Abstract

Bose–Einstein condensation is a gaseous, superfluid state of matter exhibited by bosons as they cool to near absolute zero, which was discovered as early as 1924 but was not experimentally realized until 1995. In 2006, Machida and Koyama developed the corresponding Ginzburg–Landau model for superfluid and Bose–Einstein condensation-spanning phenomena. We mainly consider the global attractor for the initial boundary value problem of the modified coupled Ginzburg–Landau equations, which come from the BCS-BEC crossover model. Combining Gronwall inequality, properties of the binomial function, with some suitable a priori estimates, we establish the existence of global attractors. The attractor results obtained in this paper can provide a strong theoretical basis for the experimental realization of the BCS-BEC spanning phenomenon, and the adopted research method can also serve as a reference for analysing other types of partial differential equation attractors.

1. Introduction

From 1924-1925, Bose–Einstein discovered a special class of particles that can be at any energy level at normal temperature, but if the temperature drops to near absolute zero, such particles plummet to the lowest energy level and gather together like a supersized particle, such as a sudden collapse of a large summer. This phenomenon is known as Bose–Einstein condensation (BEC) [1, 2]. Due to the peculiar nature of this phenomenon, it has attracted the interest of a wide range of experts and scholars who wish to visualize its dynamic behavior. Thus, in 2006, Machida and Koyama constructed a time-dependent Ginzburg–Landau theory, which described the crossover from a weak-coupling BCS state to a strong-coupling BEC state of superfluid atomic Fermi gases near the Feshbach resonance [3]. The coupled Ginzburg–Landau equations they constructed are as follows:where , coupling coefficients are constants, is the chemical potential, is the threshold energy of the Feshbach resonance, and generally is a complex number with .

The Ginzburg–Landau equation has been particularly favored by scientists for its ability to effectively capture various types of features in the model and has yielded fruitful results [410]. These results are both interpretations of the existence results for the solutions [4, 5, 79] and explorations of the attractor problem for the Ginzburg–Landau model [6, 10]. Particularly interesting is the long-time behavior, i.e., the limiting graph of the state points moving in phase space as time changes and eventually converging, also known as attractors. The famous “butterfly effect” is the simplification of the convection equation in the atmospheric equations to obtain the trajectory shape of the attractor similar to a butterfly. The study of attractors is very interesting. In this paper, we focus on the attractor of the Ginzburg–Landau equation model from the BCS-BEC span phenomenon.

Noting that the attractors can describe the long-term dynamic behavior of the system, in recent years, many scientists have turned their attention to the study of the attractor theory. However, the study of attractors is based on the existence of the weak solutions to the equations, and people have found that the Ginzburg–Landau equation is becoming more and more important. Then, more and more scientists have devoted themselves to studying the attractor of the Ginzburg–Landau equation and obtained fruitful results.

Most of these results are obtained for the single Ginzburg–Landau equation. For example, the complex Ginzburg–Landau equation

Doering C R, Gibbon J D, and Holm D D combined linearised sideband stability analysis with the theory of inertial manifolds to estimate Lyapunov and Hausdorff dimension of the attractor generated by equation (11). By replacing the last term of the above equation with , Li and Guo showed the existence of global solution and attractor of the equation in unbounded domain by introducing weigh function [12].

For the following derivative Ginzburg–Landau equation,

When

Boling and Bixiang discussed the existence of the global attractor for the equation and estimated the dimension of attractor [13]; Gao et al. improved the above results and proved the existence of the maximal attractor whenand the dependence of the dimension of the attractor on the parameters on the equation is explored [14].

In the following equation,

When

Gao and Guo obtained the existence of the solution and the global attractor of the equation and estimated the dimension of the attractor [15]; Gao proved the existence of exponential attractor [16].

The results for the set of Ginzburg–Landau equations coupled by two equations are very few, and most of the coupled equations are real coefficient equations [17, 18] or consist of at least one that is a real coefficient equation. The results for the set of Ginzburg–Landau equations coupled by two equations that are both complex coefficient equations are even fewer. According to the information available to the authors, only existential results obtained by the set of equations (1) and (2) considered in this paper under specific conditions were found [5, 19]. To further explore the properties of the coupled Ginzburg–Landau system of equations (1) and (2), the authors further analysed the overall attractor problem for the system of equations (1) and (2) under specific initial value conditions [10]. Due to the special structure of the system of equations (1) and (2), the attractor results cannot be obtained according to the existing research methods on the attractor problem if appropriate conditions are not given. For this reason, Fang [6] et al. added appropriate viscosity terms to the system of equations (1) and (2) to achieve the goal that attractor theory can be established using the existing research methods on attractors. Subsequently, Jiang et al. [20] generalized the results of Fang [6] by correcting the energy inequality.

