# Global Analysis for Rough Solutions to the Davey-Stewartson System

Authors: Han Yang; Xiaoming Fan; Shihui Zhu
Journal: Abstract and Applied Analysis
 (2012)

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## 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 inHs with s > 2/5, then there exists a global solution in time, and the Hs 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 Lt,x4 estimate for nonlinear equation with the relatively general defocusing power nonlinearity.


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## Body

## 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)(1.1)iut+uxx+μuyy=a|u|2u+buvx,cvxx+vyy=(|u|2)x,
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
(1.2)elliptic-elliptic:μ>0,c>0,(1.3)elliptic-hyperbolic:μ>0,c<0,(1.4)hyperbolic-ellipti:μ<0,c>0,(1.5)hyperbolic-hyperbolic:μ<0,c<0.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 H5/2(ℝ2) whose L2 norm is sufficiently small. Tsutsumi [4] obtained the Lp-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 Lp-Lq 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 Hs(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),
(1.6)iut+uxx+uyy=a|u|2u+buvx,(1.7)vxx+vyy=(|u|2)x,(1.8)u(0,x)=u0(x).
As is well known, the system (1.6)–(1.8) enjoys two useful conservation laws: one is the energy conservation law:
(1.9)H(u,v)=∫ℝ2‍(|ux|2+|uy|2+a|u|4+b(vx2+vy2)2)dxdy,
and the other is the mass conservation law:
(1.10)M(u)=∫ℝ2‍|u|2dxdy.
For details one can see Ghidaglia and Saut [2]. Moreover, one can easily establishes
(1.11)∥u(T)∥Hs≤C(s,∥u0∥Hs,T)
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 denoteu,v as the solutions of (1.7)–(1.9). It follows from (1.8) that
(1.12)vx=ℱ-1ξ12|ξ|2ℱ|u|2,
where ℱ and ℱ-1 denote the Fourier transform and its inversion. For brevity we denote
(1.13)E(ψ)=ℱ-1ξ12|ξ|2ℱψ.
Combining (1.8) and (1.9), (1.6)–(1.8) are changed to
(1.14)iut+Δu=a|u|2u+bE(|u|2)u,(1.15)u(0,x)=u0(x).It is conjectured that the system (1.14)-(1.15) is globally well posed in Hs 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∈Hs(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 Hs(s>1) norm is similarly bounded, but if the initial data is only in Hs(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 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 be merely in Hs(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)=eitΔu0 is the linear group, Bourgain’s method shows addition that (St-S(t))u0∈H1(ℝ2) for all time provided u0∈Hs, s>3/5. Thus, he showed that the solution is globally well posed with initial data in Hs(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 Hs norm of the solution u(t) with the energy. In order to overcome this difficulty, we introduce the following multiplier IN:
(1.16)INf^(ξ):=m(ξ)f(ξ),
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)H(INu)=∫ℝ2‍(|INux|2+|INuy|2+(a|INu|4+bE(|INu|2)INu2)2)dxdy.
Note that if u(t)∈Hs(ℝ2) then H(INu)<∞. 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 ofs 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∈Hs(ℝ2), s>2/5 and a+b>0. Moreover, the solution satisfies the following estimate:
(1.18)supt∈[0,T]∥u(t)∥Hs≤C(1+T)(3s(1-s))/(2(5s-2))+,
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 Hs(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 Lx,t4 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 notationA≲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 use𝒮(ℝ4) to denote the Schwartz space and 𝒮′(ℝ4) to denote its topological dual space. We use Lxr(ℝ2) to denote the usual Lebesgue space of functions f: ℝ2→ℂ whose norm
(2.1)∥f∥Lxr:=(∫ℝ2‍|f|rdx)1/r
is finite, with the usual modification in the case r=∞. We also define the space-time space LtqLxr by
(2.2)∥u∥Lt∈JqLxr:=(∫J‍∥u∥Lxrqdt)1/q
for any space-time slab J×ℝ2, with the usual modification when either q or r are infinity. When q=r we abbreviate LtqLxq by Lt,xq.Definition 2.1.
