Dynamics of second grade fluids: the Lagrangian approach. Recent trends in dynamical systems

This article is devoted to the mathematical analysis of the second grade fluid equations in the two-dimensional case. We first begin with a short review of the existence and uniqueness results, which have been previously proved by several authors. Afterwards, we show that, for any size of the material coefficient α > 0, the second grade fluid equations are globally well posed in the space V 3, p of divergence-free vector fields, which belong to the Sobolev space W3,p(T2)2 , 1 0, there exists β(α) > 0, such that Aα belongs to V 3 + β(α), p if the forcing term is in W1+β(α)(T2)2 . We also show that this attractor is contained in any Sobolev space V 3 + m, p provided that α is small enough and the forcing term is regular enough. The method of proof of the existence and regularity of the compact global attractor is new and rests on a Lagrangian method. The use of Lagrangian coordinates makes the proofs much simpler and clearer