Global attractors of the 3He-4He system in Hα\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$H_{\alpha}$\end{document} spaces

In this paper, the existence of a global attractor for the 3He-4He system is investigated. By using an iteration procedure, combining with the classical existence theorem of global attractors, we prove that this system possesses a global attractor, which attracts any bounded set of Hα\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$H_{\alpha}$\end{document} in Hα\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$H_{\alpha}$\end{document}-norm.


Introduction
Superfluidity is a phase of matter in which 'unusual' effects are observed when liquids, typically of helium- or helium-, overcome friction by surface interaction when at a stage, known as the 'lambda point' for helium-, at which viscosity of the liquid becomes zero.
Experiments have indicated that helium atoms have two stable isotopes  He and  He, and liquid  He and  He can be dissolved into each other. The general features of phase separation and superfluidity in mixtures of liquid  He and  He have been known for some time [-].
Up to now, we find several mathematical results on model for liquid mixture of  He- He. In [, ], Ma and Wang introduced a dynamical Ginzburg-Landau phase transition/separation model for the mixture of liquid helium- and helium-, and studied phase separations between liquid helium- and liquid helium- from both the modeling and analysis points of view. The analysis leads to three critical length scales L  < L  < L  and the corresponding λ-transition and phase separation diagrams. In [], Ma studied existence of solutions to model for liquid mixture of  He- He by using spatial sequence techniques and linear operator theories. In [], Luo and Pu obtained the existence and regularity of solutions to  He- He system by using the Galerkin method.
As we know, the dynamical properties of the  He- He system such as the global asymptotical behaviors of solutions and existence of global attractors, are important for the study of phase transition and separation for mixture of liquid  He and  He, which ensure the stability of phase transition and provide the mathematical foundation for the study of phase transition dynamics.
In this article, we concerned with the global asymptotical behaviors of solutions and existence of global attractors of the  He- He system, to wit the following initial-boundary problem: where the unknown function u represents the mol fraction of liquid  He, and the complexvalued function ψ = ψ  + iψ  describes the phase transition of liquid  He between the normal and superfluid states. is the Laplace operator, ⊂ R n ( ≤ n ≤ ) is a C ∞ bounded domain. Due to the Landau average field theory (also see [, ]), we have We shall use the regularity estimates for the linear semigroups, combining with the classical existence theorem of global attractors, to prove that the  He- He system (.) possesses, in space H α , a global attractor, which attracts any bounded set of H α in H α -norm.
Attractor in H α have studied by some authors, we refer the reader to [-]. In [], Luo studied the extended Fisher-Kolmogorov equation, to wit the following initial-boundary problem: and obtained the existence of global attractors of (.) in H k spaces by using an iteration procedure. Although [] and this paper all use the iteration procedure, in this paper we not only consider the iteration procedures of the unknown functions u and ψ, but also the interaction between u and ψ. Because of that and because the unknown function ψ is a complex-valued function, the iteration procedure in this paper is more difficult than in [].
The rest of the paper is arranged as follows. In Section , we will iterate some notations and theorems for the abstract nonlinear evolution equation. In Section , we will state and prove our main result.

Preliminaries and auxiliary results
In this section, we iterate some notations, abstract theorems, and auxiliary results, which are important for getting our main result. Let In this case, we say that attracts U. Especially, if attracts any bounded set of H, is called a global attractor of S(t).
For a set D ⊂ H, we define the ω-limit set of D as follows: where the closure is taken in the H-norm.
Lemma . is the classical existence theorem of global attractors by Temam [].

for any bounded set U ⊂ H and some T >  sufficiently large, the set t≥T S(t)U is compact in H. Then the ω-limit set A = ω(B) of B is a global attractor of (.), and A is connected provided B is connected.
We used to assume that the linear operator L in (.) is a sectorial operator which generates an analytic semigroup e tL . It is known that there exists a constant λ ≥  such that L -λI generates the fractional power operator L α and fractional order space H α for α ∈ R  , where L = -(L -λI). Without loss of generality, we assume that L generates the fractional power operators L α and fractional order space H α as follows: For sectorial operators, we also have the following properties, which can be found in [].

Lemma . Let L : H  → H be a sectorial operator which generates an analytic semigroup T(t) = e tL .
If all eigenvalues λ of L satisfy Re λ < -λ  for some real number λ  > , then for where δ >  and C α >  are constants only depending on α, () the H α -norm can be defined by For convenience, we introduce the following result.
here the space H is defined as follows: Next we convert (.) into the abstract form (.). To do so, we need the following space: We define the operators L, G : where U = (u, ψ). Then the  He- He system (.) can be written in the abstract form (.).
It is easy to see that is a sectorial operator and L generates the fractional power operators L α and fractional order space H α as follows: where the space H α is given by Especially, the fractional power operator (-L)

Main result
In this section we state and prove our main result; some of our important ideas come from Ma's recent books [, ].
The main result in this article is given by the following theorem, which provides the existence of global attractors of the  He- He system (.) in space H α . ThenL,L generate the fractional power operatorsL α ,L α and fractional order spacesH α , H α as follows: Next, according to Lemma ., we prove Theorem . in the following five steps.
Step . We prove that for any bounded set W ⊂ H   there is a constant C >  such that the solution U(t, ϕ) of system (.) is uniformly bounded by the constant C for any ϕ ∈ W and t ≥ , i.e., To do that, we firstly check that system (.) has a global Lyapunov function as follows: Choosing ε  >  such   b  -ε  > ,   α  -ε  > , we get Combining with (.) yields where C  , C  , and C are positive constants. C only depends on ϕ. SetG Next, we show that for any bounded set W ⊂ H   there exists C >  such that We claim that g : we have Hence, by (.) and Lemma . we deduce that where β = α +   ( < β < ). We claim thatḠ : which implies thatḠ : H  ( ) × H  ( , C) → L  ( , C) is bounded. Hence, by (.) and Lemma . we deduce that which implies that Hence, (.) holds.
Step . We prove that for any bounded set W ⊂ H α ( ≤ α <   ) there is a constant C >  such that In fact, by the embedding theorems of fractional order spaces (see Pazy []): where ψ = ψ  + iψ  . Hence for α >   we see that Therefore, it follows from (.) and (.) that Then, by using the same method as that in Step , we get from (.) which implies that Therefore, it follows from (.) and (.) that Then, by using the same method as that in Step , we get from (.) which implies Hence, (.) holds.
Step . We prove that for any bounded set In fact, by the embedding theorems of fractional order spaces (see Pazy []): we deduce that where ψ = ψ  + iψ  . Hence for α ≥   we see that Therefore, it follows from (.) and (.) that Then, by using the same method as that in Step , we get from (.) Property () is obviously true, we now prove () in the following. It is easy to check if DF(U) = , U is a solution of the following equations: Taking the scalar product of (.) with (u, ψ), then we derive that Using the Hölder inequality and the above inequality, we have where C >  is a constant. Thus, property () is proved. Now, we show that system (.) has a bounded absorbing set in H α for any α ≥   , i.e., for any bounded set W ⊂ H α there are T >  and a constant C >  independent of ϕ such that U(t, ϕ) H α ≤ C, ∀t ≥ T, ϕ ∈ W .