A pair of exponent(q,r) is called admissible in ℝ2 if
(2.3)1q+1r=12,2≤q,r≤∞.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
(2.4)iut+Δu-f(x,t)=0,(t,x)∈J×ℝ2u(x,0)=u0(x).
Then one has the estimate
(2.5)∥u∥LtqLxr(J×ℝ2)≲∥u0∥L2(ℝ2)+∥f∥Ltq~′Lxr~′(J×ℝ2)
with the prime exponents denoting Hölder dual exponents.We also define the fractional differential operator|∇|α for any real α by
(2.6)|∇|αu^(ξ):=|ξ|αu^(ξ)
and analogously
(2.7)〈∇〉αu^(ξ):=〈ξ〉αu^(ξ),
where 〈a〉:=1+|a|2. We then define the homogeneous Sobolev space H˙s and the inhomogeneous Sobolev space Hs by
(2.8)∥u∥H˙s=∥|∇|su∥Lx2;∥u∥Hs=∥〈∇〉su∥Lx2.We also need some Littlewood-Paley theory. Specifically, letφ(ξ) be a smooth bump supported in |ξ|≤2 and equalling one on |ξ|≤1. For each number N∈2ℤ we define the Littlewood-Paley operators:
(2.9)P≤Nf^(ξ):=φ(ξN)f^(ξ),P>Nf^(ξ):=[1-φ(ξN)]f^(ξ),PNf^(ξ):=[φ(ξN)-φ(2ξN)]f^(ξ).
Similarly, we can define P<N,P≥N, and PM<·≤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 any1≤2≤q≤∞ and s>0, one has
(2.10)∥P≥Nf∥Lxp≲N-s∥|∇|sP≥Nf∥Lxp,∥|∇|sP≤Nf∥Lxp≲Ns∥P≤Nf∥Lxp,∥|∇|±sPNf∥Lxp~N±s∥PNf∥Lxp,∥P≤Nf∥Lxq≲N1/p-1/q∥P≤Nf∥Lxp,∥PNf∥Lxq≲N1/p-1/q∥PNf∥Lxp.We collect the basic properties ofIN into the following.Lemma 2.4.
Let1<p<∞ and 0≤s<1. Then
(2.11)∥INf∥Lp≲∥f∥Lp,(2.12)∥|∇|sP>Nf∥Lp≲Ns-1∥∇INf∥Lp,(2.13)∥f∥Hxs≲∥INf∥Hx1≲N1-s∥f∥Hxs.Proof.
For the proof one can see Colliander et al. [25].Now we define the Strichartz norm of functionsu(2.14)∥u∥ST0:=sup(q,r)admissible∥u∥Lt∈[0,T]qLxr.
Then we introduce the following bilinear smoothing property due to Bourgain [18].Lemma 2.5.
Letψ∈L2(ℝ2) such that
(2.15)ψ1=PNjψ,ψ2=PNkψ.
Then, for Nj≤Nk, the following inequality holds:
(2.16)∥eitΔψ1·eitΔψ2∥Lt,x2(ℝ2×ℝ)≤C(NjNk)(1/2)-∥ψ1∥2∥ψ2∥2.
That is to say, suppose u solves (1.15)–(1.18) on the time interval [0,T]. Let uj=PNju, for j=1,2 with N1>N2. Then
(2.17)∥u1u2∥LT2Lx2≤C(N2N1)(1/2)-∥u∥ST02.
The estimate (2.17) will be also valid if uj is replaced by uj¯.We also have the local well posedness result.Proposition 2.6.
Let us define quantity(2.18)μ([0,T]):=∫0T‍∫ℝ2‍|INu|4dxdt.
