# Local Well-posedness for the Kinetic MMT Model

Pierre Germain  Pierre Germain, Department of Mathematics, Huxley Building,
South Kensington Campus, Imperial College London, London SW7 2AZ, United
Kingdom<EMAIL_ADDRESS>,  Joonhyun La  Joonhyun La, June E Huh Center for
Mathematical Challenges, Korea Institute for Advanced Study, 85 Hoegiro
Dongdaemun-gu, Seoul 02455, Republic of Korea<EMAIL_ADDRESS>and
Katherine Zhiyuan Zhang  Katherine Zhiyuan Zhang, Department of Mathematics,
Northeastern University, 43 Leon St., Boston, MA 02115, United States of
America<EMAIL_ADDRESS>
###### Abstract.

The MMT equation was proposed by Majda, McLaughlin and Tabak as a model to
study wave turbulence. We focus on the kinetic equation associated to this
Hamiltonian system, which is believed to give a way to predict turbulent
spectra. We clarify the formulation of the problem, and we develop the local
well-posedness theory for this equation. Our analysis uncovers a surprising
nonlinear smoothing phenomenon.

###### Key words and phrases:

MMT model, kinetic wave equation, local well-posedness, nonlinear smoothing
effect

###### 2000 Mathematics Subject Classification:

82C99, 45G10

###### Contents

  1. 1 Introduction
  2. 2 First properties of the equation
  3. 3 The collision operator for $\alpha=\frac{1}{2}$
  4. 4 The collision operator in weighted $L^{p}$ spaces
  5. 5 Local well-posedness near scale invariant power data

##  1\. Introduction

The MMT equation, which was proposed as a toy model in [24], has become very
popular in the Physics and Applied Mathematics communities as a numerical
testing ground for the predictions of wave turbulence theory. This model has a
number of advantages: it is analytically tractable, easy to simulate, and yet
it displays a very rich range of phenomena.

The aim of this paper is to establish the foundations of the mathematical
analysis of the associated kinetic wave equation, in particular the existence
and uniqueness of solutions.

###  1.1. The MMT model and its associated kinetic equation

We consider the MMT model proposed by Majda, McLaughlin and Tabak [24]:

(1.1)
$i{\partial_{t}}\psi=|D|^{\alpha}\psi+\lambda|D|^{\beta/4}\big{(}\big{|}|D|^{\beta/4}\psi\big{|}^{2}|D|^{\beta/4}\psi\big{)},\
\lambda=\pm 1,$

where $\alpha\in(0,1),\beta\in\mathbb{R}$ and $\psi=\psi(x,t)$ is a complex-
valued wave field (a word of caution for the reader: our convention for the
sign of $\beta$ is the opposite of [24], but agrees with [32]).

The corresponding kinetic wave equation is given (see [24, 32]) by

(KWE) ${\partial_{t}}n(t,k)=\mathcal{C}(n(t))(k),$

with the collision operator given by

(1.2)
$\begin{split}&\mathcal{C}(n)(k)=\int_{\mathbb{R}^{3}}|k_{1}k_{2}k_{3}k|^{\beta/2}(n_{1}n_{2}n_{3}+n_{1}n_{2}n_{k}-n_{1}n_{3}n_{k}-n_{2}n_{3}n_{k})\\\
&\qquad\qquad\qquad\qquad\qquad\qquad\delta(k_{1}+k_{2}-k_{3}-k)\delta(\omega_{1}+\omega_{2}-\omega_{3}-\omega)\,dk_{1}\,dk_{2}\,dk_{3},\end{split}$

where we denoted

$\displaystyle n_{k}=n(t,k),\quad n_{i}=n(t,k_{i}),\quad i=1,2,3,$
$\displaystyle\omega=|k|^{\alpha},\quad\omega_{i}=|k_{i}|^{\alpha},\quad
i=1,2,3.$

The dependence of this model on two parameters, $\alpha$ and $\beta$, makes it
possible to explore different regimes of wave turbulence. Like all authors in
this subject, we will restrict most of the discussion to $\alpha=\frac{1}{2}$,
which simplifies greatly the algebra.

Having set $\alpha=\frac{1}{2}$, what are significant values of $\beta$? The
case $\beta=0$ is deemed by Zakharov and collaborators [31, 32] to be
particularly interesting for two reasons: first, there is no ”nonlinear
frequency shift” occuring in the Hamiltonian system; second, the signs of the
energy and mass fluxes for the Kolmogorov spectrum suggest (see the discussion
there) that the wave turbulence predictions should be most easily realized for
this model. Furthermore, several authors [28, 7] have focused on this case to
run numerical experiments.

###  1.2. Physics of wave turbulence in the MMT model

Classical introductions to wave turbulence can be found in the textbooks [33,
25, 19].

The MMT model was introduced by Majda, McLaughlin and Tabak in [24], with the
intention of verifying the predictions of wave turbulence theory on an easily
computable 1D model. In some regimes however, their observations seemed to
contradicted the classical theory, with the appearance of a new, so-called
”MMT spectrum”, see also [5, 6].

This discrepancy with the standard theory was examined analytically and
numerically by Zakharov and collaborators, who concluded to the influence of
nonlinear rigid structures, such as solitary waves, on the spectrum [32, 31,
27]. Other authors [28, 7] emphasized the presence of features associated to
fully developed turbulence for sufficiently strong forcing.

Very recently, Bühler and Du [4] performed new numerical experiments, and
reported that the exponent predicted by the theory of wave turbulence is
observed for the nonlinear spectrum, provided that the inertial range is
sufficiently broad.

###  1.3. Mathematical approaches to kinetic wave turbulence

There was recently major progress on the derivation of the kinetic wave
equation from Hamiltonian dynamics [3, 9, 10, 11] culminating in the first
rigorous proof on a non-trivial kinetic time scale [14, 12, 13, 30, 22].

These remarkable developments establish for the first time the validity of the
kinetic wave equation, which had been a matter of debate amongst physicists.
As a consequence, the kinetic wave equation does describe turbulent behavior
(in the appropriate regime), and further progress on turbulent dynamics could
follow from a deeper understanding of the kinetic wave equation.

As far as rigorous Mathematics are concerned, the theory of the kinetic wave
equation (in its many variants) is still in its infancy. Local well-posedness
could be proved in some cases [26, 20], and global weak solutions have been
constructed in other cases [18, 29]. Of particular interest are the
Kolmogorov-Zakharov solutions, whose stability was considered in [16, 17, 8].

In the present article, we focus on the kinetic MMT equation, with several
motivations: first, this is a standard model in the Physics and Applied
Mathematics community. Second, it is arguably the simplest model for wave
turbulence: the geometry of resonances and of the resonant manifold is very
simple since it is one-dimensional, and the homogeneity of the equation allows
to use scaling. Finally, it is the first kinetic wave equation to be studied
in dimension one, opening the door to further developments in this setting.

###  1.4. Main results and plan of the article

Section 2 is dedicated to the review of some basic properties of the equation,
such as scaling and monotone quantities; all computations are formal in this
section.

In Section 3, we turn to giving a rigorous meaning to the collision operator
$\mathcal{C}$, and obtaining a convenient parameterization. As is customary in
the literature, this is done under the simplifying assumptions that
$\alpha=\frac{1}{2}$, and that the function $n$ is even in $k$, which allows
one to think of it as a function of the dispersion $\omega$: $n=n(\omega)$.
These assumptions are made henceforth. The parameterization obtained by
Zakharov-Guyenne-Pushkarev-Dias [32] is first derived rigorously in (3.17); it
is then observed that this expression can be symmetrized to obtain (3.25).

In Section 4, we investigate the boundedness of the collision operator on the
most simple and natural class of functional spaces, namely weighted $L^{p}$
spaces given by the norm

$\|f(\omega)\|_{L^{p}_{\theta,\gamma}(0,\infty)}=\|m(\omega)f(\omega)\|{L^{p}(0,\infty)},\quad\mbox{where
$m(\omega)=|\omega|^{-\theta}\mathbf{1}_{\omega<1}+|\omega|^{\gamma}\mathbf{1}_{\omega>1}$}.$

This leads to the following theorem (which follows from Lemma 4.1 and Lemma
4.3); as far as positive results go, it suffers from the limitation to the
range $\beta\in[-\frac{1}{2},0]$.

######  Theorem 1.1.

  * (i)

The operator $\mathcal{C}$ is bounded on $L^{\infty}_{\theta,\gamma}$ if
$-1/2\leq\beta\leq 0$, $\gamma>2\beta+2$ and $\theta>-2\beta-1$

  * (ii)

The operator $\mathcal{C}$ is not bounded on $L^{p}_{\theta,\gamma}$ if $p<3$
and $\gamma,\theta\in\mathbb{R}$.

######  Corollary 1.2.

The equation (KWE) is locally well posed in $L^{\infty}_{\theta,\gamma}$ if
$-1/2\leq\beta\leq 0$, $\gamma>2\beta+2$ and $\theta>-2\beta-1$.

More specifically, for data in $L^{\infty}_{\theta,\gamma}$, there exists
$T>0$ and a unique solution in $\mathcal{C}([0,T],L^{\infty}_{\theta,\gamma})$
which depends continuously on the data. The equation
$\partial_{t}n=\mathcal{C}(n)$ is understood in the sense of equality in
$\mathcal{C}([0,T],L^{\infty}_{\theta,\gamma})$.

To go further, we consider in Section 5 smoother data, namely such that

$N(\omega)=\omega^{2\beta+\frac{3}{2}}n(\omega)$

belongs to

$X=\\{N\in L^{\infty}((0,\infty)),\,DN(\omega):=\omega N^{\prime}(\omega)\in
L^{2p_{0}}((0,\infty))\cap L^{\infty}((0,\infty))\\}$

for $\beta$ and $p_{0}$ satisfying (5.5). Note that the exponent
$2\beta+\frac{3}{2}$ appearing in the definition of $N$ corresponds to the
scaling of the equation, so that the functional framework is close to being
scaling invariant, and as such, optimal.

We first state the a priori estimate.

