# On the Analyticity for the Generalized Quadratic Derivative Complex Ginzburg-Landau Equation

Authors: Chunyan Huang
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

We study the analytic property of the (generalized) quadratic derivative Ginzburg-Landau equation(1/2⩽α⩽1) in any spatial dimension n⩾1 with rough initial data. For 1/2<α⩽1, we prove the analyticity of local solutions to the (generalized) quadratic derivative Ginzburg-Landau equation with large rough initial data in modulation spaces Mp,11-2α(1⩽p⩽∞). For α=1/2, we obtain the analytic regularity of global solutions to the fractional quadratic derivative Ginzburg-Landau equation with small initial data in B˙∞,10(ℝn)∩M∞,10(ℝn). The strategy is to develop uniform and dyadic exponential decay estimates for the generalized Ginzburg-Landau semigroup e-a+it-Δα to overcome the derivative in the nonlinear term.


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## Body

## 1. Introduction

In this paper, we are interested in the Cauchy problem of the following generalized quadratic derivative complex Ginzburg-Landau equation (GDGL):(1)ut+(a+i)(-Δ)αu-γ→·∇(u2)=0,u(0,x)=u0(x),
where u is a complex valued function of (t,x)∈ℝ+×ℝn, ℝ+=[0,∞), n⩾1. a>0 is the dissipative coefficient, 1/2⩽α⩽1. γ→ is a given complex vector in ℝn. u0(x) is a given complex valued function of x∈ℝn. ut=∂u/∂t  and  (-Δ)α denotes the fractional Laplacian defined by (-Δ)αu^(t,ξ)=|ξ|2αu^(t,ξ). It is well known that (1) can be rewritten into an integral equation as follows:
(2)u(t)=G2α(t)u0+𝒜2α(γ→·∇(u2)),
where
(3)G2α(t)≔e-(a+i)t(-Δ)α=ℱ-1e-(a+i)t|ξ|2αℱ,(𝒜2αf)(t):=∫0t‍G2α(t-τ)f(τ,x)dτ.Complex Ginzburg-Landau type equation is one of the most-studied equations in physics. It describes a lot of phenomena including nonlinear waves and the evolution of amplitudes of unstable modes for any process exhibiting a Hopf bifurcation. GDGL (1) is also called derivative fractional Ginzburg-Landau equation. For details of physical backgrounds of the fractional Ginzburg-Landau equation (1), one can refer to [1–3]. Equation (1) is both dissipative and dispersive. If α=1, (1) is the quadratic derivative Ginzburg-Landau equation (QDGL):
(4)ut-(a+i)Δu-γ→·∇(u2)=0,u(0,x)=u0(x).
If α=1, a=0, (1) reduces to the well-known quadratic derivative Schrödinger equation (DNLS):
(5)ut-iΔu-γ→·∇(u2)=0,u(0,x)=u0(x).For DNLS (5), Christ [4] proved that when space dimension n=1, the flow map u0→u is not continuous in any Sobolev space Hs(ℝ1) with any exponent s for any short time in the sense that ∥u0∥Hs≪1 but ∥u(t)∥Hs≫1 after an arbitrarily short time. For (5), Stefanov [5] established in one space dimension the existence of local solution in H1 with small total disturbance u0 in H1(ℝ1)∩L1(ℝ1)∩{f:sup⁡x|∫-∞x‍f(y)dy|⩽ɛ}. Han et al. [6] showed that (4) and (5) are locally well-posed in modulation space M1,13(ℝn) under the small condition of ∥u0∥L1 norm and they obtained the inviscid limit behavior between the solutions of (4) and (5) with initial data in M1,13(ℝn) as the dissipative parameter a→0. For general α, to the knowledge of the author, there are few results on (1). In this paper, we will study the analyticity of solutions of (1) for 1/2⩽α⩽1 with rough initial data in certain modulation space. In the case 1/2<α⩽1, we prove that the local solution of (1) is real analytic with initial data in Mp,11-2α(1⩽p⩽∞); in the case α=1/2, we show that (1) is globally well-posed with small initial data in M∞,10∩B˙∞,10 and moreover the global solution of (1) is real analytic for any t>0.We now briefly sketch the idea of the proof. The basic strategy is to choose the working space to be some time dependent type exponential modulation space, sayL~∞(I;Ep,qs) with s>0, and consider the map:
(6)𝒯:u(t)⟶G2α(t)u0+𝒜2α(γ→·∇)u2;
then use the standard contraction mapping method to prove that there exists a unique solution in this space. Due to the nice property of Ep,qs, the solution is naturally analytic for any s>0. However, the main obstacle comes from the derivative in the nonlinear term. To resolve this difficulty, our idea is to make full use of the strong dissipative property of GDGL (1) when a>0. Motivated by the work in [7, 8], we prove two exponential decay estimates of the generalized Ginzburg-Landau semigroup G2α(t)=e-(a+i)t(-Δ)α combined with frequency uniform decomposition operator □k and frequency dyadic decomposition operator Δj:
(7)∥□kG2α(t)f∥p≲e-2cat|k|2α∥□kf∥p,∥ΔjG2α(t)f∥p≲e-cat22αj∥Δjf∥p,1⩽p⩽∞,
for all f∈Lp. Then we gain 2α derivative in space from (7) in suitable space time norm which is sufficient to balance the one order derivative in the nonlinear term. More precisely, when 1/2<α⩽1, we choose the resolution space as 𝒟={u:L~2α/(2α-1)(I;Ep,1cat)⩽δ} and establish some linear estimates of G2α(t) and 𝒜2αf in this space like
(8)∥G2α(t)u0∥L~2α/(2α-1)(I,𝔸;Ep,1cat)≲∑k∈𝔸〈k〉1-2α∥□ku0∥p,0000000000000000000000000000𝔸⊂ℤn∖{0},∥□0G2α(t)u0∥Lt∈I2α/(2α-1)Lxp≲|I|(2α-1)/(2α)∥□0u0∥p.
