Global Analysis for Rough Solutions to the Davey-Stewartson System

Abstract

The global well-posedness of rough solutions to the Cauchy problem for the Davey-Stewartson system is obtained. It reads that if the initial data is in 𝐻^𝑠 with s > 2/5, then there exists a global solution in time, and the 𝐻^𝑠 norm of the solution obeys polynomial-in-time bounds. The new ingredient in this paper is an interaction Morawetz estimate, which generates a new space-time 𝐿⁴_𝑡,𝑥 estimate for nonlinear equation with the relatively general defocusing power nonlinearity.

1. 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|²𝑢+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,elliptic-hyperbolic∶𝜇>0,𝑐<0,hyperbolic-ellipti∶𝜇<0,𝑐>0,hyperbolic-hyperbolic∶𝜇<0,𝑐<0. (1.2)(1.3)(1.4)(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 𝑢₀∈𝐻¹(ℝ²), 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(ℝ²) whose 𝐿² norm is sufficiently small. Tsutsumi [4] obtained the 𝐿^𝑝-decay estimates of solutions to the systems (1.1) in the elliptic-hyperbolic case (1.3). Hayashi and Saut [5] and Linares and Ponce [6] studied some generalized Davey-Stewartson systems in different spaces, and their main tools are 𝐿^𝑝−𝐿^𝑞 estimates of solutions to linear Schrödinger equations. These estimates are usually named generalized Strchartz inequality. Guo and Wang [7] studied the Cauchy problem for a generalized Davey-Stewartson system in the elliptic-elliptic case (1.2), and they proved the global well-posedness results for initial data 𝑢₀ in 𝐻^𝑠(1≤𝑠≤2). Recently, Shu and Zhang [8] and Gan and Zhang [9] obtained the sharp conditions of global existence for Davey-Stewartson system in the elliptic-elliptic case (1.2) by constructing a type of cross-constrained variational problem and establishing so-called cross-invariant manifolds generated by the evolution flow. Zhang and Zhu [10] obtained a more precisely sharp criteria of blow-up and global existence. C. Sulem and P. L. Sulem [11] obtained some numerical observations on blow-up solutions. Richards [12] showed the mass concentration phenomenon of blow-up solutions. Li et al. [13] obtained some dynamical properties of blow-up solutions. Recently, Babaoglu and Erbay [14] proposed a generalized Davey-Stewartson system, which was studied in [15–17].

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|²𝑢+𝑏𝑢𝑣_𝑥,𝑣_𝑥𝑥+𝑣_𝑦𝑦=|𝑢|²_𝑥,𝑢(0,𝑥)=𝑢₀(𝑥). (1.6)(1.7)(1.8) As is well known, the system (1.6)–(1.8) enjoys two useful conservation laws: one is the energy conservation law: 𝐻(𝑢,𝑣)=_ℝ2𝑢_𝑥²+𝑢_𝑦²+𝑎|𝑢|⁴+𝑏𝑣²_𝑥+𝑣²_𝑦2𝑑𝑥𝑑𝑦, (1.9) and the other is the mass conservation law: 𝑀(𝑢)=_ℝ2|𝑢|²𝑑𝑥𝑑𝑦. (1.10) For details one can see Ghidaglia and Saut [2]. Moreover, one can easily establishes ‖𝑢(𝑇)‖_𝐻^𝑠≤𝐶𝑠,𝑢_0𝐻𝑠,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 𝑣_𝑥=ℱ⁻¹𝜉²₁𝜉²ℱ|𝑢|², (1.12) where ℱ and ℱ⁻¹ denote the Fourier transform and its inversion. For brevity we denote 𝐸𝜓=ℱ⁻¹𝜉²₁𝜉²ℱ𝜓. (1.13) Combining (1.8) and (1.9), (1.6)–(1.8) are changed to 𝑖𝑢_𝑡+Δ𝑢=𝑎|𝑢|²𝑢+𝑏𝐸|𝑢|²𝑢()𝑢,0,𝑥=𝑢₀(𝑥). (1.14)(1.15)

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 𝑢₀∈𝐻^𝑠(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(𝑠>1) norm is similarly bounded, but if the initial data is only in 𝐻^𝑠(𝑠<1) then the 𝐻¹(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 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(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 𝑆_𝑡 as the nonlinear flow and 𝑆(𝑡)=𝑒^𝑖𝑡Δ𝑢₀ is the linear group, Bourgain’s method shows addition that (𝑆_𝑡−𝑆(𝑡))𝑢₀∈𝐻¹(ℝ²) for all time provided 𝑢₀∈𝐻^𝑠, 𝑠>3/5. Thus, he showed that the solution is globally well posed with initial data in 𝐻^𝑠(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.

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 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: 𝐼_𝑁𝑓(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, 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: 𝐻𝐼_𝑁𝑢=_ℝ2𝐼_𝑁𝑢_𝑥²+𝐼_𝑁𝑢_𝑦²+𝑎𝐼_𝑁𝑢⁴+𝑏𝐸𝐼_𝑁𝑢²𝐼_𝑁𝑢²2𝑑𝑥𝑑𝑦. (1.17) Note that if 𝑢(𝑡)∈𝐻^𝑠(ℝ²) then 𝐻(𝐼_𝑁𝑢)<∞. 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 𝑢₀∈𝐻^𝑠(ℝ²), 𝑠>2/5 and 𝑎+𝑏>0. Moreover, the solution satisfies the following estimate: sup_{[]𝑡∈0,𝑇}‖𝑢(𝑡)‖_𝐻^𝑠≤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 in 𝐻^𝑠(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 low- and high-frequency parts. The new ingredient in our proof is a priori interaction Morawetz-type estimate, which generates a new space-time 𝐿⁴_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 𝐿²-critical 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. ℜ𝑧 and ℑ𝑧 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 𝐿^𝑟_𝑥(ℝ²) to denote the usual Lebesgue space of functions 𝑓: ℝ²→ℂ whose norm 𝑓_𝐿𝑟𝑥∶=_ℝ2𝑓^𝑟𝑑𝑥^1/𝑟 (2.1) is finite, with the usual modification in the case 𝑟=∞. We also define the space-time space 𝐿^𝑞_𝑡𝐿^𝑟_𝑥 by ‖𝑢‖_𝐿^𝑞_{𝑡∈𝐽𝐿}^𝑟_𝑥∶=_𝐽‖𝑢‖^𝑞_𝐿𝑟𝑥𝑑𝑡^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 𝐿^𝑞_𝑡𝐿^𝑞_𝑥 by 𝐿^𝑞_𝑡,𝑥.

