Existence of weak solutions to the elastic string equations in three dimensions
Author:
Andrea M. Reiff
Journal:
Quart. Appl. Math. 60 (2002), 401-424
MSC:
Primary 35L65; Secondary 74H20
DOI:
https://doi.org/10.1090/qam/1914433
MathSciNet review:
MR1914433
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Abstract: Many applied problems resulting in hyperbolic conservation laws are nonstrictly hyperbolic. As of yet, there is no comprehensive theory to describe the solutions of these systems. We examine the equations modeling an elastic string of infinite length in three-dimensional space, restricted to possess non-simple eigenvalues of constant multiplicity. We show that there exists a weak solution of the nonstrictly hyperbolic conservation law when the total variation of the initial data is sufficiently small. The proof technique is similar to Glimm’s classical existence for hyperbolic conservation laws, but necessarily departs from Glimm’s proof by not requiring strict hyperbolicity.
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George F. Carrier, On the non-linear vibration problem of the elastic string, Quart. Appl. Math. 3, 157–165 (1945)
N. Cristescu, Dynamic Plasticity, North-Holland Publ. Co., Amsterdam, 1967
J. L. Doob, Measure Theory, Springer-Verlag, New York, 1994
Heinrich Freistühler, Linear degeneracy and shock waves, Mathematische Zeitschrift 207, 583–596 (1991)
James Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Communications on Pure and Applied Mathematics 18, 95–105 (1965)
R. V. Iosue, A Case Study of Shocks in Non-linear Elasticity, Ph.D. thesis, Adelphi University, 1971
Barbara Lee Keyfitz and Herbert C. Kranzer, A system of non-strictly hyperbolic conservation laws arising in elastic theory, Archive for Rational Mechanics and Analysis 72, 219–241 (1980)
Peter D. Lax, Hyperbolic systems of conservation laws, II, Communications on Pure and Applied Mathematics X, 537–566 (1957)
Peter D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, Society for Industrial and Applied Mathematics, Philadelphia, 1973
Randall J. LeVeque, Numerical Methods for Conservation Laws, Birkäuser, Boston, 1992
A. M. Reiff and Anthony Kearsley, Existence of weak solutions to a class of nonstrictly hyperbolic conservation laws with non-interacting waves, to appear in “Pacific Journal of Mathematics"
Joel A. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, New York, 1983
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© Copyright 2002
American Mathematical Society