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Journal of Function Spaces and Applications
Volume 2013 (2013), Article ID 692879, 12 pages
Schauder-Tychonoff Fixed-Point Theorem in Theory of Superconductivity
Department of Mathematics, Rzeszów University of Technology, al.Powstańców Warszawy 6, 35-959 Rzeszów, Poland
Received 25 April 2013; Accepted 5 June 2013
Academic Editor: Józef Banaś
Copyright © 2013 Mariusz Gil and Stanisław Wędrychowicz. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We study the existence of mild solutions to the time-dependent Ginzburg-Landau ((TDGL), for short) equations on an unbounded interval. The rapidity of the growth of those solutions is characterized. We investigate the local and global attractivity of solutions of TDGL equations and we describe their asymptotic behaviour. The TDGL equations model the state of a superconducting sample in a magnetic field near critical temperature. This paper is based on the theory of Banach space, Fréchet space, and Sobolew space.
The objective of the paper is to investigate the existence and asymptotic behaviour of mild solutions on an unbounded interval of time-dependent Ginzburg-Landau equations (TDGL, for short) in superconductivity.
In the Ginzburg-Landau theory of phase transitions , the state of a superconducting material near the critical temperature is described by a complex-valued order parameter , a real vector-valued vector potential , and, when the system changes with time, a real-valued scalar potential . The latter is a diagnostic variable; and are prognostic variables, whose evolution is governed by a system of coupled differential equations: The supercurrent density is a nonlinear function of and , The system of (1)–(3) must be satisfied everywhere in , the region occupied by the superconducting material, and at all times . The boundary conditions associated with the differential equations have the form on , where is the boundary of and is the local outer unit normal to . They must be satisfied at all times .
We prove that the systems of (1)–(5) can be reduced to a semilinear equation; to use the appropriate theorem, we investigate the local and global attractivity of solutions of equations in question and describe their asymptotic behaviour.
In this paper, we consider the existence and asymptotic behaviour of mild solutions on an unbounded interval of the semilinear evolution equation of the following form: where the operator generates a -semigroup and is a real Banach space.
Recently, a lot of papers have appeared that deal with the same or similar equations on a bounded interval (see [2–21]). However, only in a few papers, problem (6)-(7) was considered on an unbounded interval [10, 22]. Additionally, in assumptions concerning the semigroup or the function , rather restrictive conditions have been imposed which frequently require the compactness of or or equicontinuity of semigroup [2–11, 13–22]. It is worthwhile mentioning that only a few papers have discussed asymptotic behaviour of solutions, mostly without the formulation of existence theorems [10, 23, 24].
Moreover, we formulate theorems about asymptotic properties and both local and global attractivity of solutions of problem (6)-(7). The existence theorems concerning that problem will be proved with the help of the technique of a family of measures of noncompactness in the Fréchet space and Schauder-Tychonoff fixed-point principle.
The paper is organized as follows. In Section 2, there are given notation and auxiliary facts that are needed further on. In Section 3, we formulate and prove a theorem on the existence of mild solution of (6) with condition (7). Moreover, the rate of the growth of those solutions is characterized.
Finally, in Section 6, we formulate the gauged TDGL equation as an abstract evolution equation in a Hilbert space. Moreover, this section is devoted to present examples of application of previously obtained theorems for TDGL equations.
Let be a real Banach space with the zero element . Denote by the closed ball in centered at and with radius . If is a subset of a linear topological space, then the symbols and stand for the closure and the convex closure of , respectively.
Let denote the Hausdorff measure of noncompactness in , defined on bounded subsets of in the following way (see ):
Further, denote by the Fréchet space consisting of all functions defined and continuous on with values in a Banach space . The space is furnished with the family of seminorms; Let us recall two facts: (*) a sequence is convergent to in if and only if is uniformly convergent to on compact subsets of ;(**) a subset is relatively compact if and only if the restrictions to of all functions from form an equicontinuous set for each and is relatively compact in for each , where .