And 𝒜2αf satisfies similar estimates. Since
(9)∥G2α(t)u0∥L~2α/(2α-1)(I,{k:|k|<J};Ep,1cat)⩽C|I|(2α-1)/2α〈J〉2α-1∥u0∥Mp,11-2α,
one can choose |I| sufficiently small to make sure that
(10)C|I|(2α-1)/(2α)〈J〉2α-1∥u0∥Mp,11-2α⩽δ4,
and finally verify that 𝒯 is a contractive map on 𝒟 which ensures that (1) has a local solution satisfying ∥u∥L~2α/(2α-1)(I;Ep,1cat)⩽δ. Moreover, we can prove that
(11)∑k‍〈k〉1-2α∥2cat|k|□ku∥Lt∈I∞Lxp≲∥u0∥Mp,11-2α+δ2,which implies that(I-Δ)(1-2α)/2u∈L~∞(I,Ep,1cat) and hence the local solution is analytic for any t>0. However, when α=1/2, it is impossible to choose I satisfying (10). To make this bound valid, one needs to impose additional small condition on ∥u0∥Mp,10. Then we will only obtain the existence of local solution with small initial data which is not ideal. So we intend to seek for different approach in this critical situation. The preferred working space would be {u:∥u∥L~∞(I;E∞,1cat)⩽δ}. But when we bound
(12)∥𝒜1(γ→·∇)u2∥L~∞(I;ℤ⋆n;E∞,1cat)≲∥(γ→·∇)u2∥L~1(I;ℤ⋆n;E∞,1cat)≲∑j=1n‍∥∂xju∥L~1(I;ℤ⋆n;E∞,1cat)∥u∥L~∞(I;ℤ⋆n;E∞,1cat),
it is easy to see that to control the right-hand side of (12), ∑j=1n‍∥∂xju∥L~1(I;ℤ⋆n;E∞,1cat) should be involved in the working space. So, the natural working space would be 𝒟={u:∥u∥Y⩽δ} where ∥u∥Y=∑i=1n‍∥∂xiu∥L~1(I;E∞,1cat)+∥u∥L~∞(I;E∞,1cat). However, the obstacle comes again. The following low frequency projection term could not be bounded in this working space:
(13)∥□0∂xi𝒜1(γ→·∇)u2∥Lt∈I1Lx∞+∥□0𝒜1(γ→·∇)u2∥Lt∈I∞Lx∞.
To overcome this difficulty, our idea is to make use of the property ∥□0f∥∞≲∥f∥∞≲∥f∥B˙∞,10 and bound (13) by
(14)∥𝒜1(γ→·∇)u2∥L~1(I;B˙∞,11)+∥𝒜1(γ→·∇)u2∥L~∞(I;B˙∞,10)≲∥(γ→·∇)u2∥L~1(I;B˙∞,10)≲∥u2∥L~1(I;B˙∞,11)≲∥u∥L~∞(I;B˙∞,10)∥u∥L~1(I;B˙∞,11).
Then the time dependent type Besov norm ∥u∥X=∥u∥L~1(I;B˙∞,11)∩L~∞(I;B˙∞,10) should be included in the working space. So, finally we choose the resolution space as
(15)𝒟={u:∥u∥X+∥u∥Y⩽δ}.
The corresponding condition imposed on the initial data would be stronger, u0∈M∞,10∩B˙∞,10, and sufficiently small. In Section 4, we develop estimates in this resolution space and combine them with the contraction mapping argument; finally we show that there exists a unique global solution to (1) in 𝒟 which is naturally analytic for t>0.Now let us recall some notations and basic facts that will be used in the sequel.C⩾1, c⩽1 will denote universal positive constants which can be different at different places. X≲Y (for X,Y>0) means that X⩽CY. For any x∈ℝn, we write |x|p=(|x1|p+⋯+|xn|p)1/p and |x|=|x|2. Now we introduce some spaces. We denote by Lp=Lp(ℝn) the Lebesgue space on which the norm is written as ∥·∥p. Let X be a Banach space. For any I⊂ℝ+, we define
(16)∥u∥Lγ(I;X)=(∫I‍∥u(t,·)∥Xγdt)1/γ
for 1⩽γ<∞ and with usual modification for γ=∞. If X=Lp, we will write ∥u∥Lt∈IγLxp=∥u∥Lγ(I;Lp) and simply denote ∥u∥LtγLxp if I=[0,+∞). Now let us recall the notation and definitions in Littlewood-Paley theory [9]. Let ψ:ℝn→[0,1] be a smooth radial cutoff function satisfying
(17)ψ(ξ)={1,|ξ|⩽1,smooth,1<|ξ|<2,0,|ξ|⩾2.
Denote φ(ξ):=ψ(ξ)-ψ(2ξ) and we introduce the function sequence φk(ξ)=φ(2-kξ), k∈ℤ. Then Δk:=ℱ-1φkℱ, k∈ℤ, are said to be the homogeneous dyadic decomposition operators and satisfying the operator identity: I=∑k=-∞+∞‍Δk. The low frequency projection operators Sk are defined by Sk:=∑j=-∞k‍Δj. It is easy to see that Sku→u as k→∞ in the sense of distributions. With this decomposition, the norms in homogeneous Besov spaces are defined as follows:
(18)∥f∥B˙p,qs={(∑j=-∞+∞‍2jsq∥Δjf∥pq)1/q,ifq<∞,sup⁡-∞<j<+∞2js∥Δjf∥p,ifq=∞.
And the space time homogenous Besov norms are defined by
(19)∥f∥L~γ(I;B˙p,qs)=(∑j=-∞+∞‍2jsq∥Δjf∥Lt∈IγLxpq)1/q<∞
with the usual modification for q=∞. Such a kind of space was first used in Chemin [10]. It is easy to see by Minkowski inequality that
(20)∥f∥L~γ(I;B˙p,qs){⩽∥f∥Lγ(I;B˙p,qs),ifγ⩽q,⩾∥f∥Lγ(I;B˙p,qs),ifγ⩾q.We now recall the definition of Modulation space which was first introduced by Feichtinger [11] in 1983 (see also Gröchenig [12]). Let σ be a smooth cutoff function with supp⁡σ⊂[-3/4,3/4]n, σk=σ(·-k), and
(21)∑k∈ℤn‍σk(ξ)≡1,∀ξ∈ℝn.
Then the frequency uniform decomposition operator □k is defined as
(22)□k=ℱ-1σkℱ,k∈ℤn.
Using this decomposition operator, for any 0<p, q⩽∞, s∈ℝ, we define
(23)∥f∥Mp,qs=(∑k∈ℤn‍〈k〉sq∥□kf∥pq)1/q,
with usual modification for q=∞. Mp,qs is said to be a modulation space and it has been successfully applied to study nonlinear evolutions in recent years [8, 13–16]. Let 0<p, q⩽∞, s⩾0; the exponential modulation space Ep,qs was introduced in [8] with the following norm:
(24)∥f∥Ep,qs=(∑k‍2qs|k|∥□kf∥pq)1/q.
