Well-Posedness of Nonlinear Schrödinger Equations Yin Yin Su Win
Acknowledgments
The author is greatly indebted to her advisor, Professor Yoshio Tsutsumi, for his constant support, guidance and valuable suggestions.  The author wishes to express her thank to Professor Hideo Takaoka for fruitful suggestions.  She also would like to thank Satoshi Masaki and other members of her Laboratory for discussions. This work is supported by the Japanese Government (MEXT) Scholarship.

ii
Notation
Dα = ∂ α 1 · · · ∂ N . for a multi-index α ∈ N N . Vu = (∂ıս, . . . , ∂ N U ). Δ = ∑ j =1 ∂ 2 . S(Rn) Schwartz space; i.e., the set of all real- or complex -valued C c  functions on R such that for every nonegative interger m and every multi-index α, Pm,α(u) = sup(1 + | x | ² ) 77 | Dαu(x) | < ∞ . xER" S(R n ) is a Fréchet space when equipped with the seminorms p m,α S'(Rn) space of tempered distributions on R n ; i.e.,the topological dual of S ( R n ). S'(Rn) is a subspace of Co (Rn). LP(R) Banach space of measurable functions u : R → R or ( u : R → C) such that ||u|| L < ∞, with
H ³ p ( R n ) (s ∈ R,1 ≤ p ≤ ∞) Banach space of elements u ∈ S ′ ( R n ) such that F-1[(1 + |ξ|²)²²²] ∈ LP(Rn). = Hs.p(Rn) is equipped with the norm
H 8 ( R n ) = H 8 ;²( R n ) H 8 P ( R n ) B 8 q (R n ) (s ∈ R, 1 ≤ p ≤ ∞) homogeneous version of the Sobolev space H s:p ( R n ) (s ∈ R,1 ≤ p ≤ ∞) Banach space of elements u ∈ S ′ ( R n ) such that || u || B ñ , q < ∞ with
where F-1(φ j û ) is the j th dyadic block of the Littlewood-Paley decompo- sition of u. B s q (R n ) (s ∈ R, 1 ≤ p, q ≤ ∞) homogeneous version of the Sobolev space B s q ( R n ). X s,b (s,b ∈ R) Fourier restriction space corresponding to the Schrödinger equation of elements u ∈ S ′ ( R ² ) such that where ⟨·⟩ = (1 + | · |²)². A ≤ B for any positive A and B means A < CB with constant C. A ~ B A ≤ B and B ≥ A. A ≪ B A ≤ cB with small constant c.
In this paper we study the Cauchy problem of nonlinear Schrödinger equations with initial data in Sobolev space
where u is a complex valued function with λ ∈ C and p > 1. There are many problems concerning the short time and the long time behaviour of solutions of nonlinear Schrödinger equations. We are particularly interested in two problems such as the unconditional well-posedness and the global well-posedness under H 1 norm. First, we briefly recall the definition of well-posedness in H 8 that we use here: Definition 1.0.1. For any initial data φ ∈ H 8 , there exists a positive time T = T(||uo|| H = ) depending on the norm of initial data such that (1.1)-(1.2) has a strong unique solution u in X ⊂ C ([−T,T],H 8 ), where X is a Banach spaces and the solution map from H s to C ([−T,T], H s ) continuously depends on the initial data. If T can be chosen arbitrarily large, we say that (1.1)-(1.2) is globally well-posed. If we can take X = C ([−T,T], H ® ), then we say that (1.1)−(1.2) is unconditionally well-posed. One may show that existence and uniqueness result by fixed point method via the following equation
where U ( t ) is a free Schrödinger evolution operator. It is known that the solution of (1.1)-(1.2) satisfies the conservation of mass
and the conservation of energy
equation with power nonlinearity. 1
Unconditionally well-posed
Kato [28] introduces the concept of unconditional well-posedness of nonlinear Schrödinger equation. He explains that the well-posedness problem needs the auxiliary condition which is related to Strichartz estimates to ensure uniqueness with two examples. First we start with the definition of admissible pair. Let (q, r) be an admissible pair such that when 2 ≤ r ≤ ππ = 2 (2 ≤ r ≤ ∞ if n = 1,2 ≤ r < ∞ if n = 2). Let us consider the equations (1.1)-(1.2) in the following two cases. (i) Problem (A) Assume p ≤ 1+ π = 2 .  There exists a unique solution u ∈ C ([0,T), H 1 (R N ))∩
L 9 ((0,T), L r (R n )), for some T ≥ 0, where (q,T) is an admissible pair. (ii) Problem (B) Assume p < 1 + ".  There exists a unique solution u ∈ C ([0, T], L 2 ( R N )) ∩ L 9 ((0,T), L r ( R n )), for some T ≥ 0, where (q, r ) is an admissible pair. In problem (A), auxiliary space L 9 ((0,T), L r ( R n )) can be removed, it is bonus as Kato points out, which may or may not appear in theorem. Hence we say that problem (A) is unconditional well-posedness. But in problem (B), auxiliary space Lg((0,T), Lr(Rn)) is essential part of the well-posedness because we might not prove the uniqueness without auxiliary conditions. Hence we say that, problem (B) is conditional well-posedness in L 2 ( R N ) with L 9 ((0 , T ), L 7 ( R n )). In this case there are infinitely many auxiliary spaces but they are consistent. The uncondi- tional uniqueness is a concept of uniqueness which does not depend on how to construct the solution. Our another interesting is the global well-posedness of nonlinear Schrödinger equations when the regularity is below H 1 .
Globally well-posed below H 1
It is known that the global well-posedness in H s ( R N ) for s ≥ 1, is obtained These two laws lead to L 2 via the laws of conservation of mass and energy. and H 1 regularities, respectively and the both norms stay bounded for all time. The question then arises whether (1.1)-(1.2) is globally well-posed or not when regularity is below H 1 . In that direction, two methods have been developed our understanding recent years, namely Fourier truncation method of Bourgain and method of almost conserved quantities or I-method of J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao. They both proceed by choosing a large frequency N which depend on time and dividing a solution into two parts of lower and higher frequencies than N . Let us consider for the particular case
with the conservation of energy
where p = 3 or 5. It is also known that this nonlinear Schrödinger equation enjoys the scaling symmetry To explain the I -method, we start with the definition of Fourier multiplier operator
where m(ξ) is an arbitrary real valued multiplier which is the identity on the low frequencies and like as a fractional integral operator of order 1 – s on the high frequencies. Then we see that
We set u = Iv by applying I to (1.3), then u solves the equation
Hence we have
Then it can be shown that there exists α > 0 such that
for all t ∈ [0, 1] which is controlled by bilinear or multi-linear estimates and usual Strichartz estimates under X s,b spaces setting. Here we divide the proof into three main parts such as low-low, high-low and high-high frequency interaction. In low-low case, E ( Iu ) is conserved because the operator I is the same as the identity. In high-high case, we get the small error in IH 1 . The most difficult case is high-low frequencies interaction. Hence we have to improve some estimates (for instance, the refinement of bilinear Strichartz estimate, see Chapter 6) to get a good decay which needs to match the error of high-high case. For instance, when p = 3, n = 2, J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao [11] show
By using time iteration and scaling, global well-posedness for H 8 with s > 4 is shown. In this subject, we shall mainly concentrate on the derivative nonlinear Schrödinger equation (DNLS) in one dimension
where u(t,x) is a complex valued function and δ = +1 or −1. It is not necessary to distinguish the defocusing and the focusing because we consider the suffuciently small initial data. In Chapter 2, we give the function spaces, several applications such as a gauge transformation on both periodic and non-periodic cases, paraproduct method, the functions with 2πλ-periodic and several properties for rescaled functions. In Chapter 3, we are concerned with the Fourier restriction spaces. In Chapter 4, we discuss the Hartree type nonlinear Schrödinger equation
where λ ∈ R and T > 0. Let 0 < γ < min(4, n ). We show that (1.6)-(1.7) is uncon- ditionally well-posed in C ([0,T]; H s ( R n )) when n ≥ 3, 0 < s < π and γ < 2s + 2. In Chapter 5, we study the derivative nonlinear Schrödiger equation as an un- conditional well-posedness in energy space. In Chapter 6 we examine the global well-posedness of the derivative nonlinear Schrödinger equation for H s with ½ < s < 1.

