# GLOBAL SMOOTHING FOR THE DAVEY-STEWARTSON
SYSTEM ON $\mathbb{R}^{2}$

Engİn Başakoğlu<EMAIL_ADDRESS>Department of Mathematics,
Boğaziçi University, Bebek 34342, Istanbul, Turkey

###### Abstract.

In this paper we study the regularity properties of solutions to the Davey-
Stewartson system. It is shown that for initial data in a Sobolev space, the
nonlinear part of the solution flow resides in a smoother space than the
initial data for all times. We also obtain that the Sobolev norm of the
nonlinear part of the evolution grows at most polynomially. As an application
of the smoothing estimate, we study the long term dynamics of the forced and
weakly damped Davey-Stewartson system. In this regard we give a new proof for
the existence and smoothness of the global attractors in the energy space.

###### 2010 Mathematics Subject Classification:

35Q55, 35B41

The author was partially supported by the TÜBİTAK grant 118F152 and the
Boğaziçi University Research Fund grant BAP-14081.

##  1\. INTRODUCTION

The Davey-Stewartson (DS) equations in dimensionless are given by couple of
equations of the form

(1)
$\displaystyle\begin{cases}i\partial_{t}u+c_{0}\partial_{x}^{2}u+\partial_{y}^{2}u=c_{1}|u|^{2}u+c_{2}u\partial_{x}\phi\\\
\partial_{x}^{2}\phi+c_{3}\partial_{y}^{2}\phi=\partial_{x}(|u|^{2})\end{cases}$

where $u=u(x,y,t)$ is complex valued and $\phi=\phi(x,y,t)$ is real valued
functions that represent amplitude and mean velocity potential respectively,
here the constants are real numbers and their signs determine the character of
the equation. The systems (1) were first derived by Davey and Stewartson [4],
Benney and Roskes [1], Djordjevic and Redekopp [5] and model the time
evolution of $2$D surface of water waves that propagate predominantly in one
direction but the wave amplitude is modulated slowly in the horizontal
directions. In [5], Djordjevic and Redekopp showed that the parameter $c_{3}$
can be negative when capillary effects are important. According to the signs
of $c_{0}$ and $c_{3}$ respectively, the system (1) is classified as follows

Elliptic-Elliptic | Elliptic-Hyperbolic | Hyperbolic-Elliptic | Hyperbolic-Hyperbolic
---|---|---|---
$(+,+)$ | $(+,-)$ | $(-,+)$ | $(-,-)$

DS systems are very well studied in terms of well-posedness and stability,
blow-up profiles, existence of standing and travelling waves. The
investigation of the system (1) in terms of well-posedness was initiated by
Ghidaglia and Saut [11] who established the local well-posedness in the
elliptic-elliptic, elliptic-hyperbolic and hyperbolic-elliptic cases. More
precisely, they studied the local and global properties of the elliptic-
elliptic and the hyperbolic-elliptic cases in $L^{2},H^{1},H^{2}$ also in the
elliptic-hyperbolic case, they obtained a global existence of weak solution of
(1) under smallness assumption for data in $L^{2}$. Linares and Ponce [19]
showed that under some smallness assumptions on the data, elliptic-hyperbolic
and hyperbolic-hyperbolic cases of (1) are locally well-posed in the spaces
$H^{s}(\mathbb{R}^{2})\cap H^{6}(\mathbb{R}^{2}:r^{6}dxdy)$ for $s\geq 12$,
and $H^{s}(\mathbb{R}^{2})\cap H^{3}(\mathbb{R}^{2}:r^{2}dxdy)$ for $s\geq 6$
respectively. As regards to the initial value problem posed on the $2$-torus
Godet [13] obtained a local well-posedness result for the hyperbolic-elliptic
problem in $H^{s}(\mathbb{T}^{2})$ for $s>1/2$ as well as a blow-up rate for
this equation. Concerning the half-plane problem, Fokas [10] studied the DS
equation on the half-plane by using the inverse scatering transform techniques
along with the formulation of a $d$-bar problem for a sectionally non-analytic
function. As for the problem of global well-posedness, it is conjectured that
the elliptic-elliptic type of (1) is globally well-posed in $H^{s}$ for all
$s\geq 0$. Towards this conjecture, Shen and Guo [20] proved that the initial
value problem of (1) in the elliptic-elliptic case (with some assumptions on
the constants) is globally well-posed for data in $H^{s}(\mathbb{R}^{2})$, for
$s>4/7$. Thereafter, Yang et al. [25] improved this result by establishing
global well-posedness in $H^{s}(\mathbb{R}^{2})$ for $s>2/5$ where they took
advantage of the $I$-method. In particular they obtained a polynomial in time
bound for the $H^{s}$ norm of the solution for $s>2/5$. Some of the other
results regarding the system (1) can be found in [14, 15, 16, 17, 22]. Upon
considering elliptic-elliptic type of the system (1) we will study the initial
value problem

(2) $\displaystyle\begin{cases}i\partial_{t}u+\Delta
u=c_{1}|u|^{2}u+c_{2}u\partial_{x}\phi,\ \ (x,y)\in\mathbb{R}^{2},\
t\in\mathbb{R},\\\
\partial_{x}^{2}\phi+\partial_{y}^{2}\phi=\partial_{x}(|u|^{2})\\\
u(x,y,0)=u_{0}(x,y)\in H^{s}(\mathbb{R}^{2}).\end{cases}$

To reformulate (2) in a better form we implement the Fourier transform in the
spatial variable to the second equation of (2) so that the system reduces to a
single equation

$\displaystyle i\partial_{t}u+\Delta u=c_{1}|u|^{2}u+c_{2}K(|u|^{2})u$

where $K$ is the pseudo-differential operator with symbol $\alpha$ given by

$\displaystyle\widehat{K(f)}(\xi)=\alpha(\xi)\hat{f}(\xi)\ \text{and}\
\alpha(\xi)=\frac{\xi_{1}^{2}}{|\xi|^{2}},\ \xi=(\xi_{1},\xi_{2})\neq(0,0).$

Therefore by the Duhamel formula, the equation (2) is equivalent to

(3) $\displaystyle
u(x,y,t)=e^{it\Delta}u_{0}-i\int_{0}^{t}e^{i(t-\tau)\Delta}(c_{1}|u|^{2}u+c_{2}K(|u|^{2})u)(\tau)d\tau$

where $e^{it\Delta}$ denotes the free solution operator of the corresponding
linear problem for the equation (2). In this paper our primary goal is to
obtain smoothing properties of (2) globally in time, therefore in order to
take advantage of global well-posedness result of Theorem 2.2 below, we will
assume the restriction that $c_{1}+c_{2}>0$. In the absence of this
restriction, smoothing argument here works in the local sense only. The
smoothing in this text means that the nonlinear part of the solution flow in
relation to (3) lies in a more regular space than the initial data belongs to.
To make it rigorous, below we state our result:

######  Theorem 1.1.

Let $c_{1}+c_{2}>0$ . Fix $s>\frac{1}{2}$ and
$a<\min(\frac{1}{2},s-\frac{1}{2})$. Consider the solution to IVP (2) on
$\mathbb{R}^{2}\times\mathbb{R}$ with data $u_{0}\in H^{s}(\mathbb{R}^{2})$.
Suppose that there is an a priori growth bound $\left\lVert
u(t)\right\rVert_{H^{s}}\leq C(\left\lVert u_{0}\right\rVert_{H^{s}})\langle
t\rangle^{\beta(s)}$ for some $\beta(s)$. Then

(4) $\displaystyle u(x,y,t)-e^{it\Delta}u_{0}\in C_{t}^{0}H^{s+a}_{x,y}$

furthermore, we have the growth bound

$\displaystyle\left\lVert u(t)-e^{it\Delta}u_{0}\right\rVert_{H^{s+a}}\leq
C(s,a,\left\lVert u_{0}\right\rVert_{H^{s}})\langle
t\rangle^{1+\beta(s)(3+\frac{2}{s})}.$

Let $I$ denote the identity operator and $K$ denote the multiplier operator
introduced as above, hence to be able to prove the Theorem 1.1 we will need
the key trilinear estimate:

######  Proposition 1.2.

For $s>\frac{1}{2}$, $a<min(\frac{1}{2},s-\frac{1}{2})$ and
$b=\frac{1}{2}+\epsilon$ for $\epsilon>0$ sufficiently small, we have

(5)
$\displaystyle||(c_{1}I+c_{2}K)(u\overline{v})w||_{X^{s+a,b-1}}\lesssim||u||_{X^{s,b}}||v||_{X^{s,b}}||w||_{X^{s,b}}.$

Note that we can replace the $X^{s,b}$ norm in the above proposition with the
time restricted version of this norm.

In the remainder part, we present an application of the smoothing estimate–an
alternative proof of the existence of a global attractor for the forced and
damped Davey-Stewartson system in the energy space $H^{1}$. In [24], Wang and
Guo obtained the existence of a global attractor in $H^{1}$ yet with a proof
based on the splitting argument that requires more regular initial data to
reach the compactness. The smoothing effect replaces the splitting method of
[24], as a result simplifying the proof and also as a byproduct, gives us that
the global attractor is indeed a compact subset of $H^{\frac{3}{2}-}$. Thus we
consider the forced and damped Davey-Stewartson system

(6) $\displaystyle\begin{cases}iu_{t}+\Delta u+i\delta
u=c_{1}|u|^{2}u+c_{2}u\partial_{x}\phi+f,\,\,(x,y)\in\mathbb{R}^{2}\\\
\Delta\phi=\partial_{x}(|u|^{2})\end{cases}$

where the forcing term $f\in L^{2}(\mathbb{R}^{2})$ is time independent,
$\delta>0$, $c_{1}\geq 0$ and $c_{1}+c_{2}\geq 0$. The use of Theorem 2.5
below which is provided by a smoothing estimate for the dissipative problem
(6) yield the following result:

######  Theorem 1.3.

Consider the forced and weakly damped Davey-Stewartson system (6) on
$\mathbb{R}^{2}\times[0,\infty)$ with the initial data $u(x,0)=u_{0}(x)\in
H^{1}(\mathbb{R}^{2})$. Then the equation (6) possesses a global attractor in
$H^{1}(\mathbb{R}^{2})$. Furthermore, for any $a\in(0,\frac{1}{2})$ the global
attractor is a compact subset of $H^{1+a}(\mathbb{R}^{2})$.

We now briefly explain the organization of the paper. In section 2, we
introduce the notations, function spaces and necessary tools for the proof of
the Theorem 1.1. Also as regards to a global attractor of the equation (6), we
give necessary definitions and a compactness theorem. In section 3, we discuss
Tao’s $[k;Z]$ multiplier method in order to prove the key trilinear estimate
(5). The proof of (5) will be given in Section 4. In Section 5, we prove the
Theorem 1.1 and finally Section 6 is devoted to prove the existence of a
global attractor for the Davey-Stewartson system.

##  2\. Notation & Preliminaries

Throughout the text we use $A\lesssim B$ to denote an estimate $A\leq CB$ for
some absolute constant C. We write $A\sim B$ if $A\lesssim B$ and $B\lesssim
A$. We also write $A\ll B$ to denote an estimate $A\leq\frac{1}{C}B$ for some
sufficiently large constant $C$. Moreover we define
$\langle\cdot\rangle=\sqrt{1+|\cdot|^{2}}$. For $s,b\in\mathbb{R}$, we require
$X^{s,b}$ spaces corresponding to the evolution $u$ of the DS system that is
defined by means of the norm

$\left\lVert
u\right\rVert_{X^{s,b}}=\left\lVert\langle\xi\rangle^{s}\langle\tau+|\xi|^{2}\rangle^{b}\widehat{u}(\xi,\tau)\right\rVert_{L^{2}_{\xi,\tau}}.$

Localized $X^{s,b}$ space is also defined by

$\left\lVert
u\right\rVert_{X^{s,b}_{\delta}}=\inf_{\tilde{u}=u,\,|t|\leq\delta}\left\lVert\tilde{u}\right\rVert_{X^{s,b}}.$

For a general discussion, let $\varphi$ be the smooth function satisfying
$\varphi(t)=0$ if $|t|\geq 2$, and $\varphi(t)=1$ for $|t|\leq 1$. Also let
$\varphi_{\delta}(t)=\varphi(t/\delta)$ for $0<\delta\leq 1$. Consider the
Cauchy problem

(7) $\displaystyle\begin{cases}iu_{t}+h(\frac{1}{i}\nabla)=F(u)\\\
u(0)=u_{0}\end{cases}$

where $h$ is a real polynomial on $\mathbb{R}^{d}$ and $F$ is a non-linear
function. Recall that in order to solve the equation (7) locally in time, say,
in the interval $[-\delta,\delta]$, we write an integral equation
corresponding to (7) as

$\displaystyle u(t)$
$\displaystyle=\varphi(t)W(t)u_{0}+i\varphi_{\delta}(t)\int_{0}^{t}W(t-s)F(u(s))ds$
$\displaystyle=:\varphi(t)W(t)u_{0}+i\varphi_{\delta}(t)[W*_{t}F(u)](t)$

where $W(t)=exp(ith(\frac{1}{i}\nabla))$ is the group of unitary operators
that solves the corresponding linear equation and $*_{t}$ denotes the
convolution with respect to the time variable restricted to the interval
$[0,t]$. With these notations we have the following lemma which will be
important in attaining Theorem 1.1.

######  Lemma 2.1 (See [12].).

Let $-\frac{1}{2}<b^{\prime}\leq 0\leq b\leq b^{\prime}+1$. Then

$\left\lVert\varphi_{\delta}(t)W*_{t}F(u)(t)\right\rVert_{X^{s,b}}\lesssim\delta^{1-b+b^{\prime}}\left\lVert
F\right\rVert_{X^{s,b^{\prime}}_{\delta}}.$

Consider the local problem (2) with the local existence time $\delta$. We will
quantify the dependence of $\delta$ to an initial data which we use in the
proof of Theorem 1.1. From the dilation symmetry of this equation, assuming
that $(u,\phi)$ solve (2) with initial data $u_{0}$ on $[0,\lambda^{-2}]$, we
come up with the symmetry solutions

$u^{\lambda}(x,y,t)=\lambda^{-1}u(x/\lambda,y/\lambda,t/\lambda^{2}),\,\,\phi^{\lambda}(x,y,t)=\lambda^{-1}\phi(x/\lambda,y/\lambda,t/\lambda^{2})$

with data $u^{\lambda}_{0}(x,y)=\lambda^{-1}u_{0}(x/\lambda,y/\lambda)$ which
solve the equation on $[0,1]$. Thus for $\lambda>1$, by comparing the $H^{s}$
norms of $u_{0}$ and $u_{0}^{\lambda}$, the solution $(u,\phi)$ can be defined
with respect to the local existence time

(8) $\displaystyle\delta\sim(C+\left\lVert
u_{0}\right\rVert_{H^{s}})^{-\frac{2}{s}}$

where $C=C(\left\lVert u_{0}\right\rVert_{L^{2}})$.