Definition 2.1. A pair of exponent (𝑞,𝑟) is called admissible in ℝ² if 1𝑞+1𝑟=12,2≤𝑞,𝑟≤∞. (2.3)

We recall the known Strichartz estimates [21] (and the reference therein).

Proposition 2.2. Let (𝑞,𝑟) and (𝑞,𝑟)̃ be any two admissible pairs 𝑖𝑢_𝑡+Δ𝑢−𝑓(𝑥,𝑡)=0,(𝑡,𝑥)∈𝐽×ℝ²𝑢(𝑥,0)=𝑢₀(𝑥). (2.4) Then one has the estimate ‖𝑢‖_𝐿^𝑞_𝑡𝐿^𝑟_{𝑥(𝐽×ℝ}²₎≲𝑢_0𝐿2(ℝ2)+𝑓_𝐿𝑞^′_𝑡𝐿𝑟̃^′_{𝑥(𝐽×ℝ2)} (2.5) with the prime exponents denoting Hölder dual exponents.

We also define the fractional differential operator |∇|^𝛼 for any real 𝛼 by |∇|^𝛼𝑢(𝜉)∶=𝜉^𝛼𝑢(𝜉) (2.6) and analogously ⟨∇⟩^𝛼𝑢(𝜉)∶=𝜉^𝛼𝑢(𝜉), (2.7) where ⟨𝑎⟩∶=1+|𝑎|². We then define the homogeneous Sobolev space 𝐻̇^𝑠 and the inhomogeneous Sobolev space 𝐻^𝑠 by ‖𝑢‖_̇𝐻^𝑠=|∇|^𝑠𝑢_𝐿2𝑥;‖𝑢‖_𝐻^𝑠=⟨∇⟩^𝑠𝑢_𝐿2𝑥. (2.8)

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: 𝑃_≤𝑁𝑓(𝜉)𝜉∶=𝜑𝑁𝑓(𝜉),𝑃_>𝑁𝑓(𝜉)𝜉∶=1−𝜑𝑁𝑓(𝜉),𝑃_𝑁𝑓(𝜉)𝜑𝜉∶=𝑁−𝜑2𝜉𝑁𝑓(𝜉). (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_≤𝑁𝑓 and similarly for the other operators. We recall the following standard Bernstein and Sobolev type inequalities.

Lemma 2.3. For any 1≤2≤𝑞≤∞ and 𝑠>0, one has 𝑃_≥𝑁𝑓_𝐿^𝑝_𝑥≲𝑁^−𝑠|∇|^𝑠𝑃_≥𝑁𝑓_𝐿^𝑝_𝑥,|∇|^𝑠𝑃_≤𝑁𝑓_𝐿^𝑝_𝑥≲𝑁^𝑠𝑃_≤𝑁𝑓_𝐿^𝑝_𝑥,|∇|^±𝑠𝑃_𝑁𝑓_𝐿𝑝𝑥∼𝑁^±𝑠𝑃_𝑁𝑓_𝐿^𝑝_𝑥,𝑃_≤𝑁𝑓_𝐿^𝑞_𝑥≲𝑁^{1/𝑝−1/𝑞}𝑃_≤𝑁𝑓_𝐿^𝑝_𝑥,𝑃_𝑁𝑓_𝐿^𝑞_𝑥≲𝑁^{1/𝑝−1/𝑞}𝑃_𝑁𝑓_𝐿^𝑝_𝑥. (2.10)

We collect the basic properties of 𝐼_𝑁 into the following.

Lemma 2.4. Let 1<𝑝<∞ and 0≤𝑠<1. Then 𝐼_𝑁𝑓_𝐿𝑝≲𝑓_𝐿𝑝,|∇|^𝑠𝑃_>𝑁𝑓_𝐿𝑝≲𝑁^𝑠−1∇𝐼_𝑁𝑓_𝐿𝑝,𝑓_𝐻𝑠𝑥≲𝐼_𝑁𝑓_𝐻1𝑥≲𝑁^1−𝑠𝑓_𝐻𝑠𝑥. (2.11)(2.12)(2.13)

Proof. For the proof one can see Colliander et al. [25].

Now we define the Strichartz norm of functions 𝑢‖𝑢‖_𝑆⁰_𝑇∶=sup_{(𝑞,𝑟)admissible}‖𝑢‖_𝐿^𝑞_{𝑡∈[0,𝑇]𝐿}^𝑟_𝑥. (2.14) Then we introduce the following bilinear smoothing property due to Bourgain [18].

Lemma 2.5. Let 𝜓∈𝐿²(ℝ²) such that 𝜓₁=𝑃_𝑁𝑗𝜓,𝜓₂=𝑃_𝑁𝑘𝜓. (2.15) Then, for 𝑁_𝑗≤𝑁_𝑘, the following inequality holds: 𝑒^𝑖𝑡Δ𝜓₁⋅𝑒^𝑖𝑡Δ𝜓_{2𝐿2𝑡,𝑥(ℝ2×ℝ)}𝑁≤𝐶^𝑗𝑁_𝑘^(1/2)−𝜓₁₂𝜓₂₂. (2.16) That is to say, suppose 𝑢 solves (1.15)–(1.18) on the time interval [0,𝑇]. Let 𝑢_𝑗=𝑃_𝑁𝑗𝑢, for 𝑗=1,2 with 𝑁₁>𝑁₂. Then 𝑢₁𝑢_{2𝐿2𝑇𝐿2𝑥}𝑁≤𝐶²𝑁₁^(1/2)−‖𝑢‖²_𝑆0𝑇. (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 𝜇([0,𝑇])∶=^𝑇_0ℝ2𝐼_𝑁𝑢⁴𝑑𝑥𝑑𝑡. (2.18) If 𝜇([0,𝑇])<𝜇₀, where 𝜇₀ 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: 𝑍_𝐼([0,𝑇])∶=sup_{𝑞,𝑟admissible}⟨∇⟩𝐼_𝑁𝑢_𝐿^𝑞_{𝑡𝐿𝑟𝑥([0,𝑡]×ℝ2)}≲⟨∇⟩𝐼_𝑁𝑢_0𝐿2. (2.19)

Proof. The proof is standard, see for example [25, 26].

