Uniqueness and stability results for non-linear Johnson-Segalman viscoelasticity and related models

In this paper we have proved exponential asymptotic stability for the corotational incompressible diffusive Johnson-Segalman viscolelastic model and a simple decay result for the corotational incompressible hyperbolic Maxwell model. Moreover we have established continuous dependence and uniqueness results for the non-zero equilibrium solution. In the compressible case, we have proved a Holder continuous dependence theorem upon the initial data and body force for both models, whence follows a result of continuous dependence on the initial data and, therefore, uniqueness. For the Johnson-Segalman model we have also dealt with the case of negative elastic viscosities, corresponding to retardation effects. A comparison with other type of viscoelasticity, showing short memory elastic effects, is given.