Next we discuss a global wellposedness result for the elliptic-elliptic
problem. We will exploit it once iterating our local result. Note that
recalling the Sobolev index range $s>1/2$ in our case, we indeed need a result
that at least covers this range. Hence rewriting the elliptic-elliptic IVP

(9) $\displaystyle\begin{cases}iu_{t}+\Delta
u=c_{1}|u|^{2}u+c_{2}K(|u|^{2})u\\\ u(x,0)=u_{0}(x)\in
H^{s}_{x}(\mathbb{R}^{2}),\end{cases}$

with a multiplier operator $K$ given by

(10) $\displaystyle
K(f)=\mathcal{F}^{-1}\frac{\xi_{1}^{2}}{|\xi|^{2}}\mathcal{F}f$

for $\xi=(\xi_{1},\xi_{2})\neq 0$, the required global well-posedness result
for (9) and (10) with a growth bound is as follows:

######  Theorem 2.2 (See [25]).

Let $c_{1}+c_{2}>0.$ For any $1>s>\frac{2}{5}$, the initial value problem (9)
$\&$ (10) is globally well-posed in $H^{s}(\mathbb{R}^{2})$. Furthermore,
there is a growth bound

$\sup_{t\in[0,T]}\left\lVert u(t)\right\rVert_{H^{s}(\mathbb{R}^{2})}\leq
C(1+T)^{\frac{3s(1-s)}{2(5s-2)}+}$

where the constant $C$ depends only on the index $s$, $\left\lVert
u_{0}\right\rVert_{L^{2}}$.

The essential ingredients of the proof of Theorem 2.2 are the interaction
Morawetz type estimate and the almost conservation of the modified energy
obtained from plugging the smoothing of order $1-s$ operator
$\widehat{Iu}(\xi)=m(\xi)\widehat{u}(\xi)$,

$\displaystyle m(\xi):=\begin{cases}\hskip 5.69046pt1,\hskip
42.67912pt|\xi|<N\\\ \Big{(}\frac{|\xi|}{N}\Big{)}^{s-1},\hskip
13.37282pt|\xi|>2N\end{cases}$

in the usual Hamiltonian energy of the equation (9). The globalizing technique
used in [25] is called the $I$-method, in view of the operator
$I:H^{s}\rightarrow H^{1}$, which was introduced by
Colliander–Keel–Staffilani–Takaoka–Tao [2]. Next by virtue of Theorem 2.2, we
reserve the following growth bound to be used later. For $s>\frac{1}{2}$,
define $\beta(s)\geq\frac{3s(1-s)}{2(5s-2)}$ so that we have the a priori
estimate

(11) $\left\lVert u(t)\right\rVert_{H^{s}}\lesssim\langle
t\rangle^{\beta(s)}=:T(t)$

for some non-decreasing function $T(t)$ (which we need in iterating the local
result), and where the implicit constant depends on the Sobolev index and
$L^{2}$ norm of the initial data.

We now consider the forced and weakly damped Davey–Stewartson equation (6)
that can be reduced to the single equation

(12) $\displaystyle iu_{t}+\Delta u+i\delta
u=c_{1}|u|^{2}u+c_{2}K(|u|^{2})u+f$

with the same multiplier operator $K$ as in (10). Below we list a few
properties of $K$ to be used in the energy calculations of (12):

  1. (i)

$K$ is a bounded linear operator on $L^{p}$, $1<p<\infty$,

  2. (ii)

$\overline{K(\psi)}=K(\overline{\psi})$,

  3. (iii)

$\int K(\psi)\varphi=\int\psi K(\varphi)$.

As a consequence of smoothing estimate, we will study the existence of global
atractors for the dissipative DS equation. For a general discussion, let $H$
be a phase space, and assuming the existence of a global wellposed problem in
$H$, we also let $U(t):H\rightarrow H$ denote the evolution operator, mapping
data to solution. Our goal is to describe the long time asymptotics of the
solution by a compact invariant subset of $H$. The following definitions and
result will give the features of this invariant subset which we need in
establishing the proof of Theorem 1.3.

######  Definition 2.3 ([21]).

We say that a compact subset $\mathcal{A}$ of the phase space $H$ is a global
attractor for the semigroup $\\{U(t)\\}_{t\geq 0}$ if
$U(t)\mathcal{A}=\mathcal{A}$ for all $t>0$, and

$\lim\limits_{t\rightarrow\infty}d(U(t)g,\mathcal{A})=0,\,\,\forall g\in H.$

Next we define a bounded subset of a phase space which is of particular
importance, called an absorbing set, and all solutions enter into this set
after a certain time. The existence of this set can be verified by using the
energy estimates for the equation.

######  Definition 2.4 ([21]).

A bounded subset $\mathcal{B}$ of a phase space $H$ is called an absorbing set
if for any bounded $\mathcal{S}\subset H$, there exists $T=T(\mathcal{S})$
such that $U(t)\mathcal{S}\subset\mathcal{B}$ for all $t\geq T$.

Note that a global attractor for a semigroup, if exists, implies the existence
of an absorbing set whereas the converse of this argument holds still true
whenever a semigroup $\\{U(t)\\}_{t\geq 0}$ satisfies certain criterion:

######  Theorem 2.5 ([21]).

Suppose that $H$ is a metric space and $U(t):H\rightarrow H$ is a continuous
semigroup defined for all $t\geq 0$. Furthermore suppose that there is an
absorbing set $\mathcal{B}$. If the semigroup $\\{U(t)\\}_{t\geq 0}$ is
asymptotically compact, that is, for every bounded sequence $\\{x_{k}\\}$ in
$H$ and every sequence of times $t_{k}\rightarrow\infty$, the set
$\\{U(t_{k})x_{k}\\}_{k}$ is relatively compact in $H$, then the
$\omega$-limit set

$\omega(\mathcal{B})=\bigcap\limits_{s\geq 0}\overline{\bigcup\limits_{t\geq
s}U(t)\mathcal{B}}$

is a global attractor, where the closure is taken on $H$.

By the above theorem, the problem of existence of a global attractor boils
down to finding an absorbing ball with which one needs to prove asymptotic
compactness.

The theory of existence of global attractors is only meaningful for systems as
$t\rightarrow\infty$. Thus to see that the equation (6) is globally wellposed
in the energy space, we can proceed in a similar fashion to the proof of the
existence of an absorbing set in section $2$ of [24] to obtain the following a
priori estimate:

(13) $\displaystyle\left\lVert u(t)\right\rVert_{H^{1}}\leq
Ae^{-Bt}+C,\,\,t>0,$

where $A=A(\left\lVert f\right\rVert_{L^{2}},\left\lVert
u_{0}\right\rVert_{H^{1}})$, $B=B(\delta)>0$, $C=C(\delta,\left\lVert
f\right\rVert_{L^{2}})$. Note that (13) implies the existence of an absorbing
ball with radius $C=C(\delta,\left\lVert f\right\rVert_{L^{2}})$ as well.

##  3\. Main Method

In this section we discuss Tao’s $[k;Z]$ multipliers method [23] so as to
study the estimate of multilinear expression associated to the elliptic-
elliptic type of Davey-Stewartson system. Suppose that $Z$ is any additive
abelian group with an invariant measure $d\xi$, as an example, one can take
$Z=\mathbb{R}^{n+1}$ with Lebesgue measure or $\mathbb{Z}^{n}\times\mathbb{R}$
with the product of counting and Lebesgue measures. Let $k\geq 2$ be any
integer, and let $\Gamma_{k}(Z)\subset Z^{k}$ denote the hyperplane

$\Gamma_{k}(Z):=\\{(\xi_{1},...,\xi_{k})\in Z^{k}:\xi_{1}+...+\xi_{k}=0\\}$

endowed with the measure

$\int_{\Gamma_{k}(Z)}f=\int_{Z^{k-1}}f(\xi_{1},...,\xi_{k-1},-\xi_{1}-...-\xi_{k-1})\,\text{d}\xi_{1}...\text{d}\xi_{k-1}.$

######  Definition 3.1.

A complex valued function $m:\Gamma_{k}(Z)\rightarrow\mathbb{C}$ is called the
$[k;Z]$ multiplier if the inequality

$\displaystyle\Big{|}\int_{\Gamma_{k}(Z)}m(\xi_{1},...,\xi_{k})\prod_{i=1}^{k}f_{i}(\xi_{i})\Big{|}\leq\left\lVert
m\right\rVert_{[k;Z]}\prod_{i=1}^{k}\left\lVert f_{i}\right\rVert_{L^{2}(Z)}$

holds for all test functions $f_{i}$ on $Z$ and the best constant, denoted by
$\left\lVert m\right\rVert_{[k;Z]}$.

Note that $\left\lVert\cdot\right\rVert_{[k;Z]}$ determines a norm on the
$[k;Z]$ multipliers $m$. Since multilinear estimates might boil down to
bilinear estimates of some sort (reason later), we emphasize the case $k=3$
which specifically associates to the bilinear estimates for our Schrödinger
type equation. So we write

(14) $\displaystyle\xi_{1}+\xi_{2}+\xi_{3}=0,\ \ \ \
\tau_{1}+\tau_{2}+\tau_{3}=0,$ (15)
$\displaystyle\lambda_{j}:=\tau_{j}+h_{j}(\xi_{j}),\ \ \ \
h_{j}(\xi_{j})=\pm|\xi_{j}|^{2}\ \ \text{for}\ \ 1\leq j\leq 3.$

Here $\lambda_{j}$ measures how close in frequency the $j^{th}$ wave is to a
free solution. It is remarkable to note that for $d\geq 2$, whenever the two
of the frequencies $\xi_{1},\xi_{2},\xi_{3}$ form an orthogonal pair,
$\lambda_{j}$’s simultaneously vanish. Thus in order to take care of this
situation we introduce the function
$h:\Gamma_{3}(\mathbb{R}^{2})\rightarrow\mathbb{R}$ defined by

(16) $\displaystyle
h(\xi_{1},\xi_{2},\xi_{3}):=h_{1}(\xi_{1})+h_{2}(\xi_{2})+h_{3}(\xi_{3})=\lambda_{1}+\lambda_{2}+\lambda_{3}$

which measures to what extent the frequencies $\xi_{1},\xi_{2},\xi_{3}$
resonate with each other so let it be referred as a resonance function. The
domain on which the resonance function may vanish depends on the sign of
Schrödinger dispersion relation $h_{j}(\xi_{j})=\pm|\xi_{j}|^{2}$. Up to
symmetry we have two possibilities: the $(+++)$ case

(17) $\displaystyle h_{1}(\xi)=h_{2}(\xi)=h_{3}(\xi)=|\xi|^{2}$

and the $(++-)$ case

(18) $\displaystyle h_{1}(\xi)=h_{2}(\xi)=|\xi|^{2},h_{3}(\xi)=-|\xi|^{2}.$

Comparing the two cases, analysis of the first one is rather easier since the
resonance function

$\displaystyle
h(\xi_{1},\xi_{2},\xi_{3})=|\xi_{1}|^{2}+|\xi_{2}|^{2}+|\xi_{3}|^{2}$

vanishes only at the origin. As for the second case, the resonance function

$\displaystyle
h(\xi_{1},\xi_{2},\xi_{3})=|\xi_{1}|^{2}+|\xi_{2}|^{2}-|\xi_{3}|^{2}$

can vanish whenever the frequencies $\xi_{1}$ and $\xi_{2}$ become orthogonal.
More precisely, for $(\xi_{1},\xi_{2},\xi_{3})\in\Gamma_{3}(\mathbb{R}^{2})$,
we write

$\displaystyle|h(\xi_{1},\xi_{2},\xi_{3})|=||\xi_{1}|^{2}+|\xi_{2}|^{2}-|\xi_{3}|^{2}|=2|\xi_{1}\cdot\xi_{2}|\sim|\xi_{1}||\xi_{2}||\pi/2-\angle(\xi_{1},\xi_{2})|$

where $\angle(\xi_{1},\xi_{2})$ denotes the angle between $\xi_{1}$ and
$\xi_{2}$. So the extent to which $\xi_{1}$ and $\xi_{2}$ get closer to being
orthogonal, the more rapidly resonance function tends to vanish. At this point
we assume that

(19) $\displaystyle|h(\xi_{1},\xi_{2},\xi_{3})|\lesssim|\xi_{1}||\xi_{2}|,$

and

(20)
$\displaystyle\angle(\xi_{1},\xi_{2})=\pi/2+O(\frac{|h(\xi_{1},\xi_{2},\xi_{3})|}{|\xi_{1}||\xi_{2}|}).$

We will estimate the $[3;\mathbb{R}^{3}]$ norm of multipliers by making use of
the dyadic decomposition of the variables $\xi_{j}$, $\lambda_{j}$ and the
function $h(\xi_{1},\xi_{2},\xi_{3})$. Thus we use the capitalized variables
$N_{j}$, $L_{j}$ and $H$ to denote the magnitude of the pieces into which the
variables $\xi_{j}$, $\lambda_{j}$ and the resonance function $h$ decomposed
respectively. Here these variables are assumed to be dyadic, that is, they
range over the numbers of the form $2^{k}$, $k\in\mathbb{Z}$. In terms of
sizes of the variables $N_{j}>0$, $j=1,2,3$, we write $N_{\min}\leq
N_{\text{med}}\leq N_{\max}$ to denote the minimum, median and maximum of
$N_{1},N_{2},N_{3}$. This, in its own right, saves us from repetitive analysis
and reduces the number of cases substantially. Likewise we define
$L_{\min}\leq L_{\text{med}}\leq L_{\max}$ for $L_{j}>0$, $j=1,2,3.$ Next we
make some assumptions on the sizes of these variables. Before doing so, we
need to state several lemmas from [23].

######  Lemma 3.2 (Comparison Principle).

Let $m$ and $M$ be $[k;Z]$ multipliers, if for all $\xi\in\Gamma_{k}(Z)$
$|m(\xi)|\leq M(\xi)$, then $\left\lVert m\right\rVert_{[k;Z]}\leq\left\lVert
M\right\rVert_{[k;Z]}$. Furthermore if $a_{1},...,a_{k}$ are real valued
functions on $Z$ and $m$ is a $[k;Z]$ multiplier, then

$\displaystyle\left\lVert
m(\xi)\prod_{i=1}^{k}a_{i}(\xi_{i})\right\rVert_{[k;Z]}\leq\left\lVert
m\right\rVert_{[k;Z]}\prod_{i=1}^{k}\left\lVert
a_{i}\right\rVert_{L^{\infty}(Z)}.$

######  Lemma 3.3.

For any $\xi_{0}\in\Gamma_{k}(Z)$ and any $[k;Z]$ multiplier $m$, we have the
translation invariance of the norm

(21) $\displaystyle\left\lVert m(\xi+\xi_{0})\right\rVert_{[k;Z]}=\left\lVert
m(\xi)\right\rVert_{[k;Z]}$

also we have the averaging estimate

(22) $\displaystyle\left\lVert m*\mu\right\rVert_{[k;Z]}\leq\left\lVert
m\right\rVert_{[k;Z]}\left\lVert\mu\right\rVert_{L^{1}(\Gamma_{k}(Z))}$

for any finite measure $\mu$ on $\Gamma_{k}(Z).$

###### Proof.

Translation invariance immediately follows by the definition of a $[k;Z]$
multipliers. For the averaging estimate, for all test functions $f_{i}$,
bearing in mind that $\mu$ is a finite measure on $\Gamma_{k}(Z)$ and then
application of Minkowski integral inequality with $p=1$ leads to

$\Big{|}\int_{\Gamma_{k}(Z)}m*\mu(\xi)\prod_{i=1}^{k}f_{i}(\xi_{i})\Big{|}=\Big{|}\int_{\Gamma_{k}(Z)}\Big{(}\int_{\Gamma_{k}(Z)}m(\xi-\eta)\,\text{d}\mu(\eta)\Big{)}\prod_{i=1}^{k}f_{i}(\xi_{i})\Big{|}\\\
\leq\Big{|}\int_{\Gamma_{k}(Z)}\Big{(}\int_{\Gamma_{k}(Z)}m(\xi-\eta)\prod_{i=1}^{k}f_{i}(\xi_{i})\Big{)}\,\text{d}\mu(\eta)\Big{|}$

Next since $m$ is a $[k;Z]$ multipliers we estimate the inner integral by
using the definition, and hence the total integral is majorized by

$\displaystyle\prod_{i=1}^{k}\left\lVert
f_{i}\right\rVert_{L^{2}(Z)}\int_{\Gamma_{k}(Z)}\left\lVert
m(\xi-\eta)\right\rVert_{[k;Z]}\,\text{d}\mu(\eta)=\left\lVert
m(\xi)\right\rVert_{[k;Z]}\left\lVert\mu\right\rVert_{L^{1}(\Gamma_{k}(Z))}\prod_{i=1}^{k}\left\lVert
f_{i}\right\rVert_{L^{2}(Z)}$

where we make use of (21) in the last line.

