Global and exponential attractors for a Ginzburg-Landau model of superfluidity

The long-time behavior of the solutions for a non-isothermal model in superuidity is investigated. The model describes the transition between the normal and the superuid phase in liquid 4He by means of a non-linear di erential system, where the concentration of the superuid phase satisfi a non-isothermal Ginzburg-Landau equation. This system, which turns out to be consistent with thermodynamical principles and whose well-posedness has been recently proved, has been shown to admit a Lyapunov functional. This allows to prove existence of the global attractor which consists of the unstable manifold of the stationary solutions. Finally, by exploiting recent techinques of semigroups theory, we prove the existence of an exponential attractor offnite fractal dimension which contains the global attractor.