######  Theorem 1.3.

Suppose that $p_{0},\beta$, and $N$ satisfy the technical assumptions
(A1)-(A4) (see Section 5). Then there exists a time $T=T(p_{0},\beta,N_{0})$
and a constant $\mathbf{C}(p_{0},\beta,N_{0})$ such that

(1.3) $\|N\|_{L^{\infty}(0,T;X)}\leq\mathbf{C}(p_{0},\beta,N_{0}).$

Also, we have the gain of regularity (5.65) and in particular, for
$\beta\in(-3/4,3/4)$,

(1.4) $\int_{0}^{T}[DN(t)]_{\beta}^{2}dt\leq C(N_{0},\beta,p_{0})<\infty,$

where

(1.5)
$\begin{split}[F]_{\beta}^{2}&:=\int_{0}^{\infty}\int_{0}^{\infty}\mathbf{1}_{[2/3,3/2]}\left(\frac{\omega^{\prime}}{\omega}\right)\frac{|F(\omega)-F(\omega^{\prime})|^{2}}{|\omega^{\prime}-\omega|^{\beta+1}}(\omega\omega^{\prime})^{\beta/2}d\omega
d\omega^{\prime}\\\
&+\int_{0}^{\infty}\int_{0}^{\infty}\mathbf{1}_{[1/2,2]}\left(\frac{\omega^{\prime}}{\omega}\right)\frac{|F(\omega)-F(\omega^{\prime})|^{2}}{|\omega^{\prime}-\omega|^{-2\beta}}(\omega\omega^{\prime})^{-(2\beta+1)/2}d\omega
d\omega^{\prime}.\end{split}$

Note that the seminorm (1.5) is reminiscent of the fractional Sobolev norm in
its integral formulation111 Recall that
$\|u\|_{H^{s}(\mathbb{R})}^{2}=\|u\|_{L^{2}}^{2}+\iint\frac{|u(x)-u(y)|^{2}}{|x-y|^{1+2s}}\,dx\,dy\qquad\mbox{if
$s<\frac{1}{2}$},$ see for example, [15]..

Next, we state the following local well-posedness theorem.

######  Theorem 1.4.

For $p_{0},\beta$ satisfying (5.5), the Cauchy problem for (5.2) with initial
data $N_{0}\in X$, $\inf_{\omega\in(0,\infty)}N_{0}(\omega)>0$ admits the
unique strong solution $N\in L^{\infty}(0,T;X)$ for a time
$T=T(N_{0},p_{0},\beta)>0$. Moreover,
$\inf_{\omega\in(0,\infty),t\in(0,T)}N(t,\omega)>0$

The solution is either understood in the mild sense, or in the sense of
pointwise equality $\partial_{t}N=\overline{\mathcal{C}}(N)$, which holds
almost everywhere in $t$ for each $\omega>0$. Indeed, the field $N$ is
continuous and almost everywhere differentiable.

Theorem 1.3 reveals an unexpected nonlinear smoothing mechanism which is tied
to the structure of the collision operator. Such a phenomenon is already known
to occur in some Boltzmann type equations (see the discussion below), but it
came as a surprise in the context of 1D kinetic wave equation. We believe it
might open interesting research directions.

On the other hand, we remark the gap between Theorem 1.4 and Theorem 1.3:
Theorem 1.4 does not conclude the gain of regularity (5.65) and (5.66). As one
can see from the proof, the main reason is that we do not know how to obtain
further regularity from these gains: more precisely, we only have weak
convergence of derivatives of approximate solutions, not pointwise. To extract
pointwise convergence, we need to employ regularity properties of the
derivative, but it is not clear at this point how to do this. Development of
regularity theory for the kinetic MMT equation would enable us to recover
(5.65) and (5.66).

###  1.5. Perspectives

Analogy with non-cutoff Boltzmann situation. There is a very interesting
analogy with the regularizing effect of Boltzmann equation with non-cutoff
kernel (for example, [1], [2], [21] and also [23] for relativistic Boltzmann.)
In addition to the global well-posedness in smooth, small data regime, the
gain of regularity has been shown in [21]. The gain of regularity was measured
in the following anisotropic norm:

(1.6) $|f|_{N^{s,\gamma}}^{2}:=\int_{\mathbb{R}^{n}}\langle
v\rangle^{\gamma+2s}|f(v)|^{2}dv+\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}(\langle
v\rangle\langle
v^{\prime}\rangle)^{\frac{\gamma+2s+1}{2}}\frac{|f(v)-f(v^{\prime})|^{2}}{d(v,v^{\prime})^{n+2s}}\mathbf{1}_{d(v,v^{\prime})\leq
1}dvdv^{\prime},$

where $\langle v\rangle=\sqrt{1+|v|^{2}}$ and

$d(v,v^{\prime})=\sqrt{|v-v^{\prime}|^{2}+\frac{1}{4}(|v|^{2}-|v^{\prime}|^{2})^{2}}$

(see [21] for further details.) They proved that if initial data is in a
certain weighted Sobolev space (in $x,v$ variable), then not only solution
propagates in that space, its derivatives gain the regularity measured by
(1.6). The norm in (1.6) is quite analogous to the norm (1.5) presented in the
paper. Besides, in a series of works including [1] and [2], the authors showed
that the solution in fact acquires $\mathcal{C}^{\infty}$-regularity.
Therefore, it would be naturally interesting to ask if similar
$\mathcal{C}^{\infty}$-smoothing effect could be shown for the kinetic wave
equation as well.

Smoothness of solutions. Based on the regularizing effect presented above, it
seems natural to conjecture that the solution to (KWE) becomes positive and
smooth for positive time - as long as it exists.

Global solutions. Obtaining global solutions for (KWE) seems like a very
challenging question. Indeed, the kernel of the collision operator is very
singular, and this seems to preclude any possibility to make sense of it for
locally $L^{1}$ fields, let alone prove local well-posedness at such a low
regularity.

###  1.6. Notations

For two quantities $A,B$, we write $A\lesssim B$ if there exists a constant
$C>0$ such that $A\leq CB$. If $A\lesssim B$ and $B\lesssim A$, we denote
$A\sim B$

### Acknowledgements.

When preparing this article, Pierre Germain was supported by the Simons
Foundation Collaboration on Wave Turbulence, a start up grant from Imperial
College, and a Wolfson fellowship. Joonhyun La was supported by June Huh
fellowship from KIAS. Katherine Zhiyuan Zhang was supported by the Simons
Foundation Collaboration on Wave Turbulence and by an AMS-Simons travel grant.
The authors thank Jin Woo Jang for fruitful discussions.

##  2\. First properties of the equation

In this section, we review some of the formal properties of the equation:
scaling, monotone quantities, explicit solutions, and deduce the expected
behavior of the equation for various ranges of parameters.

###  2.1. Monotone quantities

For any function $F(k)$, we get after symmetrizing that

$\displaystyle\frac{d}{dt}\int F(k)n_{k}\,dk$
$\displaystyle=-\frac{1}{4}\int_{\mathbb{R}^{4}}|k_{1}k_{2}k_{3}k|^{\beta/2}(n_{1}n_{2}n_{3}+n_{1}n_{2}n_{k}-n_{1}n_{3}n_{k}-n_{2}n_{3}n_{k})$
$\displaystyle\qquad\delta(k_{1}+k_{2}-k_{3}-k)\delta(\omega_{1}+\omega_{2}-\omega_{3}-\omega)(F_{1}+F_{2}-F_{3}-F)\,dk_{1}\,dk_{2}\,dk_{3}\,dk.$

Choosing $F(k)=1,k,|k|^{\alpha}$, this gives the conservation of the mass,
momentum, and energy

$\int n_{k}\,dk,\qquad\int kn_{k}\,dk,\qquad\int\omega_{k}n_{k}\,dk.$

Similarly, we get after symmetrizing that

$\displaystyle\frac{d}{dt}\int
F(n_{k})\,dk=-\frac{1}{4}\int_{\mathbb{R}^{4}}|k_{1}k_{2}k_{3}k|^{\beta/2}(n_{1}n_{2}n_{3}+n_{1}n_{2}n_{k}-n_{1}n_{3}n_{k}-n_{2}n_{3}n_{k})$
$\displaystyle\qquad\delta(k_{1}+k_{2}-k_{3}-k)\delta(\omega_{1}+\omega_{2}-\omega_{3}-\omega)(F^{\prime}(n_{1})+F^{\prime}(n_{2})-F^{\prime}(n_{3})-F^{\prime}(n_{k}))\,dk_{1}\,dk_{2}\,dk_{3}\,dk.$

Choosing $F(n)=\log n$, this gives the H-theorem

$\displaystyle\frac{d}{dt}\int\log n_{k}\,dk\geq 0.$

###  2.2. Changing the independent variable from $k$ to $\omega$

It is customary and convenient to adopt $\omega=\sqrt{k}$ as the independent
variable. For notational simplicity in this new set of coordinates, we will
assume from now on that $n(k)$ is even, so that we can choose $n(\omega)$,
$\omega>0$, as the unknown function.

We will still denote $n$ and $\mathcal{C}$ for the unknown function, and the
collision operator; it will be obvious from the context whether these are
considered as functions of $k$ or $\omega$.

Up to time rescaling by a constant, the equation is now given by

${\partial_{t}}n(t,\omega)=\mathcal{C}(n(t))(\omega),$

with the collision operator

(2.1)
$\begin{split}\mathcal{C}(n)(\omega)&=\sum_{\epsilon_{1},\epsilon_{2},\epsilon_{3}\in\\{\pm
1\\}}\int_{\mathbb{R}_{+}^{3}}\omega^{\frac{\beta}{2\alpha}}(\omega_{1}\omega_{2}\omega_{3})^{\frac{\beta}{2\alpha}-1+\frac{1}{\alpha}}(n_{1}n_{2}n_{3}+n_{1}n_{2}n_{\omega}-n_{1}n_{3}n_{\omega}-n_{2}n_{3}n_{\omega})\\\
&\qquad\qquad\qquad\delta(\epsilon_{1}\omega_{1}^{1/\alpha}+\epsilon_{2}\omega_{2}^{1/\alpha}+\epsilon_{3}\omega_{3}^{1/\alpha}-\omega^{1/\alpha})\delta(\omega_{1}+\omega_{2}-\omega_{3}-\omega)\,d\omega_{1}\,d\omega_{2}\,d\omega_{3},\end{split}$

where we denoted

$\displaystyle n_{\omega}=n(t,\omega),\quad n_{i}=n(t,\omega_{i}),\quad
i=1,2,3.$

In these new coordinates, the conserved quantities become

$\int_{\mathbb{R}_{+}}n(\omega)\omega^{\frac{1}{\alpha}-1}\,d\omega,\qquad\int_{\mathbb{R}_{+}}n(\omega)\omega^{\frac{1}{\alpha}}\,d\omega$

(the momentum being identically zero by parity).