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:
(2.19)ZI([0,T]):=sup(q,r)admissible∥〈∇〉INu∥LtqLxr([0,t]×ℝ2)≲∥〈∇〉INu0∥L2.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
(2.20)∥〈∇〉IuN1〈∇〉IuN2∥LT2Lx2≤C(N2N1)(1/2)-∥〈∇〉Iu∥ST02.


## 3. Almost Conservation Laws

In this section we prove the almost conservation of the modified energyH(INu(t)).Proposition 3.1.
If the initial datau0∈Hs 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)supt∈[0,T]|H(INu(t))|≤|H(INu(0))|+CN-(3/2)-∥IN〈∇〉u(0)∥Lx24+CN-2-∥IN〈∇〉u(0)∥Lx26.
In particular when ∥IN〈∇〉u(0)∥Lx2≲1 one has
(3.2)supt∈[0,T]|H(INu(t))|≲N-(3/2)-.Proof.
In light of (2.19), it suffices to control the energy increment |H(INu(t))-H(INu(0))| for t∈[0,T] in terms of ZI([0,T]). Applying the IN operator to the system (1.14)-(1.15):
(3.3)iINut+ΔINu=aIN(|u|2u)+bIN(E(|u|2)u),INu(0,x)=INu0(x).
From now on, we abbreviate IN 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:
(3.4)H1=|∫0t∫ξ1+ξ2+ξ3+ξ4=0(1-m(ξ2+ξ3+ξ4)m(ξ2)m(ξ3)m(ξ4))ΔIu-^(ξ1)Iu^(ξ2)Iu-^(ξ3)Iu^(ξ4)|,(3.5)H2=|(a+b)∫0t‍∫ξ1′+ξ4+ξ5+ξ6=0‍(1-m(ξ1′)m(ξ4)m(ξ5)m(ξ6))I(|u|2u-)^(ξ1′)Iu^(ξ4)Iu-^(ξ5)Iu^(ξ6)|.
Here we used the properties of operator E for 1<p<∞: (i)E∈ℒ(Lp,Lp), where ℒ(Lp,Lp) denotes the space of bounded linear operators from Lp to Lp; (ii)ifu∈Hs,then E(u)∈Hs,s∈ℝ.
We estimateH1 first. We use uNj to denote PNju. When ξj is dyadically localized to {|ξ|~Nj} and we will write m(ξj) by mj. 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
≫
N
2
). 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.
N
2
≳
N
≫
N
3
>
N
4, for ξ1+ξ2+ξ3+ξ4=0, we have N1~N2. Applying the mean value theorem, we deduce that
(3.6)1-m(ξ2+ξ3+ξ4)m(ξ2)m(ξ3)m(ξ4)=m(ξ2)-m(ξ2+ξ3+ξ4)m(ξ2)≤∇m(ξ2)(ξ3+ξ4)m(ξ2)≤N3N2.
Moreover, H1 is controlled by
(3.7)N3N2(N3N1)1/2(N4N2)1/2〈N1〉〈N2〉〈N3〉〈N4〉∥Iu∥ZI([0,T])4≤N-(3/2)-(N1N2〈N3〉〈N4〉)0-ZI([0,T])4.
Case III:
(
N
2
≥
N
3
≥
N
). We have the bound on the symbol:
(3.8)|1-m(ξ2+ξ3+ξ4)m(ξ2)m(ξ3)m(ξ4)|≤m(ξ1)m(ξ2)m(ξ3)m(ξ4).
If N1~N2≥N3≳N, then we bounded H1 by renormalizing the derivatives and multiplier, paring uN1uN3 and uN2uN4 and using Lemma 2.5:
(3.9)m1m2m3m4(N3N1)1/2(N4N2)1/2〈N1〉〈N2〉〈N3〉〈N4〉ZI([0,T])4.
We write this bound as
(3.10)m1m2N21/2m3N31/2m4N41/2N11/2ZI([0,T])4.
