Exponential stability of an incompressible non-Newtonian fluid with delay

Linfang Liu; Tomás Caraballo; Xianlong Fu

doi:10.3934/dcdsb.2018138

Article Contents

2018, Volume 23, Issue 10: 4285-4303. Doi: 10.3934/dcdsb.2018138

This issue Previous Article Dynamics of weak solutions for the three dimensional Navier-Stokes equations with nonlinear damping Next Article Finite dimensionality of global attractor for the solutions to 3D viscous primitive equations of large-scale moist atmosphere

Exponential stability of an incompressible non-Newtonian fluid with delay

1.
Department of Mathematics, Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai 200241, China
2.
Dpto. Ecuaciones Diferenciales y Análisis Numérico, Fac. Matemáticas, Universidad de Sevilla, c/Tarfia s/n, 41012-Sevilla, Spain

* Corresponding author: caraball@us.es
* Corresponding author: caraball@us.es

Received: November 30, 2017

Revised: December 31, 2017

Early access: April 2018

Published: September 2018

This research was partially supported by the projects MTM2015-63723-P (MINECO/FEDER, EU) and P12-FQM-1492 (Junta de Andalucía), and by NSF of China (Nos. 11671142 and 11771075), Science and Technology Commission of Shanghai Municipality (No. 13dz2260400) and Shanghai Leading Academic Discipline Project (No. B407), respectively.

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Abstract

The existence and uniqueness of stationary solutions to an incompressible non-Newtonian fluid are first established. The exponential stability of steady-state solutions is then analyzed by means of four different approaches. The first is the classical Lyapunov function method, while the second one is based on a Razumikhin type argument. Then, a method relying on the construction of Lyapunov functionals and another one using a Gronwall-like lemma are also exploited to study the stability, respectively. Some comments concerning several open research directions about this model are also included.

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Mathematics Subject Classification: Primary: 34D20, 34K20, 34K21, 37L15, 76A05.

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