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

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.

Exponential stability of an incompressible non-Newtonian fluid with delay

“…where constant λ 1 > 0 denotes the first eigenvalue of the operator A. Let us recall a result ensuring existence and uniqueness of solution to our problem which was stated and proved in [17].…”

“…The existence and uniqueness of solution, the existence of maximal compact attractor and global (or pullback) attractor for non-Newtonian equations have been studied in [1,2,3,13,17,23,24,25], while Guo et al analyzed in [12] the martingale stationary solutions for some stochastic non-Newtonian fluids without delay. However, to the best of our knowledge, there are no available works concerning the 4286 LINFANG LIU, TOMÁS CARABALLO AND XIANLONG FU local stability analysis of incompressible non-Newtonian fluids containing hereditary characteristics (constant, distributed or variable delay, memory, etc).…”

“…Enlightened by the analysis carried out in [6] in the case of 2D-Navier-Stokes equations, in the current paper we focus on the asymptotic behavior of the stationary solution of the following incompressible non-Newtonian fluid with delay in a 2D bounded domain Ω ⊂ R 2 whose existence and uniqueness of solutions have already been analyzed in [17] (see…”

“…In this respect, it is worth mentioning that Guo and Lin studied in [11] the existence and uniqueness of stationary solutions of non-Newtonian viscous incompressible fluids without delay, but this reference does not contain a complete proof for the existence of such stationary solution, a gap which is solved in our current paper since the result in [11] can be obtained as a particular case of the analysis we are doing in this paper by just taking h = 0. We would also like to recall that the existence and uniqueness of solutions, and the existence of pullback attractors of our delay model have been investigated in our previous work [17].…”

“…Several options, for instance, the theories of skewproduct and uniform attractor are available, but we would like to emphasize that the theory of pullback attractors may allow more general non-autonomous terms in the models. In this respect, the existence of pullback attractor of an incompressible non-Newtonian fluids with bounded delay has been established in [17].…”