∎

It is remarkable that by the finiteness assumption of the measure $\mu$, the
mapping $m\rightarrow m*\mu$ can be regarded as an averaging of $m$. Therefore
by an averaging over unit time scales, implementation of the lemmas above
allow us to restrict the multiplier $m(\xi_{1},\xi_{2},\xi_{3})$ to the region

$\displaystyle|\lambda_{j}|\gtrsim 1,\ \ j=1,2,3.$

Furthermore since we deal with a multiplier with no singularities for
$|\xi_{j}|\ll 1$, by using the similar reason we may assume that

$\displaystyle\max(|\xi_{1}|,|\xi_{2}|,|\xi_{3}|)\gtrsim 1$

Therefore, through a decomposition of the variables we may assume without loss
of generality that

$\displaystyle N_{\max}\gtrsim 1,\ \ L_{\min}\gtrsim 1.$

We now fix some summation conventions in order to be used in the rest. Any
summation of the form $L_{\max}\sim...$ is a sum over the three dyadic
variables $L_{1},L_{2},L_{3}$, for example,

$\displaystyle\sum_{L_{\max}\sim H}:=\sum_{L_{1},L_{2},L_{3}\gtrsim
1:L_{\max}\sim H}$

Furthermore any summation of the form $N_{\max}\sim...$ is a sum over the
three dyadic variables $N_{1},N_{2},N_{3}>0$, for instance,

$\displaystyle\sum_{N_{\max}\sim N_{\text{med}}\sim
N}:=\sum_{N_{1},N_{2},N_{3}>0:N_{\max}\sim N_{\text{med}}\sim N}$

Letting $m$ be a $[3;\mathbb{R}^{2}\times\mathbb{R}]$ multiplier, in the
subsequent discussion, we intend to study the problem of controlling

(23) $\displaystyle\left\lVert
m((\xi_{1},\tau_{1}),(\xi_{2},\tau_{2}),(\xi_{3},\tau_{3}))\right\rVert_{[3;\mathbb{R}^{2}\times\mathbb{R}]},$

so by a dyadic decomposition of the support of $m$ in the variables $\xi_{j}$,
$\lambda_{j}$ together with a dyadic decomposition of the resonance function
$h$ we have

(24) $\displaystyle\eqref{3Zmult}\lesssim\left\lVert\sum_{N_{\max}\gtrsim
1}\sum_{H}\sum_{L_{1},L_{2},L_{3}}m((N_{1},L_{1}),(N_{2},L_{2}),(N_{3},L_{3}))X_{N_{1},N_{2},N_{3};H;L_{1},L_{2},L_{3}}\right\rVert_{[3;\mathbb{R}^{2}\times\mathbb{R}]}$

where $X_{N_{1},N_{2},N_{3};H;L_{1},L_{2},L_{3}}$ is the multiplier

$\displaystyle
X_{N_{1},N_{2},N_{3};H;L_{1},L_{2},L_{3}}(\xi,\tau):=\chi_{|{h(\xi)|\sim
H}}\prod_{j=1}^{3}\chi_{|\xi_{j}|\sim N_{j}}\chi_{|\lambda_{j}|\sim L_{j}}.$

Note that $N_{j}$ and $L_{j}$, in turn, measure the size of the frequency of
the $j^{th}$ wave and how closely it approximates a free solution, whereas $H$
measures the amount of resonance. From (14), (15) and (16) it can be deduced
that $X_{N_{1},N_{2},N_{3};H;L_{1},L_{2},L_{3}}$ vanishes unless

(25) $\displaystyle N_{\max}\sim N_{\text{med}}$

and

(26) $\displaystyle L_{\max}\sim\max(L_{\text{med}},H).$

Therefore using (25), (26) and implementing Schur’s test [23] (which enables
us to replace sum with a supremum) to the sums in $N_{\max}$ and
$N_{\text{med}}$, we obtain

$\eqref{normoftriplesum}\lesssim\sup_{N\gtrsim 1}\Bigl{\|}\sum_{N_{\max}\sim
N_{\text{med}}\sim
N}\sum_{H}\sum_{L_{max}\sim\max{(L_{\text{med}},H)}}m((N_{1},L_{1}),(N_{2},L_{2}),(N_{3},L_{3}))\\\
\times
X_{N_{1},N_{2},N_{3};H;L_{1},L_{2},L_{3}}\Bigl{\|}_{[3;\mathbb{R}^{2}\times\mathbb{R}]}.$

Therefore, by the triangle inequality and (26) it suffices to control

(27) $\sum_{N_{\max}\sim N_{\text{med}}\sim N}\sum_{L_{1},L_{2},L_{3}\gtrsim
1}m((N_{1},L_{1}),(N_{2},L_{2}),(N_{3},L_{3}))\left\lVert
X_{N_{1},N_{2},N_{3};L_{\max};L_{1},L_{2},L_{3}}\right\rVert_{[3;\mathbb{R}^{2}\times\mathbb{R}]}$

or

(28) $\sum_{N_{\max}\sim N_{\text{med}}\sim N}\sum_{L_{\max}\sim
L_{\text{med}}}\sum_{H\ll
L_{\max}}m((N_{1},L_{1}),(N_{2},L_{2}),(N_{3},L_{3}))\\\ \times\left\lVert
X_{N_{1},N_{2},N_{3};H;L_{1},L_{2},L_{3}}\right\rVert_{[3;\mathbb{R}^{2}\times\mathbb{R}]}$

for all $N\gtrsim 1$. The following lemma gives a sharp bound for the quantity

(29) $\displaystyle\left\lVert
X_{N_{1},N_{2},N_{3};H;L_{1},L_{2},L_{3}}\right\rVert_{[3;\mathbb{R}^{2}\times\mathbb{R}]}.$

######  Lemma 3.4 ([23]).

Suppose that $N_{1},N_{2},N_{3}>0$, $L_{1},L_{2},L_{3}>0$ and $H>0$ satisfy
(25) and (26).

  1. (1)

In the case $(+++)$, let the dispersion relations be given by (17) so we may
assume that $H\sim N_{\max}^{2}$. Then

(30) $\displaystyle\eqref{charblock}\lesssim
L_{\min}^{1/2}N_{\max}^{-1/2}N_{\min}^{1/2}\min(N_{\max}N_{\min},L_{\text{med}})^{1/2}$

  2. (2)

In the case $(++-)$, let the dispersion relations be given by (18) and from
(19) $H\lesssim N_{1}N_{2}$. Then

     * •

$((++)\ \text{case})$ If $N_{1}\sim N_{2}\gtrsim N_{3}$ then (29) vanishes
unless $H\sim N_{1}^{2}$ in this case we have

(31) $\displaystyle\eqref{charblock}\lesssim
L_{\min}^{1/2}N_{\max}^{-1/2}N_{\min}^{1/2}\min(N_{\max}N_{\min},L_{\text{med}})^{1/2}$

     * •

$((+-)\ \text{coherence})$ If $N_{1}\sim N_{3}\gtrsim N_{2}$ and $H\sim
L_{2}\gg L_{1},L_{3},N_{2}^{2}$ then

(32) $\displaystyle\eqref{charblock}\lesssim
L_{\min}^{1/2}N_{\max}^{-1/2}N_{\min}^{1/2}\min(H,\frac{H}{N_{\min}^{2}}L_{\text{med}})^{1/2}$

The same estimate holds with the roles of $1$ and $2$ reversed.

     * •

In all other cases,

(33) $\displaystyle\eqref{charblock}\lesssim
L_{\min}^{1/2}N_{\max}^{-1/2}N_{\min}^{1/2}\min(H,L_{\text{med}})^{1/2}\min(1,\frac{H}{N_{\min}^{2}})^{1/2}.$

Also the lemma below demonstrates that higher order multilinear estimates
might be reduced to the lower ordered ones and by this means the analysis of
the whole multiplier splits up.

######  Lemma 3.5 ([23]).

If $k_{1},k_{2}\geq 1$ and $m_{1}$ and $m_{2}$ are functions on $Z^{k_{1}}$
and $Z^{k_{2}}$ respectively, then we have the composition estimate

(34) $\left\lVert
m_{1}(\xi_{1},...,\xi_{k_{1}})m_{2}(\xi_{k_{1}+1},...,\xi_{k_{1}+k_{2}})\right\rVert_{[k_{1}+k_{2};Z]}\\\
\leq\left\lVert
m_{1}(\xi_{1},...,\xi_{k_{1}})\right\rVert_{[k_{1}+1;Z]}\left\lVert
m_{2}(\xi_{1},...,\xi_{k_{2}})\right\rVert_{[k_{2}+1;Z]}.$

In particular, for every $m:Z^{k}\rightarrow\mathbb{R}$ we have the $TT^{*}$
identity

(35) $\displaystyle\left\lVert
m(\xi_{1},...,\xi_{k})\overline{m(-\xi_{k+1},...,-\xi_{2k})}\right\rVert_{[2k;Z]}=\left\lVert
m(\xi_{1},...,\xi_{k})\right\rVert_{[k+1;Z]}^{2}.$

###### Proof.

We prove only the identity (34). So for $s=1,2$, let $L_{s}$ be the
$k_{s}$-linear operator defined by

$\displaystyle\langle
L_{s}(f_{1},...,f_{k_{s}}),f_{k_{s}+1}\rangle=\int_{\Gamma_{k_{s}+1}(Z)}m_{s}(\xi_{1},...,\xi_{k_{s}})\prod_{j=1}^{k_{s}+1}f_{j}(\xi_{j}).$

By duality this entails that

(36) $\displaystyle\left\lVert
L_{s}(f_{1},...,f_{k_{s}})\right\rVert_{L^{2}}=\sup_{\left\lVert
f_{k_{s}+1}\right\rVert_{L^{2}}=1}|\langle
L_{s}(f_{1},...,f_{k_{s}}),f_{k_{s}+1}\rangle|\leq\left\lVert
m_{s}\right\rVert_{[k_{s}+1;Z]}\prod_{j=1}^{k_{s}}\left\lVert
f_{j}\right\rVert_{L^{2}}.$

Thus Cauchy-Schwarz inequality and (36) lead to

$\int_{\Gamma_{k_{1}+k_{2}}(Z)}m_{1}(\xi_{1},...,\xi_{k_{1}})m_{2}(\xi_{k_{1}+1},...,\xi_{k_{1}+k_{2}})\prod_{j=1}^{k_{1}+k_{2}}f_{j}(\xi_{j})\\\
=\int_{\Gamma_{k_{1}+k_{2}}(Z)}m_{1}(\xi_{1},...,\xi_{k_{1}})\prod_{j=1}^{k_{1}}f_{j}(\xi_{j})\Big{(}m_{2}(\xi_{k_{1}+1},...,\xi_{k_{1}+k_{2}})\prod_{j=k_{1}+1}^{k_{1}+k_{2}}f_{j}(\xi_{j})\Big{)}\\\
=\langle
L_{1}(f_{1},...,f_{k_{1}}),L_{2}(f_{k_{1}+1},...,f_{k_{1}+k_{2}})\rangle\leq\left\lVert
L_{1}(f_{1},...,f_{k_{1}})\right\rVert_{L^{2}}\left\lVert
L_{2}(f_{k_{1}+1},...,f_{k_{1}+k_{2}})\right\rVert_{L^{2}}\\\ \leq\left\lVert
m_{1}\right\rVert_{[k_{1}+1;Z]}\left\lVert
m_{2}\right\rVert_{[k_{2}+1;Z]}\prod_{j=1}^{k_{1}+k_{2}}\left\lVert
f_{j}\right\rVert_{L^{2}}.$

∎

##  4\. Proof of Proposition 1.2: Trilinear $X^{s,b}$ Estimate

Before getting start with the estimate, we suitably arrange the $X^{s,b}$ norm
of the non-linearity so as to sort out the trilinear form. By duality we may
write the norm of the non-linearity as follows

$\left\lVert c_{1}|u|^{2}u+c_{2}K(|u|^{2})u\right\rVert_{X^{s+a,b-1}}=\\\
\sup_{\left\lVert
d\right\rVert_{X^{-(s+a),1-b}}=1}\Big{|}\int_{\mathbb{R}^{2}\times\mathbb{R}}\big{(}c_{1}|u|^{2}u+c_{2}K(|u|^{2})u\big{)}(x,t)d(x,t)\,\text{d}x\,\text{d}t\Big{|}$

