Global Analysis for Rough Solutions to the Davey-Stewartson System

and Applied Analysis 3 For details one can see Ghidaglia and Saut 2 . Moreover, one can easily establishes ‖u T ‖Hs ≤ C s, ‖u0‖Hs, T 1.11 for s 1 with bounds uniformly in T , and with some additional arguments one can deduce the same claim for s > 1. The mass conservation law 1.10 also gives 1.11 for s 0, but unfortunately this does not immediately imply any results for s > 0 except in the small mass case. To make the statement more precise, we denote u, v as the solutions of 1.7 – 1.9 . It follows from 1.8 that vx F−1 ξ2 1 |ξ| F|u|, 1.12 where F and F−1 denote the Fourier transform and its inversion. For brevity we denote E ( ψ ) F−1 ξ 2 1 |ξ| Fψ. 1.13 Combining 1.8 and 1.9 , 1.6 – 1.8 are changed to iut Δu a|u|u bE ( |u| ) u, 1.14 u 0, x u0 x . 1.15 It is conjectured that the system 1.14 1.15 is globally well posed inH for all s ≥ 0 and in particular 1.11 holds for all s > 0. This conjecture remains open now. In this paper we aim to prove that the Cauchy problem for 1.1 is globally well posed below the energy norms. That is, we will prove the global well-posedness for initial data u0 ∈ H R2 with s < 1 sufficiently close to one, then we meet the obstacle that there is no conservation law. Indeed, if the initial data is in H1 R2 then it is bounded in H1 R2 for all time and hence the H s > 1 norm is similarly bounded, but if the initial data is only in H s < 1 then theH1 R2 norm may be infinite, and also the conservation of the Hamiltonian appears to be useless. Conservation of the L2 norm also appears to be unhelpful for this particular problem. For solutions below the energy threshold the first result was established by Bourgain for nonlinear Schrödinger equation with critical nonlinearity in space dimension two see 18 . Bourgain came up with the idea of introducing a large frequency parameter N by dividing the solution into the low-frequency portion ulow when |ξ| ≤ N and the high frequency portion uhigh when |ξ| ≥ N . The main tool is an extrasmoothing estimate, which shows that if the high frequencies would bemerely inH R2 for some s < 1, then interactions arising from high frequencies were significantly smooth. In fact, they were in the energy class H1. Moreover, if we denote St as the nonlinear flow and S t eu0 is the linear group, Bourgain’s method shows addition that St −S t u0 ∈ H1 R2 for all time provided u0 ∈ H, s > 3/5. Thus, he showed that the solution is globally well posed with initial data inH R2 for any s > 3/5. Recently, Kenig, Ponce, Fonseca, Ginibre, Molinet, Pecher 19–23 , and Miao and Zhang 24 have developed this methods to study different evolution systems. 4 Abstract and Applied Analysis Another improvement was given by Colliander et al. in 25, 26 , where the authors used the “I-method” that we state below. If s < 1, then the energy is infinite and one cannot compare theH norm of the solution u t with the energy. In order to overcome this difficulty, we introduce the following multiplier IN : ÎNf ξ : m ξ f ξ , 1.16 where m ξ : η ξ/N , η is a smooth, radial, nonincreasing in |ξ| such that: η ξ 1, |ξ| ≤ 1; η ξ 1/|ξ| 1−s, |ξ| ≥ 2, andN 1 is a dyadic number playing the role of a parameter to be chosen. Then we plug this multiplier into the energy which generates to so-called modified energy: H INu ∫ R2 ⎛ ⎜⎝|INux|2 ∣INuy ∣∣2 ( a|INu| bE ( |INu| ) INu 2 ) 2 ⎞ ⎟⎠dx dy. 1.17 Note that if u t ∈ H R2 then H INu < ∞. Note also that asN goes to infinity, the multiplier I “approaches” the identity operator. Therefore the variant of this smoothed energy is expected to be slow as N goes to infinity. This is the “I-method”, originally invented by Colliander et al. 25 to prove global existence for semilinear Schrödinger equations with rough data. In this paper we design it for the Davey-Stewartson system. The