Rate-independent systems in Banach spaces

Abstract

In the Banach-space setting, we assume that the topologies are given by either the weak or the strong topology. Throughout this chapter, we will assume that the topological spaces \(\mathcal{Y}\) and \(\mathcal{Z}\) of Chapter 2 are given via separable reflexive Banach spaces \(\boldsymbol{Y }\) and \(\boldsymbol{Z}\) equipped with their weak topologies, unless stated otherwise explicitly. Thus \(\mathcal{Y}\subset \boldsymbol{Y }\), \(\mathcal{Z}\subset \boldsymbol{Z}\), and \(\mathcal{Q}:= \mathcal{Y}\times \mathcal{Z}\) from ( 2.0.1) is a subset of $$\displaystyle\begin{array}{rcl} \boldsymbol{Q} = \boldsymbol{Y \times \boldsymbol{Z}\quad \mbox{ with separable, reflexive Banach spaces $\boldsymbol{Y }$ and $\boldsymbol{Z}$.}& &{}\end{array}$$ (3.0.1) In Banach spaces, we have two important additional tools deriving from the linear structure. First, the functionals at hand may have differentials or subdifferentials such that it is possible to formulate force balances, such as $$\displaystyle{ \partial _{\dot{q}}\mathcal{R}(q(t),\dot{q}(t)) + \partial _{q}\mathcal{E}(t,q(t)) \ni 0\qquad \mbox{ for a.a. } t\! \in \! [0,T], }$$ (3.0.2) and to formulate rate equations rather than compare energies, as in the energetic formulation. Second, we can employ convexity and duality methods such as the Legendre–Fenchel transform, as indicated in Section 1.3.4 Here we use the symbol ∂ for the Frechet subdifferential (cf. Section 3.3.1 for the definition), which generalizes the convex subdifferential and the Frechet derivative, and \(\partial _{a}\mathcal{J} (a,b)\) or \(\partial _{b}\mathcal{J} (a,b)\) denotes the partial Frechet subdifferentials, where b or a is kept fixed, respectively.