MURPHYS-HSFS-2014 - 7th International Workshop on Multi-Rate Processes & Hysteresis, 2nd International Workshop on Hysteresis and Slow-Fast Systems, April 7-11, 2014 - Abstract

Visintin, Augusto

Strong and weak formulation of models of hysteresis

Hysteresis occurs in continuum mechanics, magnetism, biology, phase transitions and so on. Hysteresis evolution has extensively been studied in the framework of function spaces, starting with the pioneering research of R. Bouc about 50 years ago, and with that of the Russian school of M.A. Krasnosel'skiĭ, A.V. Pokrovskiĭ and others in the 1970s.
Hysteresis processes in space-distributed systems have been modeled by coupling an (either continuous or discontinuous) hysteresis relation between $u$ and $w$ with PDEs like begingather fracpartialpartial t (u+ w) -Delta u =f qquad mbox quasilinear parabolic, in $bf R^Ntimes ]0,T[$, labelone tag1
fracpartial^2partial t^2 (u+ w) -Delta u=f qquad 2^mboxndmbox-order quasilinear hyperbolic, in $bf R^Ntimes ]0,T[$, labeltwo tag2
fracpartialpartial t (u+ w) + fracpartial upartial x =f qquad 1^hboxstmbox-order quasilinear hyperbolic, in $bf Rtimes ]0,T[$. labelthree tag3 endgather medskip This talk will survey some of these models, distinguishing between: beginitemize item[(i)] a strong formulation of bf continuous/ hysteresis operators, typically in $C^0([0,T])$; item[(ii)] a weak formulation of bf discontinuous/ hysteresis operators (relays and Preisach models), in terms of it nonvariational/ inequalities; item[(iii)] a variational formulation of Prandtl-Ishlinskiĭ models in terms of variational inequalities. enditemize The main results for scalar problems include: beginitemize item[(i)] well-posedness of a strong formulation of (refone), for continuous hysteresis, item[(ii)] well-posedness of a weak formulation of (refone), for discontinuous hysteresis, item[(iii)] existence of a weak solution of (reftwo), for discontinuous hysteresis, item[(iv)] well-posedness of a weak formulation of (refthree), for discontinuous hysteresis (with a Kružkov-type entropy condition). enditemize Some of these results take over to vector hysteresis.
The weak formulation will be illustrated.