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
(3.11)m1m2N21/2m3N31/2m4N41/2N11/2≤1N(3/2)+,
here we used the fact that miNi1/2≥m(N)N1/2=N1/2 for i=2,3. For s<1/2, by using the definition of m(ξ):
(3.12)m1m2N21/2m3N31/2m4N41/2N11/2≤N21/2-sN31/2-sN41/2-sN2-2sN11/2N11-s≤N11/2-2sN2-2sN11/2.
Using the facts 1/2>s>1/4 and N2≥N3≥N,
(3.13)N11/2-2sN2-2sN11/2≤N-(3/2)-.
Hence
(3.14)H1≤N-(3/2)-(N1N2N3〈N4〉)0-ZI([0,T])4.
IfN2~N3≥N, then paring uN1uN2 and uN3uN4 and using Lemma 2.5 again, a similar analysis leads to the bound:
(3.15)m1m2m3m4(N1N2)1/2(N4N3)1/2〈N1〉〈N2〉〈N3〉〈N4〉ZI([0,T])4≤N-3/2-(N1N2N3〈N4〉)0-ZI([0,T])4.
Now we turn to give the bound for the termH2. 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
≫
N
4
). On ξ1′+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.
(
N
4
≳
N
≥
N
5
). On ξ1′+ξ4+ξ5+ξ6=0, N123~N4 in this case. By the mean value theorem,
(3.16)|[1-m123m4m5m6]|=|m4-m456m4|≤N5N4.
Applying the Cauchy-Schwartz inequality and the above multiplier bound to (3.5), we deduce that
(3.17)(N4〈N5〉)-1N5N4∥P123I(uN1uN2uN3)IuN6∥Lt,x2∥I〈∇〉uN4I〈∇〉uN5∥Ltx2.
By Hölder inequality and Lemma 2.5, we control the above expression by
(3.18)(N4〈N5〉)-1(N5N4)3/2∥P123I(uN1uN2uN3)∥Lt,x2∥IuN6∥Lt,x∞ZI([0,t])2.
By the Sobolev’s inequality, we have
(3.19)∥IuN6∥Lt,x∞≤ZI([0,t]).
It follows from Colliander et al. in [26] that
(3.20)∥P123I(uN1uN2uN3)∥Lt,x2≤〈N1〉-1/2ZI([0,t])3.
We use (3.7)–(3.15) to complete the Case II analysis. H2 is bounded by
(3.21)(N4〈N5〉)-1(N5N4)3/2〈N1〉-1/2ZI([0,t])6≤N-2-(∏i=16‍〈Ni〉)0-ZI([0,t])6.
Case III
(
N
4
≥
N
5
≥
N
). We have the bound on the symbol
(3.22)|1-m123m4m5m6|≤m123m4m5m6.
Similar steps leads to the bound
(3.23)H2≤N-2-(∏i=16‍〈Ni〉)0-ZI([0,t])6.
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 Proposition3.1 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.
Ifβ 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:
(4.1)∫0t‍∫ℝ4‍(-ΔΔβ)|u(x,t)|2dxdt+∫0t‍∫ℝ4‍(Δβ)|u(x,t)|4dxdt≲supt∈[0,T]|Mβ(t)|,
where Ma(t) is the Morawetz action given by
(4.2)Mβ(t)=2∫ℝ2‍∇β(x)·𝒥(u-∇u)dx.Proof.
Sinceβ 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.
Letβ:ℝ4→ℝ be convex and u be a smooth solution to the equation:
(4.3)iut+Δu=𝒩.
Then, the following inequality holds:
(4.4)∫0t‍∫ℝ4‍(-ΔΔβ)|u(x,t)|2dxdt+2∫0t‍∫ℝ4‍∇β·{𝒩,u}pdxdt≲supt∈[0,T]|Mβ(T)-Mβ(t)|,
where Mβ(t) is the Morawetz action corresponding to u and {·}p is the momentum bracket defined by
(4.5){f,g}p=ℛ(f∇g¯-g∇f¯).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.
