# American Institute of Mathematical Sciences

September  2014, 19(7): 1815-1835. doi: 10.3934/dcdsb.2014.19.1815

## Viscoelastic fluids: Free energies, differential problems and asymptotic behaviour

 1 Dipartimento di Matematica, Università di Pisa, Pisa, Italy, Italy 2 Dipartimento di Scienze di Base e Applicate, per l'Ingegneria - Sezione Matematica, Sapienza Università di Roma, Rome, Italy 3 School of Mathematical Sciences, Dublin Institute of Technology, Kevin Street, Dublin 8, Ireland

Received  March 2013 Revised  March 2014 Published  August 2014

Some expressions for the free energy in the case of incompressible viscoelastic fluids are given. These are derived from free energies already introduced for other viscoelastic materials, adapted to incompressible fluids. A new free energy is given in terms of the minimal state descriptor. The internal dissipations related to these different functionals are also derived. Two equivalent expressions for the minimum free energy are given, one in terms of the history of strain and the other in terms of the minimal state variable. This latter quantity is also used to prove a theorem of existence and uniqueness of solutions to initial boundary value problems for incompressible fluids. Finally, the evolution of the system is described in terms of a strongly continuous semigroup of linear contraction operators on a suitable Hilbert space. Thus, a theorem of existence and uniqueness of solutions admitted by such an evolution problem is proved.
Citation: Giovambattista Amendola, Sandra Carillo, John Murrough Golden, Adele Manes. Viscoelastic fluids: Free energies, differential problems and asymptotic behaviour. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 1815-1835. doi: 10.3934/dcdsb.2014.19.1815
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