Writing Fourier transform of nonlinear pieces as a convolution, we have

$\displaystyle\widehat{u\overline{v}w}(\xi,\tau)$
$\displaystyle=\widehat{u}*\widehat{\overline{v}}*\widehat{w}(\xi,\tau)=\underset{\begin{subarray}{c}\xi_{1}+\xi_{2}+\xi_{3}-\xi=0\\\
\tau_{1}+\tau_{2}+\tau_{3}-\tau=0\end{subarray}}{\int}\widehat{u}(\xi_{1},\tau_{1})\widehat{\overline{v}}(\xi_{2},\tau_{2})\widehat{w}(\xi_{3},\tau_{3})\,\text{d}\xi_{1}\,\text{d}\xi_{2}\,\text{d}\tau_{1}\,d\tau_{2}$
$\displaystyle=\underset{\begin{subarray}{c}\xi_{1}+\xi_{2}+\xi_{3}-\xi=0\\\
\tau_{1}+\tau_{2}+\tau_{3}-\tau=0\end{subarray}}{\int}\widehat{u}(\xi_{1},\tau_{1})\overline{\widehat{v}(-\xi_{2},-\tau_{2})}\widehat{w}(\xi_{3},\tau_{3})\,\text{d}\xi_{1}\,\text{d}\xi_{2}\,\text{d}\tau_{1}\,d\tau_{2},$

and the second term in the non-linearity is dealt as follows

$\displaystyle\widehat{K(u\overline{v})w}(\xi,\tau)$
$\displaystyle=\alpha(\xi)\widehat{u\overline{v}}*\widehat{w}(\xi,\tau)=\underset{\begin{subarray}{c}\xi_{1}+\xi_{2}-\xi=0\\\
\tau_{1}+\tau_{2}-\tau=0\end{subarray}}{\int}\alpha(\xi_{1})\widehat{u\overline{v}}(\xi_{1},\tau_{1})\widehat{w}(\xi_{2},\tau_{2})\,\text{d}\xi_{1}\,\text{d}\tau_{1}$
$\displaystyle=\underset{\begin{subarray}{c}\xi_{1}+\xi_{2}-\xi=0\\\
\tau_{1}+\tau_{2}-\tau=0\end{subarray}}{\int}\alpha(\xi_{1})\Big{[}\underset{\begin{subarray}{c}\xi_{1}-\xi_{3}-\xi_{4}=0\\\
\tau_{1}-\tau_{3}-\tau_{4}=0\end{subarray}}{\int}\widehat{u}(\xi_{3},\tau_{3})\widehat{\overline{v}}(\xi_{4},\tau_{4})\,\text{d}\xi_{3}\,\text{d}\tau_{3}\Big{]}\widehat{w}(\xi_{2},\tau_{2})\,\text{d}\xi_{1}\,\text{d}\tau_{1}$
$\displaystyle=\underset{\begin{subarray}{c}\xi_{1}+\xi_{2}-\xi=0\\\
\tau_{1}+\tau_{2}-\tau=0\end{subarray}}{\int}\alpha(\xi_{1})\Big{[}\underset{\begin{subarray}{c}\xi_{1}-\xi_{3}-\xi_{4}=0\\\
\tau_{1}-\tau_{3}-\tau_{4}=0\end{subarray}}{\int}\widehat{u}(\xi_{3},\tau_{3})\overline{\widehat{v}(-\xi_{4},-\tau_{4})}\,\text{d}\xi_{3}\,\text{d}\tau_{3}\Big{]}\widehat{w}(\xi_{2},\tau_{2})\,\text{d}\xi_{1}\,\text{d}\tau_{1}$
$\displaystyle=\underset{\begin{subarray}{c}\xi_{2}+\xi_{3}+\xi_{4}-\xi=0\\\
\tau_{2}+\tau_{3}+\tau_{4}-\tau=0\end{subarray}}{\int}\alpha(\xi_{3}+\xi_{4})\widehat{w}(\xi_{2},\tau_{2})\widehat{u}(\xi_{3},\tau_{3})\overline{\widehat{v}(-\xi_{4},-\tau_{4})}\,\text{d}\xi_{3}\,\text{d}\tau_{3}\,\text{d}\xi_{4}\,\text{d}\tau_{4}.$

In the last line we have taken account of the relations regarding the
integration region so as to replace $\text{d}\xi_{1}\,\text{d}\tau_{1}$ by
$\text{d}\xi_{4}\,\text{d}\tau_{4}$. Rename the variables by writing
$\xi_{1},\xi_{2},\xi_{3}$ ; $\tau_{1},\tau_{2},\tau_{3}$ in substitution for
$\xi_{3},\xi_{4},\xi_{2}$ ; $\tau_{3},\tau_{4},\tau_{2}$ respectively to get
that

$\displaystyle\widehat{K(u\overline{v})w}(\xi,\tau)=\underset{\begin{subarray}{c}\xi_{1}+\xi_{2}+\xi_{3}-\xi=0\\\
\tau_{1}+\tau_{2}+\tau_{3}-\tau=0\end{subarray}}{\int}\alpha(\xi_{1}+\xi_{2})\widehat{u}(\xi_{1},\tau_{1})\overline{\widehat{v}(-\xi_{2},-\tau_{2})}\widehat{w}(\xi_{3},\tau_{3})\,\text{d}\xi_{1}\,\text{d}\tau_{1}\,\text{d}\xi_{2}\,\text{d}\tau_{2}.$

Therefore, as Fourier transform preserves the inner product we obtain

$\int_{\mathbb{R}^{2}\times\mathbb{R}}\big{(}c_{1}u\overline{v}w+c_{2}K(u\overline{v})w\big{)}(x,t)d(x,t)\,\text{d}x\,\text{d}t\\\
=\int_{\mathbb{R}^{2}\times\mathbb{R}}(c_{1}\mathcal{F}(u\overline{v}w)+c_{2}\mathcal{F}(K(u\overline{v})w))(\xi,\tau)\mathcal{F}^{-1}d(\xi,\tau)\,\text{d}\xi\,\text{d}\tau\\\
=\underset{\begin{subarray}{c}\xi_{1}+\xi_{2}+\xi_{3}-\xi=0\\\
\tau_{1}+\tau_{2}+\tau_{3}-\tau=0\end{subarray}}{\int}[c_{1}+c_{2}\alpha(\xi_{1}+\xi_{2})]\widehat{u}(\xi_{1},\tau_{1})\overline{\widehat{v}(-\xi_{2},-\tau_{2})}\widehat{w}(\xi_{3},\tau_{3})\widehat{d}(-\xi,-\tau)\,\text{d}\xi_{1}\,\text{d}\tau_{1}\,\text{d}\xi_{2}\,\text{d}\tau_{2}\,\text{d}\xi\,\text{d}\tau\\\
=\underset{\begin{subarray}{c}\xi_{1}+\xi_{2}+\xi_{3}+\xi_{4}=0\\\
\tau_{1}+\tau_{2}+\tau_{3}+\tau_{4}=0\end{subarray}}{\int}[c_{1}+c_{2}\alpha(\xi_{1}+\xi_{2})]\widehat{u}(\xi_{1},\tau_{1})\overline{\widehat{v}(-\xi_{2},-\tau_{2})}\widehat{w}(\xi_{3},\tau_{3})\widehat{d}(\xi_{4},\tau_{4})\,\text{d}\xi_{1}\,\text{d}\xi_{2}\,\text{d}\xi_{3}\,\text{d}\tau_{1}\,\text{d}\tau_{2}\,\text{d}\tau_{3}$

where, in the last equality, first we replace $\text{d}\xi\,\text{d}\tau$ with
$\text{d}\xi_{3}\,\text{d}\tau_{3}$ by using the relations of integral region
and then swap the variables $\xi$, $\tau$ with $-\xi_{4}$, $-\tau_{4}$
respectively. By using the fact that the spaces $X^{s,b-1}_{\tau=h(\xi)}$ and
$X^{-s,1-b}_{\tau=-h(-\xi)}$ are dual to one another, we define

$\displaystyle
f_{1}(\xi_{1},\tau_{1})=\langle\xi_{1}\rangle^{s}\langle\tau_{1}+|\xi_{1}|^{2}\rangle^{b}|\widehat{u}(\xi_{1},\tau_{1})|$
$\displaystyle
f_{2}(\xi_{2},\tau_{2})=\langle\xi_{2}\rangle^{s}\langle\tau_{2}-|\xi_{2}|^{2}\rangle^{b}|\widehat{v}(-\xi_{2},-\tau_{2})|$
$\displaystyle
f_{3}(\xi_{3},\tau_{3})=\langle\xi_{3}\rangle^{s}\langle\tau_{3}+|\xi_{3}|^{2}\rangle^{b}|\widehat{w}(\xi_{3},\tau_{3})|$
$\displaystyle
f_{4}(\xi_{4},\tau_{4})=\langle\xi_{4}\rangle^{s+a}\langle\tau_{4}-|\xi_{4}|^{2}\rangle^{1-b}|\widehat{d}(\xi_{4},\tau_{4})|$

and

$m((\xi_{1},\tau_{1}),(\xi_{2},\tau_{2}),(\xi_{3},\tau_{3}),(\xi_{4},\tau_{4}))\\\
=\frac{[c_{1}+c_{2}\alpha(\xi_{1}+\xi_{2})]\langle\xi_{4}\rangle^{s+a}}{\prod_{j=1}^{3}\langle\xi_{j}\rangle^{s}\langle\tau_{1}+|\xi_{1}|^{2}\rangle^{b}\langle\tau_{2}-|\xi_{2}|^{2}\rangle^{b}\langle\tau_{3}+|\xi_{3}|^{2}\rangle^{b}\langle\tau_{4}-|\xi_{4}|^{2}\rangle^{1-b}},$

thus the required estimate takes the form

$\displaystyle\Big{|}\int_{\Gamma_{4}(\mathbb{R}^{2}\times\mathbb{R})}m((\xi_{1},\tau_{1}),(\xi_{2},\tau_{2}),(\xi_{3},\tau_{3}),(\xi_{4},\tau_{4}))\prod_{j=1}^{4}f_{j}(\xi_{j},\tau_{j})\Big{|}\leq\left\lVert
m\right\rVert_{[4;\mathbb{R}^{2}\times\mathbb{R}]}\prod_{j=1}^{4}\left\lVert
f_{j}\right\rVert_{L^{2}(\mathbb{R}^{2}\times\mathbb{R})}$

where

$\Gamma_{4}(\mathbb{R}^{2}\times\mathbb{R})=\\{((\xi_{1},\tau_{1}),(\xi_{2},\tau_{2}),(\xi_{3},\tau_{3}),(\xi_{4},\tau_{4}))\in(\mathbb{R}^{2}\times\mathbb{R})^{4}:\sum_{j=1}^{4}(\xi_{j},\tau_{j})=0\\}.$

Hence it suffices to show that

$\displaystyle\left\lVert
m((\xi_{1},\tau_{1}),(\xi_{2},\tau_{2}),(\xi_{3},\tau_{3}),(\xi_{4},\tau_{4}))\right\rVert_{[4;\mathbb{R}^{2}\times\mathbb{R}]}\lesssim
1.$

###### Proof of Proposition 1.2.

We may suppose without loss of generality that

$|\xi_{4}|\sim\max_{1\leq j\leq 4}|\xi_{j}|$

because other cases are easier and follow immediately. In this case the
structure of the hyperplane $\Gamma_{4}(\mathbb{R}^{2}\times\mathbb{R})$
suggests three cases to consider:

  * •

Case 1 $|\xi_{4}|\sim|\xi_{1}|$

  * •

Case 2 $|\xi_{4}|\sim|\xi_{2}|$

  * •

Case 3 $|\xi_{4}|\sim|\xi_{3}|$

We begin with examining the Case $2$. In this case, the multiplier is
estimated by

$m((\xi_{1},\tau_{1}),(\xi_{2},\tau_{2}),(\xi_{3},\tau_{3}),(\xi_{4},\tau_{4}))\lesssim\frac{\langle\xi_{4}\rangle^{a}\langle\xi_{3}\rangle^{-s}}{\langle\tau_{3}+|\xi_{3}|^{2}\rangle^{b}\langle\tau_{4}-|\xi_{4}|^{2}\rangle^{1-b}}\times\frac{\langle\xi_{1}\rangle^{-s}}{\langle\tau_{1}+|\xi_{1}|^{2}\rangle^{b}\langle\tau_{2}-|\xi_{2}|^{2}\rangle^{b}}\\\
=:m_{3,4}((\xi_{3},\tau_{3}),(\xi_{4},\tau_{4}))\times
m_{1,2}((\xi_{1},\tau_{1}),(\xi_{2},\tau_{2})).$

Using Lemma 3.2 and Lemma 3.5 we have the bound

$||m((\xi_{1},\tau_{1}),(\xi_{2},\tau_{2}),(\xi_{3},\tau_{3}),(\xi_{4},\tau_{4}))||_{[4;\mathbb{R}^{2}\times\mathbb{R}]}\lesssim||m_{1,2}((\xi_{1},\tau_{1}),(\xi_{2},\tau_{2}))||_{[3;\mathbb{R}^{2}\times\mathbb{R}]}\\\
\times||m_{3,4}((\xi_{3},\tau_{3}),(\xi_{4},\tau_{4}))||_{[3;\mathbb{R}^{2}\times\mathbb{R}]}.$

As a result the proof reduces to showing the following bilinear inequalities

(37) $\displaystyle\left\lVert
m_{1,2}((\xi_{1},\tau_{1}),(\xi_{2},\tau_{2}))\right\rVert_{[3;\mathbb{R}^{2}\times\mathbb{R}]}\lesssim
1,$ (38) $\displaystyle\left\lVert
m_{3,4}((\xi_{3},\tau_{3}),(\xi_{4},\tau_{4}))\right\rVert_{[3;\mathbb{R}^{2}\times\mathbb{R}]}\lesssim
1.$

At this juncture we introduce the dummy variables $\xi_{\text{d}2}$,
$\xi_{\text{d}3}$, $\tau_{\text{d}2}$, $\tau_{\text{d}3}$ satisfying

(39)
$\displaystyle(\xi_{1},\tau_{1})+(\xi_{2},\tau_{2})+(\xi_{\text{d}3},\tau_{\text{d}3})=0,$
(40)
$\displaystyle(\xi_{\text{d}2},\tau_{\text{d}2})+(\xi_{3},\tau_{3})+(\xi_{4},\tau_{4})=0$

so that (39) and (40) correspond to the hyperplanes with regards to (37) and
(38) respectively. As discussed in the previous section, after decomposing the
variables $\xi_{j_{i}}$, $\lambda_{j_{i}}=\tau_{j_{i}}+h_{j_{i}}(\xi_{j_{i}})$
and the resonance function
$h(\xi_{j_{1}},\xi_{j_{2}},\xi_{j_{3}})=h_{j_{1}}(\xi_{j_{1}})+h_{j_{2}}(\xi_{j_{2}})+h_{j_{3}}(\xi_{j_{3}})$
into dyadics $N_{j_{i}}$, $L_{j_{i}}$ and $H$ respectively, it suffices to
establish the following inequalities

$\displaystyle\sum_{N_{max}\sim N_{med}\sim N}\sum_{L_{max}\sim
L_{med}}\sum_{H\ll
L_{max}}m_{i_{1},i_{2}}((\xi_{i_{1}},\tau_{i_{1}}),(\xi_{i_{2}},\tau_{i_{2}}))\left\lVert
X_{{N_{\text{d}j}},N_{i_{1}},N_{i_{2}};H;L_{\text{d}j},L_{i_{1}},L_{i_{2}}}\right\rVert_{[3;\mathbb{R}^{2}\times\mathbb{R}]}\lesssim
1$

and

$\displaystyle\sum_{N_{max}\sim N_{med}\sim
N}\sum_{L_{\text{d}j},L_{i_{1}},L_{i_{2}}\gtrsim
1}m_{i_{1},i_{2}}((\xi_{i_{1}},\tau_{i_{1}}),(\xi_{i_{2}},\tau_{i_{2}}))\left\lVert
X_{{N_{\text{d}j}},N_{i_{1}},N_{i_{2}};L_{max};L_{\text{d}j},L_{i_{1}},L_{i_{2}}}\right\rVert_{[3;\mathbb{R}^{2}\times\mathbb{R}]}\lesssim
1$

for all $N\gtrsim 1$ and where, in turn, $(i_{1},i_{2},j)=(1,2,3),(3,4,2)$
associate to the bilinear inequalities (37) and (38). Therefore starting with
demonstrating (38), we have to handle four cases in terms of magnitudes of the
dyadic variables corresponding to relevant frequency variables:

  1. (1)

$N_{\text{d}2}\sim N_{3}\sim N_{4}$

  2. (2)

$N_{3}\sim N_{4}\gg N_{\text{d}2}$

  3. (3)

$N_{\text{d}2}\sim N_{3}\gg N_{4}$

  4. (4)

$N_{\text{d}2}\sim N_{4}\gg N_{3}$.