2.1
Basic function spaces and fundamental prop-
erties
In this section we present some well-known spaces and their properties. Let C ′∞ ( = C ′∞ ( R n , C )) be the linear space of infinitely differentiable functions equipped with the topology induced by the semi-norms;
is finite for all m and α = (α 1 , . . . ,α n ) and D α is the differential operator
where |α| = α 1 + · · + α n ≥ 0 and m ∈ N ∪ {0}. A C ∞ function is of class S( R n ) if f and all its partial derivatives are rapidly decreasing.  Any functions in C ∞ ( R n ) belong to S ( R n ), it is a larger class of functions. We also know that S ( R n ) is a Fŕechet space and closed under dif- ferentiation. Moreover, S ( R n ) is closed under translations and multiplication by complex potentials eix& Let S ′ ( R n ) denote the set of all tempered distributions, the topological dual of S(Rn). We set f(g) = ⟨f,g⟩ for f ∈ S′(Rn), g ∈ S(Rn). Let T and T ′ be the linear operators from S( R n ) to S( R n ) such that the adjoint identity
for ψ, g ∈ S(R n ).  We define ⟨T f, g⟩ = ⟨f, T'g⟩ for any f ∈ S'. Here we note that in Hilbert space theory the above identity requires complex conjugates while in distribution theory we do not take complex conjugates even if the function is a complex valued function. Let f ∈ S(Rn), we define
then F : S(R n ) → S(R n ) is an isomorphism with inverse
and the identity
holds. The Plancherel formula is a simple consequence of this identity.  We take f(x) = Fg(x). We have f(x) = fg(ξ)eixξdξ = F-1σ(x) then Ff(x) = FF-1σ = g(x). Hence the identity becomes
Now it is turn to define the Fourier transformation
If f is any integrable function, we could define the Fourier transform of f directly.
2.1.1
Sobolev and Besov spaces on Rn
To introduce Sobolev spaces and Besov spaces we need some special systems of functions contained in S(Rn). Let φ ∈ S(Rn) be a real bump function with
with k ∈ Z. Let f ∈ S ′ ( R n ), s ∈ R. We shall denote two operators J 8 and I 8 , both from S'(R n ) to S'(R n ) are defined by
where 〈·〉＝ (1 + | · |²)¹/². The operators J− 8 and I− 8 are called the Bessel and Riesz potential of order s respectively.  Then we give some simple properties of operators J 8 and I 8 . Lemma 2.1.1. Let f ∈ S ′ ( R n ), 1 ≤ p ≤ ∞ , s ∈ R . Assume that φ k ∗ f ∈ L ρ ( R n ), then there exist a consant C > 0 such that
and if ψ * f ∈ L P ( R n )
Definition 2.1.2. Let s ∈ R, 1 ≤ p,q ≤ ∞. We define the (generalized) Sobolev space H p (R n ) and the Besov space B p,q(R n ) by
endowed with the norms
H ξ ( R n ) and B ξ ,q ( R n ) are norm linear spaces with norm ‖ · ‖ H ξ and ‖ · ‖ B ξ ,q respectively. Moreover, they are complete and therefore Banach spaces. We denote that A 1 → A 2 is continuous embedding , i.e there is a constant C such that
Remark 2.1.3 (Dilation of H p (R n ) and B p q (R n )). Let s > n max(0, p − 1). Let f(·) → f(λ·), λ > 0. Then there exist a constant C > 0 which does not depend on λ and f such that
for all f ∈ H p ( R n ) and f ∈ B p,q ( R n ) respectively. Theorem 2.1.4. If s 1 ≤ s 2 and 1 ≤ p,q ≤ ∞ we have
Moreover, S(Rn) is dense in H s (Rn) and B s q(Rn) respectively. 7 Theorem 2.1.5 (Sobolev's embedding for H ξ ( R n )). Let 1 ≤ p ≤ p 1 < ∞ , and S1, S2 E R. H 8 (R n ) → H 81 (R n ) if s − π = s 1 − π . In particular, if 1 ≤ p < ∞ , and 0 < s < π then we have
If p > 1 and s > " then H"(R") → L"(R"). Theorem 2.1.6 (Sobolev's embedding for B s ,q ). Let 1 ≤ p ≤ p 1 < ∞ , 1 ≤ q ≤ q1 < ∞, and s, s1 ∈ R.
Remark 2.1.7. Next we give some relation between H s ( R n ) and B s ,q ( R n ). (i) If 1 < p ≤ 2, then B s,p (R n ) → H s (R n ) → B s,2 (R n ). (ii) If 2 < p < ∞, then B s,2(R n ) → H s (R n ) → B s,p(R n ). (iii) In particular B ž, 2 (R n ) = H ž (R n ) = H ŝ (R n ). Remark 2.1.8. Let s 0 < s < s 1 . There exists a constant C > 0 such that
for f ∈ H p 1 ( R n ) and f ∈ B p ,q ( R n ) respectively. Theorem 2.1.9. J° is an isomorphism between H ξ ( R n ) and H ξ − σ ( R n ), B ξ ,q ( R n ) and B p,q (R n ) respectively. Corollary 2.1.10 (Duality). Let s ∈ R. If 1 ≤ p,q < ∞, we have
where f + f = 1 and f + f = 1. We have a several interpolations corresponding to the indices s, p, q. Theorem 2.1.11 (Interpolation). Let 0 < θ < 1.  Assume s 0 , s, s 1 ∈ R, 0 ≤ Po,P,P1,90,9,91 ≤ ∞ are satisfy the following formulas
Then, we have

2.1.2
The Homogeneous Sobolev and Besov spaces
We now introduce the homogeneous Sobolev spaces H ξ ( R n ) and the homogeneous Besov spaces B s ,q ( R n ). The function φ k is defined as a previous subsection. Let s ∈ R, 1 ≤ p ≤ ∞. For all f ∈ S ′ ( R n ), we define
We if the series Γ κ k=-∞ F-1(|ξ| 8 Fφk * f) converges in S ′ to an L P function. note that ˙˙˙˙ s ( R n ) is a semi-normed space and ∥ f ∥ ˙ ˙ s ( R n ) = 0 if and only if f is a polynomial. Next, s ∈ R and 1 ≤ p,q ≤ ∞. For all f ∈ S ′ ( R n ), We write
When the norm ‖ƒ‖ βξ,q ( R n ) is finite, we call ˙ B ρξ,q ( R n ) is a homogeneous Besov space. We again note that B f,q ( R n ) is a semi-normed space and ‖ f ‖ B f,q ( R n ) = 0 if and only if supp f = {0}, i.e. if and only if f is a polynomial. Furthermore, for 0 < s < 1 and q < ∞, one defines
and when q = ∞
Several results of H p ( R n ) and B p,q ( R n ) carry to the homogeneous spaces H 8 (R n ), B 8 g (R n ). For instance,
9
Theorem 2.1.12. If s 1 < s 2 and 1 ≤ p,q ≤ ∞ we have
Moreover S(Rn) is dense in H ξ (Rn) and B ξ q (Rn) respectively. Theorem 2.1.13. J° is an isomorphism between H p ( R n ) and H p − σ ( R n ), B p … ( R n ) and B 8-9 (R n ). Corollary 2.1.14 (Duality). Let s ∈ R. If 1 ≤ p,q < ∞, we have
where f + f = 1 and f + f = 1.
2.1.3
Strichartz's estimates
In this section we present the Strichartz's estimates which are useful to study the nonlinear Schrödinger equation in the fractional order Sobolev spaces H s ( R n ). The estimates are introduced by Strichartz [36] as a Fourier restriction theorem. Let consider the nonlinear Schrödinger equation
satisfy the solution u(t) = U(t)u 0 when F(u) = 0. It is known that the free Schrödinger evolution operator U(t) = eit △ is a uni- tary over L 2 but not stable on L P spaces. We now start with dispersive inequality. Proposition 2.1.15. If p ∈ [2,00] and t̸= 0, then U(t) maps from L p ′ ( R n ) continuous to LP(Rn);
for all uo ∈ Lp' ( Rn ). This decay estimate is useful for the long time theory of nonlinear Schrödinger equations, when the dimension n is large and the initial data u 0 has good inte- grability properties. But in many situations, the initial data is assumed in L 2 based Sobolev spaces such as H ž ( R n ). For that direction, Strichartz estimates are introduced by combining the dispersive estimates with some duality arguments. We continue with the notion of admissible pair. Definition 2.1.16. We say that the exponent pair (q,r) is admissible if where
Proposition 2.1.17 (Strichartz's estimates). If n ≥ 1, s ∈ R, (p,q) and (p,q) are admissible and f + f = 1, f + f = 1, then we have the homogeneous Strichartz estimate (2.2) ||U(t)uo||Lq(Lz)(Rn+1) ≤ C(n,q,r)||uo||Lz(Rn), the dual homogeneous Strichartz estimate
and the inhomogeneous Strichartz estimate
for all test function u 0 , F on Rn, Rn+1 respectively, for all t > 0 . We notice that Strichartz's estimates fail in a bounded domain Ω ⊂ R n .
2.2
Paraproduct method
In this section, we study the product of two functions with paraproduct method. This method is easier to estimate when the derivatives are involved because they identify which of the factor is high frequency and which is low frequency. We begin with the definition of Fourier multipliers. It is an important concept of paraproduct method. For k, k 1 ∈ Z , we define the Fourier multipliers
where the smooth function φ k is defined in Subsection 2.1.1. Hence P<k, P>k, Pk are smoothed out projections to the region |ξ| ≤ 2 k , |ξ| > 2 k , |ξ| ～ 2 k respectively. By the Littlewood-Paley decomposition, we have
In addition we shall use
for all f ∈ S ′ with dyadic numbers 2 k 1 We write the L P norm by using the Littlewood-Paley inequality
where 1 < p < ∞. Similarly, by Plancherel theorem, we have
and
We often use the Bernstein inequalities for R n with s ≥ 0.  It is useful to estimate when the frequency is localized. Applying the Bernstein inequality on the low and median frequency to improve low Lebesgue integrability to high Lebesgue integrability is the best . It can be seen in Chapter 4 (Lemma 4.2.1 and Lemma 4.2.3). Lemma 2.2.1. Assume s ≥ 0, 1 ≤ p ≤ q ≤ ∞. Then the following inequalities hold:
Lemma 2.2.2. For all 1 < p < ∞, the following inequality holds
This lemma is useful when we want the projection P<k and Pk to throw away. See [42] (Lecture notes 3). Proposition 2.2.3. If s ≥ 0, then we have || f g|| h 2 (Rn) ≤ C(s,n)|| f|| h 2 (Rn)|| g|| L∞(Rn) + || f || h 2 (Rn)|| g|| L∞(Rn) for all f,g ∈ H s (R n ) ∩ L ∞ (R n ). In particular, if s ≥ n , we see that H s space is closed under multiplication, i.e.
Proof.  Fix p = 2.  The case where s = 0 is trivial.  For s > 0.  By Littlewood-Paley decomposition, we write
We consider
It is clear that P k f has Fourier support in 2 k ′−1 ≤ |ξ| ≤ 2 k ′+1 and P k rg has Fourier support in 2 k ′′− 1 ≤ |ξ| ≤ 2 k ′′+¹ so that P k ((P k f)(P k ′′g) has Fourier support in the sum of these two annuli. This sum needs to intersect the annulus 2k-1 ≤ |ξ| ≤ 2k+1. If k' < k − 5 then the sum vanishes unless k − 3 < k'' < k + 3. Similarly if k" < k − 5 then the sum vanishes unless k − 3 < k' < k + 3. If k' > k − 5 then the sum vanishes unless k' − 3 < k'' < k' + 3, then
The first term of (2.7) is clear for s ≥  n because
(2.9) by Hölder and Sobolev embedding. For the second term of (2.7), we first consider the low-high frequency interaction
by Lemma 2.2.2. If | P ≤ k − 5 f ( x ) | ≤ M ( f ), (2.10) is bounded by
where M(f)(x) is the Hardy-Littlewood maximal operator with the ball B cen- tered at x and the radius r, that is
Hence we have to show | P<k-5f(x)| < M(f). We start with
The proof of high-low and low-low frequencies interaction are same as above. We remain to consider the high-high interaction;
Let k' = k + a, k'' = k + b. (2.11) becomes
It is difficult to handle both high-high frequencies interaction simultaneously, hence we use the triangle inequality to isolate the contribution of individual a
14
and b, then
if |Pk+αƒ(x)| ≥ M(f). We can estimate that
Then (2.12) becomes
The proof is completed. ● Let 
Proposition 2.2.4. [35] Let s ≥ 0, 1 ≤ p < ∞, s < π and π − π ≤ π ≤ π
t = − π − π − σ ) . Then there exists a constant C > 0 such that for all f, g ∈ H π s+2( " -s)

Gauge transformation
2.3
In this section we discuss the gauge transformation for non periodic and periodic cases. It is known that for the Schrödinger equation with power nonlinearity, the nonlinearity is controlled by L P − L q (Strichartz's estimate) estimate but the derivative nonlinear Schrodinger equations (DNLS) has loss of derivative in nonlinearity. It seems unlikely that L P − L q estimate is helpful as in the power nonlinearity case. To overcome this difficulty N. Hayashi and T. Ozawa [24], [25] introduce for non-periodic and S. Herr [26] adjusts for periodic case to derive from the derivative nonlinearity to some new nonlinearity. It is one way to avoid the loss of derivative in the nonlinearity among the several ways.
2.3.1
Non-periodic case
Definition 2.3.1. Let G be the nonlinear map from L 2 ( R ) to L 2 ( R ) by
The inverse transform of f is given by
Proposition 2.3.2. The gauge transformation is a bi-continuous operator from H¹ to H¹.
Taking L 2 norm on both sides and applying the Hölder inequality, we have
Again we consider the derivative of gauge transform,
Then we get
Similarly, we can prove the inverse of gauge transformation.