main purpose of this paper is to study that we can lower the value of s to what extent which can also grantees the global existence. In this paper we will prove the following. Theorem 1.1. The Cauchy problem 1.14 1.15 is globally well posed for all u0 ∈ H R2 , s > 2/5 and a b > 0. Moreover, the solution satisfies the following estimate: sup t∈ 0,T ‖u t ‖Hs ≤ C 1 T 3s 1−s / 2 5s−2 , 1.18 where the constant C depends only on the index s and ‖u0‖L2 . Remark 1.2. We view this result as an incremental step towards the conjecture that 1.14 1.15 is globally well posed inH R2 for all s ≥ 0. Remark 1.3. We improve the results obtained by Shen and Guo 27 , in which they demonstrated the global existence for s > 4/7 for the Cauchy problem 1.14 1.15 . The technique in their proof mainly depends on the Fourier restriction norm method of Bourgain by showing a generalized estimates of Strichartz type and splitting the data into lowand high-frequency parts. The new ingredient in our proof is a priori interaction Morawetz-type estimate, which generates a new space-time Lx,t estimates for the “approximate solution” Iu to the nonlinear equation with the relatively general defocusing power nonlinearity, and this technique is motivated by the work given by Colliander et al. in 26 . Remark 1.4. It is worth to remark that Dodson 28–30 proves a frequency-localized interaction Morawetz estimate similar to the estimate made in 31 for considering an L2-critical Abstract and Applied Analysis 5 initial value problem for cubic nonlinear Schrodinger equation. The major difference between the cubic nonlinear Schrodinger equation and the elliptic-elliptic Davey-Stewartson system 1.14 is the singular integral operator E |u|2 in 1.14 , which may result in some new difficulties to establish the corresponding frequency localized interaction Morawetz estimate. We hope to solve this problem in a forthcoming paper from the arguments derived by Dodson.and Applied Analysis 5 initial value problem for cubic nonlinear Schrodinger equation. The major difference between the cubic nonlinear Schrodinger equation and the elliptic-elliptic Davey-Stewartson system 1.14 is the singular integral operator E |u|2 in 1.14 , which may result in some new difficulties to establish the corresponding frequency localized interaction Morawetz estimate. We hope to solve this problem in a forthcoming paper from the arguments derived by Dodson. 2. Notations and Preliminaries In this paper, we will often use the notation A B whenever there exist some constants K such that A ≤ KB. Similarly, we will use A ∼ B if A B A. We use A B if A ≤ cB for some small constant c > 0. We use k± to denote the real number k±ε for any sufficiently small ε > 0. z and z are the real part and imaginary part of the complex number z, respectively. We use S R4 to denote the Schwartz space and S′ R4 to denote its topological dual space. We use Lx R 2 to denote the usual Lebesgue space of functions f : R2 → C whose norm ∥f∥Lrx : (∫ R2 ∣∣f∣∣rdx )1/r 2.1 is finite, with the usual modification in the case r ∞. We also define the space-time space L q t L r x by ‖u‖Lqt∈JLrx : (∫ J ‖u‖qLrxdt )1/q 2.2 for any space-time slab J × R2, with the usual modification when either q or r are infinity. When q r we abbreviate Lqt L q x by L q t,x. Definition 2.1. A pair of exponent q, r is called admissible in R2 if 1 q 1 r 1 2 , 2 ≤ q, r ≤ ∞. 2.3 We recall the known Strichartz estimates 21 and the reference therein . Proposition 2.2. Let q, r and q̃, r̃ be any two admissible pairs iut Δu − f x, t 0, t, x ∈ J × R2 u x, 0 u0 x . 2.4 Then one has the estimate ‖u‖Lqt Lx J×R2 ‖u0‖L2 R2 ∥∥f∥∥ L q̃′ t L r̃′ x J×R2 2.5 with the prime exponents denoting Hölder dual exponents. 6 Abstract and Applied Analysis We also define the fractional differential operator |∇|α for any real α by ̂ |∇|u ξ : |ξ|û ξ 2.6


