Young‐Measure Solutions for a Viscoelastically Damped Wave Equation with Nonmonotone Stress‐Strain Relation

Abstract.

We study the viscoelastically damped wave equation \ddot u =\pl_x \left( \sigma(\pl_x u) + \beta \pl_x \dot u\right) - \alpha u, \quad \alpha \geq 0, \, \beta > 0,\; x \in (0,1) 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.