2.3.2
Periodic case
We now discuss the gauge transformation for the periodic setting which is ad- justed by S. Herr [26]. Before we give definition, we start with some proposition. Proposition 2.3.3. Let T > 0, s ≥ 0. Then the following translations are con- tinuous.
Proposition 2.3.4. Let u ∈ C ([−T,T],L²(T)), define µ(u) = ½‖u(0)‖ž²(T)· Then, for any s ≥ 0, the following translations are homeomorphisms.
Definition 2.3.5. Let f ( x ) be a 2π - periodic function.  We define the gauge transformation such that G(f) : L 2 (T) → L 2 (T) by
where J(u)(x) :=ﾆπ ſ 2 π ſ π |u(y)|² −ﾆπ ||u|| π 2 dydθ . We have seen that Jf(x) is a 2π periodic function since | f(y)|² − ½ || f(t)|| ζ 2 is zero mean value function. Hence Gf ( x ) is also 2 π periodic function. Proposition 2.3.6. [26] For s ≥ 0, r > 0 and there exists c > 0 such that
the map G is locally Lipschitz from H 8 (T) to H 8 (T), i.e.,
The inverse map
and G-1 also satisfies (2.15) on Br, hence G is bi-Lipschitz on bounded subsets. Proof. First, we consider the plus sign
We fix for s > 1. Then we take the H 8 norm on both sides, left hand side becomes
We note that
We note that ‖‖ƒ|²‖н«-¹ ≤ c‖√ƒ|²║∫¬¬ ≤ c‖ƒ║²˛¬ ≤ c‖ƒ║²˛¬ by Sobolev embed- ding.  Then
17 choosing ε ≤ 2s. Similarly
Then combining (2.16)-(2.20),
For s = 0, it can be proved that
Similarly, we can prove minus sign.

Scaling and 2πλ-periodic functions
2.4
In this section, we prepare for Chapter 6, for the proof of global well-posedness in periodic setting. Let u ( t, x ) be a large spacial periodic on R × R/2πλZ . We recall that the Cauchy problem (1.1)-(1.2) is L 2 invariant under scaling ( t, x ) → λ−1/2( , , , ).  Hence u(t, x) solve (1.1)-(1.2) on R × R/2πλZ then
is a solution of (1.1)-(1.2). We recall the Fourier transform of spacial periodic with a large period λ. Define (dξ) λ to be the renormalized measure on Z / λ : (dξ)λ is the counting measure when λ = 1 and weakly converge to the Lebesgue measure when λ → ∞. For a 2πλ-periodic function f , we define the Lebesgue measure of its function for 1 ≤ p ≤ ∞ (usual modification, if p = ∞). The spatial Fourier transform is defined as
and the transformation is inverted by
18 When we consider ∂ π , m ∈ Z , we may doing Fourier inverse transform
Moreover, the following properties hold for large period: (i) ||ƒ||||çç = ||ƒ|||||ç),),

2.5
Notes and references
In this chapter we present the well-known properties of function spaces and some notion which can be found in several text books. For Section 2.1, we refer to the text books of T. Cazenave [6] (Chapter 1-2), J. Bergh -J. Löfström [1] (Chapter 6) , T. Runst - W. Sickel [35] (Chapter 2- 3) and R. Strichartz [37]. In Subsection 2.1.3, Strichartz estimate is introduced by R. Strichartz [36] as a Fourier restriction theorem. The end point Strichartz estimates are created by M. Keel and T. Tao [29]. The application of Strichartz estimates can be seen in several papers, for instance T. Kato [28], J. Ginibre and G. Velo [17], G. Furioli and E. Terraneo [14] etc. See also T. Tao [41] (Chapter 2, Section 2.3). In Section 2.2, we present the paraproduct method which is based on T. Tao [41] (Chapter A, Lemma A.8) and his Lecture notes (Math 254A (Time-frequency analysis)), T. Runst and W. Sickel [35] (Chapter 4, Section 4.4.4, Theorem 1).  In Sectin 2.3 we discuss the gauge transformation for both periodic and non-periodic cases and we refer to some papers such as N. Hayashi and T. Ozawa [24], [25], [34] for non-periodic and S. Herr [26] for periodic. For Section 2.4 we consult the papers of J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao [10], [12], D. De Silva, Nataša Pavlović, G. Staffilani and N. Tzirakis [13] and L. Molinet [33].
In this chapter, we define the two parameter family spaces which come from the linear dispersive equations associated to the space time Fourier transform. We study the some fundamental properties of Fourier restriction space with respect to the Schrödinger equation of both periodic and non-periodic cases and it will be applied in Chapter 5 and 6.
3.1
Definitions and fundamental properties
Let h(ξ) : R n → R be a phase function corresponding to the dispersive linear equation ∂tu = Lu,  uo(0, x) = uo where L = ih(△/i). U(t) denote the group of unitary opertor F x U(t)u 0 (ξ) = e-ith(ξ) F x u 0 (ξ). The space-time Fourier transform of solution u will be supported in the region {(τ,ξ);τ + h(ξ) = 0}. We can localize the solution in time, assume that u(t, x) = η(t)U(t)uo(x) where η ∈ C ∞ with 0 ≤ η ≤ 1. η(t) = 1 on |t| ≤ 1, η(t) = 0 on |t| ≥ 2. Then Fu(τ,ξ) = ˆ η(τ − ˆ h(ξ))Fxuo(ξ). We see that the Fourier transform of ηū will be focus in the region of {(τ,ξ); τ + h(ξ) = O(1)}. Example 3.1.1 (The free Schrödinger equation). Consider ∂tu − i∆u = 0 with h(ξ) = −ξ², n = 1, h(ξ) = − |ξ|², n ∈ N . Example 3.1.2 (The Airy equation). Consider ∂tu + ∂tu = 0 with h(ξ) = ξ³, n = 1. Definition 3.1.3 (For R). Let h : R → R be a continuous function. For s,b ∈ R. Let Xs,b(R × R) be the closure of the Schwartz functions St,x(R × R n ) under norm
Definition 3.1.4 (For T). Let s,b ∈ R.  Let X s,b (R × T) be the closure of the Schwartz functions St,x (R × T) with Xs,b := {u ∈ S'(R × T) : u(t,x + 2π) = u(t,x), ||u||xs,b < +∞} under norm We call X τ = h (ξ) is Fourier restriction spaces, Bourgain spaces or dispersive Sobolev spaces.  For any time interval I , we define the restricted space X s,b ( I × T ) (X s,b (I × R)) by ||u|| x ｡,ь(1×T) := inf{||ũ|| x ｡,ь(R×T) :ﾃ||1×T = u}. Remark 3.1.5. When b = 0, the dispersion relation τ = h(ξ) is irrelevant, X s,0 space represent L 2 H 8 .  When h(ξ) = 0 (h(k) = 0), X 7 =h(ε) space is simply represented the product space H f 😊 H f Remark 3.1.6. There are some fundamental properties of the space X 7 =h(ε) (i) X τ=h(ξ) = X s,b, X 0,0 = L 2 L 2 ξ (same norms), (ii) ||ū|| χ ī =-ﾊ(-ε) = || u|| χ ī =ﾊ(ε) (complex conjugation), (iii) (X 7 =h(ξ)) ⁄ = X 7 =−h(−ξ) (duality), (iv) Xs1,b1 - Xs2,b2 if s1 > s2, b1 > b2 (trivial embedding), (v) (X s1,b1,X s2,b2)[9] = X s,b if 0 < θ < 1, s,s1,s2,b,b1,b2 ∈ R (interpolation). Remark 3.1.7 (Sobolev embedding). We have the following well-known embed- ding,
for some constant C > 0. We consider the relation between X s,b norms and the Schrödinger evolution operator U(t), t ∈ R. Define H s,b space under norm
Since
we see that
Lemma 3.1.8. Let s ∈ R. Let Y be a Banach space of on R × R n with
for all f ∈ H 8 ( R n ), τ ∈ R. Then we have
for b> 1
Take Y norm, using (3.1) and the Minkowski inequality
→ Consequently, we apply to Y = C t ( R, H z ), then we get the embedding X s,b → C(R, H ž ) for all b > ž . We use only b = ž on both periodic and non-periodic case, the other space is needed to ensure the embedding of C ( R, H 2 ). Definition 3.1.9. We define the function space Y s,b which is slightly smaller than X s,1, and it is the closure of the Schwartz function space S t,x (R × R) under the norm
For any time interval I , we define the restricted space Y s,b(I × T) (Y s,b(I × R)) by
Similarly, we define the restricted space of Z s for any time interval I such that
Moreover, we have the embedding for any s ∈ R
We note that X s,b space is an invariant under time localization. Lemma 3.1.10. Let s,b ∈ R, u ∈ S(R × R). Let η(t) be a Schwartz function in time. Then
Proof. It is clear that
● We will often use the following well-known lemma; see [15], Lemma 4.2. Lemma 3.1.11. Let 0 ≤ a ≤ b such that a + b > ½. Then
either a or b is less then 1
3.2
Linear estimates
Let u(t) = U(t)u 0 be the solution of free Schrodinger equation
for all t ∈ [0, T]. We have the usual Strichartz estimate
where (q,r) is the admissible pair. It can be adopted by Bourgain space setting as follows
for any ϵ > 0. More precisely, we have ‖ u‖ L ∞ ≤ ‖ û‖ L ₁ and the right hand side of (3.5) becomes
Here we shall discuss the bilinear estimates in large periodic setting. The first lemma is for L 4 Strichartz estimate which is improved by J. Bourgain [2]. See also A. Grünrock [18] for alternative method. Lemma 3.2.1 (Large periodic case). [18] If u 1 = u 1 ( x,t ) and u 2 = u 2 ( x,t ) are 2πλ periodic which frequencies are supported on {ξι : |ξι| ~ N 1 } and {ξι : |ξι| ~ N2} with N1 ～ N2 respectively, then
where b > 3. Proof. We have
and ‖ u N 1 ‖ L į L 2 ＝ ‖ u 1 ‖ X x x x .  By the Plancherel theorem, for λ > 1, it is sufficient to show that
We denote
and we have to show that S(τ,ξ) ≥ 1. By Lemma 4.2 of [15], we see that
By [18], let k = ξ 1 − ξ 2, ξ = ξ 1 + ξ 2, hence k + ξ = 2ξ 1, k − ξ = − 2ξ 2. Then S( τ, ξ) becomes
We take 2 τ + ξ ² = x 2 , if | k − x 0 | < 1 or | k + x 0 | < 1, the number of k is at most 4. On the other cases, by Cauchy-Schwarz inequality,
with b > 318 □ Now it is turn to discuss the refinement of Strichartz estimate. It is very useful to get a good decay in improvement of global well-posedness. Proposition 3.2.2 (The refinement of Strichartz estimate). [13] Let ϕ 1 and ϕ 2 be 2πλ-periodic functions which frequnecies are supported on {ξι : |ξι| ~ N1} and {ξ2 : |ξ2| ～ N2} with N1 ≫ N2 respectively. Then (3.8) ||η(t)(U x (t)ϕ 1 )η(t)(U x (t)ϕ 2 )|| L ¥ x ≥ C (λ, N 1 )||ϕ 1 || L 2 ||ϕ 2 || L 2
Moreover, even if N 1 ～ N 2 the estimate (3.8) holds when ξ 1 and ξ 2 have same sign. It is see that

