Weak Solutions for a Class of Nonlinear Systems of Viscoelasticity

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.