Remark 2.7. A modification (2.17) follows using the space-time estimate (2.19): For 𝑁₁>𝑁₂, and for solution 𝑢 of (1.15)–(1.18) satisfies ⟨∇⟩𝐼𝑢_𝑁1⟨∇⟩𝐼𝑢_{𝑁2𝐿2𝑇𝐿2𝑥}𝑁≤𝐶²𝑁₁^(1/2)−‖⟨∇⟩𝐼𝑢‖²_𝑆0𝑇. (2.20)

3. Almost Conservation Laws

In this section we prove the almost conservation of the modified energy 𝐻(𝐼_𝑁𝑢(𝑡)).

Proposition 3.1. If the initial data 𝑢₀∈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 sup_{𝑡∈[0,𝑇]}𝐻𝐼_𝑁𝑢(𝑡)≤𝐻𝐼_𝑁𝑢(0)+𝐶𝑁^−(3/2)−𝐼_𝑁⟨∇⟩𝑢(0)⁴_𝐿2𝑥+𝐶𝑁⁻²⁻𝐼_𝑁⟨∇⟩𝑢(0)⁶_𝐿2𝑥. (3.1) In particular when ‖𝐼_𝑁⟨∇⟩𝑢(0)‖_𝐿2𝑥≲1 one has sup_{𝑡∈[0,𝑇]}𝐻𝐼_𝑁𝑢(𝑡)≲𝑁^−(3/2)−. (3.2)

Proof. In light of (2.19), it suffices to control the energy increment |𝐻(𝐼_𝑁𝑢(𝑡))−𝐻(𝐼_𝑁𝑢(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_𝑁𝑢_𝑡+ΔI_𝑁𝑢=𝑎𝐼_𝑁|𝑢|²𝑢+𝑏𝐼_𝑁𝐸|𝑢|²𝑢,𝐼_𝑁𝑢(0,𝑥)=𝐼_𝑁𝑢₀(𝑥). (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: 𝐻₁=^𝑡_{0𝜉1+𝜉2+𝜉3+𝜉4=0}𝑚𝜉1−²+𝜉₃+𝜉₄𝑚𝜉₂𝑚𝜉₃𝑚𝜉₄Δ𝐼𝑢𝜉₁𝐼𝑢𝜉₂𝐼𝑢𝜉₃𝐼𝑢𝜉₄𝐻₂,=(𝑎+𝑏)^𝑡_{0𝜉′1+𝜉4+𝜉5+𝜉6=0}𝑚𝜉1−^′1𝑚𝜉₄𝑚𝜉₅𝑚𝜉₆𝐼|𝑢|²𝑢𝜉^′₁𝐼𝑢𝜉₄𝐼𝑢𝜉₅𝐼𝑢𝜉₆. (3.4)(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)i𝑓𝑢∈𝐻^𝑠,then 𝐸(𝑢)∈𝐻^𝑠,𝑠∈ℝ.
We estimate 𝐻₁ first. We use 𝑢_𝑁𝑗 to denote 𝑃_𝑁𝑗𝑢. 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 𝑁₂≥𝑁₃≥𝑁₄.
Case I. (𝑁≫𝑁₂). Since the convolution hypersurface is 𝜉₁+𝜉₂+𝜉₃+𝜉₄=0, we have 𝑁₁≪𝑁 as well. Hence we get 𝑚(𝜉₂+𝜉₃+𝜉₄)/𝑚(𝜉₂)𝑚(𝜉₃)𝑚(𝜉₄)=1, and the bound holds trivially.
Case II. 𝑁₂≳𝑁≫𝑁₃>𝑁₄, for 𝜉₁+𝜉₂+𝜉₃+𝜉₄=0, we have 𝑁₁∼𝑁₂. Applying the mean value theorem, we deduce that 𝑚𝜉1−²+𝜉₃+𝜉₄𝑚𝜉₂𝑚𝜉₃𝑚𝜉₄=𝑚𝜉₂−𝑚𝜉₂+𝜉₃+𝜉₄𝑚𝜉₂≤∇𝑚𝜉₂𝜉₃+𝜉₄𝑚𝜉₂≤𝑁₃𝑁₂. (3.6) Moreover, 𝐻₁ is controlled by 𝑁₃𝑁₂𝑁₃𝑁₁^1/2𝑁₄𝑁₂^1/2𝑁₁𝑁₂𝑁₃𝑁₄‖𝐼𝑢‖⁴_{𝑍𝐼([0,𝑇])}≤𝑁^−(3/2)−𝑁₁𝑁₂𝑁₃𝑁₄⁰⁻𝑍_𝐼([0,𝑇])⁴. (3.7)
Case III: (𝑁₂≥𝑁₃≥𝑁). We have the bound on the symbol: 𝑚𝜉1−²+𝜉₃+𝜉₄𝑚𝜉₂𝑚𝜉₃𝑚𝜉₄≤𝑚𝜉₁𝑚𝜉₂𝑚𝜉₃𝑚𝜉₄. (3.8) If 𝑁₁∼𝑁₂≥𝑁₃≳𝑁, then we bounded 𝐻₁ by renormalizing the derivatives and multiplier, paring 𝑢_𝑁1𝑢_𝑁3 and 𝑢_𝑁2𝑢_𝑁4 and using Lemma 2.5: 𝑚₁𝑚₂𝑚₃𝑚₄𝑁₃𝑁₁^1/2𝑁₄𝑁₂^1/2𝑁₁𝑁₂𝑁₃𝑁₄𝑍_𝐼([0,𝑇])⁴. (3.9) We write this bound as 𝑚₁𝑚₂𝑁₂^1/2𝑚₃𝑁₃^1/2𝑚₄𝑁₄^1/2𝑁₁^1/2𝑍_𝐼([0,𝑇])⁴. (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 𝑚₁𝑚₂𝑁₂^1/2𝑚₃𝑁₃^1/2𝑚₄𝑁₄^1/2𝑁₁^1/2≤1𝑁^(3/2)+, (3.11) here we used the fact that 𝑚_𝑖𝑁_𝑖^1/2≥𝑚(𝑁)𝑁^1/2=𝑁^1/2 for 𝑖=2,3. For 𝑠<1/2, by using the definition of 𝑚(𝜉): 𝑚₁𝑚₂𝑁₂^1/2𝑚₃𝑁₃^1/2𝑚₄𝑁₄^1/2𝑁₁^1/2≤𝑁₂^1/2−𝑠𝑁₃^1/2−𝑠𝑁₄^1/2−𝑠𝑁^2−2𝑠𝑁₁^1/2𝑁₁^1−𝑠≤𝑁₁^1/2−2𝑠𝑁^2−2𝑠𝑁₁^1/2. (3.12) Using the facts 1/2>𝑠>1/4 and 𝑁₂≥𝑁₃≥𝑁, 𝑁₁^1/2−2𝑠𝑁^2−2𝑠𝑁₁^1/2≤𝑁^−(3/2)−. (3.13) Hence 𝐻₁≤𝑁^−(3/2)−𝑁₁𝑁₂𝑁₃𝑁₄⁰⁻𝑍_𝐼([0,𝑇])⁴. (3.14)
If 𝑁₂∼𝑁₃≥𝑁, then paring 𝑢_𝑁1𝑢_𝑁2 and 𝑢_𝑁3𝑢_𝑁4 and using Lemma 2.5 again, a similar analysis leads to the bound: 𝑚₁𝑚₂𝑚₃𝑚₄𝑁₁𝑁₂^1/2𝑁₄𝑁₃^1/2𝑁₁𝑁₂𝑁₃𝑁₄𝑍_𝐼([0,𝑇])⁴≤𝑁^−3/2−𝑁₁𝑁₂𝑁₃𝑁₄⁰⁻𝑍_𝐼([0,𝑇])⁴. (3.15)
Now we turn to give the bound for the term 𝐻₂. it required 6-linear estimate for (3.5). We write 𝑚₁₂₃ to denote 𝑚(𝜉₁+𝜉₂+𝜉₃) and use 𝑁₁₂₃ to denote the size of 𝜉₁+𝜉₂+𝜉₃. By symmetry, we may assume 𝑁₄≥𝑁₅≥𝑁₆. We carry out a case by case analysis.
Case I. (𝑁≫𝑁₄). On 𝜉^′₁+𝜉₄+𝜉₅+𝜉₆=0, this forces 𝑁₁₂₃∼𝑁₄ as well. In this case, 𝑚(𝜉₂+𝜉₃+𝜉₄)/𝑚(𝜉₂)𝑚(𝜉₃)𝑚(𝜉₄)=1, and the bound holds trivially.
Case II. (𝑁₄≳𝑁≥𝑁₅). On 𝜉^′₁+𝜉₄+𝜉₅+𝜉₆=0, 𝑁₁₂₃∼𝑁₄ in this case. By the mean value theorem, 𝑚1−¹²³𝑚₄𝑚₅𝑚₆=𝑚₄−𝑚₄₅₆𝑚₄≤𝑁₅𝑁₄. (3.16) Applying the Cauchy-Schwartz inequality and the above multiplier bound to (3.5), we deduce that 𝑁₄𝑁₅⁻¹𝑁₅𝑁₄𝑃₁₂₃𝐼𝑢_𝑁1𝑢_𝑁2𝑢_𝑁3𝐼𝑢_{𝑁6𝐿2𝑡,𝑥}𝐼⟨∇⟩𝑢_𝑁4𝐼⟨∇⟩𝑢_{𝑁5𝐿2𝑡𝑥}. (3.17) By Hölder inequality and Lemma 2.5, we control the above expression by 𝑁₄𝑁₅⁻¹𝑁₅𝑁₄^3/2𝑃₁₂₃𝐼𝑢_𝑁1𝑢_𝑁2𝑢_{𝑁3𝐿2𝑡,𝑥}𝐼𝑢_{𝑁6𝐿∞𝑡,𝑥}𝑍_𝐼([0,𝑡])². (3.18) By the Sobolev’s inequality, we have 𝐼𝑢_{𝑁6𝐿∞𝑡,𝑥}≤𝑍_𝐼([0,𝑡]). (3.19) It follows from Colliander et al. in [26] that 𝑃₁₂₃𝐼𝑢_𝑁1𝑢_𝑁2𝑢_{𝑁3𝐿2𝑡,𝑥}≤𝑁₁^−1/2𝑍_𝐼([0,𝑡])³. (3.20)
We use (3.7)–(3.15) to complete the Case II analysis. 𝐻₂ is bounded by 𝑁₄𝑁₅⁻¹𝑁₅𝑁₄^3/2𝑁₁^−1/2𝑍_𝐼([0,𝑡])⁶≤𝑁⁻²⁻⁶_𝑖=1𝑁_𝑖⁰⁻𝑍_𝐼([0,𝑡])⁶. (3.21)
Case III (𝑁₄≥𝑁₅≥𝑁). We have the bound on the symbol 𝑚1−¹²³𝑚₄𝑚₅𝑚₆≤𝑚₁₂₃𝑚₄𝑚₅𝑚₆. (3.22) Similar steps leads to the bound 𝐻₂≤𝑁⁻²⁻⁶_𝑖=1𝑁_𝑖⁰⁻𝑍_𝐼([0,𝑡])⁶. (3.23) Combine the estimates for 𝐻₁ and 𝐻₂, 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.