We remark that when s>0, this space can be viewed as modulation space with analytic regularity and when s=0, it reduces to normal modulation space Mp,q0.Letα=(α1,…,αn), α!=α1!…αn! and ∂α=∂x1α1…∂xnαn. Recall that the Gevrey class is defined as follows:
(25)G1,p={Mα!ρ|α|f∈C∞(ℝn):∃ρ,M>0s.t.∥∂αf(x)∥p⩽Mα!ρ|α|,∀α∈ℤ+n,x∈ℝn}.It is proved thatG1,∞ is the Gevrey 1-class and any function in this space is real analytic [17]. One can easily check that G1,p1⊂G1,p2 for p1⩽p2. Therefore, any function in G1,p(0<p⩽∞) is real analytic. There is a very nice relationship between Gevrey class and exponential modulation spaces which is shown in Huang and Wang [7].Lemma 1.
Let0<p, q⩽∞. Then
(26)G1,p=⋃s>0‍Ep,qs.Remark 2.
From this property we easily see that if we can prove the solution in exponential modulation spaceEp,qs with positive regularity, then it is naturally analytic.Inspired by (19), we define the following space time exponential modulation norm:
(27)∥f∥L~γ(I;Ep,qs)=(∑k∈ℤn‍∥2s|k|□kf∥Lt∈IγLxpq)1/q
with usual modification if q=∞.In the end, let us recall the definition of multiplier spaceMp [18, 19]. Let ρ∈𝒮'. If there exists a C>0 such that ∥ℱ-1ρℱf∥Lp⩽C∥f∥Lp holds for all f∈𝒮, then ρ is called a Fourier multiplier on Lp. The linear space of all multipliers on Lp is denoted by Mp and the norm on which is defined as ∥ρ∥Mp=sup⁡{∥ℱ-1ρℱf∥Lp:f∈𝒮,∥f∥Lp=1}. Concerning the multipliers, there holds the following famous inequality which is also called the multiplier theorem.Proposition 3 (see [9], Nikol’skij’s inequality).
LetΩ⊂ℝn be a compact set, 0<r⩽∞. Denote σr=n(1/(r∧1)-1/2) and assume that s>σr. Then there exists a constant C>0 such that
(28)∥ℱ-1φℱf∥r⩽C∥φ∥Hs∥f∥r
holds for all f∈LΩr:={f∈Lr:supp⁡f^⊂Ω} and φ∈Hs. In particular, if r⩾1, then (28) holds for all f∈Lr.The remaining part of this paper is organized as follows. In Section2, we develop two decay estimates (7) associated with the generalized Ginzburg-Landau semigroup. In Section 3, we prove the analytic regularity property of the solutions to (1) when 1/2<α⩽1. In Section 4, we deal with the analytic property of (1) in the critical case α=1/2. Finally, a short conclusion is given in Section 5.


## 2. Decay Estimates for GCGL Semigroup

In this part, we will set up some decay estimates for the generalized Ginzburg-Landau semigroupG2α(t)=e-(a+i)t(-Δ)α together with the frequency uniform decomposition operator □k and the dyadic operators Δj. As explained in the introduction, these estimates are crucial to the proof of the main theorems.Proposition 4 (uniform decay estimate).
Suppose that1⩽p⩽∞, 1/2⩽α⩽1, a>0. Then there exists c>0 (say 0<c⩽2-10) such that
(29)∥□kG2α(t)f∥p≲e-2cat|k|2α∥□kf∥p
holds for all f∈Lp and k∈ℤn.Proof.
First, we choose a smooth cutoff functionσ~:ℝn→[0,1] satisfying σ~(ξ)=1 for |ξ|∞⩽3/4 and σ~(ξ)=0 for |ξ|∞>7/8. It is easy to see that σ~ equals 1 on the support of σ and has similar property as σ. Applying this fact, we deduce that
(30)∥□kG2α(t)f∥p=∥ℱ-1σ(ξ-k)σ~(ξ-k)e-t(a+i)|ξ|2αℱf∥p=∥ℱ-1σ~(ξ-k)e-t(a+i)|ξ|2αℱℱ-1σ(ξ-k)ℱf∥p≲∥σ~(ξ-k)e-t(a+i)|ξ|2α∥Mp∥□kf∥p.
In view of Nikol’skij’s inequality,
(31)∥σ~(ξ-k)e-t(a+i)|ξ|2α∥Mp≲∥σ~(ξ)e-t(a+i)|ξ+k|2α∥HL,0000000000000000000000jjjj000l0000000L>n2.
Since for |k|⩾1, applying Leibniz’s Rule, we infer that
(32)∂ξiL(σ~(ξ)e-t(a+i)|ξ+k|2α)=∑L1+L2=L‍CLβγ∂ξiL1σ~(ξ)∂ξiL2e-t(a+i)|ξ+k|2α,
where
(33)|∂ξiL2e-(a+i)t|ξ+k|2α|≲|e-(a+i)t|ξ+k|2α|×∑1⩽q⩽γβ1+⋯+βq=L2|∂ξiβ1[(a+i)t|ξ+k|2α]0000000000000000⋯∂ξiβq[(a+i)t|ξ+k|2α]|≲e-at|ξ+k|2α/2.
Due to the support property supp σ~(ξ)⊂[-7/8,7/8]n, (30)–(33), we conclude that, for |k|⩾1,
(34)∥σ~(ξ-k)e-t(a+i)|ξ|2α∥Mp≲e-at|k|2α/64.
Note that (34) also holds for k=0. Hence, we complete the proof of (29).Proposition 5 (dyadic decay estimate).
Suppose that1⩽p⩽∞, 1/2⩽α⩽1, a>0. Then there exists c>0 (say 0<c⩽2-10) such that
(35)∥ΔjG2α(t)f∥p≲e-cat22αj∥Δjf∥p
holds for all f∈Lp.Proof.
Whena>0, Ginzburg-Landau semigroup e-t(a+i)(-Δ)2α is strong dissipative. Using the exponential decay property of e-at|ξ|2α and Nikol’skij’s inequality, we deduce that, for L>n/2,
(36)∥ΔjG2α(t)f∥p⩽∥φj(ξ)e-(a+i)t|ξ|2α∥Mp∥f∥p=∥φ(ξ)e-(a+i)t22αj|ξ|2α∥Mp∥f∥p≲∥φ(ξ)e-(a+i)t22αj|ξ|2α∥HL∥f∥p≲e-cat22αj∥f∥p.
