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Viscoelastic fluids: free energies, differential problems and asymptotic behaviour. (English) Zbl 1302.74037

Summary: 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.

MSC:

74D05 Linear constitutive equations for materials with memory
76A10 Viscoelastic fluids
35B40 Asymptotic behavior of solutions to PDEs
30E20 Integration, integrals of Cauchy type, integral representations of analytic functions in the complex plane
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