Balanced viscosity (BV) solutions to infinite-dimensional rate-independent systems
- Mielke, Alexander
- Rossi, Riccarda
- Savaré, Giuseppe
2010 Mathematics Subject Classification
- 49Q20 58E99
- Doubly nonlinear equations, generalized gradient flows, rate-independent systems, vanishing-viscosity limit, variational Gamma convergence, energy-dissipation balance, arclength parameterized solutions
Balanced Viscosity solutions to rate-independent systems arise as limits of regularized rate-independent ows by adding a superlinear vanishing-viscosity dissipation. We address the main issue of proving the existence of such limits for innite-dimensional 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 dierentiability properties, and of their equivalent representation by time rescaling is also presented. Our techniques rely on a suitable chain-rule inequality for functions of bounded variation in Banach spaces, on rened lower semicontinuity-compactness arguments, and on new BVestimates that are of independent interest.
- J. Eur. Math. Soc. (JEMS), 18 (2016), pp. 2107--2165.