Local attractor for \(n-D\) Navier-Stokes system (Q1268693)

The authors study the Navier-Stokes system on a bounded subset \(\Omega\subset\mathbb{R}^n\) \((n\geq 3)\) by writing it as an abstract evolution equation of the form \[ {du\over dt}+ A_ru= F_ru+ P_rf,\quad t>0,\quad u(0)= u_0. \] It is well known that this problem has a local solution for \(f\in L^r(\Omega)^n\) and \(u_0\in X^\alpha_r\) (\({1\over 2}\leq\alpha<1\), \(r>n\)), where \(X^\alpha_r= D(A^\alpha_r)\) and \(A_r\) is the Stokes operator. The authors prove the existence of a metric space \(V\subset X^\alpha_r\) such that the nonlinear semigroup \((T(t))\) restricted to \(V\) has a global attractor \({\mathcal A}_{\alpha,r}\). Furthermore, \({\mathcal A}_{\alpha,r}\) is shown to be a local attractor in a neighborhood of zero. The set \(V\) is constructed via a uniform estimate on the norm of \(T(t)u_0\) in \(X^{{1\over 2}}_r\) under the assumption that \(\| f\|_{L^r}\) is subject to a smallness condition.