Weak Solutions for a Class of Nonlinear Systems of Viscoelasticity |
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Authors: | Sophia Demoulini |
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Institution: | Mathematical Institute?University of Oxford ?currently at:?CMS?/?DPMMS?University of Cambridge?Wilberforce Road?Cambridge CB3 OWB?UK?e-mail: S.Demoulini@dpmms.cam.ac.uk, UK
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Abstract: | The principal focus of the article is the construction of classical weak solutions of the initial value problem for a class
of systems of viscoelasticity in arbitrary spatial dimension. The class of systems studied is large enough to incorporate
certain requirements dictated by frame indifference and also has a structure which allows for a variational treatment of the
time-discretized problem. Weak solutions for this system are constructed under certain monotonicity hypotheses and are shown
to satisfy various a priori estimates, in particular giving improved regularity for the time derivative. Also measure-valued solutions are obtained under
a uniform dissipation condition, which is much weaker than monotonicity. A special case of the viscoelastic system is the
gradient flow of a non-convex potential, for which measure-valued solutions are here obtained, a new result in the vectorial
case. Furthermore, in this setting it is possible to show that these measure-valued solutions satisfy a certain property which
ensures they coincide with the classical weak solution when this exists, as for example in the convex case where existence
and uniqueness are well known.
Accepted July 1, 2000?Published online December 6, 2000 |
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