Structural stability of rate-independent nonpotential flows

Augusto Visintin

doi:10.3934/dcdss.2013.6.257

Article Contents

2013, Volume 6, Issue 1: 257-275. Doi: 10.3934/dcdss.2013.6.257

This issue Previous Article Quasistatic damage evolution with spatial $\mathrm{BV}$-regularization Next Article

Structural stability of rate-independent nonpotential flows

Augusto Visintin^1,

1.
Dipartimento di Matematica, Università di Trento, via Sommarive 14, 38050, Povo di Trento

Received: April 2011

Revised: August 2011

Published: February 2013

Abstract / Introduction Related Papers Cited by

Abstract

Several phenomena may be represented by doubly-nonlinear equations of the form $$ \alpha(D_tu) - \nabla\cdot \gamma(\nabla u)\ni h, $$ 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 $\Omega$.

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Mathematics Subject Classification: 35K60, 47H05, 49J40, 58E.

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