Affiliation:
Departamento de Matemàticas, Centro de Investigatión y de Estudios Avanzados del IPN, Apartado Postal 14-740, 07000 México, D.F., México
Get access Rights & Permissions [Opens in a new window]

Synopsis

We show that the linear viscous damping Δut, is so strong that it altogether prevents propagation of singularities of the gradient of solutions to the system of viscoelasticity. Moreover, no creation or annihilation of singularities is possible in finite time.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 127 , Issue 5 , 1997 , pp. 1067 - 1074
Copyright
Copyright © Royal Society of Edinburgh 1997

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Ball, J. M. and James, R. D.. Fine phase mixtures as minimizers of energy.Arch. Rational Mech. Anal. 100 (1987), 13–52.CrossRefGoogle Scholar
2Friesecke, G. and Dolzmann, G.. Implicit time discretization and global existence for a quasilinear evolution equation with nonconvex energy. SIAM J. Math. Anal. 28 (1997).CrossRefGoogle Scholar
3Fujiwara, D. and Morimoto, H.. An L r-theorem of the Helmholtz decomposition of vector fields. J. Fac. Sci. Univ. Tokyo, Sect. 1A Math. 24 (1977), 685–700.Google Scholar
4Hörmander, L.. The analysis of linear partial differential operators, vol. III (Berlin: Springer, 1985).Google Scholar
5Klouček, P. and Luskin, M.. The computation of the dynamics of the martensitic transformation. Contin. Mech. Thermodyn. 6 (1994), 209–40.CrossRefGoogle Scholar
6Klouček, P. and Luskin, M.. Computational modeling of the martensitic transformation with surface energy. Math. Comput. Modelling 20 (1994), 101–21.CrossRefGoogle Scholar
7Pego, R.. Phase transitions in one-dimensional viscoelasticity: admissibility and stability. Arch. Rational Mech. Anal. 97 (1987), 353–94.CrossRefGoogle Scholar
8Rybka, P.. Dynamical modelling of phase transitions by means of viscoelasticity in many dimensions. Proc. Roy. Soc. Edinburgh Sect. A 121 (1992), 101–38.CrossRefGoogle Scholar
9Rybka, P.. A priori estimates for gradients of solutions to systems of viscoelasticity in many dimensions. Topol. Methods Nonlinear Anal. 3 (1994), 235–56.CrossRefGoogle Scholar
10Swart, P. J. and Holmes, Ph.. Energy minimization and the formation of microstructure in dynamic anti-plane shear. Arch. Rational Mech. Anal. 121 (1992), 37–85.Google Scholar