In other words, by adding to equation (2) and additionally adding the external force terms and to equations (1) and (2), respectively, and then using the trajectory decomposition method, Jiang et al. studied the existence of a global attractor and estimated the fractal dimension of the global attractor [20].

Inspired by the above researches, we consider the global attractor problem of the modified time-dependent Ginzburg–Landau equations with the damped viscous term as the following:where is a bounded region of , , coupling coefficients are all real constants, is the chemical potential, is the threshold energy of the Feshbach resonance, is a damping parameter, is generally the complex number with , , the external force terms and are real-valued functions, and its are uniformly bounded with respect to .

Compared with the above results, our result has the following features:(1)The external force terms and are not only related to the space variable but also related to the time variable . This means that when dealing with the external force term, we should not only solve the difficulties caused by the space variable but also deal with the troubles caused by the time variable .(2)The value range of damping coefficient here is larger, and the result obtained here is also broader.(3)In particular, the resonance coefficient of Feshbach resonance in the equations considered in this paper is less than 0, which means the usual method to obtain the energy inequalities is invalidate. In order to overcome these difficulties, this paper focuses on the energy inequalities for and first and then using Poincare’s theory to deduce the estimate for and finally we obtain the following result:

Theorem 1. assume that and are the global weak solutions of equations (9)–(12), , , , , , , , , .

Then, equations (9)–(12) exist a global attractor , such that(1);(2), for any bound set , where

And is a semigroup operator generated by the weak solutions of the initial boundary problem equations (9)–(12), with

Finally, we get the distribution of this paper. This paper is composed of five parts. The first part is the introduction: it introduces the origin of the model, the background of the attractor research, the main advantages of the problem studied in this paper, and the main results of this paper. The second part is the basic lemmas: it introduces the basic definitions and basic lemmas used in the proof of the paper. The third part is the a priori estimation: it proves the various energy inequalities needed to establish the main results of this paper. The fourth part is the proof of the attractor: it is mainly the proof of the existence of the overall attractor. The fifth part is the conclusion: it mainly summarizes the main results obtained in this paper and presents the problems to be considered in the future.

2. Fundamental Lemma

In this section, we mainly introduce some basic inequalities and lemmas needed in the proof of the main result.

Lemma 1 (Gronwall’s inequality [21]). Let be a non-negative continuous differentiable function, for some positive constant , and , if

Then,

Definition 1 (see [22]). We assume that is a complete metric space and is a continuous semigroup on , we call the subset of is a global attractor of , if satisfies the following conditions:
Compactness: is a compact set.
Invariance: , .
Attractiveness: For any bounded set in , we havewhere the was defined as

Lemma 2 (see [23]). Let be a Banach space, is a set of semigroup operators, , it satisfies , , where is an identity operator, and assume that the semigroup operator satisfies the following conditions:(1)Operator is uniformly bounded on . In other words, for any , there exist a constant , such that when , for any , we have .(2)There exists a bounded absorbing set on , namely, for any bounded set , there exists a constant , such that when , we have .(3)When , is a completely continuous operator.

Then, the semigroup operator aa!37BE! % MathType! End!1!1! has a compact global attractor.

3. Priori Estimate

In order to prove Theorem 1, we need to establish some suitable priori estimates for global weak solutions and to equations (9)–(12). To simplify, we write and convert the equations (9) and (10) to the following equations (19) and (20).

In order to establish the attractor theory of the initial boundary value problem equations (9)–(12), we should give the existence theorem of the weak solutions to the problem equations (9)–(12) first.

Theorem 2. assume that , , , , , , , , , , and . Then, there exists a pair of global weak solutions to equations (9)–(12), and

Theorem 2 can be proved by the standard Faedo–Galerkin method. Then, the proof is omitted here.

With the existence of weak solutions, we can proceed to establish some suitable energy inequalities for weak solutions. Since in this paper, the scattering direction coefficient , which make us cannot obtain the appropriate energy inequalities by the conventional method. For example, we cannot get the estimation of the norm for the weak solutions directly. To overcome this difficulty, we do this in reverse and estimate the norm of the weak solution’s gradient at first and then extrapolate the properties of the weak solutions by the estimation of the gradient of the weak solutions. After then, we get the following result.

Lemma 3. assume that and are global weak solutions to equations (9)–(12), , , , , , , , , and .

Then, there exist some positive constants , , and , which are independent on , such thatwhere are Poincare’s coefficients.