Letu be solution to the Cauchy problem of (1.14)-(1.15), then the following Lx,t4 space-time estimate holds
(4.6)∥u∥LT4Lx44≲T1/3supt∈[0,T]∥u∥Lt∞Lx23∥u∥Lt∞H˙x1+T1/3∥u∥Lt∞Lx24.Proof.
The proof of the Proposition4.3 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:
(4.7)iut+Δ4u=f(u),(t,x)∈[0,T]×ℝ4,
here
(4.8)f(u)=a|u1|2u+bE(|u1|2)u+a|u2|2u+bE(|u2|2)u,
and Δ4 is the Laplace in ℝ4=ℝ2×ℝ2.
Then we conclude that(4.9)∫0T‍∫ℝ2×ℝ2‍(-ΔΔβ(x1,x2))|u1(x1,t)|2|u2(x2,t)|2dx1dx2dt≲2sup[0,T]|Mβ⊗2(t)|,
where
(4.10)Mβ⊗2(t)=2∫ℝ2⊗ℝ2‍∇β(x)·𝒥(u1⨂u2¯(x)∇u1⨂u2(x))dx.
Note that the definition of β(x1,x2) implies
(4.11)-ΔΔβ(x1,x2)=2Mδx1=x2+O(1M3).
Thus
(4.12)∫0T‍∫ℝ2×ℝ2‍(-ΔΔβ(x1,x2))|u(x1,t)|2|u(x2,t)|2dx1dx2dt=2M∫0T‍∫ℝ4‍|u(x,t)|4dxdt+O(1M3)∫0T‍∫ℝ2×ℝ2‍|u(x1,t)|2|u(x2,t)|2dx1dx2dt.
It follows from the Fubuni’s theorem that
(4.13)CM3∫0T‍∫ℝ2×ℝ2‍|u(x1,t)|2|u(x2,t)|2dx1dx2dt≲CTM3∥u∥Lt∞Lx24.
On the other hand,
(4.14)sup[0,T]|Mβ⨂2(t)|≲supt∈[0,T]∥u∥Lt∞Lx23∥u∥Lt∞H˙x1.
Picking M~T1/3, we get
(4.15)∥u∥LT4Lx44≲T1/3supt∈[0,T]∥u∥Lt∞Lx23∥u∥Lt∞H˙x1+T1/3∥u∥Lt∞Lx24.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
(4.16)-ΔΔβ(x1,x2)=2Mδx1=x2+O(1M3).
This idea first appeared in [31].Now, we consider the solutionIu of
(4.17)iIut+ΔIu=aI(|u|2u)+bI(E(|u|2)u).
If Iu does not solve (4.7) but the following equation:
(4.18)iIut+ΔIu=a|Iu|2Iu+bE(|Iu|2)Iu,
then the calculations that we have done above would reveal that
(4.19)∥Iu∥LT4Lx44≲T1/3sup[0,T]∥Iu∥Lt∞Lx23∥Iu∥Lt∞H˙x1+T1/3∥Iu∥Lt∞Lx24,
of course this is not the case. We may rewrite (4.17) as
(4.20)iIut+ΔIu=a(|Iu|2Iu)+b(E(|Iu|2)Iu)+I(a|u|2u+b(E(|u|2)u))-a(|Iu|2Iu)-b(E(|Iu|2)Iu)=G(Iu)+(IG(u)-G(Iu)).
For what follows we abbreviate ui=u(xi) where ui is the solution of
(4.21)iut+Δu=a|u|2u+bE(|u|2)u,(t,xi)∈[0,T]×ℝ2.
We aim to prove the following theorem.Theorem 4.5.
LetIu be a solution to (4.7), then
(4.22)∥Iu∥LT4Lx44≲T1/3sup[0,T]∥Iu∥Lx23∥Iu∥H˙x1+T1/3∥u0∥Lx24+T1/3∫0T‍∫ℝ2×ℝ2‍∇β·{𝒩bad,Iu(x1,t)Iu(x2,t)}pdx1dx2dt,
where
(4.23)𝒩bad=∑i-12‍(IG(ui)-G(Iui))∏j=1,i≠j2‍Iuj.