Introduction
The Davey-Stewartson system has their origin in fluid mechanics, where it appears as mathematical models for the evolution of weakly nonlinear water waves having one predominant direction of travel, but in which the wave amplitude is modulated slowly in two horizontal directions see 1 . In dimensionless they read as the following system for the complex amplitude u t, x, y and the real mean velocity potential v t, x, y iu t u xx μu yy a|u| 2 u buv x , cv xx v yy |u| 2 x , 1.1 where i √ −1, u u t, x, y : 0, T × R × R → C and 0 < T ≤ ∞; the parameters μ, a, b, c are real constants. According to signs of the μ and c, these systems may be classified as elliptic-elliptic : μ > 0, c > 0, 1 Abstract and Applied Analysis elliptic-hyperbolic : μ > 0, c < 0, 1.3 hyperbolic-ellipti : μ < 0, c > 0, 1.4 hyperbolic-hyperbolic : μ < 0, c < 0. 1.5 In the last two decades, the Cauchy problem for the Davey-Stewartson system 1.1 has focused on intense mathematical research. In 1990, Ghidaglia and Saut 2 established the local well-posedness for the Cauchy problem of 1.1 in the cases of 1. 2 -1.4 . It reads that for u 0 ∈ H 1 R 2 , the systems 1.1 have a local solution in time. Hayashi and Hirata 3 studied the initial value problem to the Davey-Stewartson system for the elliptic-hyperbolic case 1.3 in the usual Sobolev space, they proved local existence and uniqueness for the initial data in H 5/2 R 2 whose L 2 norm is sufficiently small. Tsutsumi  In this paper, we consider the Cauchy problem of 1.1 in the elliptic-elliptic case without loss of generality, we may take λ μ c 1 for simplify , iu t u xx u yy a|u| 2 u buv x , As is well known, the system 1.6 -1.8 enjoys two useful conservation laws: one is the energy conservation law: It is conjectured that the system 1.14 -1.15 is globally well posed in H s for all s ≥ 0 and in particular 1.11 holds for all s > 0. This conjecture remains open now. In this paper we aim to prove that the Cauchy problem for 1.1 is globally well posed below the energy norms. That is, we will prove the global well-posedness for initial data u 0 ∈ H s R 2 with s < 1 sufficiently close to one, then we meet the obstacle that there is no conservation law. Indeed, if the initial data is in H 1 R 2 then it is bounded in H 1 R 2 for all time and hence the H s s > 1 norm is similarly bounded, but if the initial data is only in H s s < 1 then the H 1 R 2 norm may be infinite, and also the conservation of the Hamiltonian appears to be useless. Conservation of the L 2 norm also appears to be unhelpful for this particular problem.
For solutions below the energy threshold the first result was established by Bourgain for nonlinear Schrödinger equation with critical nonlinearity in space dimension two see 18 . Bourgain came up with the idea of introducing a large frequency parameter N by dividing the solution into the low-frequency portion u low when |ξ| ≤ N and the high frequency portion u high when |ξ| ≥ N . The main tool is an extrasmoothing estimate, which shows that if the high frequencies would be merely in H s R 2 for some s < 1, then interactions arising from high frequencies were significantly smooth. In fact, they were in the energy class H 1 . Moreover, if we denote S t as the nonlinear flow and S t e itΔ u 0 is the linear group, Bourgain's method shows addition that S t − S t u 0 ∈ H 1 R 2 for all time provided u 0 ∈ H s , s > 3/5. Thus, he showed that the solution is globally well posed with initial data in H s R 2 for any s > 3/5. Recently, Kenig 25, 26 , where the authors used the "I-method" that we state below. If s < 1, then the energy is infinite and one cannot compare the H s norm of the solution u t with the energy. In order to overcome this difficulty, we introduce the following multiplier I N : where m ξ : η ξ/N , η is a smooth, radial, nonincreasing in |ξ| such that: η ξ 1, |ξ| ≤ 1; η ξ 1/|ξ| 1−s , |ξ| ≥ 2, and N 1 is a dyadic number playing the role of a parameter to be chosen. Then we plug this multiplier into the energy which generates to so-called modified energy: 1.17 Note that if u t ∈ H s R 2 then H I N u < ∞. Note also that as N goes to infinity, the multiplier I "approaches" the identity operator. Therefore the variant of this smoothed energy is expected to be slow as N goes to infinity. This is the "I-method", originally invented by Colliander et al. 25 to prove global existence for semilinear Schrödinger equations with rough data. In this paper we design it for the Davey-Stewartson system. The main purpose of this paper is to study that we can lower the value of s to what extent which can also grantees the global existence. In this paper we will prove the following. Theorem 1.1. The Cauchy problem 1.14 -1.15 is globally well posed for all u 0 ∈ H s R 2 , s > 2/5 and a b > 0. Moreover, the solution satisfies the following estimate: where the constant C depends only on the index s and u 0 L 2 .
Remark 1.2. We view this result as an incremental step towards the conjecture that 1.14 -1.15 is globally well posed in H s R 2 for all s ≥ 0.