In the first of these cases $N_{\min}\sim N_{\max}\sim N\gtrsim 1$, so by the
estimate (33) of Lemma 3.4 we have

$\displaystyle\left\lVert
X_{{N_{\text{d}2}},N_{3},N_{4};H;L_{\text{d}2},L_{3},L_{4}}\right\rVert_{[3;\mathbb{R}^{2}\times\mathbb{R}]}\lesssim
L_{\min}^{1/2}N^{-1/2}N^{1/2}$
$\displaystyle\min(H,L_{\text{med}})^{1/2}\min(1,\frac{H}{N^{2}})^{1/2}$
$\displaystyle\lesssim
L_{\min}^{1/2}H^{2\epsilon}L_{\text{med}}^{1/2-2\epsilon}.$

Thus

$\displaystyle\frac{\langle N_{4}\rangle^{a}\langle
N_{3}\rangle^{-s}}{L_{3}^{b}L_{4}^{1-b}}\left\lVert
X_{{N_{\text{d}2}},N_{3},N_{4};H;L_{\text{d}2},L_{3},L_{4}}\right\rVert_{[3;\mathbb{R}^{2}\times\mathbb{R}]}\lesssim\frac{N^{-s+a}L_{\min}^{1/2}L_{\text{med}}^{1/2-2\epsilon}H^{2\epsilon}}{L_{\min}^{{1/2}+\epsilon}L_{\text{med}}^{1/2-\epsilon}}=\frac{N^{-s+a}H^{2\epsilon}}{L_{\min}^{\epsilon}L_{{\text{med}}}^{\epsilon}}.$

It follows that for if $H\sim L_{\max}$, then since $H\lesssim N_{\max}^{2}$,
we have

$\sum_{N_{max}\sim N_{med}\sim N}\sum_{L_{\text{d}2},L_{3},L_{4}\gtrsim
1}\frac{N^{-s+a}H^{2\epsilon}}{L_{\min}^{\epsilon}L_{{\text{med}}}^{\epsilon}}\
\ \sim\sum_{N_{max}\sim N_{med}\sim N}\sum_{L_{\text{d}2},L_{3},L_{4}\gtrsim
1}\frac{N^{-s+a}H^{3\epsilon}}{L_{\min}^{\epsilon}L_{{\text{med}}}^{\epsilon}L_{\max}^{\epsilon}}\\\
\lesssim\sum_{N_{max}\sim N_{med}\sim N}\sum_{L_{\text{d}2},L_{3},L_{4}\gtrsim
1}\frac{N^{-s+a+6\epsilon}}{L_{\min}^{\epsilon}L_{{\text{med}}}^{\epsilon}L_{\max}^{\epsilon}}\lesssim
1$

provided $a<s$. But if $H\ll L_{\max}$ then $L_{\text{med}}\sim L_{\max}$, so
summing in $H$ first we get

$\sum_{N_{max}\sim N_{med}\sim N}\sum_{L_{max}\sim L_{med}}\sum_{H\ll
L_{max}}\frac{N^{-s+a}H^{2\epsilon}}{L_{\min}^{\epsilon}L_{{\text{med}}}^{\epsilon}}\lesssim\sum_{N_{max}\sim
N_{med}\sim N}\sum_{L_{max}\sim
L_{med}}\frac{N^{-s+a+4\epsilon}}{L_{\min}^{\epsilon}L_{{\text{med}}}^{\epsilon/2}L_{\max}^{\epsilon/2}}\lesssim
1$

provided again that $a<s$. For the second case, we write $N_{3}\sim N_{4}\sim
N_{\max}\sim N_{\text{med}}$ and $N_{\text{d}2}\sim N_{\min}$. We consider
estimates (32) and (33) in Lemma 3.4 in this case. Since $N_{3}\sim N_{4}\sim
N\gtrsim 1$, we may establish the estimates corresponding to the estimate (33)
by following just the same lines of the previous case. So it suffices to make
use of the estimate (32) merely, where we have $H\sim L_{\text{d}2}\gg
L_{3},L_{4},N_{\text{d}2}^{2}\sim N_{\min}^{2}.$

$\displaystyle\left\lVert
X_{{N_{\text{d}2}},N_{3},N_{4};L_{\max};L_{\text{d}2},L_{3},L_{4}}\right\rVert_{[3;\mathbb{R}^{2}\times\mathbb{R}]}\lesssim
L_{\min}^{1/2}N^{-1/2}N_{\min}^{1/2}\min(H,\frac{H}{N_{\min}^{2}}L_{\text{med}})^{1/2}$

by which it follows that

$\sum_{N_{max}\sim N_{med}\sim N}\sum_{L_{\text{d}2},L_{3},L_{4}\gtrsim
1}\frac{\langle N_{4}\rangle^{a}\langle
N_{3}\rangle^{-s}}{L_{3}^{b}L_{4}^{1-b}}\left\lVert
X_{{N_{\text{d}2}},N_{3},N_{4};L_{\max};L_{\text{d}2},L_{3},L_{4}}\right\rVert_{[3;\mathbb{R}^{2}\times\mathbb{R}]}\\\
\lesssim\sum_{N_{max}\sim N_{med}\sim N}\sum_{L_{\text{d}2},L_{3},L_{4}\gtrsim
1}\frac{N^{-s+a}L_{\min}^{1/2}N^{-1/2}N_{\min}^{1/2}\min(H,\frac{H}{N_{\min}^{2}}L_{\text{med}})^{1/2}}{L_{\min}^{1/2+\epsilon}L_{\text{med}}^{1/2-\epsilon}}\\\
\lesssim\sum_{N_{max}\sim N_{med}\sim N}\sum_{L_{\text{d}2},L_{3},L_{4}\gtrsim
1}\frac{N^{-s+a-1/2}N_{\min}^{1/2}(H^{1/2+\epsilon}(\frac{HL_{\text{med}}}{N_{\min}^{2}})^{1/2-\epsilon})^{1/2}}{L_{\min}^{\epsilon}L_{\text{med}}^{1/2-\epsilon}}\\\
=\sum_{N_{max}\sim N_{med}\sim N}\sum_{L_{\text{d}2},L_{3},L_{4}\gtrsim
1}\frac{N^{-s+a-1/2}N_{\min}^{\epsilon}H^{1/2}}{L_{\min}^{\epsilon}L_{\text{med}}^{1/4-\epsilon/2}}.$

Since $L_{\max}\sim H\lesssim N^{2}$ and $N\gtrsim 1$, using

$1=\frac{L_{\max}^{\epsilon/2}\langle
N_{\min}\rangle^{2\epsilon}}{L_{\max}^{\epsilon/2}\langle
N_{\min}\rangle^{2\epsilon}}\lesssim\frac{N^{3\epsilon}}{L_{\max}^{\epsilon/2}\langle
N_{\min}\rangle^{2\epsilon}}$

in majorizing the sum above we obtain the bound

$\displaystyle\sum_{N_{max}\sim N_{med}\sim
N}\sum_{L_{\text{d}2},L_{3},L_{4}\gtrsim
1}\frac{N^{-s+a+1/2+3\epsilon}N_{\min}^{\epsilon}}{L_{\min}^{\epsilon}L_{\text{med}}^{1/4-\epsilon/2}L_{\max}^{\epsilon/2}\langle
N_{\min}\rangle^{2\epsilon}}\lesssim 1$

which is possible as long as $a<s-1/2$. Note at this point that for a non-
trivial smoothing argument it is necessary to make the assumption that
$s>1/2$. In the third and fourth cases, due to its effect in the use of Lemma
3.4, we have to decide the sign in the quantity
$\lambda_{\text{d}2}=\tau_{\text{d}2}\pm|\xi_{\text{d}2}|^{2}$. No selection
would lose the generality though, we prefer to set
$\lambda_{\text{d}2}=\tau_{\text{d}2}+|\xi_{\text{d}2}|^{2}$. Hence for the
third case, by this choice of the dummy modulation variable, we fall under the
$(++)$ case which leads to $H\sim N_{\max}^{2}$. Thus the estimate (31) in
Lemma 3.4 is to be used. Once $H\sim L_{\max}$,

$\displaystyle\left\lVert
X_{{N_{\text{d}2}},N_{3},N_{4};L_{\max};L_{\text{d}2},L_{3},L_{4}}\right\rVert_{[3;\mathbb{R}^{2}\times\mathbb{R}]}$
$\displaystyle\lesssim
L_{\min}^{1/2}N^{-1/2}N_{\min}^{1/2}\min(NN_{\min},L_{\text{med}})^{1/2}$
$\displaystyle\lesssim
L_{\min}^{1/2}(N^{3\epsilon}N_{\min}^{3\epsilon}L_{\text{med}}^{1-3\epsilon})^{1/2}=L_{\min}^{1/2}L_{\text{med}}^{1/2-3\epsilon/2}N^{3\epsilon/2}N_{\min}^{3\epsilon/2}$

implies that

$\sum_{N_{max}\sim N_{med}\sim N}\sum_{L_{\text{d}2},L_{3},L_{4}\gtrsim
1}\frac{\langle N_{4}\rangle^{a}\langle
N_{3}\rangle^{-s}}{L_{3}^{b}L_{4}^{1-b}}\left\lVert
X_{{N_{\text{d}2}},N_{3},N_{4};L_{\max};L_{\text{d}2},L_{3},L_{4}}\right\rVert_{[3;\mathbb{R}^{2}\times\mathbb{R}]}\\\
\lesssim\sum_{N_{max}\sim N_{med}\sim N}\sum_{L_{\text{d}2},L_{3},L_{4}\gtrsim
1}\frac{{N}^{-s+a+3\epsilon/2}N_{\min}^{3\epsilon/2}}{L_{min}^{\epsilon}L_{\text{med}}^{\epsilon/2}}.$

Since $L_{\max}\sim H\sim N^{2}$, using the inequality
$1=\frac{L_{\max}^{\epsilon/2}\langle
N_{\min}\rangle^{2\epsilon}}{L_{\max}^{\epsilon/2}\langle
N_{\min}\rangle^{2\epsilon}}\lesssim\frac{N^{3\epsilon}}{L_{\max}^{\epsilon/2}\langle
N_{\min}\rangle^{2\epsilon}}$ in the above estimate we attain

$\displaystyle\sum_{N_{max}\sim N_{med}\sim
N}\sum_{L_{\text{d}2},L_{3},L_{4}\gtrsim
1}\frac{{N}^{-s+a+9\epsilon/2}N_{\min}^{3\epsilon/2}}{L_{min}^{\epsilon}L_{\text{med}}^{\epsilon/2}L_{\max}^{\epsilon/2}\langle
N_{\min}\rangle^{2\epsilon}}\lesssim 1$

as long as $a<s$. Otherwise if $H\ll L_{\max}$, then as $H\sim N^{2}$ and
$L_{\text{med}}\sim L_{\max}$

$\sum_{N_{max}\sim N_{med}\sim N}\sum_{L_{max}\sim L_{med}}\sum_{H\ll
L_{max}}\frac{N^{-s+a+3\epsilon/2}N_{\min}^{3\epsilon/2}}{L_{\min}^{\epsilon}L_{{\text{med}}}^{\epsilon/2}}\\\
\sim\sum_{N_{max}\sim N_{med}\sim N}\sum_{L_{max}\sim L_{med}}\sum_{H\ll
L_{max}}\frac{N^{-s+a+\epsilon}H^{\epsilon/4}N_{\min}^{3\epsilon/2}}{L_{\min}^{\epsilon}L_{{\text{med}}}^{\epsilon/8}L_{\max}^{3\epsilon/8}}.$

Summing this in $H$ we get the bound

$\displaystyle\sum_{N_{max}\sim N_{med}\sim N}\sum_{L_{max}\sim
L_{med}}\frac{N^{-s+a+\epsilon}N_{\min}^{3\epsilon/2}}{L_{\min}^{\epsilon}L_{{\text{med}}}^{\epsilon/8}L_{\max}^{\epsilon/8}}\lesssim
1$

so long as $a<s$. In the last case, we have $N_{\text{d}2}\sim N_{4}\sim
N_{\max}\sim N$ and $N_{3}\sim N_{\min}$. Sign analysis of the modulations
$\lambda_{\text{d}2}$ (which is set in the previous case) and $\lambda_{4}$
suggests utilizing (32) and (33) in Lemma 3.4. In the separate case $H\sim
L_{\max}\sim L_{3}\gg L_{\text{d}2},L_{4},N_{3}^{2}$, the bound (32) gives
rise to the estimate

$\sum_{N_{max}\sim N_{med}\sim N}\sum_{L_{\text{d}2},L_{3},L_{4}\gtrsim
1}\frac{\langle N_{4}\rangle^{a}\langle
N_{3}\rangle^{-s}}{L_{3}^{b}L_{4}^{1-b}}\left\lVert
X_{{N_{\text{d}2}},N_{3},N_{4};L_{\max};L_{\text{d}2},L_{3},L_{4}}\right\rVert_{[3;\mathbb{R}^{2}\times\mathbb{R}]}\\\
\lesssim\sum_{N_{max}\sim N_{med}\sim N}\sum_{L_{\text{d}2},L_{3},L_{4}\gtrsim
1}\frac{N^{a-1/2}N_{\min}^{1/2}L_{\min}^{1/2}\min({H,\frac{H}{N_{\min}^{2}}L_{\text{med}}})^{1/2}}{\langle
N_{\min}\rangle^{s}L_{\max}^{1/2+\epsilon}L_{\min}^{1/2-\epsilon}}\\\
\lesssim\sum_{N_{max}\sim N_{med}\sim N}\sum_{L_{\text{d}2},L_{3},L_{4}\gtrsim
1}\frac{N^{a-1/2}N_{\min}^{1/2}H^{1/2}L_{\min}^{\epsilon}}{\langle
N_{\min}\rangle^{s}L_{\max}^{1/2+\epsilon}}.$

Since $L_{\max}\sim H\lesssim N^{2}$ writing $H\lesssim
L_{\max}^{1-\epsilon}N^{2\epsilon}$ we bound the above sum by

$\sum_{N_{max}\sim N_{med}\sim N}\sum_{L_{\text{d}2},L_{3},L_{4}\gtrsim
1}\frac{N^{a-1/2+\epsilon}N_{\min}^{1/2}L_{\min}^{\epsilon}}{\langle
N_{\min}\rangle^{s}L_{\max}^{3\epsilon/2}}\\\ \lesssim\sum_{N_{max}\sim
N_{med}\sim N}\sum_{L_{\text{d}2},L_{3},L_{4}\gtrsim
1}\frac{N^{a-1/2+\epsilon}N_{\min}^{\epsilon}}{\langle
N_{\min}\rangle^{s-1/2+\epsilon}L_{\max}^{\epsilon/6}L_{\text{med}}^{\epsilon/6}L_{\min}^{\epsilon/6}}$

which is summable provided that $a<1/2$. For the situations with $L_{\max}\gg
H$ where the estimate (33) is available, we have the bound