In particular, on a time interval Tk where the local well-posedness Proposition 2.6 holds one has that
(4.24)∫Tk‍∫ℝ2×ℝ2‍∇β·{𝒩bad,Iu(x1,t)Iu(x2,t)}pdx1dx2dt≲1N1-ZI6(Jk).Proof.
According to Corollary4.2,
(4.25)∫0T‍∫ℝ2‍(-ΔΔβ)|Iu(x,t)|2dxdt≲2sup[0,T]|∫ℝ2‍∇β·𝒥(Iu¯(x)∇Iu)dx|+|∫0T‍∫ℝ2‍∇β·{IG(ui)-G(Iui)}pdxdt|.
Set
(4.26)IU=I⨂I(u(x1,t)⨂u(x2,t))=∏j=12‍Iu(xj,t).
If u solves (4.3) for n=2, then IU solves (4.3) for n=4, with right-hand side 𝒩I given by
(4.27)𝒩I=∑i=12‍(I(𝒩i)∏j=1,i≠j2‍Iuj).
Now we decompose 𝒩I 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)∥Iu∥LT4Lx44≲T1/3sup[0,T]∥Iu∥Lx23∥Iu∥H˙x1+T1/3∥u0∥Lx24+T1/3∫0T‍∫ℝ2×ℝ2‍∇β·{𝒩bad,Iu(x1,t)Iu(x2,t)}pdx1dx2dt,
where we have used the fact that ∥Iu∥L2≲∥u∥L2=∥u0∥L2. Remark that ∇β is a real valued, thus
(4.29)∇β·ℛ(f∇g--g∇f-)=ℛ(∇β·(f∇g--g∇f-))
and that ∇=(∇x1,∇x2). We now compute the dot product under the integral in (4.20), that is,
(4.30)ℛ{∑i=12‍∇xiβ(𝒩bad(∇xi(Iu1Iu2¯))-Iu1Iu2∇xi𝒩¯bad)}.
Recall that
(4.31)𝒩bad=∑i-12‍(IG(ui)-G(Iui))∏j=1,i≠j2‍Iuj.
Using the definition of 𝒩bad and the fact that ∇x1 acts only on Iu1, we have
(4.32)𝒩bad(∇xi(Iu1Iu2¯))-Iu1Iu2(∇xi𝒩¯bad)=[(I(𝒩1)-𝒩(Iu1))∇x1Iu1¯-∇x1(I𝒩1-𝒩-(Iu1))Iu1]|Iu2|2.
Analogously, we can see that the second part is given by(4.33)ℛ{∇x2β[(I(𝒩2)-𝒩(Iu2))∇x2Iu2¯-∇x2(I𝒩-2-𝒩-(Iu2))Iu2]|Iu1|2}.
We have
(4.34)∫0T‍∫ℝ4‍|∇x1β||I(𝒩1)-𝒩(Iu1)||∇x1Iu1||Iu2|2dx1dx2dt≤∫0T‍∫ℝ2‍|I(𝒩1)-𝒩(Iu1)||∇x1Iu1|dx1dt∥Iu∥Lt∞Lx22≤(∫0T‍∫ℝ2‍|I(𝒩1)-𝒩(Iu1)||∇x1Iu1|dxdt)ZI2≤∥I(𝒩1)-𝒩(Iu1)∥Lt1Lx2∥∇Iu∥Lt∞Lx2ZI2≤∥I(𝒩1)-𝒩(Iu1)∥Lt1Lx2ZI3.
Here we used the fact that the pair (∞,2) is admissible and |∇x1β|≲1. By a similar way we can deduce that
(4.35)∫0T‍∫ℝ4‍|∇x1β||∇x1I(𝒩1)-𝒩(Iu1)||Iu1||Iu2|2dx1dx2dt≤∥∇x(I(𝒩)-𝒩(Iu)∥Lt1Lx2ZI3.