###  2.3. Scaling considerations

####  2.3.1. Scaling around the origin

A computation shows that the collision operator satisfies the scaling relation

$\mathcal{C}(n(\lambda\cdot))(\omega)=\lambda^{-2\frac{\beta}{\alpha}+1-\frac{2}{\alpha}}\mathcal{C}(n)(\lambda\omega).$

Thus, the following transformation leaves the set of solutions invariant

$n(t,\omega)\mapsto\lambda^{\frac{\beta}{\alpha}+\frac{1}{\alpha}-\frac{1}{2}}n(t,\lambda\omega)$

(there are also scaling transformations involving the time variable, but they
are less relevant, and we ignore them here).

####  2.3.2. Formation of a condensate

This scaling law suggests that the formation of a condensate at frequency 0 is
only possible if $2\beta<\alpha$.

The heuristic argument is as follows: suppose that, along a sequence of times
$(t_{n})$ converging to 1, $n(\omega)$ converges (locally) to a Dirac $\delta$
at the origin. If this happens in a self-similar way, there exists a profile
$\psi$ such that $n(t_{n},\omega)=\ell_{n}^{1/\alpha}\psi(\ell_{n}\omega)$,
where $\ell_{n}\to\infty$. If the profile $\psi$ is, say, smooth, there exists
a smooth solution $N$ on $[0,T]$, with $T>0$, of (KWE) with data $\psi$. We
now use the scaling law above to get a solution on $[t_{n},t_{n}+T]$ for (KWE)
with data
$\ell_{n}^{\frac{\beta}{\alpha}+\frac{1}{\alpha}-\frac{1}{2}}\psi(\ell_{n}\omega)$.
As long as
$\frac{1}{\alpha}<\frac{\beta}{\alpha}+\frac{1}{\alpha}-\frac{1}{2}$, which is
equivalent to $2\beta>\alpha$, the smaller data
$\ell_{n}^{1/\alpha}\psi(\ell_{n}\omega)$ should also have a smooth solution
on $[t_{n},t_{n}+T]$; but this contradicts the formation of a singularity at
time $1$.

###  2.4. Stationary solutions

A family of stationary solutions is provided by the Rayleigh-Jeans spectra

$\frac{1}{c_{1}|k|^{\alpha}+c_{2}}=\frac{1}{c_{1}\omega+c_{2}},\qquad
c_{1},c_{2}\geq 0.$

While these solutions correspond to equilibria, the Kolmogorov-Zakharov
spectra give (direct or inverse) cascades of mass or energy. They are the
following power laws [24, 32]

$\omega^{-\frac{2\beta+3-\alpha}{3\alpha}},\qquad\omega^{-\frac{2\beta+3}{3\alpha}}.$

##  3\. The collision operator for $\alpha=\frac{1}{2}$

From now on, we focus on the case $\alpha=1/2$. The main advantage of this
particular case is that it becomes possible to find an explicit
parameterization of the collision operator, which will simplify its treatment.
The aim of this section is to derive this explicit parameterization; we will
also explain carefully how to give a precise meaning to the expression giving
the collision operator.

Recall that the collision operator is given by

(3.1)
$\begin{split}\mathcal{C}(n)(\omega)&=\sum_{\epsilon_{1},\epsilon_{2},\epsilon_{3}\in\\{\pm
1\\}}\int_{\mathbb{R}_{+}^{3}}\omega^{\beta}(\omega_{1}\omega_{2}\omega_{3})^{\beta+1}(n_{1}n_{2}n_{3}+n_{1}n_{2}n_{\omega}-n_{1}n_{3}n_{\omega}-n_{2}n_{3}n_{\omega})\\\
&\qquad\qquad\qquad\delta(\epsilon_{1}\omega_{1}^{2}+\epsilon_{2}\omega_{2}^{2}+\epsilon_{3}\omega_{3}^{2}-\omega^{2})\delta(\omega_{1}+\omega_{2}-\omega_{3}-\omega)\,d\omega_{1}\,d\omega_{2}\,d\omega_{3},\end{split}$

###  3.1. Around the coarea formula

The integral giving $\mathcal{C}$ features a product of $\delta$ functions; to
show that it is well-defined, and interpret it appropriately, the coarea
formula turns out to be the key tool. The discussion will be simplified by
introducing some shorthands: denoting

$\displaystyle
F_{1}(\omega_{1},\omega_{2},\omega_{3})=\epsilon_{1}\omega_{1}^{2}+\epsilon_{2}\omega_{2}^{2}+\epsilon_{3}\omega_{3}^{2}-\omega^{2}$
$\displaystyle
F_{2}(\omega_{1},\omega_{2},\omega_{3})=\omega_{1}+\omega_{2}-\omega_{3}-\omega$
$\displaystyle\mathcal{I}(\omega_{1},\omega_{2},\omega_{3})=n_{1}n_{2}n_{3}+n_{1}n_{2}n_{\omega}-n_{1}n_{3}n_{\omega}-n_{2}n_{3}n_{\omega},$

we are dealing with a sum of terms of the type

(3.2)
$\int\mathcal{I}(\omega_{1},\omega_{2},\omega_{3})\delta(F_{1}(\omega_{1},\omega_{2},\omega_{3}))\delta(F_{2}(\omega_{1},\omega_{2},\omega_{3}))\,d\omega_{1}\,d\omega_{2}\,d\omega_{3}$

(thinking of $\omega$ as being fixed; the dependence of $F_{2}$ on
$\epsilon_{1},\epsilon_{2},\epsilon_{3}$ is implicit). Assuming that
$n\in\mathcal{C}_{0}^{\infty}(0,\infty)$ for simplicity (even though a
sufficiently fast decay at $0$ and $\infty$ for $\varphi$ would of course be
sufficient), the integrand $\mathcal{I}$ is smooth and compactly supported. By
the coarea formula222see the Encyclopedia of Mathematics,
http:encyclopediaofmath.orgindex.php?title=Coarea_formula&oldid=29308 , this
allows to write the above as

(3.3) $\eqref{aterm}=\int_{F_{1}^{-1}(\\{0\\})\cap
F_{2}^{-1}(\\{0\\})}\mathcal{I}(\omega_{1},\omega_{2},\omega_{3})\frac{1}{|J(F_{1},F_{2})|}d\mathcal{H}^{1},$

where the Jacobian of $(F_{1},F_{2})$ is given by

$J(F_{1},F_{2})=\left[\det\begin{pmatrix}|\nabla F_{1}|^{2}&\nabla
F_{1}\cdot\nabla F_{2}\\\ \nabla F_{1}\cdot\nabla F_{2}&|\nabla
F_{2}|^{2}\end{pmatrix}\right]^{\frac{1}{2}}$

the Jacobian of $(F_{1},F_{2})$ (which does not vanish), and
$d\mathcal{H}^{1}$ is the measure induced on $F_{1}^{-1}(\\{0\\})\cap
F_{2}^{-1}(\\{0\\})$ by the Lebesgue measure in $\mathbb{R}^{3}$. This is
well-defined as long as $\nabla F_{1}$ and $\nabla F_{2}$ do not align, or in
other words as long as the manifolds $\\{F_{1}=0\\}$ and $\\{F_{2}=0\\}$
intersect transversally.

The discussion above guarantees that the collision operator is well-defined;
but to compute its parametrization, it is more convenient to first use
$\delta(F_{2})$ to replace $\omega_{3}$ by $\omega_{1}+\omega_{2}-\omega$:

$\eqref{aterm}=\int\widetilde{\mathcal{I}}(\omega_{1},\omega_{2})\delta(\widetilde{F_{1}}(\omega_{1},\omega_{2}))\,d\omega_{1}\,d\omega_{2}.$

Here, the integration variables are $\omega_{1},\omega_{2}$, and
$\widetilde{\mathcal{I}}(\omega_{1},\omega_{2})=\mathcal{I}(\omega_{1},\omega_{2},\omega_{1}+\omega_{2}-\omega)$,
similarly for $\widetilde{F_{1}}$. By the coarea formula, this becomes

(3.4)
$\eqref{aterm}=\int_{\widetilde{F_{1}}^{-1}(\\{0\\})}\widetilde{\mathcal{I}}(\omega_{1},\omega_{2})\frac{1}{|\nabla\widetilde{F_{1}}|}d\mathcal{H}^{1}.$

Suppose now that $\varphi(u)=(\omega_{1}(u),\omega_{2}(u))$, $u\in[a,b]$ is a
parametrization of $\widetilde{F_{1}}^{-1}(0)$. Then we claim that

(3.5)
$\int_{\widetilde{F_{1}}^{-1}(0)}\widetilde{\mathcal{I}}(\omega_{1},\omega_{2})\frac{d\mathcal{H}^{1}}{|\nabla\widetilde{F_{1}}|}=\int_{a}^{b}\widetilde{\mathcal{I}}(u)\frac{|\omega_{1}^{\prime}(u)|}{|\partial_{\omega_{2}}\widetilde{F_{1}}|}du=\int_{a}^{b}F(u)\frac{|\omega_{2}^{\prime}(u)|}{|\partial_{\omega_{1}}\widetilde{F_{1}}|}du.$

We first note that
$\frac{d\mathcal{H}^{1}}{|\nabla\widetilde{F_{1}}|}=\frac{|\varphi^{\prime}(u)|}{{|\nabla\widetilde{F_{1}}|}}du$.
However,
$\frac{d}{du}\widetilde{F_{1}}(\omega_{1}(u),\omega_{2}(u))=\varphi^{\prime}\cdot\nabla\widetilde{F_{1}}=0$
since $\varphi$ is embedded in the level set of $\widetilde{F_{1}}$.
Therefore, $\nabla\widetilde{F_{1}}$ is parallel to
$(\varphi^{\prime})^{\perp}=(-\omega_{2}^{\prime},\omega_{1}^{\prime})$, and
in particular,

$\frac{|\varphi^{\prime}(u)|}{|\nabla\widetilde{F_{1}}|}=\frac{|\varphi^{\prime}(u)|^{2}}{|(\varphi^{\prime})^{\perp}\cdot\nabla\widetilde{F_{1}}|}.$

However, from $\varphi^{\prime}\cdot\nabla\widetilde{F_{1}}=0$ we have
$(\varphi^{\prime})^{\perp}\cdot\nabla\widetilde{F_{1}}=\frac{|\varphi^{\prime}(u)|^{2}}{\omega_{1}^{\prime}(u)}\partial_{\omega_{2}}\widetilde{F_{1}}$,
and the result follows.

