Existence results for the flow of viscoelastic fluids with a differential constitutive law

where 2 is the (symmetric) extra-stress tensor (the total stress is given by g = -pL + I, where p is the hydrodynamic pressure); 4 is the rate of deformation tensor, g[u] = *(Vu + Vu’), u being the velocity field; A, is the relaxation time, A2 the retardation time, 0 I A, < A1 ; and Q is the fluid viscosity. The symbol a),/d)t denotes an objective (frame indifferent) derivative [I, 181. More precisely,

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