Remark 1.3.
We improve the results obtained by Shen and Guo 27 , in which they demonstrated the global existence for s > 4/7 for the Cauchy problem 1.14 -1.15 . The technique in their proof mainly depends on the Fourier restriction norm method of Bourgain by showing a generalized estimates of Strichartz type and splitting the data into low-and high-frequency parts. The new ingredient in our proof is a priori interaction Morawetz-type estimate, which generates a new space-time L 4 x,t estimates for the "approximate solution" Iu to the nonlinear equation with the relatively general defocusing power nonlinearity, and this technique is motivated by the work given by Colliander et al. in 26 .

Remark 1.4.
It is worth to remark that Dodson 28-30 proves a frequency-localized interaction Morawetz estimate similar to the estimate made in 31 for considering an L 2 -critical Abstract and Applied Analysis 5 initial value problem for cubic nonlinear Schrodinger equation. The major difference between the cubic nonlinear Schrodinger equation and the elliptic-elliptic Davey-Stewartson system 1.14 is the singular integral operator E |u| 2 in 1.14 , which may result in some new difficulties to establish the corresponding frequency localized interaction Morawetz estimate. We hope to solve this problem in a forthcoming paper from the arguments derived by Dodson.

Notations and Preliminaries
In this paper, we will often use the notation A B whenever there exist some constants K We use k± to denote the real number k ±ε for any sufficiently small ε > 0. z and z are the real part and imaginary part of the complex number z, respectively.
We use S R 4 to denote the Schwartz space and S R 4 to denote its topological dual space. We use L r x R 2 to denote the usual Lebesgue space of functions f: R 2 → C whose norm is finite, with the usual modification in the case r ∞. We also define the space-time space for any space-time slab J × R 2 , with the usual modification when either q or r are infinity. When q r we abbreviate L q t L q x by L q t,x .
We recall the known Strichartz estimates 21 and the reference therein .

2.4
Then one has the estimate with the prime exponents denoting Hölder dual exponents.

Abstract and Applied Analysis
We also define the fractional differential operator |∇| α for any real α by |∇| α u ξ : |ξ| α u ξ 2.6 and analogously where a : 1 |a| 2 . We then define the homogeneous Sobolev spaceḢ s and the inhomogeneous Sobolev space H s by We also need some Littlewood-Paley theory. Specifically, let ϕ ξ be a smooth bump supported in |ξ| ≤ 2 and equalling one on |ξ| ≤ 1. For each number N ∈ 2 Z we define the Littlewood-Paley operators:

2.9
Similarly, we can define P <N , P ≥N , and P M<·≤N : P ≤N − P ≤M , whenever M and N are dyadic numbers. We will frequently write f ≤N for P ≤N f and similarly for the other operators. We recall the following standard Bernstein and Sobolev type inequalities.

2.10
We collect the basic properties of I N into the following.
Abstract and Applied Analysis 7 Lemma 2.4. Let 1 < p < ∞ and 0 ≤ s < 1. Then Proof. For the proof one can see Colliander et al. 25 .
Now we define the Strichartz norm of functions u 2.14 Then we introduce the following bilinear smoothing property due to Bourgain 18 .
Then, for N j ≤ N k , the following inequality holds: That is to say, suppose u solves 1.15 -1.18 on the time interval 0, T . Let u j P N j u, for j 1, 2 with N 1 > N 2 . Then

2.17
The estimate 2.17 will be also valid if u j is replaced by u j .
We also have the local well posedness result.

Proposition 2.6. Let us define quantity
If μ 0, T < μ 0 , where μ 0 is some universal constant then for any s > 0 the initial value problem 1.15 -1.18 is locally well-posed and the following estimate is true:

Almost Conservation Laws
In this section we prove the almost conservation of the modified energy H I N u t .
Proposition 3.1. If the initial data u 0 ∈ H s with s > 1/4, and u t, x solves 1.14 -1.15 for all t ∈ 0, T where T is the time that Proposition 2.6 applies, then