###  3.2. Trivial resonances

First, we claim that the above expression vanishes in either of the following
cases

$\displaystyle(\epsilon_{1},\epsilon_{2},\epsilon_{3})=(-,-,-),\;(+,-,+),\;(-,+,+),\;(+,+,-).$

This follows from examining the resonant manifold

(3.6)
$\epsilon_{1}\omega_{1}^{2}+\epsilon_{2}\omega_{2}^{2}+\epsilon_{3}\omega_{3}^{2}-\omega^{2}=\omega_{1}+\omega_{2}-\omega_{3}-\omega=0,\qquad\omega_{1},\omega_{2},\omega_{3},\omega\geq
0.$

Indeed:

  1. (1)

If $(\epsilon_{1},\epsilon_{2},\epsilon_{3})=(-,-,-)$, then
$\omega_{1}^{2}+\omega_{2}^{2}+\omega_{3}^{2}+\omega^{2}=0$ only has a trivial
solution.

  2. (2)

If $(\epsilon_{1},\epsilon_{2},\epsilon_{3})=(+,-,+)$, then (3.6) implies that
$(\omega_{1},\omega_{2})=(\omega,\omega_{3})$, in which case the integrand of
(3.1) vanishes.

  3. (3)

The case $(\epsilon_{1},\epsilon_{2},\epsilon_{3})=(-,+,+)$ is symmetrical.

  4. (4)

If $(\epsilon_{1},\epsilon_{2},\epsilon_{3})=(+,+,-)$, then (3.6) implies that
$\\{\omega_{1},\omega_{2}\\}=\\{\omega,\omega_{3}\\}$, and once again the
integrand of (3.1) vanishes.

This discussion was mostly formal; we now resort to (3.3) to give it a more
rigorous framework. In the case (2), it is easy to see that the gradients of
$F_{1}(\omega_{1},\omega_{2},\omega_{3})$ and
$F_{2}(\omega_{1},\omega_{2},\omega_{3})=\omega_{1}^{2}-\omega_{2}^{2}+\omega_{3}^{2}-\omega^{2}$
are never colinear on the resonant manifold where
$(\omega_{1},\omega_{2})=(\omega_{3},\omega_{4})$. The above conclusion is
therefore verified.

However, in the case (4), the gradients of
$F_{1}(\omega_{1},\omega_{2},\omega_{3})$ and
$F_{2}(\omega_{1},\omega_{2},\omega_{3})=\omega_{1}^{2}+\omega_{2}^{2}-\omega_{3}^{2}-\omega^{2}$
are colinear at the point $\omega_{1}=\omega_{2}=\omega_{3}=\omega_{4}$, which
belongs to the resonant manifold. Furthermore, the singularity caused by the
Jacobian in the denominator in (3.3) is not integrable; even though the
integrand is zero, this term appears ill-defined. To settle matters, we will
use a regularization procedure, and show that it has zero limit.

More precisely, we will interpret
$\delta(\omega_{1}+\omega_{2}-\omega_{3}-\omega_{4})$ as

$\delta(\omega_{1}+\omega_{2}-\omega_{3}-\omega_{4})=\lim_{\delta\rightarrow
0^{+}}\frac{1}{2\delta}\mathbf{1}_{(-\delta,\delta)}(\omega_{1}+\omega_{2}-\omega_{3}-\omega_{4}).$

We will furthermore denote

$\omega_{1}+\omega_{2}-\omega_{3}-\omega_{4}=t$

and, under this constraint,

$\Phi_{t}=\omega_{1}^{2}+\epsilon_{2}\omega_{2}^{2}-\epsilon_{3}\omega_{3}^{2}-\omega^{2}$

is a function of two of $(\omega_{1},\omega_{2},\omega_{3})$. By the coarea
formula, we are thus led to investigate the limit as $\delta$ goes to zero of

$\lim_{\delta\rightarrow
0^{+}}\frac{1}{2\delta}\int_{-\delta}^{\delta}\int_{\Phi_{t}^{-1}(0)}(\omega_{1}\omega_{2}\omega_{3})^{\beta+1}\left[n_{1}n_{2}n_{3}+n_{1}n_{2}n_{\omega}-n_{1}n_{3}n_{\omega}-n_{2}n_{3}n_{\omega}\right]\frac{d\mathcal{H}^{1}}{|\nabla_{\omega_{i}}\Phi_{t}|}dt,$

assuming for simplicity that $n\in\mathcal{C}_{0}^{\infty}$, where
$d\mathcal{H}^{1}$ is the measure for the curve
$\\{\Phi_{t}=0\\}\cap\\{\sum_{i=1}^{4}(-1)^{i}\omega_{i}=t\\}$ and
$\nabla_{\omega_{i}}\Phi_{t}$ is the derivative of $\Phi_{t}$ with respect to
the two variables on which it depends.

The following formula will prove useful: if $(\omega_{1}(u),\omega_{2}(u))$ is
a parameterization of $\Phi_{t}^{-1}(\\{0\\})$, then

$\int_{\Phi_{t}^{-1}(0)}F(\omega_{1},\omega_{2})\frac{d\mathcal{H}^{1}}{|\nabla_{\omega_{1},\omega_{2}}\Phi_{t}|}=\int_{a}^{b}F(u)\frac{|\omega_{1}^{\prime}(u)|}{|\partial_{\omega_{2}}\Phi_{t}|}du=\int_{a}^{b}F(u)\frac{|\omega_{2}^{\prime}(u)|}{|\partial_{\omega_{1}}\Phi_{t}|}du;$

it can be proved like (3.5). Observe that

(3.7) $|\nabla_{\omega_{2}}\Phi_{t}|=2|\omega_{2}-\omega_{3}|.$

We have the following:

(3.8) $\begin{split}|\omega_{2}-\omega_{3}|&=|\omega_{1}-\omega-t|,\\\
|\omega_{2}^{2}-\omega_{3}^{2}|&=|\omega_{1}^{2}-\omega^{2}|,\end{split}$

which implies the following:

(3.9)
$|\omega_{1}-\omega-t|=\frac{\omega_{1}+\omega}{\omega_{2}+\omega_{3}}|\omega_{1}-\omega|.$

An important consequence is that since $n$ is compactly supported,
$|\omega_{1}-\omega-t|$ cannot be too small: if $|\omega_{1}-\omega-t|<ct$ for
a sufficiently small $c<1$, $|\omega_{1}-\omega|>C^{\prime}t$ for some
$C^{\prime}>1/2$, so
$\omega_{2}+\omega_{3}=|\omega_{1}-\omega|(\omega_{1}+\omega)/|\omega_{1}-\omega-t|>C^{\prime\prime}\omega$:
so if $c<1$ is sufficiently small, both $\omega_{2}$ and $\omega_{3}$ will lie
outside the support of $n$ (since
$|\omega_{2}-\omega_{3}|=|\omega_{1}-\omega-t|$ small), and thus
$\sum_{j=1}^{4}(-1)^{j}\frac{\prod_{i=1}^{4}n_{i}}{n_{j}}=0$ there.
Furthermore, $|\omega_{1}-\omega|$ cannot be too large as well: we have

(3.10)
$(\omega_{1}-\omega)=\begin{cases}\frac{(\omega_{2}+t)^{2}-\omega_{2}^{2}}{2(\omega+\omega_{2}+t)},\text{
case }\qquad(+,+,-),\\\
\frac{\omega_{2}^{2}-(\omega_{2}+t)^{2}}{2(\omega_{1}-\omega_{2}-t)},\text{
case }\qquad(+,-,+),\end{cases}$

from the following calculation:

(3.11)
$\begin{split}0&=\omega_{1}^{2}+\omega_{2}^{2}-\omega_{3}^{2}-\omega^{2}\\\
&=(\omega_{1}-\omega)(\omega_{1}+\omega)-(\omega_{1}-\omega-t)(2\omega_{3}+\omega+t-\omega_{1}),\\\
-(2\omega_{2}t+t^{2})&=(\omega_{1}-\omega)\big{[}2\omega_{1}-2(\omega_{3}+t)\big{]}\end{split}$

which gives the result. If
$|\omega_{1}-\omega_{2}-t|\geq|\omega_{3}-\omega_{2}-t|$, since
$\omega_{2}-\omega_{3}-t=-\omega-\omega_{1}$, we have

(3.12) $|\omega_{1}-\omega|\leq\frac{C|t|}{|\omega_{1}-\omega|},$

that is, $|\omega_{1}-\omega|\leq C\sqrt{|t|}$. In both cases, for a
sufficiently small $t$, we have $|\omega_{1}-\omega-t|\leq C\sqrt{|t|}$.
Therefore, if $(\omega_{1},\omega_{2},\omega_{3})\in\Phi_{t}^{-1}(0)$, then
only those $\omega_{1}$ such that

(3.13) $c|t|<|\omega_{1}-\omega-t|<C\sqrt{|t|}$

yield nonzero integrands. On the other hand, we have

(3.14)
$\left|n_{1}n_{2}n_{3}+n_{1}n_{2}n_{\omega}-n_{1}n_{3}n_{\omega}-n_{2}n_{3}n_{\omega}\right|\leq
n_{1}n_{4}|n_{2}-n_{3}|+n_{2}n_{3}|n_{1}-n_{4}|\leq
C(|\omega_{1}-\omega-t|+|t|).$

Finally, we remark that from the upper bound in (3.13), both $\omega$ and
$\omega_{1}$ are in $[c^{-1},c]$ for some $c>1$ for
$(\omega_{1},\omega_{2},\omega_{3})\in\Phi_{t}^{-1}(0)$ for small $t$. This
implies that $|\omega_{3}-\omega_{2}|=|\omega_{1}-\omega-t|\in[0,C]$ for some
$C>0$, and therefore either $\omega_{3},\omega_{2}\in[0,C]$ for some $C>0$ or
$\omega_{3},\omega_{2}$ is greater than the upper bound for $\text{supp }n$.

Summarizing the above, we see that if we parametrize $\Phi_{t}^{-1}(0)$ with
$\omega_{1}$, we have

(3.15)
$\begin{split}&\left|\int_{\Phi_{t}^{-1}(0)}(\omega_{1}\omega_{2}\omega_{3})^{\beta+1}\left[n_{1}n_{2}n_{3}+n_{1}n_{2}n_{\omega}-n_{1}n_{3}n_{\omega}-n_{2}n_{3}n_{\omega}\right]\frac{d\mathcal{H}^{1}}{|\nabla_{\omega_{i}}\Phi_{t}|}\right|\\\
&\qquad\lesssim\int_{c|t|<|\omega_{1}-\omega-t|<C\sqrt{|t|}}\frac{|\omega_{1}-\omega-t|+|t|}{|\omega_{1}-\omega-t|}d\omega_{1}<C\sqrt{|t|},\end{split}$

which gives the desired result after integrating over $[-\delta,\delta]$ and
letting $\delta\to 0$.

