Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On the local regularity of solutions in linear viscoelasticity of several space dimensions

Author: Jong Uhn Kim
Journal: Trans. Amer. Math. Soc. 346 (1994), 359-398
MSC: Primary 35L10; Secondary 35B65, 35R10, 45K05, 73F15
MathSciNet review: 1270666
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we discuss the local regularity of solutions of a nonlocal system of equations which describe the motion of a viscoelastic medium in several space dimensions. Our main tool is the microlocal analysis combined with MacCamy's trick and the argument of the classical energy method.

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

  • [1] R. Beals, Characterization of pseudodifferential operators and applications, Duke Math. J. 44 (1977), 45-57; correction 46 (1979), p. 215. MR 0435933 (55:8884)
  • [2] B. D. Coleman, M. E. Gurtin, and I. R. Herrera, Waves in materials with memory, Arch. Rational Mech. Anal. 19 (1965), 1-19, 239-265. MR 0195336 (33:3538)
  • [3] C. M. Dafermos, An abstract Volterra equation with applications to linear viscoelasticity, J. Differential Equations 7 (1970), 554-569. MR 0259670 (41:4305)
  • [4] G. F. D. Duff, The Cauchy problem for elastic waves in an anisotropic medium, Philos. Trans. Roy. Soc. London Ser. A 252 (1960), 249-273. MR 0111293 (22:2157)
  • [5] J. M. Greenberg, L. Hsiao and R. C. MacCamy, A model Riemann problem for Volterra equations, Volterra and Functional Differential Equations (K. Hannsgen et al., eds.), Marcel Dekker, New York, 1982, pp. 25-43. MR 703531 (84j:45049)
  • [6] G. Gripenberg, S-O. Londen, and O. Staffans, Volterra integral and functional equations, Cambridge Univ. Press, Cambridge, 1990. MR 1050319 (91c:45003)
  • [7] K. B. Hannsgen and R. L. Wheeler, Behavior of the solutions of a Volterra equation as a parameter tends to infinity, J. Integral Equations 7 (1984), 229-237. MR 770149 (86b:45004)
  • [8] L. Hörmander, On the existence and the regularity of solutions of linear pseudo-differential equations, Enseign. Math. (2) 17 (1971), 99-163. MR 0331124 (48:9458)
  • [9] -, The analysis of linear partial differential operators. Vol. 3, Springer-Verlag, Berlin, 1985.
  • [10] W. J. Hrusa and M. Renardy, On wave propagation in linear viscoelasticity, Quart. Appl. Math. 43 (1985), 237-254. MR 793532 (86j:45022)
  • [11] J. U. Kim, Local regularity of the one-dimensional motion of a viscoelastic medium, SIAM J. Math. Anal. (to appear). MR 1325912 (96b:35139)
  • [12] R. C. MacCamy, A model Riemann problem for Volterra equations, Arch. Rational Mech. Anal. 82 (1983), 71-86. MR 684414 (84e:45007)
  • [13] M. Renardy, Some remarks on the propagation and non-propagation of discontinuities in linearly viscoelastic liquids, Rheology Acta 21 (1982), 251-254. MR 669374 (83j:76007)
  • [14] M. Renardy, W. J. Hrusa, and J. A. Nohel, Mathematical problems in viscoelasticity, Longman, New York, 1986. MR 919738 (89b:35134)
  • [15] M. E. Taylor, Pseudodifferential operators, Princeton Univ. Press, Princeton, NJ, 1981.
  • [16] -, Rayleigh waves in linear elasticity as a propagation of singularities phenomenon, Partial Differential Equations and Geometry (Proc. Conf., Park City, UT, 1977; C. I. Byrnes, ed.), Marcel Dekker, New York, 1979, pp. 273-291. MR 535598 (80i:73016)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 35L10, 35B65, 35R10, 45K05, 73F15

Retrieve articles in all journals with MSC: 35L10, 35B65, 35R10, 45K05, 73F15

Additional Information

Keywords: Local regularity, MacCamy's trick, propagation of singularities, energy method, microlocal regularity, bicharacteristic strip, bicharacteristic curve, singular support
Article copyright: © Copyright 1994 American Mathematical Society

American Mathematical Society