Hence, we only need to estimate ∥∇x(I(𝒩)-𝒩(Iu)∥Lt1Lx2. Observe that 𝒩=a(|u|2u)+b(E(|u|2)u), and
(4.36)|∇x(I(𝒩)-𝒩(Iu)^(ξ))|≤C∫ξ=ξ1+ξ2+ξ3‍|ξ||m(ξ)-m(ξ1)m(ξ2)m(ξ3)|u^(ξ1)u^(ξ2)u^(ξ3)dξ1dξ2dξ3.
Using the fact that a+b>0 and the properties of operator E, we have
(4.37)∥∇x(I(𝒩)-𝒩(Iu))∥Lt1Lx2=∥∇x(I(𝒩)-𝒩(Iu))∥Lt1Lξ2≤C∑N1,N2,N3‍∥∫ξ=ξ1+ξ2+ξ3‍|ξ||m(ξ)-m(ξ1)m(ξ2)m(ξ3)|u^(ξ1)u^(ξ2)u^(ξ3)dξ1dξ2dξ3∥Lt1Lξ2=C∑N1,N2,N3‍∥∫ξ=ξ1+ξ2+ξ3‍|ξ||m(ξ)-m(ξ1)m(ξ2)m(ξ3)|m(ξ1)m(ξ2)m(ξ3)I^u(ξ1)I^u(ξ2)I^u(ξ3)dξ1dξ2dξ3∥Lt1Lξ2.
There is symmetry under interchange of the indices 1, 2, 3. We may assume that
(4.38)N1≥N2≥N3.
Let
(4.39)σ(ξ1,ξ2,ξ3)=|ξ||m(ξ)-m(ξ1)m(ξ2)m(ξ3)|m(ξ1)m(ξ2)m(ξ3).
We carry out a case by case analysis for (4.28).
Case I.
N
1
≪
N, this force σ(ξ1,ξ2,ξ3)=0, then there no contribution to (4.25).
Case II.
N
1
≳
N
≫
N
2, we have
(4.40)∥∫ξ=ξ1+ξ2+ξ3‍σ(ξ1,ξ2,ξ3)Iu^(ξ1)Iu^(ξ2)Iu^(ξ3)dξ1dξ2dξ3∥Lt1Lξ2=1N∥∫ξ=ξ1+ξ2+ξ3‍Nξ1ξ2σ(ξ1,ξ2,ξ3)∇Iu^(ξ1)∇Iu^(ξ2)Iu^(ξ3)dξ1dξ2dξ3∥Lt1Lξ2≲1N∥∇Iu1∥Lt3Lx6∥∇Iu2∥Lt3Lx6∥Iu1∥Lt3Lx6,
where we have used the fact that |ξ|~N1 and by mean value theorem that
(4.41)|m(ξ)-m(ξ1)m(ξ2)m(ξ3)|m(ξ1)m(ξ2)m(ξ3)≲N3N2.
Therefore,
(4.42)Nξ1ξ2σ(ξ1,ξ2,ξ3)≲Nξ1ξ2N2N1ξ≤1.
Case III
(
N
1
≥
N
2
≳
N
≫
N
3
). We also have
(4.43)∥∫ξ=ξ1+ξ2+ξ3‍σ(ξ1,ξ2,ξ3)Iu^(ξ1)Iu^(ξ2)Iu^(ξ3)dξ1dξ2dξ3∥Lt1Lξ2=1N∥∫ξ=ξ1+ξ2+ξ3‍Nξ1ξ2σ(ξ1,ξ2,ξ3)∇Iu^(ξ1)∇Iu^(ξ2)Iu^(ξ3)dξ1dξ2dξ3∥Lt1Lξ2≲1N∥∇Iu1∥Lt3Lx6∥∇Iu2∥Lt3Lx6∥Iu1∥Lt3Lx6.
