Young‐Measure Solutions for a Viscoelastically Damped Wave Equation with Nonmonotone Stress‐Strain Relation |
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Authors: | Florian Theil |
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Institution: | Institut für Angewandte Mathematik, Welfengarten 1, Universit?t Hannover, 30167 Hannover, Germany, DE
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Abstract: | We study the viscoelastically damped wave equation with a nonmonotone stress‐strain relation σ. This system describes the dynamics of phase transitions, which is closely related to the creation of microstructures. In
order to analyze the dynamic behavior of microstructures we consider highly oscillatory initial states. Two questions are
addressed in this work: How do oscillations propagate in space and time? What can be said about the long‐time behavior? An
appropriate tool to deal with oscillations are Young measures. They describe the local distribution or one‐point statistics
of a sequence of fast fluctuating functions. We demonstrate that highly oscillatory initial states generate in a unique fashion
an evolution in the space of Young measures and we derive the determining equations. Further on we prove a generalized dissipation
identity for Young‐measure solutions. As a consequence, it is shown that every low‐energy solution converges to a Young‐measure
equilibrium as t→∞. This is a generalization of G. Friesecke's & J. B. McLeod's FM96] convergence result for classical solutions to the case
of Young‐measure solutions.
(Accepted November 12, 1997) |
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