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

Mielke, Alexander

Modeling jumps in rate-independent systems using balanced-viscosity solutions

Balanced Viscosity solutions to rate-independent systems arise as limits of regularized rate-independent flows by adding a superlinear vanishing-viscosity dissipation. The simplest equation is of the form [ 0in mathrmSign(dot u) varepsilon dot u -Delta u u-u^3 ell(t,x).] We address the main issue of proving the existence of the limit $varepsilon to 0$ for infinite-dimensional generalized gradient systems and of characterizing them by a couple of variational properties that combine a local stability condition and a balanced energy-dissipation identity. A careful description of the jump behavior of the solutions, of their differentiability properties, and of their equivalent representation by time rescaling is also presented. This is joint research with Riccarda Rossi, Giuseppe Savaré, and Sergey Zelik. small begindescriptionitemsep-0.1em item scshape A. Mielke. newblock Differential, energetic, and metric formulations for rate-independent processes. In em Nonlinear PDE's and Applications, Springer 2011, pp. 87--170. item scshape ---, R. Rossi, G. Savaré. newblock Modeling solutions with jumps for rate-independent systems on metric spaces. newblock em DCDS A, 25(2), 585--615, 2009. item scshape ---, S. Zelik. newblock On the vanishing viscosity limit in parabolic systems with rate-independent dissipation terms. newblock em Ann. Sc. Norm. Sup. Pisa Cl. Sci. (5), 2014. newblock To appear. WIAS preprint 1500. item scshape ---, R. Rossi, G. Savaré. newblock BV solutions and viscosity approximations of rate-independent systems. newblock em ESAIM Control Optim. Calc. Var., 18(1), 36--80, 2012. item scshape ---, R. Rossi, G. Savaré. newblock Variational convergence of gradient flows and rate-independent evolutions in metric spaces. newblock em Milan J. Math., 80, 381--410, 2012. item scshape ---, R. Rossi, G. Savaré. newblock Balanced-viscosity (BV) solutions to infinite-dimensional rate-independent systems. newblock em WIAS preprint 1845, 2013. enddescription