3.1
In particular when Proof. In light of 2.19 , it suffices to control the energy increment |H I N u t − H I N u 0 | for t ∈ 0, T in terms of Z I 0, T . Applying the I N operator to the system 1.14 -1.15 : iI N u t ΔI N u aI N |u| 2 u bI N E |u| 2 u ,

3.3
From now on, we abbreviate I N as I for simplicity. An elementary calculation shows that |H Iu t − H Iu 0 | is controlled by the sum of the space-time integrals: 3.5 Here we used the properties of operator E for 1 < p < ∞: i E ∈ L L p , L p , where L L p , L p denotes the space of bounded linear operators from L p to L p ; ii if u ∈ H s , then E u ∈ H s , s ∈ R.
We estimate H 1 first. We use u N j to denote P N j u. When ξ j is dyadically localized to {|ξ| ∼ N j } and we will write m ξ j by m j . The analysis will not rely upon the complex conjugate structure in the left side of 3.4 . Thus, there is symmetry under the interchange of the indices 2-4, and We may assume that N 2 ≥ N 3 ≥ N 4 .

3.7
Case III: N 2 ≥ N 3 ≥ N . We have the bound on the symbol: If N 1 ∼ N 2 ≥ N 3 N, then we bounded H 1 by renormalizing the derivatives and multiplier, paring u N 1 u N 3 and u N 2 u N 4 and using Lemma 2.5: We write this bound as

3.10
Since m x is bounded from above by 1 and m x x p for p > 1 − s is nondecreasing and bounded from above by 1, for s ≥ 1/2, we bound Abstract and Applied Analysis here we used the fact that m i N 1/2 i ≥ m N N 1/2 N 1/2 for i 2, 3. For s < 1/2, by using the definition of m ξ :

3.14
If N 2 ∼ N 3 ≥ N, then paring u N 1 u N 2 and u N 3 u N 4 and using Lemma 2.5 again, a similar analysis leads to the bound:

3.15
Now we turn to give the bound for the term H 2 . it required 6-linear estimate for 3.5 . We write m 123 to denote m ξ 1 ξ 2 ξ 3 and use N 123 to denote the size of ξ 1 ξ 2 ξ 3 . By symmetry, we may assume N 4 ≥ N 5 ≥ N 6 . We carry out a case by case analysis.

3.17
By Hölder inequality and Lemma 2.5, we control the above expression by

3.18
By the Sobolev's inequality, we have It follows from Colliander et al. in 26 that

3.20
We use 3.7 -3.15 to complete the Case II analysis. H 2 is bounded by

3.21
Case III N 4 ≥ N 5 ≥ N . We have the bound on the symbol Similar steps leads to the bound

3.23
Combine the estimates for H 1 and H 2 , we can complete the proof of Proposition 3.1.

Remark 3.2.
One can see that the proof of Proposition 3.1 closely follows the proof from Colliander 32 . However, the proof in this paper provides some clarity to the final stages of the proof in 32 and the necessary restrictions on s.

The Interaction Morawetz Inequality
In this section we develop a prior two-particle interaction Morawetz inequality of solutions to the Cauchy problem 1.14 -1.15 . This prior control will be fundamental to our analysis. We first recall the generalized viriel identity 33 .

Proposition 4.1.
If β is convex and real valued, and u is a smooth solution to 1.14 -1.15 on 0, T × R 4 , then the following inequality holds: where M a t is the Morawetz action given by Proof. Since β is convex and real valued and a b > 0, by the fundamental theorem of calculus we can easily deduce the result. In the case of a solution to an equation with a nonlinearity which is not associated to a defocusing potential, the following corollary holds.
Then, the following inequality holds: where M β t is the Morawetz action corresponding to u and {·} p is the momentum bracket defined by Now we give the interaction Morawetz inequality, although the results presented here are well known to experts, it seems to us that simple, self-contained proofs are often difficult to locate, so we present them for the convenience of the reader.
Proof. The proof of the Proposition 4.3 is similar to that in Colliander et al. 26 . Now we choose β x 1 , then, β x 1 , x 2 is smooth and convex for all x ∈ R 2 . We apply the generalized viriel identity with the weight β x 1 , x 2 and the tensor product u x 1 , where u 1 x 1 , t , u 2 x 2 , t are solutions with x 1 , x 2 ∈ R 2 × R 2 to 1.14 -1.15 . It is not hard to see that the tensor product satisfies the equation: and Δ 4 is the Laplace in R 4 R 2 × R 2 .
Abstract and Applied Analysis