By the support property of dyadic decomposition operator Δj, there holds the following identity:
(37)Δj=Δj(Δj-1+Δj+Δj+1),
so
(38)∥ΔjG2α(t)f∥p≲e-cat22αj∥Δjf∥p.
This completes the proof as desired.


## 3. Analytic Regularity for GDGL:1/2<α⩽1

The main results of this paper are the following theorems.Theorem 6 (analyticity for QDGL).
Letα=1, 1⩽p⩽∞, a>0, and n⩾1. Assume that u0∈Mp,1-1. Then there exists a Tmax⁡=Tmax⁡(u0)>0 such that (1) has a unique solution u∈L~loc2(0,Tmax⁡;Ep,1cat), c=2-10. Moreover, the solution enjoys the following properties.(i)
(I-Δ)-1/2u∈L~∞([0,Tmax⁡);Ep,1cat).(ii)
IfTmax⁡<∞, we have ∥u∥L~2(0,Tmax⁡;Ep,1cat)=∞.Theorem 7 (analyticity for GDGL (I)).
Let1/2<α<1, 1⩽p⩽∞, a>0, n⩾1. Suppose that u0∈Mp,11-2α. There exists a Tmax⁡=Tmax⁡(u0)>0 such that (1) has a unique solution u∈L~loc2α/(2α-1)(0,Tmax⁡;Ep,1cat), c=2-10. Moreover, the solution satisfies the following properties.(i)
(I-Δ)(1-2α)/2u∈L~∞([0,Tmax⁡);Ep,1cat).(ii)
IfTmax⁡<∞, then ∥u∥L~2α/(2α-1)(0,Tmax⁡;Ep,1cat)=∞.Theorems6 and 7 tell us that when 1/2<α⩽1, (1) is locally well-posed with any initial data in Mp,11-2α and moreover the local solution is analytic. However, the method used for Theorems 6 and 7 does not work for the critical case: α=1/2. We need to impose stronger conditions on the initial data, that is, u0∈B˙∞,10∩M∞,10 and be sufficiently small. We remark that there is no inclusion between B˙∞,10 and M∞,10 since S1B˙∞,1-1⊂S1M∞,1-1 while (I-S1)M∞,1-1⊂(I-S1)B˙∞,1-1.Theorem 8 (analyticity for GDGL (II)).
Letα=1/2, a>0, and n⩾1. Assume that u0∈B˙∞,10∩M∞,10 is sufficiently small; then there exists a unique global solution u to (1) satisfying u∈L~∞(ℝ+;E∞,1s(t)) and ∂xiu∈L~1(ℝ+;E∞,1s(t)) with s(t)=2-5(a∧at).Theorem8 states that (1) is globally well-posed with small initial data in u0∈B˙∞,10∩M∞,10 and the solution u(t) is actually real analytic for any t>0.In this section, we unify the proof of Theorems6 and 7 in one part. The proof of Theorem 8 is left to Section 4. For convenience, we denote, for any 𝔸⊂ℤn,
(39)∥f∥L~q~(I,𝔸;Ep,qs)=(∑k∈𝔸‍∥2s|k|□kf∥Lt∈Iq~Lxpq)1/q.
We first build up some linear estimates for G2α(t) and 𝒜2αf(t,x)=∫0t‍G2α(t-τ)f(τ,x)dτ.Proposition 9.
Let1⩽p⩽∞, 1/2<α⩽1, and a>0. There exists a constant c>0 (0<c⩽2-10) such that, for 0<t0<∞ and I=[0,t0],(40)∥G2α(t)u0∥L~2α/(2α-1)(I,𝔸;Ep,1cat)≲∑k∈𝔸‍〈k〉1-2α∥□ku0∥p,00000000000000000000000ll00000𝔸⊂ℤn∖{0},(41)∥□0G2α(t)u0∥Lt∈I2α/(2α-1)Lxp≲|I|(2α-1)/(2α)∥□0u0∥p,
holds for all u0∈Mp,11-2α.Proof.
For any𝔸⊂ℤn∖{0}, multiplying (29) by 2cat|k| and taking Lt2α/(2α-1) norm, we get
(42)∥2cat|k|□kG2α(t)u0∥Lt∈I2α/(2α-1)Lxp≲∥2-cat|k|2α2cat(|k|-|k|2α)∥Lt∈I2α/(2α-1)∥□ku0∥Lxp≲∥2-cat|k|2α∥Lt∈I2α/(2α-1)∥□ku0∥Lxp≲〈k〉1-2α∥□ku0∥Lxp.
Taking the sequence l1 norm on set 𝔸 to inequality (42), we obtain the estimate in (40). For k=0, from (29), we see that
(43)∥□0G2αu0∥Lxp≲∥□0u0∥Lxp,
takingLt2α/(2α-1)(I) norm on (43) which implies (41).Now let0<t0⩽∞. We consider the estimate of 𝒜2αf. Again using uniform decay estimate (29), we have
(44)∥□k𝒜2αf∥Lxp≲∫0t‍2-2ca(t-τ)|k|2α∥□kf(τ)∥Lxpdτ.
It follows from (44) that for |k|≠0(45)2cat|k|∥□k𝒜2αf∥Lxp≲∫0t‍2-2ca(t-τ)|k|2α2cat|k|∥□kf(τ)∥Lxpdτ≲∫0t‍2-ca(t-τ)|k|2α2-ca(t-τ)|k|2cat|k|∥□kf(τ)∥Lxpdτ=∫0t‍2-ca(t-τ)|k|2α2caτ|k|∥□kf(τ)∥Lxpdτ.
Taking Lt2α/(2α-1) norm on (45) and applying Young’s inequality, we get
(46)∥2cat|k|□k𝒜2αf∥Lt∈[0,t0]2α/(2α-1)Lxp≲∥2-cat|k|2α∥Lt2α∥2cat|k|□kf∥Ltα/(2α-1)Lxp≲〈k〉-1∥2cat|k|□kf∥Ltα/(2α-1)Lxp.
In view of ∥∂xi□kf∥Lxp≲〈ki〉∥□kf∥Lxp, we deduce that
(47)∥2cat|k|□k𝒜2α(γ→·∇)f∥Lt∈[0,t0]2α/(2α-1)Lxp≲∥2cat|k|□kf∥Lt∈[0,t0]α/(2α-1)Lxp.So, taking the sequencelq norm on both sides of (47), we have the following.Proposition 10.