###  3.3. Non-trivial resonances: the Zakharov-Guyenne-Pushkarev-Dias
parameterization

In the previous subsection, we learned that trivial resonances can be
eliminated from the sum defining $\mathcal{C}$, which becomes therefore

(3.16)
$\begin{split}\partial_{t}n_{\omega}&=\int_{\mathbb{R}_{+}^{3}}\omega^{\beta}(\omega_{1}\omega_{2}\omega_{3})^{\beta+1}(n_{1}n_{2}n_{3}+n_{1}n_{2}n_{\omega}-n_{1}n_{3}n_{\omega}-n_{2}n_{3}n_{\omega})\\\
&\qquad\cdot\delta(\omega_{1}+\omega_{2}-\omega_{3}-\omega)[\delta(\omega_{1}^{2}+\omega_{2}^{2}+\omega_{3}^{2}-\omega^{2})+\delta(\omega_{1}^{2}+\omega_{2}^{2}-\omega_{3}^{2}+\omega^{2})\\\
&\qquad\qquad+\delta(\omega_{1}^{2}-\omega_{2}^{2}+\omega_{3}^{2}+\omega^{2})+\delta(-\omega_{1}^{2}+\omega_{2}^{2}+\omega_{3}^{2}+\omega^{2})]\,d\omega_{1}\,d\omega_{2}\,d\omega_{3}\\\
&=\mathcal{C}_{1}(n,n,n)+\mathcal{C}_{2}(n,n,n)+\mathcal{C}_{3}(n,n,n)+\mathcal{C}_{4}(n,n,n).\\\
\end{split}$

In the present subsection, we turn to obtaining a parameterization for the
collision operator: we will show that

(3.17)
$\mathcal{C}=\mathcal{S}_{1}+\mathcal{S}_{2}+\mathcal{S}_{3}+\mathcal{S}_{4}$

where

$\displaystyle\mathcal{S}_{1}(n)(\omega)$
$\displaystyle=\omega^{4\beta+3}\int_{0}^{1}\frac{u^{2\beta+2}(u+1)^{2\beta+2}}{(1+u+u^{2})^{3\beta+4}}(f_{1}f_{2}f_{3}+f_{1}f_{2}f_{\omega}-f_{1}f_{\omega}f_{3}-f_{\omega}f_{2}f_{3})\,du$
$\displaystyle\mathcal{S}_{2}(n)(\omega)$
$\displaystyle=\omega^{4\beta+3}\int_{0}^{1}\frac{(1+u+u^{2})^{\beta+1}(u+1)^{2\beta+2}}{u^{2\beta+3}}(f_{1}f_{2}f_{3}+f_{1}f_{2}f_{\omega}-f_{1}f_{\omega}f_{3}-f_{\omega}f_{2}f_{3})\,du$
$\displaystyle\mathcal{S}_{3}(f)(\omega)$
$\displaystyle=\omega^{4\beta+3}\int_{0}^{1}\frac{(1+u+u^{2})^{\beta+1}u^{2\beta+2}}{(u+1)^{2\beta+3}}(f_{1}f_{2}f_{3}+f_{1}f_{2}f_{\omega}-f_{1}f_{\omega}f_{3}-f_{\omega}f_{2}f_{3})\,du$
$\displaystyle\mathcal{S}_{4}(f)(\omega)$
$\displaystyle=\omega^{4\beta+3}\int_{0}^{1}\frac{(1+u+u^{2})^{\beta+1}}{u^{2\beta+3}(u+1)^{2\beta+3}}(f_{1}f_{2}f_{3}+f_{1}f_{2}f_{\omega}-f_{1}f_{\omega}f_{3}-f_{\omega}f_{2}f_{3})\,du.$

For each of the above integrals, the notation $f_{i}$ stands for
$f(u_{i}\omega)$, where the $u_{i}$ are given as follows

$\displaystyle\mbox{for $\mathcal{S}_{1}$}:\quad
u_{1}=\frac{1+u}{1+u+u^{2}},\quad u_{2}=\frac{u(1+u)}{1+u+u^{2}},\quad
u_{3}=\frac{u}{1+u+u^{2}}$ $\displaystyle\mbox{for $\mathcal{S}_{2}$}:\quad
u_{1}=u+1,\quad u_{2}=\frac{u+1}{u},\quad u_{3}=\frac{1+u+u^{2}}{u}$
$\displaystyle\mbox{for $\mathcal{S}_{3}$}:\quad
u_{1}=\frac{1+u+u^{2}}{u+1},\quad u_{2}=\frac{u}{u+1},\quad u_{3}=u$
$\displaystyle\mbox{for $\mathcal{S}_{4}$}:\quad
u_{1}=\frac{1+u+u^{2}}{(u+1)u},\quad u_{2}=\frac{1}{u+1},\quad
u_{3}=\frac{1}{u}.$

######  Remark 3.1.

Note that the some misprints occured in the formulas in [31] and [32]: In
$\mathcal{S}_{1}$, a factor of $\frac{1}{2}(u+1)$ is missing. In
$\mathcal{S}_{2}$, a factor of $(u+1)$ is missing. In the formula for
$\mathcal{S}_{4}$ in [32], there is a mistake in the second term in the
integral and a factor of $\frac{1}{u+1}$ is missing in the front. In [31],
this factor of $\frac{1}{u+1}$ is also missing in the formula for
$\mathcal{S}_{4}$.

Before delving into the computations which will yield the above formulas,
notice that in all four cases making up nontrivial resonances, namely

$(\epsilon_{1},\epsilon_{2},\epsilon_{3})=(+,+,+),\;(-,-,+),\;(-,+,-),\;(+,-,-),$

the gradients of $F_{1}$ and $F_{2}$ are never aligned (this follows from a
simple computation relying on the parametrization of the resonant manifold
provided in the following subsections). Therefore, the corresponding integral
is well-defined through (3.3); but we will rather rely on (3.5) to derive
explicit expressions.

####  3.3.1. The case $(+,+,+)$

The first step is to consider the equations

(3.18) $\left\\{\begin{array}[]{l}u_{3}+1=u_{1}+u_{2}\\\
u_{1}^{2}+u_{2}^{2}+u_{3}^{2}=1.\end{array}\right.\qquad u_{1},u_{2},u_{3}\geq
0,\qquad u_{2}\leq u_{1}.$

The solutions in $\mathbb{R}_{+}^{3}$ of this system can be parameterized as
follows333To check this statement, first consider the system
$\begin{cases}u_{1}+u_{2}=a\\\ u_{1}^{2}+u_{2}^{2}=b\end{cases}$. It has
solutions in $\mathbb{R}_{+}^{2}$ if and only if $a\geq 0$ and $2b\geq
a^{2}\geq b$. For us, $a=1+u_{3}$, $b=1-u_{3}^{2}$, and the conditions $a\geq
0$, $2b\geq a^{2}\geq b$, $u_{3}\geq 0$ are equivalent to $0\leq
u_{3}\leq\frac{1}{3}$. In this range, $u_{3}$ can be parameterized by
$\frac{u}{1+u+u^{2}}$, with $u\in[0,1]$. Finally, we check that the proposed
parameterization (3.19) works.

(3.19) $u_{1}=\frac{1+u}{1+u+u^{2}},\quad u_{2}=\frac{u(1+u)}{1+u+u^{2}},\quad
u_{3}=\frac{u}{1+u+u^{2}}\quad 0\leq u\leq 1.$

We start with the expression

$\int\mathcal{I}(\omega_{1},\omega_{2},\omega_{3})\delta(F_{1}(\omega_{1},\omega_{2},\omega_{3}))\delta(F_{2}(\omega_{1},\omega_{2},\omega_{3}))\,d\omega_{1}\,d\omega_{2}\,d\omega_{3},$

where $F_{1}=\omega_{1}^{2}+\omega_{2}^{2}+\omega_{3}^{2}-\omega^{2}$.
Changing coordinates by setting $\omega_{i}=\zeta_{i}\omega$, the above
becomes

$\int\mathcal{I}(\omega_{1},\omega_{2},\omega_{3})\delta(F_{1}(\zeta_{1},\zeta_{2},\zeta_{3}))\delta(F_{2}(\zeta_{1},\zeta_{2},\zeta_{3}))\,d\zeta_{1}\,d\zeta_{2}\,d\zeta_{3};$

using then $\delta(F_{2})$ to eliminate the variable
$\zeta_{3}=\zeta_{1}+\zeta_{2}-1$, we obtain

$\displaystyle\int\widetilde{\mathcal{I}}(\omega_{1},\omega_{2})\delta(\widetilde{F_{1}}(\zeta_{1},\zeta_{2}))\,d\zeta_{1}\,d\zeta_{2},$

where
$\widetilde{F_{1}}(\zeta_{1},\zeta_{2})=F_{1}(\zeta_{1},\zeta_{2},\zeta_{1}+\zeta_{2}-1)$.
Here, it is possible to restrict to the case $\zeta_{1}\geq\zeta_{2}$, since
the case $\zeta_{2}\geq\zeta_{1}$ is symmetric and will be accounted for by a
factor 2. We now want to apply (3.4) and (3.5) with
$\varphi(u)=(u_{1},u_{2})$. A small computation shows that

$\displaystyle\partial_{2}\widetilde{F_{1}}=2\zeta_{1}+4\zeta_{2}-2=\frac{2u^{2}+4u}{1+u+u^{2}},\;\;\;\;|\varphi_{1}^{\prime}(u)|=\frac{u^{2}+2u}{1+u+u^{2}}\;\implies\;\left|\frac{\varphi_{1}^{\prime}(u)}{\partial_{2}\widetilde{F_{1}}}\right|=\frac{1}{2(1+u+u^{2})}.$

Therefore,

(3.20) $\begin{split}\mathcal{S}_{1}(f,f,f)&=\mathcal{C}_{1}(f,f,f)\\\
&=2\omega^{4\beta+3}\int_{0}^{1}\left(\frac{1+u}{1+u+u^{2}}\right)^{\beta+1}\left(\frac{u}{1+u+u^{2}}\right)^{\beta+1}\left(\frac{u(1+u)}{1+u+u^{2}}\right)^{\beta+1}\\\
&\qquad\qquad\qquad\cdot(f_{1}f_{2}f_{3}+f_{1}f_{2}f_{\omega}-f_{1}f_{\omega}f_{3}-f_{\omega}f_{2}f_{3})\frac{1}{2(1+u+u^{2})}du\\\
&=\omega^{4\beta+3}\int_{0}^{1}\frac{u^{2\beta+2}(u+1)^{2\beta+2}}{(1+u+u^{2})^{3\beta+4}}(f_{1}f_{2}f_{3}+f_{1}f_{2}f_{\omega}-f_{1}f_{\omega}f_{3}-f_{\omega}f_{2}f_{3})du,\end{split}$

