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**Navier-Stokes equations on thin 3D domains. I: Global attractors and global regularity of solutions.**
*(English)*
Zbl 0787.34039

This work is devoted to the regularity of solutions to the Navier-Stokes equations (NSE) in a thin 3D domain \(Qx(0,\varepsilon)\), where \(\varepsilon\) is a small parameter and \(Q\) is a bounded region on the plane. Particularly is treated the case when \(Q\) is a rectangle and the boundary conditions for the spatial region are periodic. As in the 2D case to the NSE exists a globally regular solution, and the fact that a thin 3D domain is “close” to a 2D one, the authors show how can use good properties of 2D NSE to study the global regularity of 3D NSE. A first difficulty appears as a consequence that 3D NSE on the thin domain is a singular perturbation of the 2D NSE on \(Q\), and a regularization of this perturbation problem is made here.

The main result in this paper is concerning the existence of a strong solution to the 3D NSE on the thin domain with periodic conditions, provided the initial condition and the forcing function belongs to given “large sets”. As a consequence of formulated and proved here \(H^ 1\) and \(H^ 2\)-Regularity Theorems, the set of strong solutions of the associated evolutionary equations has a local attractor in \(H^ 1\) (compact in \(H^ 2\) for indicated cases) that is a global attractor for the Leray solutions to 3D NSE on the thin domain. In order to give facilities to the reader in the technical terminology, the authors give a review about skew-product flows including local and global attractors for such semiflows. In the paper is commented the case in which Dirichlet conditions on the thin boundary of the parallelepiped are assumed and, at the same time, periodic boundary conditions on the two (not thin) parallel faces. In this case the validity of the results concerning the existence of the attractors is stated. When the boundary conditions are exclusively of Dirichlet type the authors give here appropriate estimates and prove the existence of a unique maximal compact (local) attractor for the corresponding skew-product semiflow and, that this set is the global attractor for the Leray solutions to the abstract nonlinear evolutionary equations.

The main result in this paper is concerning the existence of a strong solution to the 3D NSE on the thin domain with periodic conditions, provided the initial condition and the forcing function belongs to given “large sets”. As a consequence of formulated and proved here \(H^ 1\) and \(H^ 2\)-Regularity Theorems, the set of strong solutions of the associated evolutionary equations has a local attractor in \(H^ 1\) (compact in \(H^ 2\) for indicated cases) that is a global attractor for the Leray solutions to 3D NSE on the thin domain. In order to give facilities to the reader in the technical terminology, the authors give a review about skew-product flows including local and global attractors for such semiflows. In the paper is commented the case in which Dirichlet conditions on the thin boundary of the parallelepiped are assumed and, at the same time, periodic boundary conditions on the two (not thin) parallel faces. In this case the validity of the results concerning the existence of the attractors is stated. When the boundary conditions are exclusively of Dirichlet type the authors give here appropriate estimates and prove the existence of a unique maximal compact (local) attractor for the corresponding skew-product semiflow and, that this set is the global attractor for the Leray solutions to the abstract nonlinear evolutionary equations.

Reviewer: M.Rodriguez Ricard (La Habana)

### MSC:

37-XX | Dynamical systems and ergodic theory |

35Q30 | Navier-Stokes equations |

35K55 | Nonlinear parabolic equations |

58B99 | Infinite-dimensional manifolds |

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