Moreover, we recall that a nonempty subset is said to be bounded if Further, the family of all nonempty and bounded subsets of will be denoted by , while the family of all nonempty and relatively compact subsets of is denoted by . Obviously .
We will use a family of measures of noncompactness in the Fréchet space which was introduced in [20, 35]. In order to define these measures, recall some quantities [25–27]. Let us fix a nonempty bounded subset of the space . For , , and denote by the modulus of continuity of the function on the interval , that is, Further, let us put Obviously, the set is equicontinuous on if and only if .
Next, let us define the functions on the family of bounded subsets of by the following formula: It can be shown that the family has the following properties: the family is nonempty and ; for ; for ; if is a sequence of closed sets from such that and if for each , then the intersection set is nonempty.
Remark 1. Observe that in contrast to the definition of the concept of a measure of noncompactness given in , our mapping may take the value . Moreover, a single mapping is not the measure of noncompactness in but the whole family can be called family of measures of noncompactness.
Remark 2. Let us notice that the intersection set described in axiom is a member of the kernel of the family of measure of noncompactness , and therefore, is compact in . In fact, the inequality for and implies that . Hence, . This property of the set will be very important in our further investigations.
Definition 3. A set of bounded linear operators on is called -semigroup if (i), for ;(ii)for all , the function is continuous in .
To prove the existence results in this paper, we need the following lemmas.
Lemma 5 (see ). If is a bounded subset of Banach space , then for each there is a sequence such that
We call a set uniformly integrable if there exists such that for and a.e. .
Lemma 6 (see ). If is uniformly integrable, then is measurable and
Lemma 7 (see ). Assume that a set is bounded. Then,(i), (ii), for , where .
Lemma 8 (Gronwall’s inequality ). Assume that the function is measurable and the function is locally integrable. Moreover, we assume that for and a measurable function satisfies the following inequality: Then,
Next, we consider the operator defined by the following formula:
3. Main Result
First, we will assume that the functions involved in (6) satisfy the following conditions: is a linear operator acting from and generates -semigroup ;the mapping satisfies the Carathéodory condition; that is, is measurable for and is continuous for a.e. ; and are locally integrable functions such that for any and a.e. . There exists a locally integrable function such that for any bounded , for a.e. .
Proof. Consider the operator defined by the following formula:
Now, let us observe that for any continuous function , in view of condition , we get the following estimate:
Next, consider the following integral equation:
Solving this equation by standard methods, we get
The function is continuous, nonnegative, and nondecreasing. Observe that the following implication is true:
Indeed, linking (29) and (32), we have
In the space , let us consider the following set: Obviously, the set is convex and closed. Moreover, in view of (32), we have that is a self-mapping of .
Using the criterion of convergence (*) in and standard techniques (see [31–33, 35]), we can show that the operator is continuous on .
Now, we consider the sequence of sets defined by induction as follows: This sequence is decreasing, that is, for .
Further, let us fix and for , let us put The sequences and are nonincreasing for all , so they have limits Moreover, each function is nondecreasing; therefore , and are measurable on for .
Now, we apply the family of measures of noncompactness defined in by formula (13). In view of the above notation, we have We show that To fix a number , and take an arbitrary number .
We know from Lemma 5 that for any , there is a sequence , such that This implies that there is a sequence , such that Hence, in view of Lemma 6, , and (36), we obtain Since is arbitrary, it follows from the above inequalities that Using (36), we have Letting , we derive the following inequality: This inequality, together with Gronwall’s Lemma 8, implies that
Next let us notice that in view of and (36), we have for a.e. , and the function is measurable. Then, in virtue of equality and Lemma 9, we get Hence, we derive Letting , we get Keeping in mind (47), we deduce that This together with (39) and (47) yields
Finally, using Remark 2 for the measure , we deduce that the set is nonempty, convex, and compact. Then, by the Schauder-Tychonoff theorem, we conclude that operator has at least one fixed-point . Obviously, the function is a solution of problem (6)-(7), and, in view of the definition of the set , the estimate holds to be true. This completes the proof.