$\sum_{N_{max}\sim N_{med}\sim N}\sum_{L_{max}\sim L_{med}}\sum_{H\ll
L_{max}}\frac{\langle N_{4}\rangle^{a}\langle
N_{3}\rangle^{-s}}{L_{3}^{b}L_{4}^{1-b}}\left\lVert
X_{{N_{\text{d}2}},N_{3},N_{4};H;L_{\text{d}2},L_{3},L_{4}}\right\rVert_{[3;\mathbb{R}^{2}\times\mathbb{R}]}\\\
\lesssim\sum_{N_{max}\sim N_{med}\sim N}\sum_{L_{max}\sim L_{med}}\sum_{H\ll
L_{max}}\frac{N^{a-1/2}N_{\min}^{1/2}L_{\min}^{1/2}\min({H,L_{\text{med}}})^{1/2}\min({1,\frac{H}{N_{\min}^{2}})^{1/2}}}{\langle
N_{\min}\rangle^{s}L_{\min}^{1/2+\epsilon}L_{\text{med}}^{1/2-\epsilon}}\\\
\lesssim\sum_{N_{max}\sim N_{med}\sim N}\sum_{L_{max}\sim L_{med}}\sum_{H\ll
L_{max}}\frac{N^{a-1/2}N_{\min}^{\epsilon}H^{2\epsilon}L_{\text{med}}^{1/2-2\epsilon}}{\langle
N_{\min}\rangle^{s-1/2+\epsilon}L_{\min}^{\epsilon}L_{\text{med}}^{1/2-\epsilon}}$

as $H\lesssim N^{2}$ and $L_{\max}\sim L_{\text{med}}$, after summing in $H$,
this is bounded by

$\displaystyle\sum_{N_{max}\sim N_{med}\sim N}\sum_{L_{max}\sim
L_{med}}\frac{N^{a-1/2+4\epsilon}N_{\min}^{\epsilon}}{\langle
N_{\min}\rangle^{s-1/2+\epsilon}L_{\min}^{\epsilon}L_{\text{med}}^{\epsilon/2}L_{\max}^{\epsilon/2}}\lesssim
1$

provided that $a<1/2$. Also for the instances with $L_{\max}\sim H$ where the
estimate (33) is available, we proceed similarly as above to obtain the bound

$\displaystyle\sum_{N_{max}\sim N_{med}\sim
N}\sum_{L_{\text{d}2},L_{3},L_{4}\gtrsim
1}\frac{N^{a-1/2+5\epsilon}N_{\min}^{\epsilon}}{\langle
N_{\min}\rangle^{s-1/2+\epsilon}L_{\min}^{\epsilon}L_{\text{med}}^{\epsilon/2}L_{\max}^{\epsilon}}\lesssim
1$

provided that $a>1/2$. This completes the proof of (38). The proof of (37) is
the repetition of done for (38) without the multiplier
$\langle\xi_{4}\rangle^{a}$ and follows by using (39) and making the
assumption $s>1/2$. Next for the Case $1$, we have

$m((\xi_{1},\tau_{1}),(\xi_{2},\tau_{2}),(\xi_{3},\tau_{3}),(\xi_{4},\tau_{4}))\lesssim\frac{\langle\xi_{4}\rangle^{a}\langle\xi_{2}\rangle^{-s}}{\langle\tau_{2}-|\xi_{2}|^{2}\rangle^{b}\langle\tau_{4}-|\xi_{4}|^{2}\rangle^{1-b}}\times\frac{\langle\xi_{3}\rangle^{-s}}{\langle\tau_{1}+|\xi_{1}|^{2}\rangle^{b}\langle\tau_{3}+|\xi_{3}|^{2}\rangle^{b}}\\\
=:m_{2,4}((\xi_{2},\tau_{2}),(\xi_{4},\tau_{4}))\times
m_{1,3}((\xi_{1},\tau_{1}),(\xi_{3},\tau_{3})).$

We pick dummy variables $\xi_{\text{d}j}$, $\tau_{\text{d}j}$ for $j=2,3$ such
that

$\displaystyle(\xi_{2},\tau_{2})+(\xi_{\text{d}3},\tau_{\text{d}3})+(\xi_{4},\tau_{4})=0$
$\displaystyle(\xi_{1},\tau_{1})+(\xi_{\text{d}2},\tau_{\text{d}2})+(\xi_{3},\tau_{3})=0$

with which we write
$\lambda_{\text{d}2}=\tau_{\text{d}2}+|\xi_{\text{d}2}|^{2}$ and
$\lambda_{\text{d}3}=\tau_{\text{d}3}-|\xi_{\text{d}3}|^{2}$. So then analysis
of these two multipliers fall into $(+++)$ case which is substantially easier
and only the estimate (30) for (29) is taken for. In the spirit of the
analysis of the subcase $(3)$ above, $[3;\mathbb{R}^{3}]$ norms of the two
multipliers $m_{1,3}$ and $m_{2,4}$ can be shown to be finite provided that
$a<1/2$ and $s>1/2.$ Lastly, Case $3$ immediately follows from Case $1$
because the variables $\xi_{1}$ and $\xi_{3}$ appear to be symmetric. ∎

In general optimality in the multilinear estimates can be exhibited by Knapp-
type examples as such in [18], [3]. In what follows we construct a suitable
such example for which the trilinear estimate (5) fails for $a>1/2$, as a
result a half derivative gain is the best possible result for (3). To see
this, set

$\widehat{u}=\chi_{Q_{1}},\widehat{v}=\chi_{Q_{2}},\widehat{w}=\chi_{Q_{2}}$

where $\chi_{S}$ is the characteristic function of the set $S$ and

$\displaystyle Q_{1}=\\{(\xi_{1},\xi_{2},\tau)\in\mathbb{R}^{3}:|\xi_{1}|\leq
1,|\xi_{2}-N|\leq 1/N,|\tau+N^{2}|\leq 1\\}$ $\displaystyle
Q_{2}=\\{(\xi_{1},\xi_{2},\tau)\in\mathbb{R}^{3}:|\xi_{1}|\leq 1,|\xi_{2}|\leq
1/\sqrt{N},|\tau|\leq 1\\}$

for $N\in\mathbb{N}$ large. Since the volume of the former set $\sim 1/N$ and
of the latter $\sim N^{-1/2}$, we have $\left\lVert
u\right\rVert_{X^{s,b}}\sim N^{s-1/2}$ and $\left\lVert
v\right\rVert_{X^{s,b}}=\left\lVert w\right\rVert_{X^{s,b}}\sim N^{-1/4}$.
Note that

$\displaystyle\widehat{u\overline{v}w}(x,t)$
$\displaystyle=\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}\chi_{Q_{1}}(x-\eta-\zeta,t-\lambda-\tau)\chi_{Q_{2}}(\eta,\lambda)\chi_{Q_{2}}({\zeta,\tau})\,\text{d}\eta\,\text{d}\lambda\,\text{d}\zeta\,\text{d}\tau$
$\displaystyle=\int_{Q_{2}}\int_{Q_{2}}\chi_{Q_{1}}(x-\eta-\zeta,t-\lambda-\tau)\,\text{d}\eta\,\text{d}\lambda\,\text{d}\zeta\,\text{d}\tau$

and

$\displaystyle(x-\eta-\zeta,t-\lambda-\tau)\in
Q_{1}\Rightarrow(\zeta+\eta,\tau+\lambda)\in(x,t)-Q_{1}$
$\displaystyle(\zeta,\tau),(\eta,\lambda)\in
Q_{2}\Rightarrow(\zeta+\eta,\tau+\lambda)\in 2Q_{2},$

where we have used the fact that $Q_{2}+Q_{2}=2Q_{2}$ as $Q_{2}$, having box
like shape, is convex. Hence $(\zeta+\eta,\tau+\lambda)\in
2Q_{2}\cap[(x,t)-Q_{1}]$ (notice
$2Q_{2}=\\{(\xi_{1},\xi_{2},\tau)\in\mathbb{R}^{3}:|\xi_{1}|\leq
2,|\xi_{2}|\leq 2N^{-1/2},|\tau|\leq 2\\}$). Therefore in order that
$2Q_{2}\cap[(x,t)-Q_{1}]\neq\emptyset$, we should have $|x_{1}-\xi_{1}|\leq
2,|x_{2}-\xi_{2}|\leq 2N^{-1/2},|t-\tau|\leq 2$ for $(\xi_{1},\xi_{2},\tau)\in
Q_{1}$ and $x=(x_{1},x_{2})$. Hence using these relations and taking $N$
sufficiently large, $(x,t)$ can be thought to belong to a set that covers
$Q_{1}$. Therefore for $(x,t)\in Q_{1}$, we can find some $C>0$ such that

$\displaystyle|\widehat{u\overline{v}w}(x,t)|\geq CN^{-1}\chi_{Q_{1}}(x,t).$

This follows

$\displaystyle||u\overline{v}w||_{X^{s+a,b-1}}$ $\displaystyle\geq
CN^{-1}\Big{(}\int_{Q_{1}}\langle\tau+|\xi|^{2}\rangle^{2(b-1)}\langle\xi\rangle^{2(s+a)}\,\text{d}\xi\,\text{d}\tau\Big{)}^{1/2}$
$\displaystyle\sim
CN^{s+a-1}\Big{(}\int_{Q_{1}}\text{d}\xi\,\text{d}\tau\Big{)}^{1/2}\geq
CN^{s+a-3/2}.$

Thus for $a>1/2$ Proposition 1.2 implies that $CN^{s+a-3/2}\lesssim
N^{s-1/2}N^{-1/4}N^{-1/4}=N^{s-1}\Rightarrow C\lesssim N^{1/2-a}$;
nevertheless as $N$ gets larger, the right hand side of this inequality
becomes arbitrarily small, while the left side is a fixed positive number.
In the sequel we make use of the ideas in [6] and [8] so as to finish the
proof of Theorem 1.1.

##  5\. Proof of Theorem 1.1

Let $\delta$ be the local existence time given by the local theory. Since
$b>\frac{1}{2}$ we use the embedding
$X^{s+a,b}\hookrightarrow{C^{0}_{t}H_{x}^{s+a}}$ along with Lemma 2.1,
Proposition 1.2 and local theory bound to obtain

(41) $\left\lVert
u(t)-e^{it\Delta}u(0)\right\rVert_{C^{0}_{t}H_{x}^{s+a}([-\delta,\delta]\times\mathbb{R}^{2})}\lesssim\left\lVert\int_{0}^{t}e^{i(t-\tau)\Delta}[c_{1}|u|^{2}u+c_{2}K(|u|^{2})u](\tau)d\tau\right\rVert_{X^{s+a,b}_{\delta}}\\\
\lesssim\left\lVert
c_{1}|u|^{2}u+c_{2}K(|u|^{2})u\right\rVert_{X^{s+a,b-1}_{\delta}}\lesssim\left\lVert
u\right\rVert_{X^{s,b}_{\delta}}^{3}\lesssim\left\lVert
u_{0}\right\rVert_{H^{s}}^{3}.$

Therefore (41) proves (4) in the local time interval $[-\delta,\delta]$, in
order to prove it for all times we have to iterate this result and then prove
the continuity argument. To do so fix $t$ large then for if $r\leq t$, from
(11) with the choice of $\beta(s)$ (remember that for $s>1/2$ this choice
makes $T(t)$ non-decreasing function of time) we have

$\displaystyle\left\lVert u(r)\right\rVert_{H^{s}}\lesssim T(r)\leq T(t).$

In particular for any $j$ with $j\delta\leq t$ we have the bound

$\displaystyle\left\lVert u((j-1)\delta)\right\rVert_{H^{s}}\lesssim
T((j-1)\delta)\leq T(t).$

Thus, under favour of global well-posedness result of Theorem 2.2, considering
the initial value problem (2) with $u((j-1)\delta)$ being the initial data,
and implementing (41) to this local problem, we obtain

$\displaystyle\left\lVert
u(j\delta)-e^{i\delta\Delta}u((j-1)\delta)\right\rVert_{H^{s+a}}\lesssim\left\lVert
u((j-1)\delta)\right\rVert_{H^{s}}^{3}\lesssim T(t)^{3}.$

By the local theory we pick $\delta\sim T(t)^{-\frac{2}{s}}$ so that, for
$J=t/\delta\sim tT(t)^{\frac{2}{s}}$, we get

$\displaystyle\left\lVert u(t)-e^{it\Delta}u_{0}\right\rVert_{H^{s+a}}$
$\displaystyle=\left\lVert u(J\delta)-e^{i\delta
J\Delta}u(0)\right\rVert_{H^{s+a}}$
$\displaystyle=\left\lVert\sum_{j=1}^{J}e^{i\delta(J-j)\Delta}u(j\delta)-e^{i\delta(J-j+1)\Delta}u((j-1)\delta)\right\rVert_{H^{s+a}}$
$\displaystyle\leq\sum_{j=1}^{J}\left\lVert
e^{i\delta(J-j)\Delta}u(j\delta)-e^{i\delta(J-j+1)\Delta}u((j-1)\delta)\right\rVert_{H^{s+a}}$
$\displaystyle=\sum_{j=1}^{J}\left\lVert
u(j\delta)-e^{i\delta\Delta}u((j-1)\delta)\right\rVert_{H^{s+a}}$
$\displaystyle\lesssim JT(t)^{3}\lesssim\langle t\rangle
T(t)^{3+\frac{2}{s}}.$

This finishes the iteration argument, in order for (4) to hold we are left to
show that the difference

$\displaystyle
D(t):=u(t)-e^{it\Delta}u_{0}=-i\int_{0}^{t}e^{i(t-\tau)\Delta}(c_{1}|u|^{2}u+c_{2}K(|u|^{2})u)(\tau)d\tau$

is continuous in $H^{s+a}$. Assume that $r$ is fixed and also without loss of
generality that $t>r$, then

$D(t)-D(r)=i\int_{0}^{r}e^{i(r-\tau)\Delta}(c_{1}|u|^{2}u+c_{2}K(|u|^{2})u)(\tau)d\tau\\\
-i\int_{0}^{t}e^{i(t-\tau)\Delta}(c_{1}|u|^{2}u+c_{2}K(|u|^{2})u)(\tau)d\tau\\\
=-i(e^{i(t-r)\Delta}-\text{Id})\int_{0}^{r}e^{i(r-\tau)\Delta}(c_{1}|u|^{2}u+c_{2}K(|u|^{2})u)(\tau)d\tau\\\
-i\int_{r}^{t}e^{i(t-\tau)\Delta}(c_{1}|u|^{2}u+c_{2}K(|u|^{2})u)(\tau)d\tau.$

Thus

$\left\lVert D(t)-D(r)\right\rVert_{H^{s+a}}\\\
\lesssim\left\lVert\langle\xi\rangle^{s+a}\mathcal{F}\Big{(}(e^{i(t-r)\Delta}-\text{Id})\int_{0}^{r}e^{i(r-\tau)\Delta}(c_{1}|u|^{2}u+c_{2}K(|u|^{2})u)(\tau)d\tau\Big{)}\right\rVert_{L^{2}_{\xi}}\\\
\hskip
65.44142pt+\left\lVert\langle\xi\rangle^{s+a}\mathcal{F}\Big{(}\int_{r}^{t}e^{i(t-\tau)\Delta}(c_{1}|u|^{2}u+c_{2}K(|u|^{2})u)(\tau)d\tau\Big{)}\right\rVert_{L^{2}_{\xi}}\\\
=:\text{I}_{1}+\text{I}_{2}.$

