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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
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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.


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  • [1] Richard Beals, Characterization of pseudodifferential operators and applications, Duke Math. J. 44 (1977), no. 1, 45–57. MR 0435933
  • [2] Bernard D. Coleman and Morton E. Gurtin, Waves in materials with memory. II. On the growth and decay of one-dimensional acceleration waves, Arch. Rational Mech. Anal. 19 (1965), 239–265. MR 0195336
  • [3] Constantine M. Dafermos, An abstract Volterra equation with applications to linear viscoelasticity, J. Differential Equations 7 (1970), 554–569. MR 0259670
  • [4] G. F. D. Duff, The Cauchy problem for elastic waves in an anistropic medium, Philos. Trans. Roy. Soc. London Ser. A 252 (1960), 249–273. MR 0111293
  • [5] J. M. Greenberg, Ling Hsiao, and R. C. MacCamy, A model Riemann problem for Volterra equations, Volterra and functional-differential equations (Blacksburg, Va., 1981), Lecture Notes in Pure and Appl. Math., vol. 81, Dekker, New York, 1982, pp. 25–43. MR 703531
  • [6] G. Gripenberg, S.-O. Londen, and O. Staffans, Volterra integral and functional equations, Encyclopedia of Mathematics and its Applications, vol. 34, Cambridge University Press, Cambridge, 1990. MR 1050319
  • [7] Kenneth B. Hannsgen and Robert L. Wheeler, Behavior of the solution of a Volterra equation as a parameter tends to infinity, J. Integral Equations 7 (1984), no. 3, 229–237. MR 770149
  • [8] Lars Hörmander, On the existence and the regularity of solutions of linear pseudo-differential equations, Enseignement Math. (2) 17 (1971), 99–163. MR 0331124
  • [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), no. 2, 237–254. MR 793532
  • [11] Jong Uhn Kim, Local regularity of the one-dimensional motion of a viscoelastic medium, SIAM J. Math. Anal. 26 (1995), no. 3, 738–749. MR 1325912, 10.1137/S0036141092227599
  • [12] R. C. MacCamy, A model Riemann problem for Volterra equations, Arch. Rational Mech. Anal. 82 (1983), no. 1, 71–86. MR 684414, 10.1007/BF00251725
  • [13] M. Renardy, Some remarks on the propagation and nonpropagation of discontinuities in linearly viscoelastic liquids, Rheol. Acta 21 (1982), no. 3, 251–254. MR 669374, 10.1007/BF01515713
  • [14] Michael Renardy, William J. Hrusa, and John A. Nohel, Mathematical problems in viscoelasticity, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 35, Longman Scientific & Technical, Harlow; John Wiley & Sons, Inc., New York, 1987. MR 919738
  • [15] M. E. Taylor, Pseudodifferential operators, Princeton Univ. Press, Princeton, NJ, 1981.
  • [16] Michael E. Taylor, Rayleigh waves in linear elasticity as a propagation of singularities phenomenon, Partial differential equations and geometry (Proc. Conf., Park City, Utah, 1977) Lecture Notes in Pure and Appl. Math., vol. 48, Dekker, New York, 1979, pp. 273–291. MR 535598

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1994-1270666-7
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