Structural stability of rate-independent nonpotential flows
Augusto Visintin · Discrete and Continuous Dynamical Systems - S · 2012
Several phenomena may be represented by doubly-nonlinear equations of the form$$\alpha(D_tu) - abla\cdot \gamma( abla u) i h,$$with $\alpha$ and $\gamma$ (possibly multivalued) maximal monotone mappings.Hysteresis effects are characterized by rate-independence, which corresponds to $\alpha$ positively homogeneous of zero degree. Fitzpatrick showed that any maximal monotone relation may be represented variationally.On this basis, an initial- and boundary-value problem associated to the equation above is here formulated as a null-minimization problem, without assuming $\gamma$ to be cyclically monotone.Existence of a solution $u\in H^1(0,T; H^1(\Omega))$ is proved, as well as its stability with respect to variations of the data, of the mapping $\gamma$, and of the domain $\Omega$.