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 \begin{aligned} \partial_tu= \nu\Delta u-\nabla p- (u\nabla)u+ f- g(t, u_t)\quad &\text{in }(\tau,\infty)\times \Omega,\\ \text{div}(u)= 0,\;u= 0\quad &\text{on }(\tau,\infty)\times \partial\Omega,\;u(\tau, x)= u_0(x),\\ u(t,x)= \phi(t- \tau,x)\quad &\text{for }t\in (\tau- h,\tau),\;x\in\Omega.\end{aligned}\tag{1} Here, $$u_t(s)= u(t+ s)$$, $$s\in (-h,0)$$; the delay is encoded in $$\phi$$, $$g$$. $$\Omega\subset\mathbb{R}^2$$ 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 $$V_0= \{u\in C^\infty_0(\Omega)$$, $$\text{div}(u)= 0\}$$, while $$V$$ is the closure of $$V_0$$ with respect to the norm $$\|\nabla u\|$$. In order to take care of the delay, spaces $$C_H$$, $$C_V$$, $$L^2_H$$ are introduced, where $$C_H= C^0([- h,0], H)$$ etc. The delay is taken into account by $$g(t,\xi)$$ which is a mapping from $$\mathbb R\times C_H$$ to $$L^2(\Omega)^2$$. The function $$g(t,\xi)$$ is subject to a series of assumption involving measurability and two kinds of Lipschitz conditions. System (1) is then put into the abstract form $\partial_t u+\nu Au+ Bu= f+ g(t, u_t),\;u(0)= u_0\tag{2}$ 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 $$\{A(t),\,t\in\mathbb{R}\}$$ 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:

 35Q30 Navier-Stokes equations 35R10 Partial functional-differential equations 47H20 Semigroups of nonlinear operators 37L30 Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems
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References:

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