25
Proof.  Let ψ = ˆ η be a positive even Schwartz function.  We rewrite the left hand side of (3.8) by Plancherel,
We estimate the τ 1 integration by
with ψ also Schwartz function. Substituting this into (3.10) and using Cauchy- Schwarz, we get
V λ by integrating in τ on ψ , using Fubini's theorem over ξ 1 , ξ 2 and then Plancherel with
We estimate M as follows:
where S = {ξ₁ ∈ ξ₁ Z : |ξ₁| ～ N₁, |ξ − ξ₁| ～ N₂,ξ₂ − 2ξ₁(ξ − ξ₁) = τ + O(1)} and #S denote the number of elements of S. When N < 1, then #S < O(λ) which implies C ( λ, N 1 ) < 1. When N > 1 rename ξ 1 = z. Then
Let zo be an element of S, i.e
We shall count the number of z's ∈ 1 Z such that z 0 + z ∈ S to obtain the upper bound of #S. Then
We rewrite the left hand side of (3.12) as
by using (3.11). Since (3.10)-(3.11),
26 On the other hand, by (3.8), we have | z 0 − ξ | ～ N 1 . Hence it is sufficient to count z's ∈ 1Z satisfying (3.11) and such that z 2 + 2 z ( z 0 − ξ ) =)(1) where z 0 satisfies (3.10). Then
this implies
Hence M < ( )
Let us consider the inhomogeneous Schrödinger equation
Then the solution u ( t ) of (3.15)-(3.16) satisfy the Duhamel formula
Lemma 3.2.4. Let u 0 ∈ H 8 (R) for some s ∈ R.  Then for any Schwartz time cut off function η(t) ∈ S(R) we have Proof.  We have
Next we consider that the convolution part of (3.17) is estimated by Xs,b norm ||(U * F)|| x s,s ≤ C || F || x s,s where ⋆ is convolution in time and it is equivalent to || LF || H ×, b  ≤ C || F || H × , b with the same constant C, and X s ′ , ν and H s ′ , ν ′ is the dual of X s, b and H s, b respectively. The operator L is defined by
Lemma 3.2.5. Let s ∈ R and the following estimates are hold.
Then (3.20) becomes
where
The first contribution is bounded by
We apply the Young inequality,
is bounded by
28 Combining them, the first contribution is bounded by
When we apply the H 2  norm on II, we see that
By using the Young inequality,
We apply the H ž norm on III and by using the Young inequality, we get
Combining H 2  norm of I, II and III, we get
Fix ξ and multiplying each by 〈ξ〉 ² ² and taking the L 2 norm over ξ . We obtain

3.3
Notes and references
The Fourier restriction norm space of Schrodinger equation is introduced by J. Bourgain [2], he also concern for KdV equation. T. Tao [41] (Chapter 2, Section 2.3) is mostly support for this chapter. The refinement of Strichartz estmate for periodic Schrodinger equation is from the paper of D. De Silva, N. Pavlović, G. Staffilani and N. Tzirakis [13], and for the case of periodic KdV equation can be found in the paper of J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao [10]. We also refer to J. Ginibre, Y. Tsutsumi and G. Velo [15], A. Grünrock [21] for this chapter.

Introduction
4.1
In this chapter we study the Hartree nonlinear Schrödinger equation
where λ ∈ R and T > 0. Let 0 < γ < min(4, n ) and s ≥ 0. Our aim is to prove the unconditional well-posedness in C ([0, T]; H s ( R n )) for 0 < s < ". The proof is based on the paraproduct method and the end point Strichartz estimates. The concept of unconditional well-posedness of Schrödinger equation is intro- duced by T. Kato [28] and he proves for power nonlinearity by using Strichartz estimates and Sobolev embedding.  More precisely, we consider the following equation,
where α > 0. We first recall Kato's result. Let s ≥ 0. If the following cases satisfy (i) s ≥ (ii) n ≥ 2, 0 ≤ s < ≥ and α < min( n − 2s , n − 2s ) (iii) n = 1, 0 ≤ s < ½ and α ≤ ½±²²²²² then (4.3) - (4.4) is unconditional well-posedness in C ([0,T); H s ( R n )). He proves the solution of (4.3) - (4.4) is unique in L ∞ ((0,T); L 2 (R n )) ∩ L ∞ ((0,T); L P (R n )). Then the result is followed by Sobolev embedding with H s ( R n ) ⊂ L p ( R n ). This result was improved by G. Furioli and E. Terraneo [14] for low dimensional case. They show that if the following cases satisfy
30

(ii) 3 ≤ n ≤ 5, 1 < α < min{ n − 2s ′ n − 2s (4.3) - (4.4) is unconditional well-posedness in C ([0, T ); H s ( R n )).  They prove the solution is unique in L ∞ ((0,T); B Σ 2 ( R n )) for σ < 0. Their proof is based on the end point Strichartz estimate, the paraproduct technique and Sobolev embedding H 8 ( R n ) ⊂ B σ 2 ( R n ). We use the same idea as in [14] on Hartree nonlinearity. Our main theorem is as follows: Theorem 4.1.1. Let n ≥ 3, 0 < s < π , γ < 2s + 2 and u,ν ∈ C ([0,T]; H s ( R n )) be two solutions of (4.1)-(4.2) with the same initial data u 0 ∈ H α ( R n ), Then the solution is unique for all t ∈ [0,T). This chapter is organized as follows. Section 4.2 contain the proof of several estimates, some are based on the paraproduct method with Besov spaces. In Section 4.3 we give the uniqueness result of our main theorem.
4.2
Lemmas
In this section, we prepare for the argument of uniqueness theorem. We start with the following lemma. Lemma 4.2.1. Let u ∈ H 8 with 0 < s < π . There exist σ, p such that s − π = σ − 2 and we assume that s + σ > 0, σ < 0 and s ≥ σ > s − 1. If w ∈ B f,2 then the following estimates hold. (i) ||u²|| Hç < ||u||²¸¸ where t = s+2(ç −s) ' (ii) ||uw|| Bg,2 ~ ||u|| H × ||w|| Bg,2 where 1 = 1 + n-2s Proof. By Proposition 2.2.4, we see that (i) is clear. Now we prove (ii) by para- product method. We may write the product of uw by Littlewood-Paley decom- position, Pk(uw) = ∑k',k'' ∈ZPk((Pk'u)(Pk''w)). It says that P k u has Fourier support in 2 k − 1 ≤ | ξ | ≤ 2 k ′ +1 and P k , ′ w has Fourier support in 2 k ′′−¹ ≤ |ξ| ≤ 2 k ′′+¹ so that P k ((P k ′u)(P k ′′w ) has Fourier support in the sum of these two annuli. This sum needs to intersect the annulus 2 k −1 ≤ |ξ| ≤ 2 k +1. If k ′ < k − 2 then the sum vanishes unless k − 1 < k ′ < k + 1. Similarly
if k ′ < k − 2 then the sum vanishes unless k − 1 < k ′ < k + 1. If k ′ > k − 2 then the sum vanishes unless k' − 1 < k'' < k' + 1, then
We consider the first term of (4.5). We take the L 9 norm on both sides and apply Hölder's inequality,
by Sobolev embedding. Similarly, we can prove the second term of (4.5), we have ||Pk(P<k-2WPku)||Lq < ||P<k-2W||Lp||Pku||L π²=" product by 2kg and then we take the l 2 norm on both sides, then
It follows that we get the desired estimate. Finally, we consider the third term of (4.5) using the triangle inequality and Bernstein inequality with ξ ≤ ξ + ξ with p > 2, then
Product by 2 kg and then we apply the l 2 norm on both sides, then
for a ～ b. Then we get the desired estimate since s + σ > 0. □ Next we have to prove the following lemma. Lemma 4.2.2. We put 0 < γ < min(4,n). Let σ < 0 < s. Let 2s < γ for (i) and 2s − σ < γ for (ii).  If u ∈ H α ( R n ) and w ∈ B α ,2, then the following estimates hold.
Proof. We first prove ( i ). Let u ∈ H s , by Lemma 4.2.1's ( i ) we have that u 2 ∈ H 8 n_. We put f = |u|² and ψ k = φ(2−kξ) − φ(2−k+¹ξ), then
by using Hardy-Littlewood inequality, we get the desired estimate. Now we prove ( ii ).  By Lemma 4.2.1's ( ii ) we know that uw ∈ Bg,2 with q = f + π=2s. We put g = uw, then
where 1 = p + n=2s and by using Hardy-Littlewood inequality.
Finally, we are ready to prove the following main estimates. Lemma 4.2.3. Let u,v ∈ H 8 with 0 < s < π and 0 < γ < min(4,n). There exist σ, p justifying s − = σ − π and the following two conditions (a)   s + σ > 0,σ < 0,σ < 0,
s ≥ σ > s − 1,   (b)    3s − γ ≥ σ ≥ 3s − γ + 1,  σ ≤  ≥ + 2s − γ  such  that if w ∈ B q 2 , then the following estimates hold.
for  2n2  < r ′  < 2. Proof. It is enough to prove (i).  We put f = |x|~7 * |u|2.  By Lemma 4.2.2, f ∈ H å n_. We may write the product of fw by Littlewood-Paley decomposition,
We consider that the Fourier support of f and w are same as u and w in Lemma 4.2.1 respectively, then
We first consider the first term of (4.8). We take the L r ′ norm on both sides and apply Hölder's inequality,
where Again, product by 2 kg and then we take l 2 norm on both sides, then
by Sobolev embedding. Similarly, we can consider the second term of (4.8), we have || Pk(P<k-2WPkf)|| Lr  < || P<k-2W|| Lr || Pkf || L ≠ ≥ ≥ , product by 2kg and then we take the l 2 norm on both sides, then
Here, 2 k − j l σ ∈ l 1 , since σ < 0
It follows that we get the desired estimate. Finally, we consider the third term of (4.8) using the triangle inequality and Bernstein inequality with ⇌ ≤ φ + ζ = since by assumption σ ≤ φ + 2s − γ, then
Product by 2 kσ and then we apply the l 2 norm on both sides, then
for a ～ b. Then we get the desired estimate since s + σ > 0. □
4.3
Uniqueness Theorem
In this section, we consider the uniqueness of solution only, since the uniqueness is a main part of unconditional well-posedness problem. Proof of Theorem 4.1.1. Let u and v be the solutions of (4.1)-(4.2) with the same initial data. By Duhamel formula, we have that
where the pair (q,r) is admissible, and (q,q'),(r,r') are dual pairs. By Lemma 4.2.3,
35 where α = 2s-7+2 < +∞ for γ < 2s + 2. Then, we conclude that
Hence, we choose T small enough so that u(t) = v(t) on [0,T).
Remark 4.3.1. If s < π , n ≥ 3, the inequality
holds if δ(ρ) + δ(r) + 2δ(ρ) = γ, where δ(ρ) = n(ž − ρ) and l + l = 1.  Since
the admissible values of ρ and r are 2 ≤ r,ρ ≤ l = z, the admissibe value of ρ are n+2−γ < p ≤ ∞. We want H 8 → L P , which requires γ < 2s + 2.

