Solution Theory of Ginzburg-Landau Theory on BCS-BEC Crossover

We establish strong solution theory of time-dependent Ginzburg-Landau (TDGL) systems on BCS-BEC crossover. By the properties of Besov, Sobolev spaces, and Fourier functions and the method of bootstrapping argument, we deduce that the global existence of strong solutions to time-dependent Ginzburg-Landau systems on BCS-BEC crossover in various spatial dimensions.

The main purpose of this paper is to establish strong solutions theory of (1)- (4).The systems are the general form of Ginzburg-Landau theory for superfluid atomic Fermi gases describing the BCS-BEC crossover near the Feshbach resonance from the Fermion-Boson model (double-channel model) [1].The BCS-BEC crossover phenomena have been found as early as in 1992 [2,3].Due to the strange feature taken on by the quantum phenomena, it attracts many scientists' attention and interest [4][5][6].Though so many results have been found, there are a few works got by mathematical framework.
Even though the Ginzburg-Landau theory can capture almost every unique feature that the superfluid exhibited macroscopically [7,8], this leads to the fact that it played an important role in the history of superfluid atomic Fermi gases research.In fact, the Ginzburg-Landau equation (single-channel model) has proved fruitful for illustration of the connection between infinite dimensional dynamics and finite dimensional dynamical systems [7][8][9][10][11].Thus, Machida-Koyama constructed a time-dependent Ginzburg-Landau theory for BCS-BEC crossover from the Fermion-Boson model [1].Then, Chen and Guo found weak solutions [12] and classical solutions [13] theories of the time-dependent Ginzburg-Landau theory.In present paper, we would further show the global strong solutions of initial value problem (1)-(4) under some suitable conditions.
In the strong solution theory, the admissible parameter values of , , , ,  and the dimension  are interrelated.This arbitrariness contrasts sharply with the theory of weak solutions [10].Luckily, we can overcome this problem by Fourier transform and using the properties of Fourier functions.However, the higher the dimension is, the harder 2 Advances in Mathematical Physics the problem is [14].The difficulty lies in obtaining a priori estimate of higher derivatives of solutions.Furthermore, the approaches in both [15,16] could not be applied to the current paper, since the crossover between the states BCS and BEC can bring some technical difficulties.In order to handle these difficulties, we have to use the properties of Besov and Sobolev spaces and the bootstrapping argument.But what make things worse is that a priori estimate even in the case of nonlinear wave [17][18][19][20] requires some restrictions on the nonlinear term, which cause the fact that one just can obtain the desired result in the case of  ≤ 11.Thus, combining all techniques of Fourier transform, Besov and Sobolev spaces, and the energy method and bootstrapping argument, we overcome these difficulties and establish the strong solutions theory of ( 1)-( 4) in various spatial dimensions.

Theorem 1.
Let Ω be a bounded domain in   , with Lipschitzian boundary.Assume that , ,   , ,  are positive coefficients and , , , ] are real numbers.For  > 0, assume that Then the problem (1)-( 4) with Remark 3. The relations between physics and mathematics are closed and they can complement each other.The physical phenomenon can validate the mathematical results; on the other hand, the mathematical results would provide the theoretical support for physical phenomena.For example, Theorem 1 means that, under suitable conditions, the transformation process of the two kinds of particles in BCS-BEC crossover region exists a smooth state.Therefore, in practice, we can control whether the smooth state should take on by adjusting the related conditions.Theorem 2 shows that, under certain conditions, we can decide that which region should take on the smooth state by adjusting parameters.

Local Existence of Strong Solutions
In this section, we would establish local strong solutions to (1)-( 4).At first, the initial boundary value problem ( 1)-( 2) may be rewritten as follows: For ⃗  = ( ) with ⃗  0 = (  0 + 0  0 ), (1)-( 4) become where Obviously,  3 is a nonnegative defined matrix.Consider the evolution of ⃗  = ⃗ () in a Banach space  to be governed by the abstract initial-value problem (10) with ⃗ (0) = ⃗  0 ∈  and the perturbation  is often a nonlinear and noncontinuous map over .Then, via the corresponding integral equation assume that  is a locally Lipschitz continuous map from  into itself; that is, where   (⋅) and  Lip (⋅) are nondecreasing functions.Then, employing the contraction mapping theorem, one can get the following basic result.
In order to elevate these mild solutions to strong solutions, we use standard bootstrapping argument.First, evoking the regularity of   , the gradient of (15) yields Then, use the  1 -estimate of ∇  with where   is a constant depending only on the dimension , applied to the successive iterates ( 16) to show that each ∇ ⃗  () lies in ((0, ], (Ω)) and that the sequence converges.It follows that ⃗  ∈ ((0, ],  1 (Ω)) with Then, if it was the case that ⃗  0 ∈  1 (Ω), then the singularity in this estimate at  = 0 disappears and one sees that ⃗  ∈ ([0, ],  1 (Ω)) and that ∇ ⃗  is a solution of where ( ⃗ ) represents the derivative of ( ⃗ ) with respect to ⃗ , where in this case ( ⃗ )∇ ⃗  = ( ∇ 1 ∇ 2 ) is given explicitly by A repetition of the above regularity argument starting from (20) shows that ⃗  is in ((0, ],  2 (Ω)).Moreover, because (8) and ( 9) relate the first time derivative to the second space derivative, the solution must also be in  1 ((0, ], (Ω)) and is therefore a strong solution of systems ( 1)-( 4) so long as it is a mild solution.
Refinements of the basic existence argument given above greatly enlarge the class of initial data that evolve into strong solutions for finite times.

