Attractors for nonautonomous 2D Navier-Stokes equations with normal external forces.

*(English)*Zbl 1083.35094The authors prove he existence of attractors for the 2d-Navier-Stokes equations with an exterior force \(f(t)\) by using the theory of processes by Babin, Vishik and others. A process is based on a Banach space \(X\), an index set \(\Sigma\) (itself a topological vector space). A process is a collection \(U_\sigma(t,\tau)\) of nonlinar operators, acting on \(X\), labeled by \(\sigma\in\Sigma\) such that
\[
U_\sigma(t,s) U_\sigma(s,\tau)= U_\sigma(t,\tau),\;t\geq s\geq \tau,\;U_\sigma(\tau, \tau)= \text{Id},\;\tau\in\mathbb{R},\;\sigma\in\Sigma.\tag{1}
\]
\(\Sigma\) is called the symbol space, \(\sigma\) a symbol. Concepts such as uniform attractor, uniform absorbing set etc. are now introduced. E.g., \(B_0\subset X\) is uniformly absorbing if given \(C\in\mathbb{R}\) and a bounded set \(B\subset X\) there is \(T_0= T_0(\tau,B)\geq \tau\) such that
\[
\bigcup_{\sigma\in\Sigma} U_\sigma(t,\tau)B\subset B_0\quad \text{for }t\geq T_0.\tag{2}
\]
The authors now prove a number of preparatory lemmas concerning properties of processes. These results are then applied to the 2d-Navier-Stokes equation on a smooth bounded domain \(\Omega\). To this end this equation is put into standard abstract form
\[
\partial_t u+\nu Au+ B(u,u)= f(t),\quad u(0)= u_0\tag{3}
\]
based on the Hilbert spaces
\[
\begin{aligned} H&= \{u\in L^2(\Omega)^2,\,\text{div}(u)= 0,\,u\cdot\vec n|_{\partial\Omega}= 0\},\text{ norm }|\;|,\\ V&= \{u\in H^1_0(\Omega)^2,\,\text{div}(u)= 0\},\text{ norm }\|\;\|. \end{aligned}
\]
The exterior force \(f(t)= \sigma(t)\), \(t\in\mathbb{R}\) is taken as symbol of the system (3), resp. of the induced process; one assumes
\[
\sup_t \int^{t+1}_t |f(s)|^2\,ds< \infty.\tag{4}
\]
After recalling global existence and uniqueness of solutions of (3) the authors proceed to prove the existence of a uniform attractor \(A_0\) of (3) and investigate its properties. The relevant Theorem 3.3 states among others (expressed somewhat losely) that if \(f(t)\), \(t\in\mathbb{R}\) has an additional property called “normal” then the \(A_0\) associated with \(f(t)\), \(t\in\mathbb{R}\) coincides with the uniform attractor \(A_b\) associated with \(f(t+b)\), \(t\in\mathbb{R}\) , for any \(b\in\mathbb{R}\). Further results of this type are obtained (Theorems 4.1, 4.2).

Reviewer: Bruno Scarpellini (Basel)

##### MSC:

35Q30 | Navier-Stokes equations |

35B40 | Asymptotic behavior of solutions to PDEs |

35B41 | Attractors |

37L30 | Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems |

76D05 | Navier-Stokes equations for incompressible viscous fluids |