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Attractors for 2D-Navier-Stokes equations with delays on some unbounded domains. (English) Zbl 1131.35059

The authors study the inhomogeneous Navier-Stokes equations (NSE) with delay on an unbounded domain \(\Omega\subseteq\mathbb R^2\), which is not required to have smooth boundary; however it is assumed to satisfy a Poincaré-inequality. The NSE with delay are given as follows: \[ \partial_t+\nu Au+B(u)=f(t)+g(t,u_t),\quad t\in (\tau,\infty).\tag{1} \]
System (1) is interpreted in a standard \(L^2\)-setting and based on the familiar Hilbert spaces \(V\), \(H\) [R. Temam, Navier-Stokes equations and nonlinear functional analysis. SIAM (1995; Zbl 0833.35110)]. While \(A=-P_\Delta\) is the Stokes operator based on homogeneous Dirichlet boundary conditions, \(B(u)=P(u\nabla)u\) is the nonlinearity and \(f(t)\) the exterior force; \(g(t,u_t)\) is the term which takes care of the delay. More explicitly, set \(C_H=C^0([-h,0];H)\) and \(L^2_X=L^2((-h,0);X)\) for \(X=H,V,V'\) and define \(u_t\) by \(u_t(s)=u(t+s)\) for \(s\in[-h,0]\). The initial conditions underlying (1) are
\[ u(\tau,x)=u_0(x),\quad u(t,x)=\varphi(t-\tau,x)\text{ for }t\in(\tau-h,\tau),\quad x\in\Omega,\tag{2} \]
where \(\varphi(s)\), \(s\in[-h,0]\) is given. The delay function \(g(\;)\) is by definition a mapping \(g:\mathbb R\times C_H\to L^2(\Omega)^2\), subject to five conditions, two of which are of Lipschitz type. Based on this frame, the authors interprete (1) and (2) in the distributional sense and recall that (1) and (2) admits a unique global solution lying in \(C^0([\tau,\infty],H)\) (Theorem 1).
Since the authors want to prove the existence of global attractors for (1) and (2) they first have to introduce some relevant concepts. Since (1) is nonautonomous, the usual solution semigroup \(S(t)\), \(t\geq 0\) has to be replaced by a process \(S(t_1,t_2)\), \(t_1\geq t_2\), acting on a space \(X\), which satisfies \(S(t,t)x=x\), \(x\in X\) and \(t\in\mathbb R\) and \(S(t_2,t_1) S(t_1,t_0)=S(t_2,t_0)\) for \(t_0\leq t_1\leq t_2\). The notion of pullback \(\mathcal D\)-asymptotically compact process is then introduced, where \(\mathcal D\) is a nonempty class of parametrized sets \(\widehat D=\{D(t)\), \(t\in\mathbb R\}\subseteq P(X)\), with \(P(X)\) the family of all nonempty subsets of \(X\). Using these notions, the authors give appropriate definitions of \(\omega\)-limit sets and attractors and prove some relevant properties.
The main result of the paper asserts that under suitable assumptions on \(f\), \(g\) there exists a unique global pullback \({\mathcal D}_m\)-attractor for the process \(S(t_1,t_2)\), \(t_1\geq t_2\) associated with (1) and (2) via Theorem 1. Here, \({\mathcal D}_m\) is a special class of parametrized sets, whose definition has to be omitted.

MSC:

35Q30 Navier-Stokes equations
37L30 Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems
35R10 Partial functional-differential equations
35B41 Attractors
76D05 Navier-Stokes equations for incompressible viscous fluids

Citations:

Zbl 0833.35110
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References:

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