Using the inequality
$|e^{i(r-t)|\xi|^{2}}-1|\lesssim\min(1,|\xi|^{2}|t-r|)\leq(|\xi|^{2}|t-r|)^{\epsilon}$
in the subsequent calculation (for a sufficiently small $\epsilon>0$), we
obtain

(42)
$\text{I}_{1}=\Bigl{|}\\!\Bigl{|}\langle\xi\rangle^{s+a}\mathcal{F}\Big{(}\mathcal{F}^{-1}\Big{(}e^{-i(t-r)|\xi|^{2}}\mathcal{F}\Big{(}\int_{0}^{r}e^{i(r-\tau)\Delta}(c_{1}|u|^{2}u+c_{2}K(|u|^{2})u)(\tau)d\tau\Big{)}\Big{)}\\\
-\int_{0}^{r}e^{i(r-\tau)\Delta}(c_{1}|u|^{2}u+c_{2}K(|u|^{2})u)(\tau)d\tau\Big{)}\Bigl{|}\\!\Bigl{|}_{L^{2}_{\xi}}\\\
=\left\lVert\langle\xi\rangle^{s+a}(e^{-i(t-r)|\xi|^{2}}-1)\mathcal{F}\Big{(}\int_{0}^{r}e^{i(r-\tau)\Delta}(c_{1}|u|^{2}u+c_{2}K(|u|^{2})u)(\tau)d\tau\Big{)}\right\rVert_{L^{2}_{\xi}}\\\
\lesssim|t-r|^{\epsilon}\left\lVert\langle\xi\rangle^{s+a+2\epsilon}\mathcal{F}\Big{(}\int_{0}^{r}e^{i(r-\tau)\Delta}(c_{1}|u|^{2}u+c_{2}K(|u|^{2})u)(\tau)d\tau\Big{)}\right\rVert_{L^{2}_{\xi}}\\\
=|t-r|^{\epsilon}\left\lVert\int_{0}^{r}e^{i(r-\tau)\Delta}(c_{1}|u|^{2}u+c_{2}K(|u|^{2})u)(\tau)d\tau\right\rVert_{H^{s+2\epsilon+a}}\\\
=|t-r|^{\epsilon}\left\lVert\sum_{j=1}^{r/\delta}\int_{(j-1)\delta}^{j\delta}e^{i(r-\tau)\Delta}(c_{1}|u|^{2}u+c_{2}K(|u|^{2})u)(\tau)d\tau\right\rVert_{H^{s+2\epsilon+a}}\\\
\leq|t-r|^{\epsilon}\sum_{j=1}^{r/\delta}\left\lVert\int_{(j-1)\delta}^{j\delta}e^{i(r-\tau)\Delta}(c_{1}|u|^{2}u+c_{2}K(|u|^{2})u)(\tau)d\tau\right\rVert_{H^{s+2\epsilon+a}}$

where we pick the same $\delta$ given by the local existence time of the
solution. The sum above contains no more than finitely many terms because
$\delta$ depends upon $\sup\limits_{t\in[0,r]}\left\lVert
u(t)\right\rVert_{H^{s}(\mathbb{R}^{2})}$ that is finite by Theorem 2.2 and
$r$ is fixed. As the length of each interval of integration is $\delta$, by
the time translation and time reversal symmetries of the solution, it suffices
just to estimate the following integral for $t\in[-\delta/2,\delta/2]$:

$\left\lVert\int_{0}^{t}e^{i(t-\tau)\Delta}(c_{1}|u|^{2}u+c_{2}K(|u|^{2})u)(\tau)d\tau\right\rVert_{H^{s+2\epsilon+a}}\\\
\leq\left\lVert\eta(t/\delta)\int_{0}^{t}e^{i(t-\tau)\Delta}(c_{1}|u|^{2}u+c_{2}K(|u|^{2})u)(\tau)d\tau\right\rVert_{L^{\infty}_{t}H^{s+2\epsilon+a}_{x,y}}$

where $\eta(t)$ is a $C^{\infty}$ function supported on $[-2,2]$ that equals
to $1$ on $[-1,1]$. Next we use the embedding,
$X^{s+2\epsilon+a,\tilde{b}}\subset{L^{\infty}_{t}H^{s+2\epsilon+a}_{x,y}}$
for $\tilde{b}>\frac{1}{2}$ (set for instance $\tilde{b}=1/2+\epsilon$ for
$\epsilon>0$), to majorize the integral above by

$\displaystyle\left\lVert\eta(t/\delta)\int_{0}^{t}e^{i(t-\tau)\Delta}(c_{1}|u|^{2}u+c_{2}K(|u|^{2})u)(\tau)d\tau\right\rVert_{X^{s+2\epsilon+a,\tilde{b}}}$

which, by means of Lemma 2.1 with $b^{\prime}=-\frac{1}{2}+2\epsilon$ and
$b^{\prime}=b-1$, can be estimated by

$\displaystyle\delta^{\epsilon}\left\lVert
c_{1}|u|^{2}u+c_{2}K(|u|^{2})u\right\rVert_{X^{s+2\epsilon+a,b-1}_{\delta}}$

Finally an application of Proposition 1.2 with $s+2\epsilon$ and the local
theory gives the bound

$\displaystyle\delta^{\epsilon}\left\lVert
u\right\rVert_{X^{s+2\epsilon,b}_{\delta}}^{3}\lesssim\delta^{\epsilon}\left\lVert
u_{0}\right\rVert_{H^{s+2\epsilon}}^{3}.$

As a result, the sum in (42) is bounded and hence $I_{1}$ converges to $0$ as
$t\rightarrow r$. Also the same result follows for $\text{I}_{2}$ by using the
Proposition 1.2 and the Lemma 2.1.

##  6\. existence of a global attractor: proof of thm 1.3

This section is devoted to give an alternative proof of the existence of a
global attractor for the forced and weakly damped DS by using smoothing
estimates. Firstly we show that the evolution operator is weakly continuous
then we exploit this in handling the corresponding energy equation to upgrade
the weak convergence, resulting from boundedness of the flow, to strong
convergence giving rise to the asymptotic compactness of the flow. Throughout
this section, $\xrightarrow{w}$ and $\xrightarrow{w^{*}}$ will denote the weak
and the weak∗ convergences respectively.

######  Lemma 6.1.

If $u_{0}^{k}\xrightarrow{w}u_{0}$ in $H^{1}$, then for all $T>0$ and
$a\in(0,\frac{1}{2})$, the linear and the nonlinear parts of the semigroup
operator $U(t)$ of (6) satisfy

$\displaystyle
L^{\delta}(t)u_{0}^{k}\xrightarrow{w}L^{\delta}(t)u_{0}\,\,\text{in}\,\,L^{2}([0,T];H^{1})$
$\displaystyle
N(t)u_{0}^{k}\xrightarrow{w}N(t)u_{0}\,\,\text{in}\,\,L^{2}([0,T];H^{1+a}).$

Moreover, for $t\in[0,T]$,

$\displaystyle
L^{\delta}(t)u_{0}^{k}\xrightarrow{w}L^{\delta}(t)u_{0}\,\,\text{in}\,\,H^{1}$
$\displaystyle N(t)u_{0}^{k}\xrightarrow{w}N(t)u_{0}\,\,\text{in}\,\,H^{1+a}.$

###### Proof.

We just verify the assertions concerning the nonlinear part as the ones for
the linear part will follow from the Fourier representation of the linear flow
at once. To make use of the smoothing result for the forced problem (6), we
shall transform the equation (6) with data $u_{0}$ by setting

$g=\big{(}\widehat{f}/\langle\xi\rangle^{2}\big{)}^{\vee}=(1-\Delta)^{-1}f\in
H^{2},$

and $v=u+g$ so that obtain the equation

(43) $\displaystyle iv_{t}+\Delta v+i\delta
v=c_{1}|v-g|^{2}(v-g)+c_{2}(v-g)\phi_{x}+(1+i\delta g),$

with $\phi_{x}=K(|v-g|^{2})$ and data $v(\cdot,0)=u_{0}(\cdot)+g(\cdot)$. Let
$L^{\delta}$ denote the semigroup for the corresponding homogeneous linear
equation

(44) $\displaystyle iw_{t}+\Delta w+i\delta w=0.$

Then the nonlinear part $n=v-w$ satisfies the equation

(45) $\displaystyle in_{t}+\Delta n+i\delta
n=c_{1}|n+w-g|^{2}(n+w-g)+c_{2}(n+w-g)K(|n+w-g|^{2})+(1+i\delta g)$

with data $n(\cdot,0)=0$. Proceeding as in Section 5, notably using the
trilinear estimate (5) with $s=1$, $a\in(0,\frac{1}{2})$, for $T\leq 1$, we
get

(46) $\displaystyle\left\lVert
n\right\rVert_{C^{0}_{t}H^{1+a}([0,T]\times\mathbb{R}^{2})}\lesssim\left\lVert
u_{0}\right\rVert_{H^{1}}^{3}+\left\lVert
g\right\rVert_{H^{a-1}}\leq\left\lVert
u_{0}\right\rVert_{H^{1}}^{3}+\left\lVert f\right\rVert_{L^{2}}.$

Note that in the above estimate we use a variant of Lemma 2.1 that replaces
the linear group for the nondissipative equation by the dissipative group
$L^{\delta}(t)=e^{it\Delta-t\delta}$. For a proof of this, see [7]. Using the
$H^{1}$ global wellposedness (a priori bound (13)) in iterating the local
result above, we obtain the global bound

(47) $\displaystyle\left\lVert n(t)\right\rVert_{H^{1+a}}\leq
C(a,\delta,\left\lVert f\right\rVert_{L^{2}}\left\lVert
u_{0}\right\rVert_{H^{1}}),$

for the details see the section $3$ of [7], or section $6$ of [9]. Lastly the
continuity in $H^{1+a}$ follows as in Section 5, so for any $T>0$ and initial
data $u_{0}\in H^{1}$, we have

(48) $\displaystyle n\in C^{0}_{t}H^{1+a}([0,T]\times\mathbb{R}^{2}).$

In order to show that $N(t)u_{0}^{k}\xrightarrow{w}N(t)u_{0}$ in spaces given
by the statement of the lemma, it suffices to show that every subsequence of
$N(t)u_{0}^{k}$ has a further subsequence which converges weakly to the same
limit. Let $w^{k}$ denote the solution to (44) with data $u_{0}^{k}$. In
relation to this denote by $n^{k}$ the nonlinear part. Since weak convergence
$u_{0}^{k}\xrightarrow{w}u_{0}$ in $H^{1}$ implies that $\sup_{k}\left\lVert
u_{0}^{k}\right\rVert_{H^{1}}\leq M$ for some $M>0$, using this in (46) and
(47), we infer that, for every $T>0$, $\\{n^{k}\\}_{k}$ is bounded in

(49) $\displaystyle C([0,T];H^{1+a})\cap C^{1}([0,T];H^{a-1})$

with a uniform bound of (47). Firstly, in conjuction with this boundedness we
infer, by Arzelà-Ascoli theorem, that $\\{n^{k}\\}_{k}$ is relatively compact
in $C([0,T];H_{loc}^{a-1})$ thanks to the uniform boundedness of the
derivatives which implies equicontinuity. Therefore interpolating between this
and (49) gives us a subsequence of $\\{n^{k}\\}_{k}$ that converges strongly
in $C([0,T];H_{loc}^{1+a})$. Secondly, by the Banach-Alaoglu theorem,
boundedness of $\\{n^{k}\\}_{k}$ in (49) yield a weak* convergent subsequence
in $L^{\infty}([0,T];H^{1+a})$. Therefore combining these two, we reach a
further subsequence, denoted also by $\\{n^{k}\\}_{k}$, convergent in the
above spaces with the corresponding type of convergences. We shall write
$n^{k}\xrightarrow{w^{*}}n$ in $L^{\infty}([0,T];H^{1+a})$, and
$n^{k}\rightarrow n$ strongly in $C([0,T];H_{loc}^{1+a})$ (weak∗ limits are
unique). The analogous arguments for the linear parts $w^{k}$ hold in $H^{1}$
as well, so denote the corresponding limit by $w$. Later we will see that the
limit $n$ is indeed the weak limit in the spaces dictated by the lemma. Using
local strong convergence above, next, we will show that $n$ is a
distributional solution. Note that $n^{k}$ satisfies the equation (45) and let
$F(p,q)=c_{1}|p+q-g|^{2}(p+q-g)+c_{2}(p+q-g)K(|p+q-g|^{2})$. Thus for any
$\varphi\in C_{c}^{\infty}([0,T];\mathbb{R}^{2})$,

$\iint\Big{(}in_{t}+\Delta n+i\delta
n-F(n,w)-(1+i\delta)g\Big{)}\varphi\,\text{d}x\,\text{d}t\\\
=\iint\Big{(}\big{[}-i\varphi_{t}+\Delta\varphi+i\delta\varphi\big{]}n-\big{[}(1+i\delta
g)+F(n,w)\big{]}\varphi\Big{)}\,\text{d}x\,\text{d}t\\\
=\lim_{k\rightarrow\infty}\iint\Big{(}\big{[}-i\varphi_{t}+\Delta\varphi+i\delta\varphi\big{]}n^{k}-\big{[}(1+i\delta
g)+F(n^{k},w^{k})\big{]}\varphi\Big{)}\,\text{d}x\,\text{d}t\\\
=\lim_{k\rightarrow\infty}\iint\Big{(}in^{k}_{t}+\Delta n^{k}+i\delta
n^{k}-F(n^{k},w^{k})-(1+i\delta)g\Big{)}\varphi\,\text{d}x\,\text{d}t=0,$

which proves that $n$ is a distributional solution. It is just left to verify
the second equality above. It suffices to show that the following identity

$\Big{|}\iint\Big{(}\big{[}-i\varphi_{t}+\Delta\varphi+i\delta\varphi\big{]}(n^{k}-n)-\big{[}F(n^{k},w^{k})-F(n,w)\big{]}\varphi\Big{)}\,\text{d}x\,\text{d}t\Big{|}\\\
\leq\Big{|}\iint\big{[}-i\varphi_{t}+\Delta\varphi+i\delta\varphi\big{]}(n^{k}-n)\text{d}x\,\text{d}t\Big{|}+\left\lVert\varphi\right\rVert_{L^{\infty}_{x,t}}\Big{|}\iint\limits_{\text{supp}\varphi}\big{[}F(n^{k},w^{k})-F(n,w)\big{]}\,\text{d}x\,\text{d}t\Big{|}$

eventually decreases to zero for increasing $k$. The first integral is clearly
decaying due to strong local convergence of $n^{k}$, whereas the integral part
of the second summand is majorized by

$\Big{|}\iint\limits_{\text{supp}\varphi}\big{(}n^{k}-n+w^{k}-w\big{)}\big{[}c_{1}I+c_{2}K\big{]}(|n^{k}+w^{k}-g|^{2})\,\text{d}x\,\text{d}t\Big{|}\\\
+\Big{|}\iint\limits_{\text{supp}\varphi}(\overline{n^{k}-n+w^{k}-w})(n^{k}+w^{k}-g)\big{[}c_{1}I+c_{2}K\big{]}(n+w-g)\,\text{d}x\,\text{d}t\Big{|}\\\
+\Big{|}\iint\limits_{\text{supp}\varphi}\big{(}n^{k}-n+w^{k}-w\big{)}\big{(}\overline{n+w-g}\big{)}\big{[}c_{1}I+c_{2}K\big{]}(n+w-g)\,\text{d}x\,\text{d}t\Big{|}$

where we have used the property $(\text{iii})$ of the operator $K$ given in
Section 2 in the computation above. Owing to the strong local convergences of
$n^{k}$ and $w^{k}$ along with the boundedness of $K$, we conclude that these
sums vanish in the limit. Moreover, by the uniqueness of the weak∗ limits, $n$
is a unique distribution belonging to $C([0,T];H^{1+a})$ yielding that
$n=N(t)u_{0}$ (since $n$ has shown to be the distributional solution of (45)).
Therefore, using the fact that $L^{2}([0,T];H^{1+a})$ embeds in the dual of
$L^{\infty}([0,T];H^{1+a})$, weak∗ convergence in $L^{\infty}([0,T];H^{1+a})$
implies the weak convergence in $L^{2}([0,T];H^{1+a})$. This finishes the
proof of the first assertion in the lemma.