Case IV
(
N
1
≥
N
2
≥
N
3
≳
N
). We estimate as follows:
(4.44)∥∫ξ=ξ1+ξ2+ξ3‍σ(ξ1,ξ2,ξ3)Iu^(ξ1)Iu^(ξ2)Iu^(ξ3)dξ1dξ2dξ3∥Lt1Lξ2≲1N2∏j=13‍∥∇Iuj∥Lt3Lx6,
where we have used the estimate
(4.45)σ(ξ1,ξ2,ξ3)N2ξ1ξ2ξ3≤N-1+3s(N1N2N3)-s≤1.
Finally, since the pair (3, 6) is admissible, we can get
(4.46)∥∇x(I(𝒩)-𝒩(Iu))∥Lt1Lx2≲1NZI3.
Combining (4.22), (4.24), and (4.40), we complete the proof Theorem 4.5.


## 5. The Proof of Theorem1.1

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
(5.1)uλ(x)=1λu(xλ,tλ2)
satisfies the same equation with initial data u0λ=(1/λ)u0(x/λ). This scaling preserves the L2 norms of u(t) as well as the Lt,x4 space-time norm. From estimate (2.13), we have
(5.2)∥Iu0λ∥H1≲N1-s∥u0λ∥Hs≲N1-sλ-s.
We can choose the parameter λ in the manner
(5.3)λ~N(1-s)/s,
so that ∥Iu0λ∥H1=O(1).Then, for anyT0 (arbitrarily large), define
(5.4)Γ={0<t<λ2T0∶∥Iuλ∥Lt4Lx4([0,t]×ℝ2)≤δN1/8t1/12},
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
(5.5)∥Iuλ∥Lt4Lx4([0,t]×ℝ2)
is a continuous function of time, there exist some T∈[0,λ2T0] with the properties,
(5.6)∥Iuλ∥Lt4Lx4([0,T]×ℝ2)≥δN1/8t1/12,(5.7)∥Iuλ∥Lt4Lx4([0,T]×ℝ2)≤2δN1/8t1/12.
Now we divide the time interval [0,T0] into subintervals Jk, k=1,…,L in such a way that
(5.8)∥Iuλ∥Lt4Lx4([0,Jk]×ℝ2)4≤μ0,
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
(5.9)L~(2δN1/8t1/12)4μ~(2δ)4N1/2T1/3μ.According to Propositions2.6 and 3.1, we have that, for any 1/3<s<1/2(5.10)supt∈[0,T]H(Iuλ)≲H(Iu0λ)+LN3/2,
for our choice of λ, H(Iu0λ)≲1. Noting that 2/5>1/4, we can apply the Proposition 3.1. In order to guarantee that
(5.11)H(Iuλ)≲1
for all t∈[0,T], we require
(5.12)L≲N3/2.
Since T≤λ2T0, this is fulfilled as long as
(5.13)(2δ)4N1/2λ2/3T01/3μ0~N3/2,
note (5.3), that is to say we have
(5.14)T01/3(2δ)4μ0~N(5s-2)/3s.
If s>2/5, we have that T0 is arbitrarily large if we send N to infinity.We use Theorem4.5 to show that (5.6) is not true. Recall the estimates (4.22)–(4.24), we have
(5.15)∥Iuλ∥Lt4Lx44≲T1/3supt∈[0,T]∥Iuλ∥H˙12∥Iuλ∥L22+T1/3∥uλ∥L24+T1/3L1N≲N1/2T1/3,
which implies that
(5.16)∥Iuλ∥Lt4Lx4([0,T]×ℝ2)≲N1/8t1/12.
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,
(5.17)∥u(T0)∥Hs≲∥u0∥L2+∥u(T0)∥H˙s=∥u0∥L2+λs∥uλ(λ2T0)∥H˙s≲λs∥Iuλ(λ2T0)∥H1≲λs≲N1-s≲T03s(1-s)/2(5s-2).
Since T0 is arbitrarily large, the priori bound on the Hs norm concludes the global well-posedness of the Cauchy Problem (1.14)-(1.15).



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*Source: 578701-2012-11-06.xml*