Let1⩽p⩽∞, a>0. There exists a constant c>0 (0<c⩽2-10) such that, for 0<t0<∞ and I=[0,t0],
(48)∥∂xi𝒜2αf∥L~2α/(2α-1)(I,𝔸;Ep,1cat)≲∥f∥L~α/(2α-1)(I,𝔸;Ep,1cat),00ll000𝔸⊂ℤn∖{0},(49)∥□0𝒜2αf∥Lt∈I2α/(2α-1)Lxp≲|I|1/(2α)∥□0f∥Lt∈Iα/(2α-1)Lxp,
holds for all f∈L~α/(2α-1)(I;Ep,1cat).For the proof of (49), one only needs to take k=0 in (44) and then Lt2α/(2α-1) norm together with Young’s inequality will imply the result.Now we recall a nonlinear mapping estimate inL~q~(I;Ep,qs(t)) which was shown in [7].Proposition 11.
Let1⩽p,p1,p2,q,q1,q2,q~,q~1,q~2⩽∞. Consider 1/p=1/p1+1/p2,1/q~=1/q~1+1/q~2 and 1/q=1/q1+1/q2-1, s(·):ℝ+→ℝ+, and I⊂ℝ+ with sup⁡t∈Is(t)<∞. Then one has
(50)∥fg∥L~q~(I;Ep,qs(t))≲sup⁡t∈I24s(t)∥f∥L~q~1(I;Ep1,q1s(t))∥g∥L~q~2(I;Ep2,q2s(t)).Now let us consider the following map:(51)𝒯:u(t)⟶G2α(t)u0+𝒜2α((γ→·∇)u2).
Let c=2-10, I=[0,t0]. For any δ>0, one can choose J:=J(u0)>0 satisfying ∑|k|⩾J‍〈k〉1-2α∥□ku0∥p⩽δ/4C. By Proposition 9,
(52)∥G2α(t)u0∥L~2α/(2α-1)(I,{k:|k|⩾J};Ep,1cat)⩽C∑|k|⩾J‍〈k〉1-2α∥□ku0∥p⩽δ4.
On the other hand, one can choose I satisfying
(53)|I|(2α-1)/(2α)〈J〉2α-1⩽δ4C∥u0∥Mp,11-2α,
so,
(54)∥G2α(t)u0∥L~2α/(2α-1)(I,{k:|k|<J};Ep,1cat)⩽C∑|k|<J‍∥∥2-2cat|k|2α□ku0∥p∥Lt∈I2α/(2α-1)⩽C|I|(2α-1)/(2α)〈J〉2α-1∑|k|<J‍〈k〉1-2α∥□ku0∥p⩽δ4.
Hence, in view of (52) and (54),
(55)∥G2α(t)u0∥L~2α/(2α-1)(I;Ep,1cat)⩽δ2.
It follows from Propositions 10 and 11 that
(56)∥𝒜2α(γ→·∇)(u2)∥L~2α/(2α-1)(I;Ep,1cat)≲(1+|I|1/(2α))∥u2∥L~α/(2α-1)(I;Ep,1cat)≲(1+|I|1/(2α))∥u∥L~2α/(2α-1)(I;Ep,1cat)∥u∥L~2α/(2α-1)(I;E∞,1cat)≲(1+|I|1/(2α))∥u∥L~2α/(2α-1)(I;Ep,1cat)2.
We can assume that |I|⩽1. Hence, collecting (55)-(56), we have
(57)∥𝒯u∥L~2α/(2α-1)(I;Ep,1cat)⩽δ2+C∥u∥L~2α/(2α-1)(I;Ep,1cat)2.
Now we can fix δ verifying Cδ⩽1/4. Put
(58)𝒟={u∈L~2α/(2α-1)(I;Ep,1cat):∥u∥L~2α/(2α-1)(I;Ep,1cat)⩽δ}.
We have 𝒯u∈𝒟 if u∈𝒟 and
(59)∥𝒯u-𝒯v∥L~2α/(2α-1)(I;Ep,1cat)⩽12∥u-v∥L~2α/(2α-1)(I;Ep,1cat),0000000000000000000lll000000000000u,v∈𝒟.
By the standard contraction mapping argument, there exists a unique solution to (1) in 𝒟 satisfying ∥u∥L~2α/(2α-1)(I;Ep,1cat)⩽δ.Next, we prove that(60)∑k‍〈k〉1-2α∥2cat|k|□ku∥Lt∈[0,t0]∞Lxp≲∥u0∥Mp,11-2α+δ2.
In order to show (60), we need the following.Proposition 12.
Let1⩽p⩽∞, a>0. There exists a constant c>0(0<c⩽2-10) such that, for 0<t0<∞ and I=[0,t0],
(61)∑k‍〈k〉1-2α∥2cat|k|□kG2α(t)u0∥Lt∈[0,t0]∞Lxp≲∥u0∥Mp,11-2α,∑k‍〈k〉1-2α∥2cat|k|□k∂xi𝒜2αf∥Lt∈[0,t0]∞Lxp≲∥f∥L~α/(2α-1)(I;Ep,1cat)
hold for allu0∈Mp,11-2α and f∈L~α/(2α-1)(I;Ep,1cat).Proof.
In view of (29), we immediately have (61).Let us apply Proposition12:
(62)∑k‍〈k〉1-2α∥2ct|k|□ku∥Lt∈[0,t0]∞Lxp≲∥u0∥Mp,11-2α+∥u2∥L~α/(2α-1)(I;Ep,1cat)≲∥u0∥Mp,11-2α+∥u∥L~2α/(2α-1)(I;Ep,1cat)2,
which implies (60).We now extend the solution fromI=[0,t0] to I1=[t0,t1] for some t1>t0. Consider the mapping
(63)𝒯1:u(t)⟶G2α(t-t0)u(t0)+∫t0t‍G2α(t-τ)(γ→·∇)(u2)(τ)dτ,
and the resolution space
(64)𝒟1={u∈L~2α/(2α-1)(I1;Ep,1cat):∥u∥L~2α/(2α-1)(I1;Ep,1cat)⩽δ1},
where δ1 will be chosen below. Taking t=t0 in (60), one has that
(65)∑k‍〈k〉1-2α∥2cat0|k|□ku∥p≲∥u0∥Mp,11-2α+δ2.
For any δ1>0, in view of (65), we can choose a sufficiently large J such that
(66)C∑|k|>J‍〈k〉1-2α∥2cat0|k|□ku∥p⩽δ14.