Proof making the inner product of equation (19) with , integrating by parts, and then taking the real parts of the result equation, with , we can find that

Applying Young’s inequality and Poincare’s inequality, we can getwhere is Poincare’s coefficient.

Thus,

Note thatwhere

Integrating by parts, and employing formula (37), we discover

According to the properties of the binomial functions and lettingbe a nonpositive definite matrix, namely, , noting that and , we can find that

Substituting the above formula into equation (25), then

Choosing and small enough, such thatthen

Similarly, making the inner product of the equation (20) with , integrating by parts, and taking the real parts of the result equation, we obtain that

Applying Young’s inequality, we can discover

Choosing and small enough such thatthen

Now, adding to (where and are arbitrary positive constants),

Lett

That is,

And let , then

Thus,

Choose positive constants , and satisfying

Then,

Forwe have

Employing Gronwall’s inequality, we can find that

Noting that , then there exist constants independent on , such thatand

This means that

Using Poincare’s inequality, we haveand

The proof of Lemma 3 is completed.

Lemma 4. assume that and are the global weak solutions of equations (9)–(12), , , , , , , , and .

Then, there exist positive constants and , such that

Proof taking the derivative of equation (19) with respect to and then multiplying on both sides, integrating by parts, we obtain

Taking the real parts of the former equality and noting thatwe have

Applying Young’s inequality, we discover

By Theorem 2, we can find that

Combining Galiardo–Nirenberg’s inequality and Poincare inequality with Lemma 3, we havewhere , are bounded constants and is Poincare’s coefficient.

Letand substituting these estimates into the inequality equation (57), we can find that

Similarly, taking the derivative of equation (20) with respect to and then multiplying on both sides, integrating by parts, we have

Taking the real parts of the former equation and applying Young’s inequality, we can get that

Thus,

Now, adding to (where and are arbitrary positive constants), we have

Choosing suitable , , such that , hence

Let

We can compute that

Assume that , then

Substituting and into the inequality equation (66), we discover that

Now, let

We have

We follow define

Then,

Employing Gronwall’s inequality,which means thatand

We further have

Now, we complete the proof of Lemma 4.

4. The Existence of Global Attractor

In order to obtain the existence of the global attractor for equations (9)–(12), we need Lemma 2 to prove Theorem 1.

Proof of Theorem 1: In order to using Lemma 2 to prove Theorem 1, we should verify the conditions of Lemma 2 in turn. Thus, we choose Banach space , and assume that are a pair of weak solutions of equations (9)–(12). We further define a map , such that is a mapping of , and . Therefore, is a semigroup operator, which is generated by the weak solutions of equations (9)–(12). Now, we can verify the conditions of Lemma 2 one by one.(1)At first, we would demonstrate that the operator is uniformly bounded on .Using the conclusions of Lemma 3, for any sphere with radius , let , we find thatandFor any , there exists a constant , such that when , we haveThus, the operator is uniformly bounded on .(2)Second, we would prove that there exists a bounded absorbing set on .According to Lemma 3 and noting that the constants and independent on , we can deduce thatSimilarly,where are bounded positive constants, which is independent on .DefineThen, for any bounded set , there exists a constant , such that when , we haveandSo, is a bounded absorbing set of operator .(3)Finally, we would prove that when , is a completely continuous operator.According to Lemma 4, we can find that there exist some constants and independent on , such that when , we haveand similarly, we also haveCombining these estimates with the results of Lemmas 3 and 4, we can deduce that the functions and are completely continuous, which means that the mapping is a completely continuous operator.Now, applying the existence theorem of attractors, Lemma 2, there exists a compact global attractor of the equations (9)–(12). Theorem 1 is proved.

5. Conclusion

In this paper, the results on the existence of the overall attractor under the external force from the modified Ginzburg–Landau model of the BCS-BEC spanning phenomenon under appropriate conditions are established mainly by means of the Gronwall Lemma, the properties of binomial-type functions, and energy inequalities. The result contains two cases in which the external force is related to both the time variable , the spatial variable , and the resonance coefficient. In the next research work, we will consider the case of the existence of the modified Ginzburg–Landau equations for the system of integral attractors from the BCS-BEC spanning model under the condition of nonequilibrium states.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

Shuai Liu designed the study and drafted the manuscript. Shuhong Chen participated in conception of the study and the amendment of the paper. All authors have read and approved the final manuscript.

Acknowledgments

This article was supported by the National Natural Science Foundation of China (No. 11571159)Foundation of Wuyi University (No. YJ202118)