Dynamics of second grade fluids: the Lagrangian approach. Recent trends in dynamical systems
Résumé
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 < p < + ∞, where T2 is the two-dimensional torus. Like previous authors, we introduce an auxiliary transport equation in the course of the proof of this existence result. Since the second grade fluid equations are globally well posed, their solutions define a dynamical system S α (t). We prove that S α (t) admits a compact global attractor Aα in V 3, p . We show that, for any α > 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.