4. 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]×ℝ⁴, then the following inequality holds: −ΔΔ𝛽|𝑢^𝑡_0ℝ4(𝑥,𝑡)|²𝑑𝑥𝑑𝑡+^𝑡_0ℝ4Δ𝛽|𝑢(𝑥,𝑡)|⁴𝑑𝑥𝑑𝑡≲sup_{𝑡∈[0,𝑇]}𝑀_𝛽(𝑡), (4.1) where 𝑀_𝑎(𝑡) is the Morawetz action given by 𝑀_𝛽(𝑡)=2_ℝ2∇𝛽(𝑥)⋅𝒥(𝑢∇𝑢)𝑑𝑥. (4.2)

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.

Corollary 4.2. Let 𝛽∶ℝ⁴→ℝ be convex and 𝑢 be a smooth solution to the equation: 𝑖𝑢_𝑡+Δ𝑢=𝒩. (4.3) Then, the following inequality holds: −ΔΔ𝛽|𝑢^𝑡_0ℝ4(𝑥,𝑡)|²𝑑𝑥𝑑𝑡+2^𝑡_0ℝ4∇𝛽⋅{𝒩,𝑢}_𝑝𝑑𝑥𝑑𝑡≲sup_{𝑡∈[0,𝑇]}𝑀_𝛽(𝑇)−𝑀_𝛽(𝑡), (4.4) where 𝑀_𝛽(𝑡) is the Morawetz action corresponding to 𝑢 and {⋅}_𝑝 is the momentum bracket defined by 𝑓,𝑔_𝑝=ℛ𝑓∇𝑔−𝑔∇𝑓. (4.5)

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.