Lemma 9 (𝐿
for some positive absolute constants Thus, one sees that Then, a direct   estimate of the mild formulation (15) for every  satisfying where  and  satisfy The idea is to recast (32) as an inequality for the ratio whence obtaining It is easily seen from (30) that whenever the condition holds.The iterates (16) contract in the space  , ([0, ], Ω) that is the completion of ([0, ],   (Ω)) for some  sufficiently small, in the norm At last, we show that Lemma 9 holds just as the condition (25) is satisfied.By the second equality of (33) and then the first one, thus, we have /2 * + (/ * ) = (/) < 1, which means that the second inequality of (36) is seen to be equivalent to the condition Again, using (33), the first inequality of (36) becomes which implies that By a direct calculation and using the condition that  < , we deduce then, one sees that (42) is satisfied by choosing  = (2 + 1); thus, condition (37) is met and the contraction mapping argument yields a unique solution ⃗  in  , ([0, ], Ω) for  = (2 + 1).
Combining the above argument with Lemmas 4 and 8 yields the desired result.

Global Existence of Strong Solutions
From Lemma 10 in the last section, one can find that it suffices to obtain global control of any   norm where  satisfies (46).Precisely, we have the following lemma.
Proof of Theorem 1. From Lemma 11, we can see that the  2 norm of ⃗  is uniformly bounded in time.Then whenever Lemma 10 then implies that the problem (1)-( 4) has global strong solutions.When (67) is violated, one must control more than the  2 norm.Thus when in addition to (50) one also has Lemma 10 then implies that the problem (1)-( 4) has global strong solutions.We can find a  > 2 satisfying both (50) and (68) provided, which means that Thus, by Lemma 10, we get the global strong solutions for (1)-( 4) with the  2 initial conditions.This completes the Proof of Theorem 1.
Proof of Theorem 2. An alternation to directly controlling the   norm is to rather first directly control the  1 norm and then control   through Sobolev estimate.As the  2 norm is already controlled by (49), it suffices to control  2 norm of ∇ ⃗ .Assuming that  ≥ 1/2 and utilizing the decomposition identity then a direct calculation shows that Similarly, as the estimate of (57), Provided that the last term of (71) will be nonpositive.And then it becomes This differential inequality shows that the  2 norm of ∇ ⃗  for ⃗  = ( +    ) is uniformly bounded in time.This then gives a priori control of the  1 norm if condition (73) is satisfied.
Control of the  1 norm yields, by a Sobolev inequality, the   norm for every  satisfying When, in addition to (73) and (78), one also has Lemma 10 implies that the problem (1)-( 4) has global strong solutions.Given that   ,   , and  satisfy (73), we can find a  satisfying both (78) and (79), provided that This encompasses all subcritical and critical nonlinearities in every spatial dimension ( ≤ 2), the supercritical cubic nonlinearity ( = 1) in dimension  = 3, and all nonlinearities in dimensions  ≤ 2.
Equations ( 65) and (77) may be combined to gain control on a set that is actually much larger than the union of the strips (50) and (73) about the axes of the (  /4|| 2 ,   /|| 2 )-plane.This is reflected in the condition of (7) in Theorem 2.
We will produce a bounds on a functional of the form Taking a linear combination of (55), ( 62), (74), and (75), while using (82), and noting that   > 0 give This differential inequality shows that  is uniformly bounded in time when −1 ≤  < 1 and is exponentially bounded in time when  = 1.This then gives a priori control  if   ,   , , , and  are such that (88) is satisfied.
Control of the  obviously gives control of the  1 norm of ⃗ .Assume that (80) and (88) are satisfied.The following choices for  and  will maximize the upper bounds (88) on .If  = 0, then for any  and   we may choose a  large enough to satisfy (88) for any fixed value of , say, 1/2, and there is no restriction, where we have used (69).If   = 0, the result is obvious.If   /4|| 2 ̸ = 0 and 4/ ≥ 0, then choose  = √4/ and there is again no restriction on .If   /4|| 2 ̸ = 0 and   /4|| 2 and /|| 2 have the opposite signs, then choose  = −4/ for any value of .The maximum of the upper bounds (88) on  is then obtained by setting  = 1 and yields (69).