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Attractors for 2D-Navier-Stokes models with delays. (English) Zbl 1068.35088

The authors consider the Navier-Stokes equations with delay in 2D. The equation in question is

t u=νΔu-p-(u)u+f-g(t,u t )in(τ,)×Ω,div(u)=0,u=0on(τ,)×Ω,u(τ,x)=u 0 (x),u(t,x)=φ(t-τ,x)fort(τ-h,τ),xΩ·(1)

Here, u t (s)=u(t+s), s(-h,0); the delay is encoded in φ, g. Ω 2 is a smooth bounded domain. In order to cast (1) into an abstract form, a standard L 2 -setting is imposed on (1). E.g., H is the L 2 -closure of V 0 ={uC 0 (Ω), div(u)=0}, while V is the closure of V 0 with respect to the norm u. In order to take care of the delay, spaces C H , C V , L H 2 are introduced, where C H =C 0 ([-h,0],H) etc. The delay is taken into account by g(t,ξ) which is a mapping from ×C H to L 2 (Ω) 2 . The function g(t,ξ) is subject to a series of assumption involving measurability and two kinds of Lipschitz conditions. System (1) is then put into the abstract form

t u+νAu+Bu=f+g(t,u t ),u(0)=u 0 (2)

along conventional lines. The authors first recall a result obtained in an earlier paper, stating that (under some assumptions) (2) admits a unique global solution in some suitable function space. The authors then proceed to show that (2) admits a global attractor.

In this context, the usual notion of attractor does not suffice; it has to be replaced by a new notion, i.e. by a family {A(t),t} called pullback attractor, related to a family U(t,s) of evolution operators associated with (2). After proving the existence of an absorbing set, the authors then show that under suitable assumptions on the data, a unique, uniformly bounded pullback attractor for (2) exists. The paper concludes with an application to the case where (2) is supplied by a forcing term.

MSC:
35Q30Stokes and Navier-Stokes equations
35R10Partial functional-differential equations
47H20Semigroups of nonlinear operators
37L30Attractors and their dimensions, Lyapunov exponents