Proposition 4.3. Let 𝑢 be solution to the Cauchy problem of (1.14)-(1.15), then the following 𝐿⁴_𝑥,𝑡 space-time estimate holds ‖𝑢‖⁴_{𝐿4𝑇𝐿4𝑥}≲𝑇^1/3sup_{𝑡∈[0,𝑇]}‖𝑢‖³_{𝐿∞𝑡𝐿2𝑥}‖𝑢‖_𝐿^∞_𝑡̇𝐻¹_𝑥+𝑇^1/3‖𝑢‖⁴_{𝐿∞𝑡𝐿2𝑥}. (4.6)

Proof. The proof of the Proposition 4.3 is similar to that in Colliander et al. [26]. Now we choose 𝛽(𝑥₁,𝑥₂)=(1/2𝑀)𝑥²(1−log(𝑥/𝑀)) if |𝑥|<𝑀/√𝑒; 𝛽(𝑥₁,𝑥₂)=50𝑥 if |𝑥|>𝑀; then, 𝛽(𝑥₁,𝑥₂) is smooth and convex for all 𝑥∈ℝ². We apply the generalized viriel identity with the weight 𝛽(𝑥₁,𝑥₂) and the tensor product 𝑢(𝑥₁,𝑥₂,𝑡)=𝑢₁(𝑥₁,𝑡)⊗𝑢₂(𝑥₂,𝑡)=𝑢₁(𝑥₁,𝑡)𝑢₂(𝑥₂,𝑡), where 𝑢₁(𝑥₁,𝑡),𝑢₂(𝑥₂,𝑡) are solutions with (𝑥₁,𝑥₂)∈ℝ²×ℝ² to (1.14)-(1.15). It is not hard to see that the tensor product satisfies the equation: 𝑖𝑢_𝑡+Δ₄𝑢=𝑓(𝑢),(𝑡,𝑥)∈[0,𝑇]×ℝ⁴, (4.7) here 𝑓(𝑢)=𝑎𝑢₁²𝑢+𝑏𝐸𝑢₁²𝑢+𝑎𝑢₂²𝑢+𝑏𝐸𝑢₂²𝑢, (4.8) and Δ₄ is the Laplace in ℝ⁴=ℝ²×ℝ².
Then we conclude that −ΔΔ𝛽𝑥^𝑇_0ℝ2×ℝ21,𝑥₂𝑢₁𝑥₁,𝑡²𝑢₂𝑥₂,𝑡²𝑑𝑥₁𝑑𝑥₂𝑑𝑡≲2sup_[]0,𝑇𝑀_𝛽^⊗2(𝑡), (4.9) where 𝑀_𝛽^⊗2(𝑡)=2_ℝ2⊗ℝ2∇𝛽(𝑥)⋅𝒥𝑢₁𝑢₂(𝑥)∇𝑢₁𝑢₂(𝑥)𝑑𝑥. (4.10) Note that the definition of 𝛽(𝑥₁,𝑥₂) implies −ΔΔ𝛽𝑥₁,𝑥₂=2𝑀𝛿_𝑥1=𝑥21+𝑂𝑀³. (4.11) Thus −ΔΔ𝛽𝑥^𝑇_0ℝ2×ℝ21,𝑥₂𝑢𝑥₁,𝑡²𝑢𝑥₂,𝑡²𝑑𝑥₁𝑑𝑥₂=2𝑑𝑡𝑀^𝑇_0ℝ⁴|𝑢(𝑥,𝑡)|⁴1𝑑𝑥𝑑𝑡+𝑂𝑀^3𝑇_0ℝ²_×ℝ²𝑢𝑥₁,𝑡²𝑢𝑥₂,𝑡²𝑑𝑥₁𝑑𝑥₂𝑑𝑡. (4.12) It follows from the Fubuni’s theorem that 𝐶𝑀^3𝑇_0ℝ²_×ℝ²𝑢𝑥₁,𝑡²𝑢𝑥₂,𝑡²𝑑𝑥₁𝑑𝑥₂𝑑𝑡≲𝐶𝑇𝑀³‖𝑢‖⁴_{𝐿∞𝑡𝐿2𝑥}. (4.13) On the other hand, sup_[0,𝑇]𝑀^⨂2_𝛽(𝑡)≲sup_{𝑡∈[0,𝑇]}‖𝑢‖³_{𝐿∞𝑡𝐿2𝑥}‖𝑢‖_{𝐿∞𝑡̇‌H}¹_𝑥. (4.14) Picking 𝑀∼𝑇^1/3, we get ‖𝑢‖⁴_{𝐿4𝑇𝐿4𝑥}≲𝑇^1/3sup_{𝑡∈[0,𝑇]}‖𝑢‖³_{𝐿∞𝑡𝐿2𝑥}‖𝑢‖_𝐿^∞_𝑡̇𝐻¹_𝑥+𝑇^1/3‖𝑢‖⁴_{𝐿∞𝑡𝐿2𝑥}. (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 −ΔΔ𝛽𝑥₁,𝑥₂=2𝑀𝛿_𝑥1=𝑥21+𝑂𝑀³. (4.16) This idea first appeared in [31].

Now, we consider the solution 𝐼𝑢 of 𝑖𝐼𝑢_𝑡+Δ𝐼𝑢=𝑎𝐼|𝑢|²𝑢+𝑏𝐼𝐸|𝑢|²𝑢. (4.17) If 𝐼𝑢 does not solve (4.7) but the following equation: 𝑖𝐼𝑢_𝑡+Δ𝐼𝑢=𝑎|𝐼𝑢|²𝐼𝑢+𝑏𝐸|𝐼𝑢|²𝐼𝑢, (4.18) then the calculations that we have done above would reveal that ‖𝐼𝑢‖⁴_{𝐿4𝑇𝐿4𝑥}≲𝑇^1/3sup_[0,𝑇]‖𝐼𝑢‖³_{𝐿∞𝑡𝐿2𝑥}‖𝐼𝑢‖_𝐿^∞_𝑡̇𝐻¹_𝑥+𝑇^1/3‖𝐼𝑢‖⁴_{𝐿∞𝑡𝐿2𝑥}, (4.19) of course this is not the case. We may rewrite (4.17) as 𝑖𝐼𝑢_𝑡+Δ𝐼𝑢=𝑎|𝐼𝑢|²𝐼𝑢+𝑏𝐸|𝐼𝑢|²𝐼𝑢+𝐼𝑎|𝑢|²𝑢+𝑏𝐸|𝑢|²𝑢−𝑎|𝐼𝑢|²𝐼𝑢−𝑏𝐸|𝐼𝑢|²𝐼𝑢=𝐺(𝐼𝑢)+(𝐼𝐺(𝑢)−𝐺(𝐼𝑢)). (4.20) For what follows we abbreviate 𝑢_𝑖=𝑢(𝑥_𝑖) where 𝑢_𝑖 is the solution of 𝑖𝑢_𝑡+Δ𝑢=𝑎|𝑢|²𝑢+𝑏𝐸|𝑢|²𝑢,𝑡,𝑥_𝑖∈[0,𝑇]×ℝ². (4.21) We aim to prove the following theorem.