Hence, in view of Proposition 9,
(67)∥G2α(t-t0)u(t0)∥L~2α/(2α-1)(I1,{k:|k|>J};Ep,1cat)⩽C∑|k|>J‍〈k〉1-2α∥2cat0|k|□ku∥p⩽δ14,
and one can choose t1>t0 verifying C|I1|(2α-1)/(2α)〈J〉2α-1(∥u0∥Mp,11-2α+δ2)⩽δ1/4, so
(68)∥G2α(t-t0)u(t0)∥L~2α/(2α-1)(I1,{k:|k|⩽J};Ep,1cat)⩽C|I1|(2α-1)/(2α)∑|k|⩽J‍∥2cat0|k|□ku∥p⩽C|I1|(2α-1)/(2α)〈J〉2α-1∑|k|⩽J‍〈k〉1-2α∥2cat0|k|□ku∥p⩽C|I1|(2α-1)/(2α)〈J〉2α-1(∥u0∥Mp,11-2α+δ2)⩽δ14.
In view of (67) and (68),
(69)∥G2α(t-t0)u(t0)∥L~2α/(2α-1)(I1;Ep,1cat)⩽δ12.
It follows from Propositions 10 and 11(70)∥∫1t‍G2α(t-τ)(γ→·∇)u2(τ)dτ∥L~2α/(2α-1)(I1;Ep,1cat)⩽C(1+|I1|1/(2α))24cat1∥u2∥L~α/(2α-1)(I1;Ep,1cat)≲C(1+|I1|1/(2α))24cat1∥u∥L~2α/(2α-1)(I1;Ep,1cat)2.
Hence, collecting (69) and (70), we have
(71)∥𝒯1u∥L~2α/(2α-1)(I1;Ep,1cat)⩽δ12+C(1+|I1|1/(2α))24cat1∥u∥L~2α/(2α-1)(I1;Ep,1cat)2.
Now we can choose δ1>0 satisfying
(72)C(1+|I1|1/(2α))24cat1δ1⩽14.
It follows that 𝒯1u∈𝒟1 for any u∈𝒟1. So, we have extended the solution from [0,t0] to [t0,t1]. Noticing that
(73)∑k‍〈k〉1-2α∥2cat|k|□kG2α(t)u0∥Lt∈[t0,t1]∞Lxp≲∑k‍〈k〉1-2α∥2cat0|k|□ku0∥p,
we easily see that the solution can be extended to [t1,t2],…,[tm,tm+1],… and finally find a Tmax⁡ satisfying the conclusions in Theorems 6 and 7.Remark 13.
As explained in the introduction, the method used here does not work for the critical caseα=1/2 and we need to find other ways to deal with this situation in the next section.


## 4. Analytic Regularity for GDGL:α=1/2

Recall that, in Section2, we verify the following frequency uniform exponential decay estimate for the generalized Ginzburg-Landau semigroup G1(t)=e-(a+i)t(-Δ)1/2:
(74)∥□kG1(t)f∥p≲2-2cat|k|∥□kf∥p
and the frequency dyadic decay estimate:
(75)∥ΔjG1(t)f∥p≲e-cat2j∥Δjf∥p.
Based on (74) and (75), we will establish some estimates for G1(t) and 𝒜1f in suitable space time exponential modulation and Besov norms, respectively. First, let us consider the linear estimates for G1(t) in space time exponential modulation norm.Proposition 14.
Assume that1⩽p,q⩽∞, a>0, I=[0,1]; then there hold
(76)∥∂xiG1(t)u0∥L~1(I;Ep,qcat)≲∥u0∥Mp,q0,∥G1(t)u0∥L~∞(I;Ep,qcat)≲∥u0∥Mp,q0.Proof.
For|k|⩾1, from (74)
(77)2cat|k|∥□k∂xiG1(t)u0∥p≲2-cat|k|〈ki〉∥□ku0∥p.
Taking Lt∈I1 norm to (77),
(78)∥2cat|k|□k∂xiG1(t)u0∥Lt∈I1Lxp≲〈k〉-1〈ki〉∥□ku0∥p≲∥□ku0∥p.
For k=0, (78) also holds. Hence taking a sequence lq norm on both sides of (78) implies the first estimate. For the second estimate, one only needs to take Lt∈I∞ norm to (74) and then a sequence lq norm.More generally, using the method in Proposition14, one can verify that, for the infinite time interval I=[1,+∞), there holds the following.Proposition 15.
Let1⩽p,q⩽∞, a>0, I=[1,+∞); then for any 𝔸⊂ℤn∖{0},
(79)∥∂xiG1(t-1)u(1)∥L~1(I;𝔸;Ep,qcat)≲∥u(1)∥Ep,qca,∥G1(t-1)u(1)∥L~∞(I;𝔸;Ep,qcat)≲∥u(1)∥Ep,qca.Next, we proceed with the estimates of𝒜1f in space time exponential modulation norm.Proposition 16.
Let1⩽p,q⩽∞, 1⩽q~1⩽q~⩽∞, a>0. Then there exists a constant c>0 such that, for 0<t0⩽∞, I=[0,t0), and 𝔸⊂ℤn∖{0},
(80)∥𝒜1f∥L~q~(I;𝔸;Ep,qcat)≲∑k∈𝔸‍〈k〉1/q~1-1/q~-1∥2cat|k|□kf∥Lt∈Iq~1Lxp
holds for all f∈L~q~(I;𝔸;Ep,qcat).Proof.
By the decay estimate (74), we easily see that
(81)∥□k𝒜1f∥Lxp≲∫0t‍2-2ca(t-τ)|k|∥□kf(τ)∥Lxpdτ.
Multiplying (81) by 2cat|k|,
(82)2cat|k|∥□k𝒜1f∥Lxp≲∫0t‍2-ca(t-τ)|k|2caτ|k|∥□kf(τ)∥Lxpdτ.
When |k|⩾1, applying Young’s inequality to (82) with q~⩾q~1,
(83)∥2cat|k|□k𝒜1f∥Lt∈Iq~Lxp≲〈k〉1/q~1-1/q~-1∥2caτ|k|□kf(τ)∥Lt∈Iq~Lxp.
So, taking the sequence lq norm on both sides of (83) implied the conclusion.In view of∥∂xi□kf∥Lxp≲〈ki〉∥□kf∥Lxp and taking special index in Proposition 16, we obtain the following.Corollary 17.