####  3.3.2. The case $(-,-,+)$

We start with the equations

(3.21) $\left\\{\begin{array}[]{l}u_{3}+1=u_{1}+u_{2}\\\
u_{3}^{2}=u_{1}^{2}+u_{2}^{2}+1\end{array},\qquad u_{0},u_{1},u_{2}\geq
0,\qquad u_{1}\leq u_{2}.\right.$

Since the application
$(u_{1},u_{2},u_{3})\mapsto(\frac{u_{1}}{u_{3}},\frac{u_{2}}{u_{3}},\frac{1}{u_{3}})$
maps solutions of (3.21) to solutions of (3.18), we see that the solutions in
$\mathbb{R}_{+}^{3}$ of (3.21) can be parameterized by

$u_{1}=u+1,\quad u_{2}=\frac{u+1}{u},\quad u_{3}=\frac{1+u+u^{2}}{u}$

with $0\leq u\leq 1$. As in the previous paragraph, we write

$\displaystyle\int\mathcal{I}(\omega_{1},\omega_{2},\omega_{3})\delta(F_{1}(\omega_{1},\omega_{2},\omega_{3}))\delta(F_{2}(\omega_{1},\omega_{2},\omega_{3}))\,d\omega_{1}\,d\omega_{2}\,d\omega_{3}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad=\int\widetilde{\mathcal{I}}(\omega_{1},\omega_{2})\delta(\widetilde{F_{1}}(\zeta_{1},\zeta_{2}))\,d\zeta_{1}\,d\zeta_{2},$

except that $F_{1}(\omega_{1},\omega_{2},\omega_{3})$ is now given by
$-\omega_{1}^{2}-\omega_{2}^{2}+\omega_{3}^{2}+\omega^{2}$; and just like in
the previous paragraph, we can restrict matters to the case
$\zeta_{1}\leq\zeta_{2}$ by multiplying by a factor 2. Before applying (3.4)
and (3.5) with $\varphi(u)=(\varphi_{1}(u),\varphi_{2}(u))$, we compute

$\partial_{2}\widetilde{F_{1}}=2\zeta_{1}-2=2u,\quad|\varphi_{1}^{\prime}(u)|=1\quad\implies\quad\left|\frac{\varphi_{1}^{\prime}(u)}{\partial_{2}\widetilde{F}_{1}}\right|=\frac{1}{2u}.$

Therefore,

(3.22) $\begin{split}&\mathcal{S}_{2}(f,f,f)=\mathcal{C}_{2}(f,f,f)\\\
&\;\;=2\omega^{4\beta+3}\int_{0}^{1}(u+1)^{\beta+1}\left(\frac{1+u+u^{2}}{u}\right)^{\beta+1}\left(\frac{u+1}{u}\right)^{\beta+1}(f_{1}f_{2}f_{3}+f_{1}f_{2}f_{\omega}-f_{1}f_{\omega}f_{3}-f_{\omega}f_{2}f_{3})\frac{1}{2u}du\\\
&\;\;=\omega^{4\beta+3}\int_{0}^{1}\frac{(1+u+u^{2})^{\beta+1}(u+1)^{2\beta+2}}{u^{2\beta+3}}(f_{1}f_{2}f_{3}+f_{1}f_{2}f_{\omega}-f_{1}f_{\omega}f_{3}-f_{\omega}f_{2}f_{3})du.\\\
\end{split}$

####  3.3.3. The case $(-,+,-)$

By symmetry between the variables $\omega_{1}$ and $\omega_{2}$, it is the
same as $(+,-,-)$ below (so $\mathcal{C}_{3}=\mathcal{C}_{4}$).

####  3.3.4. The case $(+,-,-)$

First consider the equations

(3.23) $\left\\{\begin{array}[]{l}u_{3}+1=u_{1}+u_{2}\\\
u_{1}^{2}=u_{2}^{2}+u_{3}^{2}+1.\end{array}\right.$

Noticing that the application
$(u_{1},u_{2},u_{3})\mapsto(\frac{1}{u_{1}},\frac{u_{3}}{u_{1}},\frac{u_{2}}{u_{1}})$
maps solutions of (3.23) to solutions of (3.18), we see that the solutions in
$\mathbb{R}_{+}^{3}$ of (3.23) can be parameterized by

$u_{1}=\frac{1+u+u^{2}}{u+1},\quad u_{2}=\frac{u}{u+1},\quad u_{3}=u$

with $0<u<\infty$. We write now

$\displaystyle\int\mathcal{I}(\omega_{1},\omega_{2},\omega_{3})\delta(F_{1}(\omega_{1},\omega_{2},\omega_{3}))\delta(F_{2}(\omega_{1},\omega_{2},\omega_{3}))\,d\omega_{1}\,d\omega_{2}\,d\omega_{3}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad=\int\widetilde{\mathcal{I}}(\omega_{1},\omega_{2})\delta(\widetilde{F_{1}}(\zeta_{1},\zeta_{2}))\,d\zeta_{1}\,d\zeta_{2},$

being understood that $F_{1}(\omega_{1},\omega_{2},\omega_{3})$ is now given
by $\omega_{1}^{2}-\omega_{2}^{2}-\omega_{3}^{2}+\omega^{2}$. In order to
apply (3.4) and (3.5) with $\varphi(u)=(\varphi_{1}(u),\varphi_{2}(u))$, we
compute

$\partial_{1}\widetilde{F_{1}}=-2\zeta_{2}+2=\frac{2}{u+1},\quad|\varphi_{2}^{\prime}(u)|=\frac{1}{(1+u)^{2}}\quad\implies\quad\left|\frac{\varphi_{2}^{\prime}(u)}{\partial_{1}\widetilde{F}_{1}}\right|=\frac{1}{2(1+u)}.$

This gives the formula:

(3.24)
$\begin{split}\mathcal{C}_{4}(f,f,f)&=\omega^{4\beta+3}\int_{0}^{\infty}\left(\frac{1+u+u^{2}}{u+1}\right)^{\beta+1}\left(\frac{u}{u+1}\right)^{\beta+1}u^{\beta+1}\\\
&\quad\qquad\qquad\qquad\cdot(f_{1}f_{2}f_{3}+f_{1}f_{2}f_{\omega}-f_{1}f_{\omega}f_{3}-f_{\omega}f_{2}f_{3})\frac{1}{2(u+1)}du\\\
&=\frac{1}{2}\omega^{4\beta+3}\int_{0}^{\infty}\frac{(1+u+u^{2})^{\beta+1}u^{2\beta+2}}{(u+1)^{2\beta+3}}(f_{1}f_{2}f_{3}+f_{1}f_{2}f_{\omega}-f_{1}f_{\omega}f_{3}-f_{\omega}f_{2}f_{3})\,du,\\\
\end{split}$

where

$\omega_{1}=\omega u_{1},\qquad\omega_{2}=\omega u_{2},\qquad\omega_{3}=\omega
u_{3}.$

Splitting the above integral into $\int_{0}^{1}$ and $\int_{1}^{\infty}$ and
making a change of variables $u\rightarrow 1/u$ for the $\int_{1}^{\infty}$
integral, we obtain

$\begin{split}\mathcal{C}_{3}(f,f,f)+\mathcal{C}_{4}(f,f,f)=\mathcal{S}_{3}(f,f,f)+\mathcal{S}_{4}(f,f,f)\end{split}$

where $\mathcal{S}_{3}$ and $\mathcal{S}_{4}$ have the desired form.

###  3.4. A more symmetrical representation formula

In this section, we derive the following representation formula for the
collision operator:

(3.25)
$\mathcal{C}(n)(\omega)=\omega^{3+4\beta}\int_{0}^{1}\sum_{j,k=1}^{4}(-1)^{j+k}\frac{1}{1+u+u^{2}}v_{k}^{-1}\left(\prod_{i=1}^{4}\frac{v_{i}}{v_{k}}\right)^{\beta+1}\frac{\prod_{i=1}^{4}n_{i,k}}{n_{j,k}}\,du,$

where

(3.26) $v_{1}=\frac{1+u}{1+u+u^{2}},\quad v_{2}=1,\quad
v_{3}=\frac{u(1+u)}{1+u+u^{2}},\quad v_{4}=\frac{u}{1+u+u^{2}},$

$v_{i,k}=v_{i}/v_{k}$, and $n_{i,k}=n(\omega v_{i,k})$.

This formula can be established from the decomposition (3.17), by observing
that $\mathcal{S}_{1}$, $\mathcal{S}_{2}$, $\mathcal{S}_{3}$,
$\mathcal{S}_{4}$ correspond to the summands for $k=2,4,1,3$ in (3.25)
respectively. The advantage of this formulation, besides its compactness, is
that it encapsulates two different cancellations: the alternating sum over $j$
corresponds to the classical cancellation between gain and loss terms in each
of the terms $\mathcal{S}_{i}$, while the alternating sum over $k$ expresses a
cancellation arising between the different terms $\mathcal{S}_{i}$.

It turns out that the combination of cancellation in $j$ and $k$ is
responsible for the gain of regularity. Also, we remark that the even symmetry
assumption $n(-k)=n(k)$ was crucial in obtaining $(-1)^{k}$ cancellation. It
would be interesting if one could obtain the cancellation structure without
even symmetry.

##  4\. The collision operator in weighted $L^{p}$ spaces

###  4.1. Boundedness in weighted $L^{\infty}$

For any $f=f(\omega)$, denote

(4.1) $\|f\|_{L^{\infty}_{\theta,\gamma}}:=\|m(\omega)f\|_{L^{\infty}},$

where

$m(\omega)=\begin{cases}|\omega|^{-\theta}&|\omega|\leq 1\\\
|\omega|^{\gamma}&|\omega|>1\end{cases}$

and $\theta$, $\gamma$ are constants.

######  Lemma 4.1.

If $-1/2\leq\beta\leq 0$, $\gamma>2\beta+2$ and $\theta>-2\beta-1$,

(4.2)
$\|\mathcal{F}(f,g,h)\|_{L^{\infty}_{\theta,\gamma}}\lesssim\|f\|_{L^{\infty}_{\theta,\gamma}}\|g\|_{L^{\infty}_{\theta,\gamma}}\|h\|_{L^{\infty}_{\theta,\gamma}}.$

######  Remark 4.2.

The ranges of the parameters $\beta$, $\theta$ and $\gamma$ in this statement
are optimal if one ignores the cancellations in the collision term, as can be
seen from the proof. The critical space given the scaling of the equation
would be $\theta=-\gamma=-2\beta-\frac{3}{2}$.