Theorem 4.5. Let 𝐼𝑢 be a solution to (4.7), then ‖𝐼𝑢‖⁴_{𝐿4𝑇𝐿4𝑥}≲𝑇^1/3sup_[0,𝑇]‖𝐼𝑢‖³_𝐿2𝑥‖𝐼𝑢‖_̇𝐻¹_𝑥+T^1/3𝑢₀⁴_𝐿2𝑥+𝑇^1/3𝑇_0ℝ2×ℝ2∇𝛽⋅𝒩_bad,𝐼𝑢𝑥₁,𝑡𝐼𝑢𝑥₂,𝑡_𝑝𝑑𝑥₁𝑑𝑥₂𝑑𝑡, (4.22) where 𝒩_bad=²_𝑖−1𝐼𝐺𝑢_𝑖−𝐺𝐼𝑢_𝑖²_{𝑗=1,𝑖≠𝑗}𝐼𝑢_𝑗. (4.23) In particular, on a time interval 𝑇_𝑘 where the local well-posedness Proposition 2.6 holds one has that ∇𝛽⋅𝒩_{𝑇𝑘ℝ2×ℝ2bad},𝐼𝑢𝑥₁,𝑡𝐼𝑢𝑥₂,𝑡_𝑝𝑑𝑥₁𝑑𝑥₂1𝑑𝑡≲𝑁¹⁻𝑍⁶_𝐼𝐽_𝑘. (4.24)