Under the same conditions as above, one has(84)∥∂xi𝒜1f∥L~1(I;𝔸;E∞,1cat)≲∥f∥L~1(I;𝔸;E∞,1cat),∥𝒜1f∥L~∞(I;𝔸;E∞,1cat)≲∥f∥L~1(I;𝔸;E∞,1cat).We now consider the linear estimates ofG1(t) and 𝒜1f in space time homogeneous Besov norms based on dyadic decay estimate (75).Proposition 18.
Assume that1⩽p,q,γ⩽∞, a>0; then
(85)∥G1(t)f∥L~γ(ℝ+;B˙p,qs)≲∥f∥B˙p,qs-1/γ.Proof.
Let us takeLtγ norm on both sides of dyadic decay estimate (75):
(86)∥ΔjG1(t)f∥LtγLxp≲∥e-cat2j∥Ltγ∥Δjf∥Lxp≲2-j/γ∥Δjf∥Lxp.
Then multiplying the above inequality by 2js and taking a sequence lq norm, we obtain the result as desired.Proposition 19.
Let1⩽p,q⩽∞, 1⩽γ1⩽γ⩽∞, a>0. Then
(87)∥𝒜1f∥L~γ(ℝ+;B˙p,qs)≲∥f∥L~γ1(ℝ+;B˙p,qs-(1+1/γ-1/γ1)).
In particular,
(88)∥𝒜1f∥L~1(ℝ+;B˙∞,11)≲∥f∥L~1(ℝ+;B˙∞,10),∥𝒜1f∥L~∞(ℝ+;B˙∞,10)≲∥f∥L~1(ℝ+;B˙∞,10).Proof.
Using dyadic decay estimate (75), we see that
(89)∥Δj𝒜1f∥Lxp⩽∫0t‍∥ΔjG1(t-τ)f(τ,x)∥Lxpdτ⩽∫0t‍e-ca(t-τ)2j∥Δjf(τ)∥Lxpdτ.
Taking Ltγ norm on both sides of the above estimate and with Young’s inequality, we get
(90)∥Δj𝒜1f∥LtγLxp⩽2-j(1+1/γ-1/γ1)∥Δjf∥Ltγ1Lxp.
Arguing as before, we multiply (90) by 2js and take a sequence lq norm and then finally get the conclusion as desired.With the preparation above, we are now ready to start the proof of Theorem8. In the following let I=[0,1]. We choose the following resolution space:
(91)𝒟={u:∥u∥X+∥u∥Y⩽δ}
with metric d(u,v)=∥u-v∥X∩Y, where ∥u∥X=∥u∥L~1(I;B˙∞,11)∩L~∞(I;B˙∞,10) and ∥u∥Y=∑i=1n‍∥∂xiu∥L~1(I;E∞,1cat)+∥u∥L~∞(I;E∞,1cat). Consider the map
(92)𝒯:u(t)⟶G1(t)u0+𝒜1[(γ→·∇)u2].
We will first carry out the computation of ∥G1(t)u0∥Y. Set c=2-5. By (74), for |k|⩾1,
(93)2cat|k|∥□k∂xiG1(t)u0∥∞≲2-cat|k|〈ki〉∥□ku0∥∞.
Denote ℤ⋆n=ℤn∖{0}. Taking Lt∈I1 norm and then sequence l1 norm on ℤ⋆n to (93), we get
(94)∥∂xiG1(t)u0∥L~1(I;ℤ⋆n;E∞,1cat)≲〈k〉-1〈ki〉∥u0∥M∞,10≲∥u0∥M∞,10.
For k=0, there holds
(95)∥□0∂xiG1(t)u0∥∞≲∥□0u0∥∞.
From (94) and (95), it is easy to see that
(96)∥∂xiG1(t)u0∥L~1(I;E∞,1cat)≲∥u0∥M∞,10.
Similarly,
(97)∥G1(t)u0∥L~∞(I;E∞,1cat)≲∥u0∥M∞,10.
Hence
(98)∥G1(t)u0∥Y=∑i=1n‍∥∂xiG1(t)u0∥L~1(I;E∞,1cat)+∥G1(t)u0∥L~∞(I;E∞,1cat)≲∥u0∥M∞,10.
By Corollary 17 and Proposition 11,
(99)∑i=1n‍∥∂xi𝒜1(γ→·∇)u2∥L~1(I;ℤ⋆n;E∞,1cat)+∥𝒜1(γ→·∇)u2∥L~∞(I;ℤ⋆n;E∞,1cat)≲∥(γ→·∇)u2∥L~1(I;ℤ⋆n;E∞,1cat)≲∑j=1n‍∥∂xju∥L~1(I;ℤ⋆n;E∞,1cat)∥u∥L~∞(I;ℤ⋆n;E∞,1cat)≲∥u∥Y2.
In view of ∥□0f∥∞≲∥f∥∞≲∥f∥B˙∞,10, we observe that
(100)∥□0∂xi𝒜1(γ→·∇)u2∥Lt∈I1Lx∞+∥□0𝒜1(γ→·∇)u2∥Lt∈I∞Lx∞≲∥𝒜1(γ→·∇)u2∥L~1(I;B˙∞,11)+∥𝒜1(γ→·∇)u2∥L~∞(I;B˙∞,10)≲∥(γ→·∇)u2∥L~1(I;B˙∞,10)≲∑i=1n‍∥∂xiu2∥L~1(I;B˙∞,10)≲∥u2∥L~1(I;B˙∞,11).
To bound the right-hand side of (100), let us recall the paraproduct decomposition [10, 20]: for any two tempered distributions f,g, fg can be decomposed as the summation of two parts:
(101)fg=∑i,j‍ΔifΔjg=∑j‍Sj-1fΔjg+∑i‍SigΔif.
From the support property in frequency space, there holds
(102)Δi(SjfΔjg)=0,ifi>j+3.
Thus
(103)Δi(fg)=∑i⩽j+3‍Δi(Sj-1fΔjg+SjgΔjf).
With this tool, we now estimate ∥u2∥L~1(I;B˙∞,11). 						(104)∥Δi(u2)∥Lx∞=∥∑i⩽j+3‍Δi(Sj-1uΔju+SjuΔju)∥Lx∞≲∑j⩾i-3‍∥Sju∥Lx∞∥Δju∥Lx∞≲∑j⩾i-3‍∑k⩽j‍∥Δku∥Lx∞∥Δju∥Lx∞.
