Regularity of the global attractor and finite-dimensional behavior for the second grade fluid equations

Abstract

This paper is devoted to the large time behavior and especially to the regularity of the global attractor of the second grade fluid equations in the two-dimensional torus. We first recall that, for any size of the material coefficient α > 0 , these equations are globally well posed and admit a compact global attractor A α in ( H 3 ( T 2 ) ) 2 . We prove that, for any α > 0 , there exists β ( α ) > 0 , such that A α belongs to ( H 3 + β ( α ) ( T 2 ) ) 2 if the forcing term is in ( H 1 + β ( α ) ( T 2 ) ) 2 . We also show that this attractor is contained in any Sobolev space ( H 3 + m ( T 2 ) ) 2 provided that α is small enough and the forcing term is regular enough. These arguments lead also to a new proof of the existence of the compact global attractor A α . Furthermore we prove that on A α , the second grade fluid system can be reduced to a finite-dimensional system of ordinary differential equations with an infinite delay. Moreover, the existence of a finite number of determining modes for the equations of the second grade fluid is established.