Proof. According to Corollary 4.2, −ΔΔ𝛽|𝐼𝑢(^𝑇_0ℝ2𝑥,𝑡)|²𝑑𝑥𝑑𝑡≲2sup_[]0,𝑇ℝ2∇𝛽⋅𝒥(𝑥)+𝐼𝑢∇𝐼𝑢𝑑𝑥_𝑇0ℝ2∇𝛽⋅𝐼𝐺𝑢_𝑖−𝐺𝐼𝑢_𝑖𝑝𝑑𝑥𝑑𝑡. (4.25) Set 𝐼𝑈=𝐼𝐼𝑢𝑥₁,𝑡𝑢𝑥₂,𝑡=²_𝑗=1𝐼𝑢𝑥_𝑗,𝑡. (4.26) If 𝑢 solves (4.3) for 𝑛=2, then 𝐼𝑈 solves (4.3) for 𝑛=4, with right-hand side 𝒩_𝐼 given by 𝒩_𝐼=²_𝑖=1𝐼𝒩_𝑖²_{𝑗=1,𝑖≠𝑗}𝐼𝑢_𝑗. (4.27) 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: ‖𝐼𝑢‖⁴_{𝐿4𝑇𝐿4𝑥}≲𝑇^1/3sup_[0,𝑇]‖𝐼𝑢‖³_𝐿2𝑥‖𝐼𝑢‖_̇𝐻¹_𝑥+𝑇^1/3𝑢₀⁴_𝐿2𝑥+𝑇^1/3𝑇_0ℝ2×ℝ2∇𝛽⋅𝒩_bad,𝐼𝑢𝑥₁,𝑡𝐼𝑢𝑥₂,𝑡_𝑝𝑑𝑥₁𝑑𝑥₂𝑑𝑡, (4.28) where we have used the fact that ‖𝐼𝑢‖_𝐿²≲‖𝑢‖_𝐿²=‖𝑢₀‖_𝐿2. Remark that ∇𝛽 is a real valued, thus ∇𝛽⋅ℛ𝑓∇𝑔−𝑔∇𝑓=ℛ∇𝛽⋅𝑓∇𝑔−𝑔∇𝑓 (4.29) and that ∇=(∇_𝑥1,∇_𝑥2). We now compute the dot product under the integral in (4.20), that is, ℛ²_𝑖=1∇_𝑥𝑖𝛽𝒩_bad∇_𝑥𝑖𝐼𝑢₁𝐼𝑢₂−𝐼𝑢₁𝐼𝑢₂∇_𝑥𝑖𝒩_bad. (4.30) Recall that 𝒩_bad=²_𝑖−1𝐼𝐺𝑢_𝑖−𝐺𝐼𝑢_𝑖²_{𝑗=1,𝑖≠𝑗}𝐼𝑢_𝑗. (4.31) Using the definition of 𝒩_bad and the fact that ∇_𝑥1 acts only on 𝐼𝑢₁, we have 𝒩_bad∇_𝑥𝑖𝐼𝑢₁𝐼𝑢₂−𝐼𝑢₁𝐼𝑢₂∇_𝑥𝑖𝒩_bad=𝐼𝒩₁−𝒩𝐼𝑢₁∇_𝑥1𝐼𝑢₁−∇_𝑥1𝐼𝒩₁−𝒩𝐼𝑢₁𝐼𝑢₁𝐼𝑢₂². (4.32)
Analogously, we can see that the second part is given by ℛ∇_𝑥2𝛽𝐼𝒩₂−𝒩𝐼𝑢₂∇_𝑥2𝐼𝑢₂−∇_𝑥2𝐼𝒩₂−𝒩𝐼𝑢₂𝐼𝑢₂𝐼𝑢₁². (4.33) We have ^𝑇_0ℝ4∇_𝑥1𝛽𝐼𝒩₁−𝒩𝐼𝑢₁∇_𝑥1𝐼𝑢₁𝐼𝑢₂²𝑑𝑥₁𝑑𝑥₂≤𝑑𝑡_𝑇0ℝ2𝐼𝒩₁−𝒩𝐼𝑢₁∇_𝑥1𝐼𝑢₁𝑑𝑥₁𝑑𝑡‖𝐼𝑢‖²_{𝐿∞𝑡𝐿2𝑥}≤^𝑇_0ℝ2𝐼𝒩₁−𝒩𝐼𝑢₁∇_𝑥1𝐼𝑢₁𝑑𝑥𝑑𝑡𝑍²_𝐼≤𝐼𝒩₁−𝒩𝐼𝑢_{1𝐿1𝑡𝐿2𝑥}‖∇𝐼𝑢‖_𝐿^∞_𝑡𝐿²_𝑥𝑍²_𝐼≤𝐼𝒩₁−𝒩𝐼𝑢_{1𝐿1𝑡𝐿2𝑥}𝑍³_𝐼. (4.34) Here we used the fact that the pair (∞,2) is admissible and |∇_𝑥1𝛽|≲1. By a similar way we can deduce that ^𝑇_0ℝ4∇_𝑥1𝛽∇_𝑥1𝐼𝒩₁−𝒩𝐼𝑢₁𝐼𝑢₁𝐼𝑢₂²𝑑𝑥₁𝑑𝑥₂𝑑𝑡≤∇_𝑥(𝐼(𝒩)−𝒩(𝐼𝑢)_{𝐿1𝑡𝐿2𝑥}𝑍³_𝐼. (4.35) Hence, we only need to estimate ‖∇_𝑥(𝐼(𝒩)−𝒩(𝐼𝑢)‖_𝐿¹_𝑡𝐿²_𝑥. Observe that 𝒩=𝑎(|𝑢|²𝑢)+𝑏(𝐸(|𝑢|²)𝑢), and ∇_𝑥𝐼(𝒩)−𝒩(𝐼𝑢)(𝜉)≤𝐶_{𝜉=𝜉1+𝜉2+𝜉3}𝜉𝑚(𝜉)−𝑚𝜉₁𝑚𝜉₂𝑚𝜉₃𝑢𝜉₁𝑢𝜉₂𝑢𝜉₃𝑑𝜉₁𝑑𝜉₂𝑑𝜉₃. (4.36) Using the fact that 𝑎+𝑏>0 and the properties of operator 𝐸, we have ∇_𝑥(𝐼(𝒩)−𝒩(𝐼𝑢))_{𝐿1𝑡𝐿2𝑥}=∇_𝑥(𝐼(𝒩)−𝒩(𝐼𝑢))_{𝐿1𝑡𝐿2𝜉}≤𝐶_{𝑁1,𝑁2,𝑁3𝜉=𝜉1+𝜉2+𝜉3}𝜉𝑚(𝜉)−𝑚𝜉₁𝑚𝜉₂𝑚𝜉₃𝑢𝜉₁𝑢𝜉₂𝑢𝜉₃𝑑𝜉₁𝑑𝜉₂𝑑𝜉_{3𝐿1𝑡𝐿2𝜉}=𝐶_{𝑁1,𝑁2,𝑁3𝜉=𝜉1+𝜉2+𝜉3}𝜉𝑚(𝜉)−𝑚𝜉₁𝑚𝜉₂𝑚𝜉₃𝑚𝜉₁𝑚𝜉₂𝑚𝜉₃𝐼𝑢̂𝜉₁𝐼𝑢̂𝜉₂𝐼𝑢̂𝜉₃𝑑𝜉₁𝑑𝜉₂𝑑𝜉_{3𝐿1𝑡𝐿2𝜉}. (4.37) There is symmetry under interchange of the indices 1, 2, 3. We may assume that 𝑁₁≥𝑁₂≥𝑁₃. (4.38) Let 𝜎𝜉₁,𝜉₂,𝜉₃=𝜉𝑚(𝜉)−𝑚𝜉₁𝑚𝜉₂𝑚𝜉₃𝑚𝜉₁𝑚𝜉₂𝑚𝜉₃. (4.39) We carry out a case by case analysis for (4.28).
Case I. 𝑁₁≪𝑁, this force 𝜎(𝜉₁,𝜉₂,𝜉₃)=0, then there no contribution to (4.25).
Case II. 𝑁₁≳𝑁≫𝑁₂, we have _{𝜉=𝜉1+𝜉2+𝜉3}𝜎𝜉₁,𝜉₂,𝜉₃𝐼𝑢𝜉₁𝐼𝑢𝜉₂𝐼𝑢𝜉₃𝑑𝜉₁𝑑𝜉₂𝑑𝜉_{3𝐿1𝑡𝐿2𝜉}=1𝑁_{𝜉=𝜉1+𝜉2+𝜉3}𝑁𝜉₁𝜉₂𝜎𝜉₁,𝜉₂,𝜉₃∇𝐼𝑢𝜉₁∇𝐼𝑢𝜉₂𝐼𝑢𝜉₃𝑑𝜉₁𝑑𝜉₂𝑑𝜉_{3𝐿1𝑡𝐿2𝜉}≲1𝑁∇𝐼𝑢_{1𝐿3𝑡𝐿6𝑥}∇𝐼𝑢_{2𝐿3𝑡𝐿6𝑥}𝐼𝑢_{1𝐿3𝑡𝐿6𝑥}, (4.40) where we have used the fact that |𝜉|∼𝑁₁ and by mean value theorem that 𝑚(𝜉)−𝑚𝜉₁𝑚𝜉₂𝑚𝜉₃𝑚𝜉₁𝑚𝜉₂𝑚𝜉₃≲𝑁₃𝑁₂. (4.41) Therefore, 𝑁𝜉₁𝜉₂𝜎𝜉₁,𝜉₂,𝜉₃≲𝑁𝜉₁𝜉₂𝑁₂𝑁₁𝜉≤1. (4.42)
Case III (𝑁₁≥𝑁₂≳𝑁≫𝑁₃). We also have _{𝜉=𝜉1+𝜉2+𝜉3}𝜎𝜉₁,𝜉₂,𝜉₃𝐼𝑢𝜉₁𝐼𝑢𝜉₂𝐼𝑢𝜉₃𝑑𝜉₁𝑑𝜉₂𝑑𝜉_{3𝐿1𝑡𝐿2𝜉}=1𝑁_{𝜉=𝜉1+𝜉2+𝜉3}𝑁𝜉₁𝜉₂𝜎𝜉₁,𝜉₂,𝜉₃∇𝐼𝑢𝜉₁∇𝐼𝑢𝜉₂𝐼𝑢𝜉₃𝑑𝜉₁𝑑𝜉₂𝑑𝜉_{3𝐿1𝑡𝐿2𝜉}≲1𝑁∇𝐼𝑢_{1𝐿3𝑡𝐿6𝑥}∇𝐼𝑢_{2𝐿3𝑡𝐿6𝑥}𝐼𝑢_{1𝐿3𝑡𝐿6𝑥}. (4.43)
Case IV (𝑁₁≥𝑁₂≥𝑁₃≳𝑁). We estimate as follows: _{𝜉=𝜉1+𝜉2+𝜉3}𝜎𝜉₁,𝜉₂,𝜉₃𝐼𝑢𝜉₁𝐼𝑢𝜉₂𝐼𝑢𝜉₃𝑑𝜉₁𝑑𝜉₂𝑑𝜉_{3𝐿1𝑡𝐿2𝜉}≲1𝑁²³_𝑗=1∇𝐼𝑢_{𝑗𝐿3𝑡𝐿6𝑥}, (4.44) where we have used the estimate 𝜎𝜉₁,𝜉₂,𝜉₃𝑁²𝜉₁𝜉₂𝜉₃≤𝑁^−1+3𝑠𝑁₁𝑁₂𝑁₃^−𝑠≤1. (4.45) Finally, since the pair (3, 6) is admissible, we can get ∇_𝑥(𝐼(𝒩)−𝒩(𝐼𝑢))_{𝐿1𝑡𝐿2𝑥}≲1𝑁𝑍³_𝐼. (4.46) Combining (4.22), (4.24), and (4.40), we complete the proof Theorem 4.5.