By taking Lt1 norm, we deduce that
(105)∥Δi(u2)∥Lt1Lx∞≲∑j⩾i-3‍∑k⩽j‍∥Δku∥Lt∞Lx∞∥Δju∥Lt1Lx∞≲∥u∥L~∞(I;B˙∞,10)∑j⩾i-3‍∥Δju∥Lt1Lx∞.
Multiplying (105) by 2i and taking a sequence l1 norm, we get
(106)∥u2∥L~1(I;B˙∞,11)≲∥u∥L~∞(I;B˙∞,10)∥∑j⩾i-3‍2-(j-i)2j∥Δju∥Lt1Lx∞∥l1≲∥u∥L~∞(I;B˙∞,10)∥u∥L~1(I;B˙∞,11),
where in the last inequality, we use Young’s inequality. From (100) and (106), we obtain
(107)∥□0∂xi𝒜1(γ→·∇)u2∥Lt∈I1Lx∞+∥□0𝒜1(γ→·∇)u2∥Lt∈I∞Lx∞≲∥u∥L~∞(I;B˙∞,10)∥u∥L~1(I;B˙∞,11)≲∥u∥X2.
Collecting (98), (99), and (107), we see that
(108)∥𝒯u∥Y≲∥u0∥M∞,10+∥u∥Y2+∥u∥X2.
On the other hand, by Propositions 18 and 19, (100), and (106), similar reasoning yields that
(109)∥𝒯u∥X≲∥u0∥B˙∞,10+∥𝒜1(γ→·∇)u2∥L~1(I;B˙∞,11)⋂‍L~∞(I;B˙∞,10)≲∥u0∥B˙∞,10+∥u∥X2.
Thus from (108) and (109), we have
(110)∥𝒯u∥X+∥𝒯u∥Y⩽C∥u0∥M∞,10∩B˙∞,10+C∥u∥X2+C∥u∥Y2.
Suppose that
(111)Cδ⩽14,C∥u0∥M∞,10∩B˙∞,10⩽δ2.By the standard contraction mapping principle, we see that there exists a unique solution to𝒯u=u in space X∩Y. In particular, there hold
(112)∥u∥L~∞(0,1;E∞,1cat)⩽δ;∥u∥L~∞(0,1;B˙∞,10)⩽δ.
So for u(1)=u(1,x),
(113)∥u(1)∥E∞,1ca⩽δ,∥u(1)∥B˙∞,10⩽δ,c=2-5.
We now extend the regularity of solutions. Consider the integral equation
(114)u(t)=G1(t-1)u(1)+∫1t‍G1(t-τ)(γ→·∇)(u2)(τ)dτ.
Let I1=[1,+∞). Denote
(115)∥u∥X1=∥u∥L~1(I1;B˙∞,11)∩L~∞(I1;B˙∞,10),∥u∥Y1=∑i=1n‍∥∂xiu∥L~1(I1;E∞,1ca)+∥u∥L~∞(I1;E∞,1ca).
Choose the resolution space to be
(116)𝒟1={u:∥u∥X1+∥u∥Y1⩽δ1}
with metric d1(u,v)=∥u-v∥X1∩Y1. δ1 is to be determined later. Similar to (109), we have
(117)∥𝒯1u∥X1⩽C∥u(1)∥B˙∞,10+C∥u∥X12.
To estimate ∥𝒯1u∥Y1, by Proposition 15(118)∥∂xiG1(t-1)u(1)∥L~1(I1,ℤ⋆n;E∞,1ca)+∥G1(t-1)u(1)∥L~∞(I1,ℤ⋆n;E∞,1ca)≲∥u(1)∥E∞,1ca.
For the remaining part concerning □0, we use ∥□0f∥≲∥f∥B˙∞,10 and Proposition 18 to bound
(119)∥□0∂xiG1(t-1)u(1)∥Lt∈I11Lx∞+∥□0G1(t-1)u(1)∥Lt∈I1∞Lx∞≲∥G1(t-1)u(1)∥L~1(I;B˙∞,11)∩L~∞(I;B˙∞,10)≲∥u(1)∥B˙∞,10.
Arguing as before, with Corollary 17 and Proposition 11, we get
(120)∥∂xi∫1t‍G1(t-τ)(γ→·∇)u2(τ)∥L~1(I1,ℤ⋆n;E∞,1ca)+∥∫1t‍G1(t-τ)(γ→·∇)u2(τ)∥L~∞(I1,ℤ⋆n;E∞,1ca)≲∥(γ→·∇)u2∥L~1(I1,ℤ⋆n;E∞,1ca)≲∥∂xiu∥L~1(I1;E∞,1ca)∥u∥L~∞(I1;E∞,1ca)≲∥u∥Y12.
Similar as in (107), we have
(121)∥□0∂xi∫1t‍G1(t-τ)(γ→·∇)u2(τ)∥Lt∈I11Lx∞+∥□0∫1t‍G1(t-τ)(γ→·∇)u2(τ)∥Lt∈I1∞Lx∞≲∥u∥L~∞(I1;B˙∞,10)∥u∥L~1(I1;B˙∞,11)≲∥u∥X12.
Then
(122)∥𝒯1u∥Y1⩽C∥u(1)∥B˙∞,10+C∥u(1)∥E∞,1ca+C∥u∥X12+C∥u∥Y12⩽Cδ+C∥u∥X1∩Y12.
Choose δ1 satisfying Cδ⩽δ1/2 and Cδ1⩽1/2; then by the contraction mapping argument, we see that 𝒯1u=u has a unique solution in 𝒟1 which completes the proof of Theorem 8.Remark 20.
The same reasoning can show that ifu0∈B˙p,10∩Mp,10(1⩽p⩽∞) is sufficiently small, then there exists a unique global solution u to (1) satisfying u∈L~∞(ℝ+;Ep,1s(t)) and ∂xiu∈L~1(ℝ+;Ep,1s(t)) with s(t)=2-5(a∧at). We omit the details here.


## 5. Conclusion

In this paper, we investigate and prove some well-posedness results and analytic regularity property of the solutions to the (fractional) quadratic derivative Ginzburg-Landau equation with rough initial data in certain critical modulation spaces and Besov type spaces. The basic strategy is via contraction mapping principle. The main obstacle that comes from the one order derivative in the nonlinearity is finally resolved by developing suitable exponential decay estimates for the (fractional) Ginzburg-Landau semigroup. The suggested approach can be used for more general fractional nonlinear partial differential equations.



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*Source: 607028-2014-04-16.xml*