4. Local and Global Attractivity
Definition 12. We say that mild solution of (6) with initial condition (7) is locally attractive if there exists a ball in the space such that for arbitrary solution of (1) with initial-value , condition (55) does hold.
In the case when the limit (55) is uniform with respect to all solutions , that is, when for each there exist such that for all being solutions of (6) with initial-value , and for , we will say that solution is uniformly locally attractive on .
Now, we formulate the main result of this section. We will consider (6) under the following conditions: is the infinitesimal generator of an exponentially stable -semigroup ; that is, there exist , such that for all ; there exist locally integrable functions , such that for and . Moreover, we assume that
Remark 13. The property () is generally satisfied in diffusion problem. A necessary and sufficient condition for () is presented in .
The main result of this section is shown in the given theorem below.
Proof. Existence of a solution is a consequence of Theorem 10. Let us fix and .
Let denote a mild solution of (6) with the initial condition . Using () and (), we get Now, let us put Taking into account Lemma 5, we obtain Further, let be the set of all mild solutions of (6) with the initial-value .
Put The estimate (61) implies that the function is well defined. Applying () and (), we get or, equivalently, Using again Lemma 5 for the above estimate (where ), we obtain Elementary calculations lead to the following equality: Hence, Applying assumption (), we derive and this proves that is locally attractive. Finally, this equality together with definition of the function implies that is globally attractive. The proof is complete.
Remark 15. In the case when is constant, the following condition means that . Observe that this condition cannot be weakened. This observation is illustrated by the following exampled.
Example 16. Let , , , , and . Then the equation (for any fixed ) has the solution expressed by the following formula: Notice that for , the solution is neither globally attractive nor locally uniformly attractive, because for each other solution with initial condition , obviously described by similar formula as , we would have a contradiction:
5. Asymptotic Behaviour
In this section, we will give a theorem describing asymptotic behaviour of mild solutions of (6) with condition (7). This theorem generalizes the result included in [72, Theorem 4.4]. First, we formulate the assumptions. This condition is almost identical with () and the only difference is that we assume the functions and are locally essentially bounded on . There exists such that there exists the limit and Moreover, there exists a number such that for and .
Remark 17. The condition () in conjunction with () ensures the existence of (see ). Clearly, () implies ().
Proof. The existence of a mild solution is guaranteed by Theorem 10. Let us put
We show that
Recall that if assumption () is fulfilled, then for each we have (see )
Using the above fact and (), we get
Linking the above equality with (), we obtain
and applying (), we derive the following inequality:
The above inequality in conjunction with Lemma 5 gives
Before proving (76), we first show that
To prove this equality, it is sufficient to show that the last component in the formula expressing , that is, , tends to zero as .
To this end, let us fix . Assumption () implies that there exists such that This inequality together with () implies Now, fix such that and Then using (86) and (88), we conclude that for we have This fact proves (85).
Further, using () and (85) and employing de L’Hospital’s rule for the fraction on the right-hand side of inequality (84), we obtain that condition (76) is satisfied. This fact completes the proof.
6. An Application to the Ginzburg-Landau Equations of Superconductivity
In this section, we formulate the gauged time-dependent Ginzburg-Landau (TDGL) equations as an abstract evolution equation in a Hilbert space. Moreover, we show applications of the above theorems to TDGL equations.
We assume that is a bounded domain in with boundary of class . That is, is an open and connected set whose boundary is a compact -manifold described by Lipschitz continuous differentiable charts. We consider two- and three-dimensional problems ( and , resp.). Assume that the vector potential takes its values in . The vector will represent the (externally) applied magnetic field, which is a function of space and time; similarly to , it takes its values in . The function is defined and satisfies Lipschitz condition on , and for . The parameters in the TDGL equations are , a (dimensionless) friction coefficient, and , the (dimensionless) Ginzburg-Landau parameter.