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), we can scale it and obtain a new solution, namely, the scale function 𝑢^𝜆(𝑥)=1𝜆𝑢𝑥𝜆,𝑡𝜆² (5.1) satisfies the same equation with initial data 𝑢^𝜆₀=(1/𝜆)𝑢₀(𝑥/𝜆). This scaling preserves the 𝐿² norms of 𝑢(𝑡) as well as the 𝐿⁴_𝑡,𝑥 space-time norm. From estimate (2.13), we have 𝐼𝑢^𝜆_0𝐻1≲𝑁^1−𝑠𝑢^𝜆_0𝐻𝑠≲𝑁^1−𝑠𝜆^−𝑠. (5.2) We can choose the parameter 𝜆 in the manner 𝜆∼𝑁^{(1−𝑠)/𝑠}, (5.3) so that ‖𝐼𝑢^𝜆₀‖_𝐻¹=𝑂(1).

Then, for any 𝑇₀ (arbitrarily large), define Γ=0<𝑡<𝜆²𝑇₀∶𝐼𝑢^𝜆_{𝐿4𝑡𝐿4𝑥([0,𝑡]×ℝ2)}≤𝛿𝑁^1/8𝑡^1/12, (5.4) where 𝛿 is a constant to be chosen later. We claim that Γ is the whole interval [0,𝜆²𝑇₀]. Indeed, if Γ is not the whole interval [0,𝜆²𝑇₀], then using the fact that 𝐼𝑢^𝜆_{𝐿4𝑡𝐿4𝑥([0,𝑡]×ℝ2)} (5.5) is a continuous function of time, there exist some 𝑇∈[0,𝜆²𝑇₀] with the properties, 𝐼𝑢^𝜆_{𝐿4𝑡𝐿4𝑥([0,𝑇]×ℝ2)}≥𝛿𝑁^1/8𝑡^1/12,𝐼𝑢^𝜆_{𝐿4𝑡𝐿4𝑥([0,𝑇]×ℝ2)}≤2𝛿𝑁^1/8𝑡^1/12. (5.6)(5.7) Now we divide the time interval [0,𝑇₀] into subintervals 𝐽_𝑘, 𝑘=1,…,𝐿 in such a way that 𝐼𝑢^𝜆4_{𝐿4𝑡𝐿4𝑥([0,𝐽𝑘]×ℝ2)}≤𝜇₀, (5.8) where 𝜇₀ 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 𝐿∼2𝛿𝑁^1/8𝑡^1/124𝜇∼(2𝛿)⁴𝑁^1/2𝑇^1/3𝜇. (5.9)

According to Propositions 2.6 and 3.1, we have that, for any 1/3<𝑠<1/2sup_{[]𝑡∈0,𝑇}𝐻𝐼𝑢^𝜆≲𝐻𝐼𝑢^𝜆₀+𝐿𝑁^3/2, (5.10) for our choice of 𝜆, 𝐻(𝐼𝑢^𝜆₀)≲1. Noting that 2/5>1/4, we can apply the Proposition 3.1. In order to guarantee that 𝐻𝐼𝑢^𝜆≲1 (5.11) for all 𝑡∈[0,𝑇], we require 𝐿≲𝑁^3/2. (5.12) Since 𝑇≤𝜆²𝑇₀, this is fulfilled as long as (2𝛿)⁴𝑁^1/2𝜆^2/3𝑇₀^1/3𝜇₀∼𝑁^3/2, (5.13) note (5.3), that is to say we have 𝑇₀^1/3(2𝛿)⁴𝜇₀∼𝑁^{(5𝑠−2)/3𝑠}. (5.14) If 𝑠>2/5, we have that 𝑇₀ is arbitrarily large if we send 𝑁 to infinity.

We use Theorem 4.5 to show that (5.6) is not true. Recall the estimates (4.22)–(4.24), we have 𝐼𝑢^𝜆4_{𝐿4𝑡𝐿4𝑥}≲𝑇^1/3sup_{𝑡∈[0,𝑇]}𝐼𝑢^𝜆2_̇𝐻1𝐼𝑢^𝜆2_𝐿2+𝑇^1/3𝑢^𝜆4_𝐿2+𝑇^1/3𝐿1𝑁≲𝑁^1/2𝑇^1/3, (5.15) which implies that 𝐼𝑢^𝜆_{𝐿4𝑡𝐿4𝑥([0,𝑇]×ℝ2)}≲𝑁^1/8𝑡^1/12. (5.16) This estimate contradicts to (5.6) for suitable choice of 𝛿 (namely, we choose 𝛿≫1). Therefore Γ=[0,𝜆²𝑇₀], and 𝑇₀ can be chosen arbitrarily large. In addition, 𝑢𝑇_0𝐻𝑠≲𝑢_0𝐿2+𝑢𝑇_0̇𝐻𝑠=𝑢_0𝐿2+𝜆^𝑠𝑢^𝜆𝜆²𝑇_0̇𝐻𝑠≲𝜆^𝑠𝐼𝑢^𝜆𝜆²𝑇_0𝐻1≲𝜆^𝑠≲𝑁^1−𝑠≲𝑇₀^{3𝑠(1−𝑠)/2(5𝑠−2)}. (5.17) Since 𝑇₀ is arbitrarily large, the priori bound on the 𝐻^𝑠 norm concludes the global well-posedness of the Cauchy Problem (1.14)-(1.15).

Acknowledgments

This work is supported by National Natural Science Foundation of China (11071177) and Fundamental Research Funds for the Central Universities (SWJTU12CX061 and SWJTU12ZT13).