Introduction
5.1
In this chapter we consider the Cauchy problem of derivative nonlinear Schrödinger equation:
where u(t, x) is a complex valued function and δ ∈ R. Our purpose is to prove the unconditional well-posedness in energy space. The proof is based on gauge transformation and Fourier restriction method. We prove the unconditional well-posedness in the following sense. Theorem 5.1.1. Let u 0 ∈ H 1 ( R ) and T > 0, assume that u and v are two solutions of (5.1)-(5.2) in L ∞ (0,T;H 1 (R)) with the same initial data. Then u(t) = v(t), t ∈ [0,T]. One may show the existence and uniqueness result by fix pointed method via the following equation
It is known that in usual Schrödinger equation, nonlinearity is controlled by LP – L9 type inequality but (5.1)-(5.2) has loss of derivative in nonlinearity. It is unlikely to be used directly in the same way as usual Schrödinger equation. In [24], [25], Hayashi and Ozawa proposed the gauge transformation to overcome this difficulty. They use two gauge transformations to convert (5.1)-(5.2) to a system equation without derivative in nonlinearity and then the local well-posedness of (5.1)-(5.2) in H 1 ( R ) is shown. Suppose u is a smooth solution to (5.1)-(5.2). Let v(x,t) = e-i ſ-∞ |u(y)|²dyu(x).
37
A strighfoward calculation yields, (5.1)-(5.2) becomes
In the periodic case, M. Tsutsumi and I. Fukuda [43] proved that the solu- tion of (5.1) - (5.2) is local existence in L ∞ (0,T;H s ) ∩ C (0,T;H s − 1 ), s > 3/2 and it is also hold for non-periodic case. Their results was improved by N. Hayashi, T. Ozawa [25] and they proved the uniqueness in C ([0,T],H 1 ( R )) ∩ L 4 c ([0, T], H 1,00 ( R )).  Moreover, N. Hayashi [24] proved the uniqueness in C ([0, T], H1(R)) ∩ L12([0, T], H1,3(R)). In the case of real line, H. Takaoka [39] uses the Fourier restriction method to handle the transformed equation containing derivative nonlinearity and shows the resulting problem is locally well-posed in H s ( R ) for s ≥みand ill-posedness for s < 1 in the sense that solution map is no longer uniformally continuous. More precisely he showed the solution of (5.1)-(5.2) is unique in C ([0,T),H*(R)) ∩ L4(W 8,00) for s ≥ 1. we start the following observation. let
We first see that if ŭ ∈ L ∞ ( R ; H ¹ ( R )) then we have ŭ ∈ L ² ( R ; H ¹ ( R )), which means ŭ ∈ X 1 0 .  On the other hand, by taking the Fourier transform on both sides of (5.1), we can see
where v(t) = ũ(t),t ∈ (0,T), i.e., v ∈ X q ,1. By using the interpolation between above two cases, we get the solution й ∈ X T 1 . Hence, if the solution is unique in X 1 , 1 , the problem (5.1)-(5.2) is unconditionally well-posed in H 1 . The unconditional uniqueness is a concept of uniqueness which does not de- pend on how to construct the solution. For example, we consider (5.1)-(5.2) with zero Drichlet boundary condition on the ball BR centered at the origin with ra- dius R > 0. Let u 0 be a compactly supported function in H 1 ( R ). If ‖ u 0‖ L 2 ( B R ) is small, we have a global solution ur ∈ L ∞ ( R ; H 1 ( B R )) such that
and
where E(u(t)) = ∥ ∂ x ∪ i 2 (B R ) − z Im ſ B R u ū u ∂ x ū d x . Indeed, because we have by the Gagliardo-Nirenberg inequality,
38 there exists a solution u ∈ L ∞ ( R, H ) ( B R )) when ∥ u 0 ∥ L 2 ( B R ) is sufficiently small. Then the passage to the limit as R → ∞ leads to a solution in L ∞ ( R; H 1 ( R )). In this case, the proof of existence does not imply that a such solution is in the auxiliary spaces. The plan of this chapter is composed as follows. In Section 5.2, we derived the estimates of nonlinear terms on Bourgain space. In Section 5.3 we prove that the solution which comes from gauge transfotmation is unique in X T 1 and the solution of (5.1)-(5.2) is unique in L ∞ (0 , T ; H 1 ).
5.2
Nonlinear Estimates
In this section we derive the nonlinear estimates in the framework of L 2 based Bourgain space. Let f be a nonnegative function such that
where i=1,2 and τ,ξ ∈ R . Here the scales of time and space are different for the Schrödinger equation. Let τ = ∑ i =1 τ i , ξ = ∑ i =1 ξ i . We consider [39],
where σ i = τ i + ξ i , i = 1, 2 and σ 3 = τ 3 − ξ i . So either of the following two cases happens
then
Proof. In the region |ξ − ξ 1 |  > 1 and |ξ − ξ 2 |  > 1.   We first assume |ξ₁| ≤ |ξ|/2 and |ξ₂| ≤ |ξ|/2.  Since |ξ| ≤ |ξ − ξ₁| + |ξ₁|,
|ξ| ≤ |ξ − ξ₂| + |ξ₂|, |ξ| ≤ c|ξ − ξ₁| ≥ |ξ − ξ₂| ≥ .  In addition, if |ξ| ≤ |ξ₃|/2, since |ξ³| ≤ |ξ − ξ¹| + |ξ − ξ²| + |ξ|, we get |ξ³| ≤ c|ξ − ξ¹|ž|ξ − ξ²|ž. We conclude that
On the other hand |ξ| << |ξ|| and |ξ| << |ξ2|, we have the identity
We can show that
Similarly, we can show that
Since |ξ³| ≤ |ξ| + |ξ₁| + |ξ₂| ≤ c(|ξ₁| + |ξ₂|) and if either |ξ₁| ≤ |ξ₂| or |ξ₂| ≤ |ξ₁| we then conclude
Lemma 5.2.2. Let u(t) be the support on {t;|t| ≤ T} with 0 < T ≤ 1 and there exists c, é ≥ 0.  Then the following estimate holds.
Proof. Consider
40 where dμ = dτdξ.
Then
By Plancherel theorem,
where P = i∂ t + ∂ xx . Then we apply the L t,x dual Strichartz estimate and Hölder inequality,
We conclude that
Next, we consider the region |ξ − ξ 1 | ≫ 1 and |ξ − ξ 2 | ≫ 1 and we divided four subcases according to the one of σ ′ s is the largest. Subcase 1. ({σ} ≥ max{{σ1}, {σ2}, {σ3}}) By Plancherel theorem, lemma 5.2.1 and Hölder inequality,
This estimate follows after applying the Sobolev embedding at both space and time for u 1 , u 2 as well as L 4 Strichartz estimate for u 3 . We use again Sobolev
By Plancherel theorem and Sobolev embedding in time,
Applying Sobolev embedding again, V
Lemma 5.2.3. There exists c, é > 0 such that for T ∈ (0,1), then the following
42 Proof. We consider in the following two cases. Case (1). In the region |ξ − ξ 1 | ≤ 1 or |ξ − ξ 2 | ≤ 1.
Applying Schwartz estimate in time, (5.24) is bounded by
where ξ ≤ ρ ≤ ξ , by case 1 of lemma 3.1, we obtain the required estimate. Case(2). In the region of |ξ − ξ 1 | > 1 or |ξ − ξ 2 | > 1. We note that [26],
(5.26) We can estimate multipliers as the previous lemma,
Next, we consider the four subcases according to the one of σ ′ s is the largest. Subcase 1. ({σ} ≥ max{{σ1}, {σ2}, {σ3}}) Hence (5.24) can be estimated by
bounded by
Since Cauchy Schwarz inequality
(5.28) is bounded by
We use again Young inequality for ξ, (5.29) is bounded by
Applying Hölder inequality and δ is chosen sufficiently small, (5.30) is bounded by
Similarly we can prove for σ i s which is the largest of σ j , i̸= j , where 1 ≤ i,j ≤ 3 by using Cauchy-Schwarz inequality for τ . The desired estimates are followed as (5.17), (5.19), (5.20) and (5.22). ● Remark 5.2.4. The prove of nonlinearity estimates are same as [26] and [39]. Lemma 5.2.5. Let s ∈ R, there exists c, έ > 0 such that for T ∈ (0,1) then the following estimate holds.
where each u i (t), i = 1, ..., 5 has support in (0,T).
Then
● where we used the duality of Strichartz's estimate and the Hölder inequality.
5.3
Uniqueness Theorem
Since the uniqueness of solution is a main part of the unconditional well-posedness, we consider the uniqueness only. Theorem 5.3.1. We assume that v 0 ∈ H 1 (R), there exists T such that 0 < T < 1. Set v = G(u). Let v,̃v ∈ X f,̃be two solutions of (5.3)-(5.4) with the same initial data in H 1 . Then there exists a unique solution in X f, f for all t ∈ (0,T). Proof. For any v 0 ∈ H 1 and let M > 0 with || v 0 || H 1 ≤ M and there exists T such that 0 < T < 1, we want to prove that the transformation
is a contraction.  Let v, v ∈ X f, f, be two solutions of (5.3)-(22)with the same initial data. Let
Assume || v || x 1 , 1 = ∥ ψτῦ∥ x ζ , ≤ M . Then for all t ∈ (0,T) with 0 < T < 1, applying lemma (2.1), (3.1), (3.2) and (3.3), we get
choose T < 2C(M²+M²) z
Hence v(t) = ψ τ ῖ (t), we conclude that v(t) = ˆ v(t) for all t ∈ [0, T ).
Proof of Theorem 5.1.1. Let u 0 ∈ H 1 . We define v 0 = G ( u 0 ) ∈ H 1 , since the gauge transformation is continuous. Let v(n) ∈ C ∞ with v(n) → v 0 in H 1 . We define
We have
We can see that
45 Then By Gagliardo -Nirenberg's inequality, then
for all v n ∈ H 1 . Similarly,
We conclude that there exists a unique solution in L ∞ (0,T;H 1 ( R )) such that u n → u in L ∞ (0,T; H 1 (R)), then we obtain the unique solution of (5.1)-(5.2).   □

