PAICU, Marius; REKALO, Andrey; RAUGEL, Geneviève

doi:10.1016/j.jde.2011.10.015

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PAICU, Marius
Institut de Mathématiques de Bordeaux [IMB]

REKALO, Andrey
Université Paris-Sud - Paris 11 [UP11]

RAUGEL, Geneviève
Laboratoire de Mathématiques d'Orsay [LMO]

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Journal of Differential Equations. 2012, vol. 252, n° 6, p. 3695-3751

Elsevier

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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 alpha > 0, these equations are globally well posed and admit a compact global attractor A(alpha) in (H-3(T-2))(2). We prove that, for any alpha > 0, there exists beta(alpha) > 0, such that A(alpha) belongs to (H3+beta(alpha) (T-2))(2) if the forcing term is in (H1+beta(alpha) (T-2))(2). We also show that this attractor is contained in any Sobolev space (H3+m(T-2))(2) provided that alpha 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(alpha). Furthermore we prove that on A(alpha), 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.< Réduire

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

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