Relating a rate-independent system and a gradient system for the case of one-homogeneous potentials (Q2097623)

The author considers generalized gradient systems \(\left(X,\mathcal{F},\mathcal{R}\right)\). Specifically, two types of such generalized gradient systems are considered: the Hilbert-space gradient system \(\left(X, \mathcal{J}, \frac{1}{2}\left\|\cdot\right\|^{2}\right)\) and the energetic rate-independent system \(\left(X,\mathcal{E},\left\|\cdot\right\|\right)\). Here \(\left(X, \left\langle \cdot,\cdot\right\rangle, \left\|\cdot\right\|\right)\) is a Hilbert space, \(\mathcal{F}\) is an energy functional, \(\mathcal{R}\) is a dissipation structure, and \(\mathcal{E}(t,u)=t\mathcal{J}(u)\), as \(\mathcal{J}:X\rightarrow[0,+\infty]\) is positively homogeneous of degree 1, convex, lower semicontinuous and has a dense domain. In any case the induced evolution equation takes the form \(0\in \partial \mathcal{R}(\dot{u}(t))+\partial \mathcal{F}(t,u(t))\). The main result is a non-trivial existence and uniqueness result for the energetic rate-independent system. The results in [\textit{A. Mielke}, in: Handbook of differential equations: Evolutionary equations. Vol. II. Amsterdam: Elsevier/North-Holland. 461--559 (2005; Zbl 1120.47062); \textit{A. Mielke} et al., Arch. Ration. Mech. Anal. 162, No. 2, 137--177 (2002; Zbl 1012.74054)] are used. Many examples are presented, and in particular one example in the case of infinite dimension is given in \(L^{2}(\mathbb{R})\), where all solutions can be calculated explicitly and an initial value \(u^{0}\) can be chosen so that \(\int_{t=0}^{1}\left\|\dot{u}\right\|dt=\infty\).