6.1
Intorduction
In this chapter, we discuss the derivative nonlinear Schrödinger equation with periodic boundary condition:
where u(t,x) is a complex valued function of (t,x) ∈ R × T, and δ =+1 or −1. For simplicity, throughout this chapter, we assume δ = 1, because it makes no difference between the cases δ = 1 and δ = −1 in the results of this paper. Our aim is to prove the global well-posedness of (6.1)-(6.2) in H s (T) for s > g with small data in L 2 .  We use the method of almost conserved energy or I -method which was introduced by J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao and the refinement of bilinear estimate. We improve global well-posedness result in the following sense. Theorem 6.1.1. The initial value problem (6.1)-(6.2) is globally well-posed for initial data u 0 ∈ H 8 (T), s > ≒ if ∥ u 0 ∥ L 2 (T) < √ 2 π . We use the gauge transformation on (6.1) - (6.2), the new nonlinearities still contain derivative like u 2 π x . We then apply the Fourier restriction method to handle the transformed equation containing derivative nonlinearity as H. Takaoka [39]. Suppose u is a smooth solution to (6.1) - (6.2). Let w(t) = G(u(t,x)) and G(u)(x) = e-iJ(u)u(x) where

47
for all time t. The strightforward calculation yields, (6.1)-(6.2) becomes
where
and μ(w) = ‖ ω(t)‖ L 2 . The linear term 2μ( w )∂ x w is cancelled by transformation
Then
We then rewrite the equation (6.4) as follows:
We note that the global well-posedness of (6.1)-(6.2) in H 8 is equivalent to that of (6.6)-(6.7) according to the properties of gauge transformation. We shall write notation as ξ ij = ξ i − ξ j for i and j are even and odd integers but in some cases ξ ij = ξ i + ξ j for both i and j are even or odd. For instance, ξ 123 = ξ 1 − ξ 2 + ξ 3 , ξ 234 = ξ 2 − ξ 3 + ξ 4 , sometime we may write ξ i − j = ξ i − ξ j . We also use if m(ξ) is a function defined on frequency space, we write m(ξ i ) = m i , m(ξij) = m(ξi − ξj) = mij for i and j are even and odd respectively. We organized the sections as follows. In Seciton 6.2 and 6.3, we define gauge transformation, conservation of energy, operator I and construct differential equa- tions associated with the I -method, which is called the I -system. In Section 6.4 we prove local well-posedness by I-syatem for s > ½. In Section 6.5 we present a main proof of global well-posedness for s > 1
6.2
Almost conserved energy
In this section we begin with the conservation of energy but we do not here observe the potential term which is not going to make any trouble. Definition 6.2.1. [8] - [9] If v ∈ H 1 (T), the energy E (v) is defined by
Now we give the lemma that the energy E ( v ) is strong enough to control by H¹. Lemma 6.2.2. Let f be a smooth and 2π- periodic function. Assume f(t) E H 1 (T) such that ‖ f (t)‖ L 2 (T) < δ, δ is sufficiently small. Then we have
Proof. Let us define
where J(f)(x) =ニʃ 2 π ʃ x  (| f(y)|² −ニ|| f || ζ 2 ) dydθ . It is easily seen that || g || ι 2 = || f || L 2 < δ . Since ∂xf(x) = e- žJ(f)(∂xg − ž(|g|² − µ)g) where µ = ž, ƒ 2 π |g|²dθ, the energy (6.8) becomes
Let h = |g|² — μ be a zero mean value function. By translation x → x + ξ such that g(ξ) = 0 for some ξ ∈ [0,2π], we have h( x ) = f ×+ξ h ′ ( y )dy and h( x ) = S 2 π ξ h ′ (y)dy. Applying Hölder inequality, we obtain
Similarly,
By Gaglirado-Nirenberg inequality and (6.10) becomes
Thank to the small data ||g||z2 < δ, we get
49 On the other hand, we may show that
Combining (6.11) and (6.12), we obtain (6.9).
We give the definition of multiplier operator I which is the identity on low frequencies and like as a fractional integral operator of order 1 – s on high fre- quencies. We then construct differential equation via (6.6) - (6.7). We also study modefied energies introduced in [8] - [9] giving the name as generation of modefied energies. Let m(ξ) be an arbitrary real valued 1- multiplier and defined by
Define the multiplier operator I : H s → H 1 such that Iv(ξ) := m(ξ)ˆ ô(ξ). Then we see that
By Plancherel, we may rewrite the equation (6.8) by A notation
Here we note that E(v) could not be directly controlled the H 1 norm like as Lemma 6.2.2.  Hence we define the substitute energy which have a very slow increment in time (in term of N ) such that
This energy make sense even if v is only in H 8 . Let m(ξ) be a symmetric multi- plier. Let I be the multiplier operator associated with m ( ξ )². We define
If m(ξ) is the multiplier in (6.13), then
We then define the E 1 ( v ) is a first generation of family of modified energy such that
Proposition 6.2.3. Let v be a solution of (6.6) - (6.7) and let Mn be a n- multiplier of order n ≥ 2. Then
where α n = Σ π =1 (−1) i+1 ξ 2  and the characteristic function
Proof. Fix n = 2. We consider only nonlinear term containing derivative which play main role.
We take Fourier coefficient in spatial variable of (6.17), the first term is cancelled. We may write by symmetry
and we consider the last term of right hand side of (6.17) as a constant. Hence we get
implies
51 On the other hand,
Using Λ notation with I -system and appropriate summetry,
We also omit ʃ τ | v | 4 dx and ʃ τ | v | 2 dx as constants for the other nonlinear terms. In the rest of this paper we shall drop v( t ) from Λ notation. □ Applying proposition 6.2.3, differentating (6.14) with respect to time, using the identity ξι − ξε + ξε − . . . + (−1) n−1ξ n = 0, we have
Note that in this derivative, Λ 2 is zero because the factor ξ 2 − ξ 2 = 0 over the set ξι − ξ 2 = 0. Next, we differentiate the contribution of Λ 4 , by using proposition (6.2.3)
with α 4̸= 0. We may define the second generation of modified energy
over the set ξ 1 − ξ 2 + ξ 3 − ξ 4 = 0 and the second modified energy still satisfies (6.14). Now we differentiate the second modified energy with respect to time by using proposition 6.2.3 and (6.19), we get
where
and

53
Proposition 6.2.4. Let v be a H¹ global solution to (6.6). Then
for any T ∈ R, δ > 0 and M 6 = M 6 + M 6 − σ 6 . Furthermore, if | ξ i | ≪ N, for all j, then the multiplier M 4 , M 6 and M 8 vanish. Then we show the lemma that the second modified energy E 2 ⁄ 2 is still strong enough to control by H 1 . Lemma 6.2.5. Assume that v satisfies ‖ v(t)‖ L ≤ √ 2π and ‖ Iv(t)‖ H ¹ ≤ ε Then
Proof.  We have
In Lemma 6.2.2, we proved that
for ‖ I V‖ L 2 (T) < √ 2 π . Hence it is enough to show that | Λ 4 ( M 4 (ξ 1 , ξ 2 , ξ 3 , ξ 4 )  — ξ 13 – 42m 1m 2m 3m 4 )|  ≤ O ( N −α )|| I v || ň 1 ( T ) for some α > 0. By a Littlewood-Paley decomposition, we may restrict the frequency of ˆ ı is as |ξ i | ～ N i with dyadically and assume that N 1 ≥ N 2 ≥ N 3 ≥ N 4 . We also assume that each of u is nonnegative. When N4 ≫ N, by equation (6.21) (see next section) we may write
Since m( N 4 )( N 4 ) 1/2 is nondecreasing and L 2 for u 1 , u 2 , L ∞ for u 3 , u 4 by Hölder and Sobolev embedding, we obtain
Next, we consider the second contribution
where we use again Hölder and Sobolev embedding. ●
6.3
Multiplier estimate.
In this section, we prove M 4 multilinear estimate. Our destination is to get the same estimate as [8] and it is a main point in our proof.  we often use the following elementary tool. Lemma 6.3.1. (Double mean-value theorem) Let a be a smooth function of real variable ξ.  Then
where |θ| ~ |ξ| and max(|η|, |λ|) ≪ |ξ|. Lemma 6.3.2. Let
where m i = m(ξ i ) is even and R valued and is defined as (6.13). Then
Proof.  Let M 4 (ξι,ξε,ξε,ξε,ξε, = m 2 ξε,ξε, − m 2 ξε,ξε,ξε, − m 2 ξε,ξε.