13
Then we conclude that Note that the definition of β x 1 , x 2 implies

4.12
It follows from the Fubuni's theorem that

4.13
On the other hand,

4.15
Remark 4.4. For the common Morawetz inequality, the nonlinear term the second term in 4.1 has played the central role in the scattering theory for the nonlinear Schrödinger equation, and the first term in 4.1 did not play a big role in these works. But now by taking advantage of the first term, we can obtain a global prior estimate for defocusing nonlinearity, and we mention that the heart of the matter is that

4.19
of course this is not the case. We may rewrite 4.17 as

4.20
For what follows we abbreviate u i u x i where u i is the solution of iu t Δu a|u| 2 u bE |u| 2 u, t, x i ∈ 0, T × R 2 .

4.21
We aim to prove the following theorem.

4.23
In particular, on a time interval T k where the local well-posedness Proposition 2.6 holds one has that Proof. According to Corollary 4.2,

4.25
Set If u solves 4.3 for n 2, then IU solves 4.3 for n 4, with right-hand side N I given by Now we decompose N I as good part and bad part. The good part creates a positive term that we ignore. The bad term produces the error term. Now we have the bound:

4.28
where we have used the fact that Iu L 2 u L 2 u 0 L 2 . Remark that ∇β is a real valued, thus and that ∇ ∇ x 1 , ∇ x 2 . We now compute the dot product under the integral in 4.20 , that is, Recall that Using the definition of N bad and the fact that ∇ x 1 acts only on Iu 1 , we have

4.32
Analogously, we can see that the second part is given by We have

4.34
Here we used the fact that the pair ∞, 2 is admissible and |∇ x 1 β| 1. By a similar way we can deduce that Hence, we only need to estimate ∇ x I N − N Iu L 1 t L 2 x . Observe that N a |u| 2 u b E |u| 2 u , and

4.36
Using the fact that a b > 0 and the properties of operator E, we have

4.37
There is symmetry under interchange of the indices 1, 2, 3. We may assume that We carry out a case by case analysis for 4.28 .

4.40
where we have used the fact that |ξ| ∼ N 1 and by mean value theorem that Therefore,

4.42
Case III N 1 ≥ N 2 N N 3 . We also have x .

4.43
Case IV N 1 ≥ N 2 ≥ N 3 N . We estimate as follows:

4.44
where we have used the estimate Finally, since the pair 3, 6 is admissible, we can get Combining 4.22 , 4.24 , and 4.40 , we complete the proof Theorem 4.5.

The Proof of Theorem 1.1
The idea is followed from 26, 34 . The first observation is the fact that if u t, x is a solution of the Cauchy problem 1.14 -1. 15  Then, for any T 0 arbitrarily large , define Γ 0 < t < λ 2 T 0 : Iu λ L 4 t L 4 x 0,t ×R 2 ≤ δN 1/8 t 1/12 ,

5.4
where δ is a constant to be chosen later. We claim that Γ is the whole interval 0, λ 2 T 0 . Indeed, if Γ is not the whole interval 0, λ 2 T 0 , then using the fact that x 0,T ×R 2 ≥ δN 1/8 t 1/12 ,

5.7
Now we divide the time interval 0, T 0 into subintervals J k , k 1, . . . , L in such a way that x 0,J k ×R 2 ≤ μ 0 ,

5.8
where μ 0 is as the same as in Proposition 2.6. This is possible because of 5.7 . Then, the number of the slices, which we will call L, is most like for all t ∈ 0, T , we require L N 3/2 .

5.12
Since T ≤ λ 2 T 0 , this is fulfilled as long as This estimate contradicts to 5.6 for suitable choice of δ namely, we choose δ 1 . Therefore Γ 0, λ 2 T 0 , and T 0 can be chosen arbitrarily large. In addition,

5.17
Since T 0 is arbitrarily large, the priori bound on the H s norm concludes the global well-posedness of the Cauchy Problem 1.14 -1.15 .