The order parameter should be thought of as the wave function of the center-of-mass motion of the “superelectrons” (Cooper pairs), whose density is and whose flux is . The vector potential determines the electromagnetic field; is the electric field and is the magnetic induction, where , the total current, is the sum of a “normal” current , the supercurrent , and the transport current . The normal current obeys Ohm’s law ; the “normal conductivity” coefficient is equal to one in the adopted system of units. The difference is known as the magnetization. The trivial solution (, , ) represents the normal state, where all superconducting properties have been lost.
Now we accept the following notion: all Banach spaces are real; the (real) dual of a Banach space is denoted by . The symbol , for , denotes the usual Lebesgue space, with norm ; is the inner product in . , for nonnegative integer , is the usual Sobolev space, with norm ; is a Hilbert space for the inner product given by for . Fractional Sobolev space , with a fractional , is defined by interpolation ([40, Chap. VII], and [41, 49, 50]). , for , with , is the space of times continuous differentiable functions on ; those th-order derivatives satisfy the Hölder condition with exponent if is a proper fraction; the norm is defined in the usual way.
The definitions extend to the space of vector-valued functions in the standard way, with the caveat that the inner product in is defined by , where the symbol indicates the scalar product in . Complex-valued functions are interpreted as vector-valued functions with two real components.
Functions that vary in space and time, like the order parameter and the vector potential, are considered as mappings from the time domain, which is a subinterval of , into spaces of complex- or vector-valued functions defined in . Let be a Banach space of functions defined in . Then, the functions are defined in . Then, the functions of space and time defined on , for , may be considered as elements of , for , or , for nonnegative , or , for , with . Detailed definitions can be found, for example, in .
Obviously, function spaces of ordered pairs , where and , play an important role in the study of the gauged TDGL equations. We therefore adapt the following special notation: , for any Banach space for the order parameter and the vector potential , respectively. A suitable framework for the functional analysis of the gauged TDGL equations is the Cartesian product , where . This space is continuously imbedded in .
Assume and . Let be a minimizer of the convex quadratic form , on the domain
We now introduce the reduced vector potential : In terms of and the gauged TDGL equations have the following form: Here, and are nonlinear functions of and : The equations are supplemented by initial data, which is in the followimg form: where and are given, and by (92), we have
In the next part we connect the evolution of the solution of the system of (93) with the initial data with the dynamics of a vector in a Hilbert, space
The following analysis is restricted to the case and the case (see ).
Let vector represent the pair , and let be the linear self-adjoint operator in associated with the quadratic form given by the following formula: on the domain The quadratic form is nonnegative. Furthermore, since , is coercive on for any constant . Hence, is positively definite in ([44, Chap. 1], equation (5,45)). If it does not lead to confusion, we use the same symbol for the restriction and of to the respective linear subspace (for ) and (for ) of .
With and , we say that are a solution of (6) and (7) on the interval , for some , if is continuous and in . A mild solution of the initial-value problems (6) and (7) defines a weak solution of the boundary value problem (93), which in turn defines a weak solution of the gauged TDGL equations, provided is sufficiently regular.
Namely, let us assume that and , and for is an unknown function, . In order to apply Theorems 10 and 14, we are not going to consider as a function of and together, but rather as a mapping of variable into the space of functions , that is, , , , .
Below, we formulate the principal theorem of this paper. This theorem is a simple consequence of Theorem 10.
Theorem 20. Under assumptions ()–(), problem (6)-(7) is equivalent to the system of (1)–(5) (TDGL equations) and has for each at least one mild solution in sense of Definition 4 . Moreover, each solution satisfies estimate (25).
We observe that in the case when is the semigroup of contractions and , for are constant, after simple calculations based on estimate (25), we get that the solution has the asymptotic characterization:
For further purposes, let us formulate the following assumption:(A) is the infinitesimal generator of an exponentially stable -semigroup .
The next result of this chapter is shown in the given theorem below.
Theorem 21. Under assumptions and –(), problem (6)-(7), which is equivalent to the systems equations (1)–(5) (TDGL equations), has a mild solution for each which is globally attractive and locally uniformly attractive.
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