Applying DMVT on the first term and mean value theorem (MVT) on remaining terms. Then
we obtain case (ii).
●
Local estimates
6.4
In this section, we study the local estimates of (6.1) - (6.2) which have already known from [26] for all s ≥ 1 Let f i be a nonnegative function such that
We use the operator J 8 = J 8 in space variable, which is defined by J 8 f(ξ) = (ξ) s f (ξ) and we will use the same notation in time as J s = J s = We recall the transformed equation of (6.6) - (6.7) that
where
In Lemma 7.1, we shall estimate the expression of Proposition 6.2.4. In that case, it is not enough to use the spatial norms such as ‖ I v ‖ H 1 , hence we shall use the space-time norms such as ‖ I v ‖ z 1 . For that purpose we prove the local existence of solution in time. Before we start local existence theorem, we set u := I v. Then (6.26) - (6.27) become
Theorem 6.4.1. Let v be a global solution of (6.26) - (6.27). Assume that for some θ > 0, ||Ivo||H1(T) ≤ θ. Then, there exist two positive constants T and C such that
where C is indepentent of N appearing in the definition of I . Theorem 6.4.1 is a consequence of the following two lemmas. The first lemma is for the estimates of homogeneous and inhomogeneous problem. See [2], [15], [26].
57
Lemma 6.4.2. Let s ∈ R. Let u 0 ∈ H 8 (T). Then ||χ(t)U(t)uo||z₁ ≥ ||uo||n₁(T),
In next lemma, we prove the multilinear estimates as follows. Lemma 6.4.3. Let v be the Schwartz function with spatial periodic. There exists ε′ > 0 such that for T ∈ (0, 1], then
Proof. We take the Fourier transform on the left hand side of (6.30) to have
Assume that each of ˆ û is nonnegative. We have to show two main parts such that
and
We first prove (6.31). We can write the left hand side of (6.31) as
by definition of operator I and ʃ , denote the integration over the measure δ( τι + 72 + 73 + 74)δ(ξ1 − ξ2 + ξ3 − ξ4). Applying the Plancherel and duality, we have to show that
58 where all of f i are real and nonnegative functions . Case(1). If
we need to show that
We apply the Cauchy Schwarz inequality for left hand side of (6.34), then
We use the L 4 dual Strichartz estimate to obtain
We finally use the L 4 Strichartz estimate and Sobolev embedding in time to get the desired estimate,
Case (2). We may assume that
on the set τ 1 + τ 2 + τ 3 + τ 4 = 0. We consider the following four subcases according to the one of 〈 τ ε + (− 1) έξ ε λ ⟩ ′ s is the largest. Case(a) if max i =1,2,3,4〈 τ ε + (− 1) έξ ε λ = 〈τ₄ + ξ 2〉 and case (b) if max i =1,2,3,,₄〈 τ i + (−1) i ξ 2 〉 = 〈 τ j + (−1) i ξ 2 〉 for some j = 1, 2, 3. We divide two regions such that |ξ₁| ≤ﾎ|ξ₁| ≤ |ξ₃| and |ξ₃| ≤ﾎ|ξ₄| ≤ |ξ₁|. In both cases we see that the inequality (See [8]-[9]),
for all ξ , . . , ξ 4 such that ξ 1 − ξ 2 + ξ 3 − ξ 4 = 0, since m(ξ)(ξ) is increasing in |ξ| and m(ξ)(ξ) ≥ 1. Hence it is enough to show that
Subcase(i). In the region |ξ₁| ≤ｶ|ξ₄| ≤ |ξ₃|. We may assume that |ξι − ξ 4 | ≤ |ξι − ξ 2 | on the set ξι − ξ 2 + ξ 3 − ξ 4 = 0. Since |ξ 4 | ≤ |ξ 1 − ξ 4 |, we have |ξ 1 | ≤ ½ ξ 1 − ξ 4 | ≤ ½ ξ 1 − ξ 2 |. We observe that with s = 1 + c,
On the other hand, we assume |ξ₁ − ξ₄| ≥ |ξ₁ − ξ₂|, then
In case B , it is clear that
Now we consider the region C , we divide in two subregion such that C = C 1 + C 2 where C 1 for |ξ1 −ξ2| ≥予約|ξ1| and C 2 for |ξ1 −ξ2| ≤予約|ξ1|. When |ξ1 −ξ2| ≥予約|ξ1|,
We next consider the region |ξ₁ −ξ₂| ≤言|ξ₁|. In this case we see |ξ₁| ～ |ξ₂| ≤ |ξ₃| and so that
since s >ニSubcase(ii). |ξ³| ≤ﾎﾎ|ξৰ| ≤ |ξ₁|. We used the same idea as subcase (i). Now it is turn to prove the following cases. Case (a). We first consider in the case of max i =1,2,3,4 ( 7 i + ( −1) i ξ i ) = ( τ a + ξ 2 ). We apply the multiplier estimates of case (i) except C 2 , hence we have to show that
by Sobolev embedding for f 1 and f 2 , L 4 Strichartz estimate for f 2 and finally we use again the Sobolev embedding in time, we get the desired estimate. Case (b). Now we consider the case of max i =1,2,3,4(7 i +(−1) i ξ ?) = ( τ j +(−1) i ξ ?) need to show that
By Cauchy Schwarz inequality, we have that
Now we need to prove that
We can rewrite the left hand side of (6.40) as
by Sobolev embedding in time for the first inequality and the second is used by Hölder's inequality with p >ミ.  We fix p = 2 to get the best estimate what we desired. We note that the proof of multilinear estimate (6.37) for C 2 region is the same as case (1). Now it is turn to prove (6.32), it is enough to prove that
We can rewrite the left hand side of (6.41) as
since m(ξ)(ξ)ﾎ≥ 1. By Young's inequality in time, (6.42) is bounded by
Then we use the Hölder's inequality for two of v is in L ε  and other in L ε . Then (6.43) is bounded by
We get the desired estimate by using the space time Sobolev embedding. Lemma 6.4.4. Let v be the Schwartz function with spatial periodic. There exists ε′ > 0 such that for T ∈ (0, 1], then
Proof.  We apply the same idea of Lemma 6.4.3, hence we need to prove two main parts as follows.
where all of f i are real and nonnegative functions and f ∗ denote the integration over the measure δ( τ 1 + τ 2 + τ 3 + τ 4 )δ(ξ 1 − ξ 2 + ξ 3 − ξ 4 ) and
Now we prove (6.45). Case(1). If
by using Cauchy Schwarz inequality in the left hand side of (6.45), then
The claim is the same as case (1) of Lemma 6.4.3, we obtain desired estimate. Case (2) We may assume that
for |ξ₁ − ξ₂| ≫ 1, |ξ₁ − ξ₄| ≫ 1. We divide two regions such that |ξ₁| ≤剥|ξ₄| ≤ |ξ₃| and |ξ₃| ≤ｶ|ξ₄| ≤ |ξ₁|. In both case we see that,
for all ξ . . . ,ξ 4 such that ξ 1 − ξ 2 + ξ 3 − ξ 4 = 0, since m(ξ)(ξ) is increasing in |ξ| and m(ξ)(ξ) ≥ 1. Hence it is enough to show that
We note by [26] that,
where δ is sufficiently small. We may estimate the Fourier multipliers as Lemma 6.4.3.  We divide the following two regions such that (i) |ξз| ≤ﾎ|ξৰ| ≤ |ξı| and (ii) |ξı| ≤ヵ|ξı| ≤ |ξı|. We first recall the the multiplier estimate of Lemma 6.4.3
Here it is enough to consider only case max i =1,2,3,4 ( τ i + (−1) i ξ ? ) = ( τ 4 + ξ ? ) and the other cases can be controlled as (6.39). Now we consider the region |ξ₃| ≤ﾎﾟ|ξ₄| ≤ |ξ₁| on the set ξ₁ − ξ₂ + ξ₃ − ξ₄ = 0. We have |ξ 4 | ≤ 2 |ξ 1 − ξ 2 |, then
If |ξ₃| ≤ |ξ₁ − ξ₄|, then
We conclude that |ξ 2 ||ξ 4 | ≤ 4 |ξ 1 − ξ 2 ||ξ 1 − ξ 4 | . We see that (6.48) becomes
This case and I 2 can be controlled as case (1). We need to show (6.47) for I 1 region,
Fix ξ and we apply Young's inequality for time, then
By Cauchy Schwarz inequality
64 By (6.51) and (6.52), left hand side of (6.50) becomes
we get the desired estimate. Similary, one can show that
→ Lemma 6.4.5. Let v be the Schwartz function with spatial periodic. There exists ε′ > 0 such that for T ∈ (0, 1], then
where n = 2, 4. Fix n = 2, taking Fourier transform, we have
where all of f i are real and nonnegative functions and f ∗ denote the integration over the measure δ( τ 1 + τ 2 + τ 3 + τ 4 )δ(ξ 1 − ξ 2 + ξ 3 − ξ 4 ). We first assume that
In this case we use Cauchy Schwarz inequality on the left hand side of (6.55), then
Thus we can show that
The desired estimate follow by Hölder inequality, L 4 Strichartz estimate and Sobolev embedding in time. m(ξ4)(ξ4)[ξ2]. > 1. Here we know that Next we may assume that Π έ = 1 m(ξ ε ) (ξ ε )
for two region such that |ξ₁| ≤ﾎ|ξ₄| ≤ |ξ₃| and |ξ₃| ≤ﾎζ₄| ≤ |ξ₁|. Hence it is enough to show that
We consider two cases. We first assume |ξ૩| = max i =1,2,3 |ξ i |, then
On the other hand, if |ξз| ≤ max i =1,2,3 |ξ i |, then
by using the fractional Leibnitz rule, Hölder and L 4 dual Strichartz estimate. Similarly, we can prove that We may prove for n = 4, if we use more Sobolev estimates. □ It is now turn to prove Theorem 6.4.1. Proof of Theorem 6.4.1. Fix T > 0, to be chosen later.  Let Iv 0 ∈ H 1 , there exists a constant θ > 0 such that ‖ I v 0‖ H 1 ≤ θ . Consider the set B = { I v ∈ Z 1 : ||Iv||Z1 ≤ 2Cθ} for some constant C > 0. We have
We assume that ‖ χ I v‖ z, ～ ‖ I v‖ z,([–T,T]×T). We then apply Lemmas 6.4.2, 6.4.3, 6.4.4 and 6.4.5 to have ||H(Iv)||Zi([-T,T]×T)    ≤   C||Ivo||H1(T) + C'Tc'(1 + ||Iv||Zi([-T,T]×T))3||Iv||Zi([-T,T]×T). We choose T small enough so that T < (8C 3 θ (1 + 2Cθ) 3 ) −1/ε ′ .  We note that T ● depend only on ‖ I v 0 ‖ H 1 . This completes the proof of Theorem 6.4.1.
6.5
Estimates of Λ n, n = 4,6,8.
In this section, we compute the following main estimates. Lemma 6.5.1. Let v i = v i (x,t) be the 2πλ-periodic function with λ < N for any δ come from local theory, then
where n = 4,6,8. Lemma 6.5.2. [8] We consider the following multiplier terms M'6(ξα, ･ ･ ･ ,ξδ) = {Ψ(ξc_d)M 4(ξα,ξb,ξcde,ξf)ξd − Ψ(ξd_c)M 4(ξα,ξb,ξc,ξdef)ξe}sym,
and
then the following inequalities are hold: (i) If |ξ 3 | ≥ N, then | M 6 (ξ 1 ,ξ 2 ,…,ξ 6 ) | ≥ m(Nmax) 2 N 2 (ii) If |ξ³| ≪ N, then |Μ6(ξ¹,ξ²,…,ξδ)| ≤ NmaxN3, (iii)  |M6"(ξ1, ･･･ ,ξ6)| ≤ Nmaxm²(Nmax)･ where M 6 = M 6 + σ 6 and {1, . . , 6} = {α, . . . , f}. Proof. The proof of (i) follows from (6.22). For the proof of case (ii), we shall consider two cases and assume that N 1 ≥ . . . ≥ N 6 . Case (1). N 1 ～ N 2 . Then
Apply (6.22) to estimate M 6 , then
The function M 4 in Σ 3 =1 are strictly smaller than  = and by (6.22) then ∑ i =1 I i < N 1 N 3. We use (6.24) and the symmetry of M 4 , then
Case(2). N 1 ～ N 3 . We need some cancellation between the large terms coming from σ 6 (ξ 1 , . . . ,ξ 6 ) and the large terms of the sum of the M 4 . From (6.57), then
We use (6.25) and the symmetry of M 4 ,
We obtain
Finally, M ′ is consequence of (6.22).
In our proof, we work with the Littlewood-Paley decomposition.  We may restrict the frequency of ˆ γ is as | ξ i | ～ N i according to the dyadic decomposition and assume that N 1 ≥ N 2 · · · ≥ N i , i= 4, 6, 8. When all frequencies are smaller than N, the multiplier M j , j = 4,6,8 vanish over the integral of Λ j . Lemma 6.5.3. Let v i = v i (t,x) be the 2πλ-periodic function with λ < N for any δ come from the local theory, then
Proof. It is enough to consider the following three cases. Case (1). N 4 ≥ N ≫ N 5 . By Lemma 6.5.2 we can rewrite the left hand side of (6.58) as
● Now we are ready to show the estimate of Λ 6 ( M 6 ). Lemma 6.5.4. Let v i = v i (t, x) be the 2πλ-periodic function with λ < N for any δ come from the local theory, then
Proof. We can rewrite the left hand side of (6.59) as
In case(1) N 3 ≫ N ≥ N 4 . Since m(N3) ～ (N3) 8–1 N1–8, then
In case(2) N 2 ≫ N ≥ N 3 . By Hölder's inequality,
Then desired estimate follows by (3.14) and Sobolev embedding,
Lemma 6.5.5. We consider the Ms multiplier
then
Proof. This lemma is a consequence of Lemma 6.21.
Lemma 6.5.6. Let v i = v i (t, x) be the 2πλ-periodic function with λ < N for any δ come from the local theory, then
Proof. It is enough to consider two cases, case (1) N 3 ≥ N ; case (2) N 3 ≤ N . When all frequencies are smaller than N, we observe that M 8 vanishes over the Ag integral. In case(1) N 3 ≫ N ≥ N 4 .
In case(2). N 2 ≫ N ≥ N 3 . By Lemma 6.5.5, and the Hölder's inequality,
Lemma 6.5.7. Assume that
We prove the following cases (i) |ξı|, |ξı|, |ξı| ≥ 1 ≫ |ξı|, (ii) |ξı|, |ξı| ≥ 1 ≫ |ξı|, |ξı| then
Proof. If |ξ i | ≪ 1 for (i=1,2,3,4), we may easily seen that σ 4 is vanish. (i). By DMVT, we obtain