###### Proof.

The proof of this result will not take advantage of any cancellation arising
from the alternating signs in the collision operators. Accordingly, we will
denote $\mathcal{F}$ for the collision operator where all signs have been
changed to $+$. The decomposition (3.17) becomes
$\mathcal{F}=\mathcal{F}_{1}+\mathcal{F}_{2}+\mathcal{F}_{3}+\mathcal{F}_{4}$.

It suffices to prove that

(4.3)
$m(\omega)|\mathcal{F}(m(\omega)^{-1},m(\omega)^{-1},m(\omega)^{-1})|\lesssim
1.$

The term $\mathcal{F}_{1}$. On the one hand, if $\omega\leq 1$, it follows
from (3.20) that

(4.4)
$\begin{split}&\omega^{-\theta}\mathcal{F}_{1}(m(\omega)^{-1},m(\omega)^{-1},m(\omega)^{-1})\lesssim\omega^{4\beta+3-\theta}\int_{0}^{1}u^{2\beta+2}(u^{2\theta}\omega^{3\theta}+u^{\theta}\omega^{3\theta})du\lesssim\omega^{4\beta+3+2\theta}\end{split}$

if

(4.5) $2\beta+3+\theta>0\quad\mbox{and}\quad 2\beta+3+2\theta>0.$

The right-hand side of (4.4) is $\lesssim 1$ if

(4.6) $4\beta+3+2\theta\geq 0.$

On the other hand, if $\omega>1$,

(4.7)
$\begin{split}&\omega^{\gamma}\mathcal{F}_{1}(m(\omega)^{-1},m(\omega)^{-1},m(\omega)^{-1})\\\
&\qquad\lesssim\omega^{4\beta+3+\gamma}\left[\int_{1/\omega}^{1}u^{2\beta+2}(u^{-\gamma}\omega^{-3\gamma}+u^{-2\gamma}\omega^{-3\gamma})du\right.+\left.\int_{0}^{1/\omega}u^{2\beta+2}(u^{2\theta}\omega^{2\theta-\gamma}+u^{\theta}\omega^{\theta-2\gamma})du\right]\\\
&\qquad\sim\omega^{4\beta+3-2\gamma}+\omega^{2\beta-\gamma}+\omega^{2\beta}.\end{split}$

if

(4.8) $2\beta+3+\theta>0\quad\mbox{and}\quad\quad 2\beta+3+2\theta>0.$

The right-hand side of (4.7) is $\lesssim 1$ provided

(4.9) $4\beta+3-2\gamma\leq 0,\quad 2\beta-\gamma\leq
0\quad\mbox{and}\quad\beta\leq 0.$

The term $\mathcal{F}_{2}$. On the one hand, if $\omega<1$,

(4.10)
$\begin{split}&\omega^{-\theta}\mathcal{F}_{2}(m(\omega)^{-1},m(\omega)^{-1},m(\omega)^{-1})\\\
&\qquad\lesssim\omega^{4\beta+3-\theta}\left[\int_{0}^{\omega}u^{-2\beta-3}(\omega^{2\theta-\gamma}u^{\gamma}+\omega^{\theta-2\gamma}u^{2\gamma})du\right.+\left.\int_{\omega}^{1}u^{-2\beta-3}(\omega^{3\theta}u^{-2\theta}+\omega^{3\theta}u^{-\theta})du\right]\\\
&\qquad\sim\omega^{4\beta+3+2\theta}+\omega^{2\beta+1+\theta}+\omega^{2\beta+1}\end{split}$

if

(4.11) $2\beta+2-\gamma<0\quad\mbox{and}\quad 2\beta+2-2\gamma<0.$

The right-hand side of (4.10) is $\lesssim 1$ provided

(4.12) $4\beta+3+2\theta\geq 0,\quad 2\beta+1+\theta\geq
0,\quad\mbox{and}\quad 2\beta+1\geq 0.$

If $\omega>1$, we have

(4.13)
$\begin{split}&\omega^{\gamma}\mathcal{F}_{2}(m(\omega)^{-1},m(\omega)^{-1},m(\omega)^{-1})\lesssim\omega^{4\beta+3+\gamma}\int_{0}^{1}u^{-2\beta-3}(\omega^{-3\gamma}u^{\gamma}+\omega^{-3\gamma}u^{2\gamma})\,du\lesssim\omega^{4\beta+3-2\gamma}\end{split}$

if

(4.14) $2\beta+2-\gamma<0\quad\mbox{and}\quad 2\beta+2-2\gamma<0.$

The right-hand side of (4.13) is $\lesssim 1$ if

(4.15) $4\beta+3-2\gamma\leq 0.$

The term $\mathcal{F}_{3}$ It can be bounded exactly like $\mathcal{F}_{1}$.

The term $\mathcal{F}_{4}$ It can be bounded exactly like $\mathcal{F}_{2}$.

Conclusion The desired statement follows from combining the conditions (4.5),
(4.6), (4.8), (4.9), (4.11), (4.12), (4.14) and (4.15). ∎

###  4.2. Unboundedness in weighted $L^{p}$ spaces for $p<3$

######  Lemma 4.3.

For any $p<3$, there exists data $(f^{\epsilon})$ in
$L^{\infty}(\frac{1}{10},2)$ such that

$\|f^{\epsilon}\|_{L^{p}}=O(1)\quad\mbox{while}\quad\|\mathcal{C}(f^{\epsilon})\|_{L^{p}(\frac{1}{2},\frac{3}{2})}\to\infty\quad\mbox{as
$\epsilon\to 0$}.$

###### Proof.

We start with the following combinatorial fact: consider the system of
equations

$\displaystyle x_{1}+x_{2}-x_{3}-x_{4}=0$ $\displaystyle
x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-x_{4}^{2}=0.$

The solutions of this system of equations such that, for some
$\\{i,j,k\\}\subset\\{1,2,3,4\\}$, there holds
$\\{x_{i},x_{j},x_{k}\\}\subset\\{1/3,2/3,1\\}$, reduce to

$(x_{1},x_{2},x_{3},x_{4})=(2/3,2/3,1/3,1).$

We now consider the data

$f^{\epsilon}(\omega)=\epsilon^{-\frac{2}{p}}\mathbf{1}(\epsilon^{-2}(\omega-1/3))+\epsilon^{-\frac{1}{p}}\mathbf{1}(\epsilon^{-1}(\omega-2/3))+\epsilon^{-\frac{2}{p}}\mathbf{1}(\epsilon^{-2}(\omega-1)),$

which is such that

$\|f^{\epsilon}\|_{L^{p}}\sim 1.$

Our aim is now to estimate the size of $\mathcal{C}(f^{\epsilon})(\omega)$ if
$|\omega-1|\ll\epsilon^{2}$. Recall from the formula (3.16) that the collision
operator $\mathcal{C}$ can be split into
$\mathcal{C}_{1}+\mathcal{C}_{2}+\mathcal{C}_{3}+\mathcal{C}_{4}$.

The contribution of $\mathcal{C}_{2}(f^{\epsilon})$. The kernel of this
operator is supported on the zero set of
$\omega_{1}+\omega_{2}-\omega_{3}-\omega$ and
$\omega_{1}^{2}+\omega_{2}^{2}-\omega_{3}^{2}+\omega^{2}$. By the
combinatorial fact above, the term $\mathcal{C}_{2}(f^{\epsilon})$ is
localized close to $1/3$, and therefore

$\mathcal{C}_{2}(f^{\epsilon})(\omega)=0\qquad\mbox{if $|\omega-1|\ll 1$}.$

The contribution of
$\mathcal{C}_{3}(f^{\epsilon})=\mathcal{C}_{4}(f^{\epsilon})$. The kernel of
this operator is supported on the zero set of
$\omega_{1}+\omega_{2}-\omega_{3}-\omega$ and
$\omega_{1}^{2}-\omega_{2}^{2}+\omega_{3}^{2}+\omega^{2}$. By the
combinatorial fact above, the term $\mathcal{C}_{3}(f^{\epsilon})$ is
localized close to $2/3$, and therefore

$\mathcal{C}_{3}(f^{\epsilon})(\omega)=\mathcal{C}_{4}(f^{\epsilon})(\omega)=0\qquad\mbox{if
$|\omega-1|\ll 1$}.$

The contribution of $\mathcal{C}_{1}(f^{\epsilon})$. The kernel of this
operator is supported on the zero set of
$\omega_{1}+\omega_{2}-\omega_{3}-\omega_{4}$ and
$\omega_{1}^{2}+\omega_{2}^{2}+\omega_{3}^{2}-\omega^{2}$. By the
combinatorial fact above, only frequencies close to
$(\omega_{1},\omega_{2},\omega_{3},\omega)=(2/3,2/3,1/3,1)$ have a nonzero
contribution to $\mathcal{C}_{1}(f^{\epsilon})$.

We now recall the formula (3.20) for $\mathcal{C}_{1}$:

$\displaystyle\mathcal{C}_{1}(f^{\epsilon})$
$\displaystyle=\omega^{4\beta+3}\int_{0}^{1}\frac{u^{2\beta+2}(u+1)^{2\beta+2}}{(1+u+u^{2})^{3\beta+4}}(f_{1}f_{2}f_{3}+f_{1}f_{2}f_{\omega}-f_{1}f_{\omega}f_{3}-f_{\omega}f_{2}f_{3})\,du$
$\displaystyle=\mathcal{C}_{1}^{1}(f^{\epsilon})+\mathcal{C}_{1}^{2}(f^{\epsilon})+\mathcal{C}_{1}^{3}(f^{\epsilon})+\mathcal{C}_{1}^{4}(f^{\epsilon})$

where

$\displaystyle\omega_{1}=\omega u_{1},\qquad\omega_{2}=\omega
u_{2},\qquad\omega_{3}=\omega u_{3}$ $\displaystyle
u_{1}=\frac{1+u}{1+u+u^{2}},\quad
u_{2}=\frac{u(1+u)}{1+u+u^{2}},\quad\mbox{and}\quad
u_{3}=\frac{u}{1+u+u^{2}}.$

The contribution of $\mathcal{C}_{1}^{1}(f^{\epsilon})$. It is given by

$\displaystyle\mathcal{C}_{1}^{1}(f^{\epsilon})=\epsilon^{-4/p}\omega^{4\beta+3}$
$\displaystyle\int_{0}^{1}\frac{u^{2\beta+2}(u+1)^{2\beta+2}}{(1+u+u^{2})^{3\beta+4}}$
$\displaystyle\qquad\qquad\mathbf{1}(\epsilon^{-1}(\omega_{1}-2/3))\mathbf{1}(\epsilon^{-1}(\omega_{2}-2/3))\mathbf{1}(\epsilon^{-2}(\omega_{3}-1/3))\,du$

