The concept of minimal attractor and maximal attractors of partial differential equations of the Kuramoto-Sivashinsky type

The first part of the paper is devoted to the definition of the smallest set that may be regarded as the attractor of a dissipative system. Simple properties of this set are discussed. In the second part some geometrical facts concerning attractors of partial differential equations are revealed. They include well-known results for the two-dimensional Navier-Stokes equation and new ones for the Kuramoto-Sivashinsky and similar equations.