Dynamics of Partial Differential Equations

Volume 14 (2017)

Number 4

Dynamics of a non-autonomous incompressible non-Newtonian fluid with delay

Pages: 375 – 402

DOI: https://dx.doi.org/10.4310/DPDE.2017.v14.n4.a4

Authors

Linfang Liu (Department of Mathematics, Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai, China; and Dpto. Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas Universidad de Sevilla, Spain)

Tomás Caraballo (Dpto. Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas Universidad de Sevilla, Spain)

Xianlong Fu (Department of Mathematics, Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai, China)

Abstract

We first study the well-posedness of a non-autonomous incompressible non-Newtonian fluid with delay. The existence of global solution is obtained by classical Galerkin approximation and the energy method. Actually, we also prove the uniqueness of solution as well as the continuous dependence on the initial value. Then we analyze the long time behavior of the dynamical system associated to the incompressible non-Newtonian fluid. Finally, we establish the existence of pullback attractors for the non-autonomous dynamical system associated to the problem.

Keywords

incompressible non-Newtonian equation, pullback attractor, delay, the energy method

2010 Mathematics Subject Classification

Primary 35B40, 35Q35. Secondary 37L55, 76D03.

Full Text (PDF format)

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 11371087), Science and Technology Commission of ShanghaiMunicipality (No. 13dz2260400) and Shanghai Leading Academic Discipline Project (No. B407), respectively.

Received 23 April 2017

Published 14 December 2017