□
Lemma 6.5.8. Let v i = v i (t, x) be the 2πλ-periodic function with λ < N for any δ come from the local theory, then
Proof. It is enough to consider two cases, case(1) N 3 ≥ N ; case (2) N 3 ≤ N . When all frequencies are smaller than N, we observe that σ 4 vanishes over the Λ 4 integral. Case (1). N 3 ≥ N ≥ N 4 . By Lemma 6.5.7, we may rewrite the left hand side of (6.61) as
by Hölder's inequality and the L 4 Strichartz estimate, we get the desired estimate. Case(2). N 1 ～ N 2 ≥ N ≫ N 3 . By Lemma 6.5.7, we may rewrite the left hand side of (6.61) as
We apply (3.14), Hölder and Strichartz estimate, then
● Now we are in a position to prove Theorem 6.1.1. Proof of Theorem 6.1.1. Let u 0 ∈ H 8 with ½ < s < 1. Let u λ (t,x) = λ − 1/2u(t/λ 2 ,x/λ), where λ > 1. By Plancherel,
We choose λ ～ N ′ = for ｶ< s < 1 which implies λ < N . Then we see that
since the initial data is in H 8 . We use Proposition 6.2.4 to have
By Lemma 6.5.1, we have that
Now we apply Theorem 6.4.1 to obtain E N (t) < E N (0) + N − 1 − λ − 1+ (θ 4 + θ 6 + θ 8 ), for any t ∈ (0, 1). We may assume θ ≥ 1 without loss of generality. By iteration, we can prove for all t ∈ [0, T],
Here we can choose the size of time T such that TN-1-X-1+ < 1 and take N = N(T) that for t ∈ [0, N 1+ X 1−],
as N → ∞, if s > ½.
Hence the proof is completed for global well-posedness with s > ½.

6.6
Notes and references
Equation (6.1)-(6.2) describe the long wavelength dynamics of dispersive Alfvén waves propagating along an ambient magnetic field and satisfies the infinitely many conservation laws [38]. It is known that the global well-posedness in H 1 regularity is obtained via the laws of conservation of mass and energy. In the real line case, N. Hayashi and T. Ozawa [24], [25], [34] prove the global well- posedness in H 1 assuming the smallness condition ∥ uo∥ λ 2 ( R ) ≤ 2 π . This result was improved by H. Takaoka [40]. He proves the global well-posedness in H 8 for s > 😉 and the argument is based on the Fourier truncation method of J. Bourgain [2], estimating separately high and low frequencies of initial data. J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao [8] improve the global result for H s , (s > ) which satisfies the small initial data in L 2 with I- method. In [9], the same authors modify the energies in almost conservation laws and extend the order of dispersive derivative operator (∂t − i∂ xx ), then the global result for H s , (s > i ) except the end point is obtained. In their proof, a bilinear refinement of Strichartz estimate is a central tool and they use the space X s,b with b > 314 In the case of periodic, S. Herr [26] adjusts the gauge transformation for pe- riodic setting and proves the same local well-posedness result as H. Takaoka [39]. He uses only the L 4 Strichartz estimate because of the absence of local smoothing and he also shows a global well-posedness in H s ( s ≥ 1) via conservation laws. The crucial differences between the periodic and non-periodic cases are the condition of gauge transformation and the dispersive properties of solution. Due to gauge transform with periodic setting the transformed equation has new non- linear terms which do not appear in the real line case. In the periodic case, one of difficulties is how to handle these new nonlinear terms of transformed equations. We use the structure of the equation to cancel some additional terms out, when we consider the differential equations associated with the almost conservation laws. See section 6.3. As the absent of local smoothing (a sense of weak dispersive properties) in periodic setting, when we consider the local well-posedness of DNLS, the L 4 Strichartz estimate with the loss of ε > 0 derivative is only available to contral derivative in nonlinear terms. In the case of global well-posedness, we found again the difficulty that the order of dispersive derivative (∂t − i∂ xx ) could not extended to get a good decay like a real line case. To get over this we consider how to handle the interaction between two functions with frequency of the same order. See section 3.3. We use the same idea of the proof of local estimate as in [23], [26] and [39]. The proof of Lemma 6.2.2 is used the same idea as Lemma 3.6 in [9]. In the statement of Proposition 6.2.3, we use the same notation as in [8]-[9]. Lemma 6.2.4 is Lemma 3.8 in [8]. The M 4 multilinear estimates of Lemma 6.3.2 and 6.3.3 are Lemma 4.1 and 4.2, M 6 multilinear estimates of Lemma 6.5.2 is Lemma 6.4 in [8]. It is known that when u solves the initial value problem (6.1)-(6.2), u λ ( t, x ) = λ ž u ( λ²t, λ x ), for all λ > 0, is also a solution.  We have || D ž u λ(0)|| L ž = λ s || D ž u 0|| L ž , see that low regularity problem is easy to extend from local to global in time by using conservation laws. When one work the Cauchy problem under scaling invariant regularities, example below L 2 and H 1 , the almost conserved quantities is essential to extend a local to global theory. Here we notice that conservation laws are based on L 2 space, hence they do not yield in general L P based norms for global.
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