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

Kim, Jong Uhn

doi:10.1090/S0002-9947-1994-1270666-7

On the local regularity of solutions in linear viscoelasticity of several space dimensions
HTML articles powered by AMS MathViewer

by Jong Uhn Kim PDF

Trans. Amer. Math. Soc. 346 (1994), 359-398 Request permission

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

Richard Beals, Characterization of pseudodifferential operators and applications, Duke Math. J. 44 (1977), no. 1, 45–57. MR 435933
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 195336, DOI 10.1007/BF00250213
Constantine M. Dafermos, An abstract Volterra equation with applications to linear viscoelasticity, J. Differential Equations 7 (1970), 554–569. MR 259670, DOI 10.1016/0022-0396(70)90101-4
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 111293, DOI 10.1098/rsta.1960.0006
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
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, DOI 10.1017/CBO9780511662805
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
Lars Hörmander, On the existence and the regularity of solutions of linear pseudo-differential equations, Enseign. Math. (2) 17 (1971), 99–163. MR 331124

The analysis of linear partial differential operators

W. J. Hrusa and M. Renardy, On wave propagation in linear viscoelasticity, Quart. Appl. Math. 43 (1985), no. 2, 237–254. MR 793532, DOI 10.1090/S0033-569X-1985-0793532-0
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, DOI 10.1137/S0036141092227599
R. C. MacCamy, A model Riemann problem for Volterra equations, Arch. Rational Mech. Anal. 82 (1983), no. 1, 71–86. MR 684414, DOI 10.1007/BF00251725
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, DOI 10.1007/BF01515713
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

Pseudodifferential operators

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