Open AccessOn the vanishing-viscosity limit in parabolic systems with rate-independent dissipation terms
Author(s) -
Alexander Mielke,
Sergey Zelik
Publication year - 2014
Publication title -
annali della scuola normale superiore di pisa. classe di scienze
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.444
H-Index - 37
eISSN - 2036-2145
pISSN - 0391-173X
DOI - 10.2422/2036-2145.201004_003
Subject(s) - dissipation , limit (mathematics) , viscosity , mathematical analysis , mathematics , physics , thermodynamics
We consider semilinear and quasilinear parabolic systems with a nonsmooth\udrate-independent and a viscous dissipation term in the limit of very slow\udloading rates, or equivalently with fixed loading and vanishing viscosity " > 0.\udBecause for nonconvex energies the solutions will develop jumps, we consider\udthe vanishing-viscosity limit for the graphs of the solutions in the extended state\udspace in arclength parametrization. Here the choice of the viscosity norm for\udparametrization is crucial to keep the subdifferential structure of the problem. A\udcrucial point in the analysis are new a priori estimates that are rate independent\udand that allow us to show that the total length of the graph remains bounded in\udthe vanishing-viscosity limit. To derive these estimates we combine parabolic\udregularity estimates with ideas from rate-independent systems