A survey on Navier-Stokes models with delays: Existence, uniqueness and asymptotic behavior of solutions

Tomás Caraballo; Xiaoying Han

doi:10.3934/dcdss.2015.8.1079

Article Contents

2015, Volume 8, Issue 6: 1079-1101. Doi: 10.3934/dcdss.2015.8.1079

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A survey on Navier-Stokes models with delays: Existence, uniqueness and asymptotic behavior of solutions

Tomás Caraballo^1, and
Xiaoying Han^2,

1.
Dpto. Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, Campus Reina Mercedes, Apdo. de Correos 1160, 41080 Sevilla
2.
221 Parker Hall, Department of Mathematics and Statistics, Auburn University, Auburn, AL 36849

Received: June 2015

Revised: September 2015

Published: December 2015

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Abstract

In this survey paper we review several aspects related to Navier-Stokes models when some hereditary characteristics (constant, distributed or variable delay, memory, etc) appear in the formulation. First some results concerning existence and/or uniqueness of solutions are established. Next the local stability analysis of steady-state solutions is studied by using the theory of Lyapunov functions, the Razumikhin-Lyapunov technique and also by constructing appropriate Lyapunov functionals. A Gronwall-like lemma for delay equations is also exploited to provide some stability results. In the end we also include some comments concerning the global asymptotic analysis of the model, as well as some open questions and future lines for research.

Keywords:

Mathematics Subject Classification: Primary: 35R10, 35B40; Secondary: 47H20, 58F39.

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