To prove the second argument, fix a $\tilde{t}\in[0,T]$. As before smoothing
estimate together with the boundedness of the initial data $u^{k}_{0}$ imply
that $\\{N(\tilde{t})u_{0}^{k}\\}_{k}$ is bounded in $H^{1+a}$. Thus there
exists a weakly convergent subsequence, still denoted by
$\\{N(\tilde{t})u_{0}^{k}\\}$, that converges in $H^{1+a}$, say to
$\widetilde{n}$. But as we know, from the previous discussion above, that
$N(t)u_{0}^{k}\xrightarrow{w^{*}}N(t)u_{0}$ in $C([0,T];H^{1+a})$. So the
uniqueness entails that $\widetilde{n}=N(\tilde{t})u_{0}$.

∎

###### Proof of Theorem 1.3.

To begin with, we note that the existence of an absorbing ball $\mathcal{B}$
of the evolution follows from (13), indeed the detailed proof was given in
[24]. Hence to attain a global attractor, it is just left to affirm that the
propogator $U(t)$ is asymptotically compact. That is, it is sufficient to show
that for any sequence of initial data $\\{u_{0,k}\\}_{k}$ in absorbing ball
and any sequence of times $t_{k}\rightarrow\infty$, the sequence
$\\{U(t_{k})u_{0,k}\\}_{k}$ has a convergent subsequence in $H^{1}$. Note that
for $u_{0}\in\mathcal{B}$, (47) implies that the nonlinear part $N(t)u_{0}$ of

$\displaystyle U(t)u_{0}=L^{\delta}(t)u_{0}+N(t)u_{0}$

is contained in a ball $B_{R}$ in $H^{1+a}$ with radius
$R=R(a,\delta,\left\lVert f\right\rVert_{L^{2}})$, $a\in(0,\frac{1}{2})$. As a
result, $\\{N(t_{k})u_{0,k}\\}_{k}\subset B_{R}$. Therefore we can find a
subsequence, still denoted by $N(t_{k})u_{0,k}$, that converges weakly in
$H^{1+a}$. Moreover since the weak and weak∗ topologies agree on a reflexive
spaces, the Banach-Alaoglu theorem yields that, up to a subsequence,
$U(t_{k})u_{0,k}$ converges weakly in $H^{1}$. As
$L^{\delta}(t_{k})u_{0,k}\rightarrow 0$ strongly in $H^{1}$ as
$t_{k}\rightarrow\infty$, $N(t_{k})u_{0,k}$ and $U(t_{k})u_{0,k}$ converge to
the same limit, say to u. Furthermore, for every $T>0$, we can find a further
subsequence so that $N(t_{k}-T)u_{0,k}$ and $U(t_{k}-T)u_{0,k}$ converge
weakly in $H^{1+a}$ and $H^{1}$ respectively. As above, by the decay of the
linear part, the limits are the same, so denote it by $u_{T}$. By Lemma 6.1,

$\displaystyle
U(t_{k}-T)u_{0,k}\xrightarrow{w}u_{T}\,\,\text{in}\,\,H^{1}\implies
U(t_{k})u_{0,k}=U(T)\big{(}U(t_{k}-T)u_{0,k}\big{)}\xrightarrow{w}U(T)u_{T}\,\,\text{in}\,\,H^{1}.$

Therefore, by the uniqueness of a weak limit, $U(T)u_{T}=u$. In a subsequent
discussion, sometimes we need to take $T\rightarrow\infty$ in order to obtain
strong convergences. So to make sense of this, we may implement a
diagonalization argument for a countable set $\\{T\in\mathbb{N}\\}$ so that,
up to a same subsequence for all $T$, $U(t_{k}-\cdot)u_{0,k}$ and
$N(t_{k}-\cdot)u_{0,k}$ converge weakly at each $T$ in the corresponding
spaces above.

Next we want to upgrade the weak $H^{1}$ convergence of the solution flow
$U(t_{k})u_{0,k}$ to a strong $H^{1}$ convergence. Firstly using the equation
(12), we can obtain

$\frac{d}{dt}\left\lVert u\right\rVert_{L^{2}}+2\,\delta\left\lVert
u\right\rVert_{L^{2}}=2\,\text{Im}\langle f,u\rangle,$

and then application of the Gronwall lemma for the evolution $U(t)$ gives that

$\displaystyle\left\lVert U(t_{k})u_{0,k}\right\rVert_{L^{2}}^{2}$
$\displaystyle=e^{-2\delta T}\left\lVert
U(t_{k}-T)u_{0,k}\right\rVert_{L^{2}}^{2}+2\,\text{Im}\int_{0}^{T}e^{-2\delta(T-\tau)}\langle
f,U(t_{k}-T+\tau)u_{0,k}\rangle_{L^{2}_{x}}\,\text{d}\,\tau$
$\displaystyle\left\lVert U(T)u_{T}\right\rVert_{L^{2}}^{2}=e^{-2\delta
T}\left\lVert
u_{T}\right\rVert_{L^{2}}^{2}+2\,\text{Im}\int_{0}^{T}e^{-2\delta(T-\tau)}\langle
f,U(\tau)u_{T}\rangle_{L^{2}_{x}}\,\text{d}\,\tau,$

that yield

$\displaystyle\left\lVert U(t_{k})u_{0,k}\right\rVert_{L^{2}}^{2}-\left\lVert
U(T)u_{T}\right\rVert_{L^{2}}^{2}$ $\displaystyle=e^{-2\delta
T}\big{(}\left\lVert U(t_{k}-T)u_{0,k}\right\rVert_{L^{2}}^{2}-\left\lVert
u_{T}\right\rVert_{L^{2}}^{2}\big{)}$
$\displaystyle+2\,\text{Im}\int_{0}^{T}e^{-2\delta(T-\tau)}\langle
f,U(t_{k}-T+\tau)u_{0,k}-U(\tau)u_{T}\rangle_{L^{2}_{x}}\,\text{d}\,\tau.$

The first summand becomes negligible by taking sufficiently large $T$ since
letting $k\rightarrow\infty$ and using the fact that $u_{0,k}\in\mathcal{B}$,
we ensure that the norms in the parantheses are finite (weak convergence in
$H^{1}$ implies that $\left\lVert
u_{T}\right\rVert_{L^{2}}\leq\lim\limits_{k}\inf\left\lVert
U(t_{k}-T)u_{0,k}\right\rVert_{L^{2}}$). The second summand vanishes in the
limit by Lemma 6.1 because, with $U(T)u_{T}=u$, we have

$U(t_{k})u_{0,k}\xrightarrow{w}u\,\,\text{in}\,\,H^{1}\implies\\\
U(t_{k}-T+\tau)u_{0,k}=U(\tau-T)\big{(}U(t_{k})u_{0,k}\big{)}\xrightarrow{w}U(\tau)u_{T}\,\,\text{in}\,\,L^{2}([0,T];H^{1}).$

As a consequence we get that

$\displaystyle\lim\limits_{k}\sup\big{(}\left\lVert
U(t_{k})u_{0,k}\right\rVert_{L^{2}}^{2}-\left\lVert
U(T)u_{T}\right\rVert_{L^{2}}^{2}\big{)}=\lim\limits_{k}\sup\big{(}\left\lVert
U(t_{k})u_{0,k}\right\rVert_{L^{2}}^{2}-\left\lVert
u\right\rVert_{L^{2}}^{2}\big{)}\leq 0,$

which, along with $U(t_{k})u_{0,k}\xrightarrow{w}u$ in $H^{1}$, implies that
$U(t_{k})u_{0,k}\rightarrow u$ strongly in $L^{2}$. This strong $L^{2}$
convergence will be important in the upcoming energy calculations. So define
the functional $E$ by

$E(u_{0})(t)=\left\lVert\nabla
U(t)u_{0}\right\rVert_{L^{2}}^{2}+\frac{c_{1}}{2}\left\lVert
U(t)u_{0}\right\rVert_{L^{4}}^{4}+\frac{c_{2}}{2}\int
K(|U(t)u_{0}|^{2})|U(t)u_{0}|^{2}\,\text{d}x\\\ +2\,\text{Re}\,\int
f\overline{U(t)u_{0}}\,\text{d}x,$

and the time derivative is as follows

$\frac{d}{dt}E(u_{0})(t)=-2\delta E(u_{0})(t)+F(u_{0})(t)$

where

$F(u_{0})(t)=-\delta c_{1}\left\lVert U(t)u_{0}\right\rVert_{L^{4}}^{4}-\delta
c_{2}\int
K(|U(t)u_{0}|^{2})|U(t)u_{0}|^{2}\,\text{d}x+2\delta\,\text{Re}\,\int
f\overline{U(t)u_{0}}\,\text{d}x.$

Gronwall lemma implies that

$E(u_{0,k})(t_{k})-E(u_{T})(T)=\sum\limits_{j=1}^{4}I_{j},$

where

$\displaystyle I_{1}$ $\displaystyle=e^{-2\delta
T}\big{(}E(u_{0,k})(t_{k}-T)-E(u_{T})(0)\big{)}$ $\displaystyle I_{2}$
$\displaystyle=-\delta c_{1}\int_{0}^{T}e^{-2\delta(T-\tau)}\big{(}\left\lVert
U(t_{k}-T+\tau)u_{0,k}\right\rVert_{L^{4}}^{4}-\left\lVert
U(\tau)u_{T}\right\rVert_{L^{4}}^{4}\big{)}\text{d}\tau$ $\displaystyle I_{3}$
$\displaystyle=-\delta c_{2}\int_{0}^{T}\int
e^{-2\delta(T-\tau)}\big{(}K(|U(t_{k}-T+\tau)u_{0,k}|^{2})|U(t_{k}-T+\tau)u_{0,k}|^{2}$
$\displaystyle\hskip
204.85974pt-K(|U(\tau)u_{T}|^{2})|U(\tau)u_{T}|^{2}\big{)}\text{d}x\,\text{d}\tau$
$\displaystyle I_{4}$
$\displaystyle=2\delta\,\text{Re}\int_{0}^{T}e^{-2\delta(T-\tau)}\langle
f,U(t_{k}-T+\tau)u_{0,k}-U(\tau)u_{0,k}\rangle_{L^{2}_{x}}\text{d}\tau.$

$I_{1}$ gets arbitrarily small by increasing $T$, and $I_{2}$ can be majorized
by

$\displaystyle\delta c_{1}\int_{0}^{T}e^{-2\delta(T-\tau)}\left\lVert
U(t_{k}-T+\tau)u_{0,k}-U(\tau)u_{T}\right\rVert_{L^{4}}$ $\displaystyle\hskip
113.81102pt\times\big{(}\left\lVert
U(t_{k}-T+\tau)u_{0,k}\right\rVert_{L^{4}}+\left\lVert
U(\tau)u_{T}\right\rVert_{L^{4}}\big{)}^{3}\text{d}\tau,$

note that by $L^{4}$ Gagliardo-Nirenberg inequality

$\left\lVert U(t_{k}-T+\tau)u_{0,k}-U(\tau)u_{T}\right\rVert_{L^{4}}\\\
\lesssim\left\lVert
U(\tau-T)\big{(}U(t_{k})u_{0,k}-U(T)u_{T}\big{)}\right\rVert_{L^{2}}^{1/2}\left\lVert
U(t_{k}-T+\tau)u_{0,k}-U(\tau)u_{T}\right\rVert_{H^{1}}^{1/2}.$

Then by the strong continuity of $U(\tau-T)$ and the strong $L^{2}$
convergence $U(t_{k})u_{0,k}\rightarrow U(T)u_{T}$, the majorant of $I_{2}$
above vanishes in the limit as weak $H^{1}$ convergence yields the $H^{1}$
boundedness of the norms. Write the third term as

$I_{3}=-\delta c_{2}\int_{0}^{T}\int
e^{-2\delta(T-\tau)}\big{[}K(|U(t_{k}-T+\tau)u_{0,k}|^{2})-K(|U(\tau)u_{T}|^{2})\big{]}|U(t_{k}-T+\tau)u_{0,k}|^{2}\text{d}x\,\text{d}\tau\\\
-\delta c_{2}\int_{0}^{T}\int
e^{-2\delta(T-\tau)}K(|U(\tau)u_{T}|^{2})\big{(}|U(t_{k}-T+\tau)u_{0,k}|^{2}-|U(\tau)u_{T}|^{2}\big{)}\text{d}x\,\text{d}\tau,$

first use the linearity and the $L^{p}$ boundedness of the operator $K$, then
proceed by using the same reasoning as above to conclude that $I_{3}$ decays
to zero. Finally, by using the weak continuity (Lemma 6.1), $I_{4}$ vanishes
in the limit. Therefore, as $U(T)u_{T}=u$, we get that

$\lim\limits_{k\rightarrow\infty}\sup\big{(}E(u_{0,k})(t_{k})-E(u)(0)\big{)}=\lim\limits_{k\rightarrow\infty}\sup\big{(}E(u_{0,k})(t_{k})-E(u_{T})(T)\big{)}\leq
0.$

This inequality together with the definition of $E$ and taking limits as above
imply that

$\lim\limits_{k\rightarrow\infty}\sup\left\lVert\nabla
U(t_{k})u_{0,k}\right\rVert_{L^{2}}^{2}\leq\left\lVert\nabla
u\right\rVert_{L^{2}}^{2}.$

Therefore this with $U(t_{k})u_{0,k}\xrightarrow{w}u$ in $H^{1}$ lead to the
$L^{2}$ strong convergence of $\nabla U(t_{k})u_{0,k}$ to $\nabla u$.
Consequently $U(t_{k})u_{0,k}\rightarrow u$ strongly in $H^{1}$. This finishes
the proof of asymptotic compactness, so the proof is complete. ∎

## Acknowledgement

The author would like to thank his Ph.D advisor T. Burak Gürel for his
guidance and many salutary discussions.

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