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

Franca Franchi; Barbara Lazzari; Roberta Nibbi

doi:10.3934/dcdsb.2014.19.2111

Article Contents

2014, Volume 19, Issue 7: 2111-2132. Doi: 10.3934/dcdsb.2014.19.2111

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

1.
Department of Mathematics, University of Bologna, Piazza di Porta S. Donato, 5, 40126 Bologna

Received: March 31, 2013

Revised: June 30, 2013

Published: August 2014

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Abstract

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 Hölder 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.

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Mathematics Subject Classification: Primary: 76A05, 35B30; Secondary: 76A10.

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