Notice that

$u_{1}(1)=\frac{2}{3}\quad\mbox{and}\quad\left.\frac{d}{du}u_{1}(u)\right|_{u=1}=-\frac{1}{3}\neq
0.$

Therefore, if $\omega$ is close to $1$, the function
$u\mapsto\mathbf{1}(\epsilon^{-1}(\omega_{1}-2/3))$ is nonzero on a set of a
length $O(\epsilon)$. Therefore,

$|\mathcal{C}_{1}^{1}(f^{\epsilon})(\omega)|\lesssim\epsilon^{1-\frac{4}{p}}.$

The contribution of $\mathcal{C}_{1}^{2}(f^{\epsilon})$. It can be estimated
just like $\mathcal{C}_{1}^{1}$: we find, for $\omega$ close to $1$,

$|\mathcal{C}_{1}^{2}(f^{\epsilon})(\omega)|\lesssim\epsilon^{1-\frac{4}{p}}.$

The contribution of $\mathcal{C}_{1}^{3}(f^{\epsilon})$. It is given by

$\displaystyle\mathcal{C}_{1}^{3}(f^{\epsilon})=\epsilon^{-5/p}\omega^{4\beta+3}$
$\displaystyle\int_{0}^{1}\frac{u^{2\beta+2}(u+1)^{2\beta+2}}{(1+u+u^{2})^{3\beta+4}}$
$\displaystyle\qquad\qquad\mathbf{1}(\epsilon^{-1}(\omega_{1}-2/3))\mathbf{1}(\epsilon^{-2}(\omega_{3}-1/3))\mathbf{1}(\epsilon^{-2}(\omega-1))\,du$

The crucial point is that

$u_{3}(1)=\frac{1}{3}\quad\mbox{and}\quad\left.\frac{d}{du}u_{3}(u)\right|_{u=1}=0.$

Therefore, if $|\omega-1|\ll\epsilon^{2}$ and $|u-1|\ll\epsilon$, there holds

$\mathbf{1}(\epsilon^{-2}(\omega-1))=\mathbf{1}(\epsilon^{-1}(\omega_{1}-2/3))=\mathbf{1}(\epsilon^{-2}(\omega_{3}-1/3))=1.$

As a result, we can estimate from below, if $|\omega-1|\ll\epsilon^{2}$,

$|\mathcal{C}^{3}_{1}(f^{\epsilon})(\omega)|\gtrsim\epsilon^{1-\frac{5}{p}}.$

The contribution of $\mathcal{C}_{1}^{4}(f^{\epsilon})$. By the same argument,

$|\mathcal{C}^{4}_{1}(f^{\epsilon})(\omega)|\gtrsim\epsilon^{1-\frac{5}{p}}$

(but this lower bound is not even necessary for the rest of the argument,
since $\mathcal{C}^{4}_{1}$ has the same sign as $\mathcal{C}^{3}_{1}$).

Conclusion Gathering the contributions of all the terms above, we find that,
if $|\omega-1|\ll\epsilon^{2}$,

$|\mathcal{C}(f^{\epsilon})(\omega)|\gtrsim\epsilon^{1-\frac{5}{p}}.$

Therefore,

$\|\mathcal{C}(f^{\epsilon})\|_{L^{p}}\gtrsim\epsilon^{1-\frac{3}{p}},$

from which the desired result follows. ∎

##  5\. Local well-posedness near scale invariant power data

In this section, we establish a priori estimate and local existence of
solutions near scale-invariant power spectrum. We set $\beta\in(-1,1)$,
$\gamma=2\beta+\frac{3}{2}$, and let

(5.1) $N(\omega)=\omega^{\gamma}n(\omega).$

In particular, $\gamma$ for $\beta=-\frac{3}{4},-\frac{1}{4}$ correspond to
the Rayleigh-Jean spectrum, and $\gamma$ for $\beta=\frac{1}{4},\frac{3}{4}$
correspond to the Kolmogorov-Zhakarov spectrum.

From (3.25), the equation for $N$ reads the following:

(5.2)
$\partial_{t}N(\omega)=\int_{0}^{1}\frac{\left(\prod_{i=1}^{4}v_{i}\right)^{-\beta-\frac{1}{2}}}{1+u+u^{2}}\sum_{j,k=1}^{4}(-1)^{j+k}v_{j}^{2\beta+\frac{3}{2}}v_{k}^{2\beta-\frac{1}{2}}\frac{\prod_{i=1}^{4}N_{i,k}}{N_{j,k}}du=:\overline{\mathcal{C}}(N)(\omega),$

where $N_{i,k}=N(\omega v_{i}/v_{k})$. Also, we recall

(5.3) $DN(\omega)=\omega\frac{d}{d\omega}N(\omega).$

One has the following useful formula:

(5.4) $D(N_{i,k})=\omega\frac{d}{d\omega}(N_{i,k})=\omega
v_{i}/v_{k}N^{\prime}(\omega v_{i}/v_{k})=(DN)_{i,k}$

and therefore the expression $DN_{i,k}$ is not ambiguous.

A priori assumptions in Section 5.1, 5.2, and 5.3. We will assume the
following:

  1. (A1)

$p_{0}$ is a positive integer, and $p_{0}$ and $\beta$ satisfy

(5.5) $\beta\in(-1,1),\qquad
p_{0}>\max\left(\frac{1}{4(1-\beta)},\frac{1}{4(1+\beta)}\right).$

  2. (A2)

$N\in L^{\infty}(0,T;X)$ is a strong solution of (5.2) with $N(0)=N_{0}\in X$,
that is, for almost every $t\in(0,T)$ $\partial_{t}N\in
L^{\infty}(0,T;L^{\infty}((0,\infty)))$, $D\partial_{t}N\in
L^{1}(0,T;L^{2p_{0}}(0,\infty)\cap L^{\infty}(0,\infty))$, and (5.2) is
satisfied for almost every $(t,\omega)$,

  3. (A3)

$\inf_{\omega\in(0,\infty)}N_{0}(\omega)>0$, and

  4. (A4)

We have

(5.6)
$\int_{0}^{T}\int_{0}^{\infty}\int_{-1/3}^{1}|DN(\omega)-DN(\omega(1+u))|^{2}\left(\mathbf{1}_{[-1/3,0]}(u)u^{-(\beta+1)}+\mathbf{1}_{[0,1]}(u)u^{2\beta}\right)dud\omega
dt<\infty.$

###  5.1. A priori estimates: Positivity of $N$

We first establish the non-negativity of $N$: suppose that $N_{0}(\omega)\geq
0$ for every $\omega>0$ and $t$ is the first time such that there is a
$\omega>0$ with $N(t,\omega)=0$. Then

(5.7)
$\partial_{t}N(t,\omega)=\overline{\mathcal{C}}(N)(t,\omega)=\int_{0}^{1}\frac{\left(\prod_{i=1}^{4}v_{i}\right)^{-\beta-\frac{1}{2}}}{1+u+u^{2}}\sum_{j,k=1}^{4}\delta_{j}^{k}(-1)^{j+k}v_{j}^{2\beta+\frac{3}{2}}v_{k}^{2\beta-\frac{1}{2}}\frac{\prod_{i=1}^{4}N_{i,k}}{N_{j,k}}du$

since these are the only terms which does not contain $N_{k,k}=N(\omega)$.
Since $N_{i,k}\geq 0$ for $i\neq k$, $N(t^{\prime},\omega)\geq 0$ for all
$\omega\in(0,\infty)$ and $t^{\prime}>t$ with $t^{\prime}$ sufficiently close
to $t$. Thus, $N$ remains nonnegative. Moreover, if we write

(5.8) $\partial_{t}N^{-1}(t,\omega)=-(N^{-1})^{2}\overline{C}(N)(t,\omega).$

We see, from Section 5.2, that

(5.9)
$\|\partial_{t}N^{-1}\|_{L^{\infty}}\lesssim\|N^{-1}\|_{L^{\infty}}^{2}\|N\|_{L^{\infty}}^{2}(\|N\|_{L^{\infty}}+\|DN\|_{L^{\infty}}),$

and in the end we have
$\|N\|_{L^{\infty}}^{2}(\|N\|_{L^{\infty}}+\|DN\|_{L^{\infty}})\lesssim 1$.
This means that $N^{-1}$ is bounded above for a positive time, or $N$ is
bounded below. To summarize, we have the following:

######  Lemma 5.1.

There exists a $T_{p}=T(N_{0},\beta,p_{0})>0$ and a constant
$C(N_{0},\beta,p_{0})$ such that

$\inf_{t\in(0,T_{p}),\omega\in(0,\infty)}N(t,\omega)\geq
C(N_{0},\beta,p_{0})>0.$

###  5.2. A priori estimates: $L^{\infty}$ estimate

For $L^{\infty}$ estimate, we use the cancellation coming from gain-loss
relation, and thus this is a losing estimate. In the next section, we will
establish a non-losing estimate. We start from the following simple but useful
observation: for $u\in[0,1]$,

(5.10) $v_{1}\sim v_{2}\sim 1,\qquad v_{3}\sim v_{4}\sim u,\qquad
v_{1}-v_{2}=v_{4}-v_{3}\sim u^{2}.$

Breaking down $\overline{\mathcal{C}}(N)$ into 4 parts, we obtain

(5.11)
$\begin{split}\overline{\mathcal{C}}(N)(\omega)&=\int_{0}^{1}W(u)\sum_{k=1}^{4}(-1)^{k}v_{k}^{2\beta-\frac{1}{2}}\times\\\
&\left[(v_{2}^{2\beta+3/2}-v_{1}^{2\beta+3/2})N_{2,k}N_{3,k}N_{4,k}+v_{2}^{2\beta+3/2}(N_{1,k}-N_{2,k})N_{3,k}N_{4,k}\right.\\\
&\left.+(v_{4}^{2\beta+3/2}-v_{3}^{2\beta+3/2})N_{1,k}N_{2,k}N_{4,k}+v_{4}^{2\beta+3/2}(N_{3,k}-N_{4,k})N_{1,k}N_{2,k}\right]du,\end{split}$

where

(5.12)
$W(u)=\frac{\left(\prod_{i=1}^{4}v_{i}\right)^{-\beta-\frac{1}{2}}}{1+u+u^{2}}\sim
u^{-2\beta-1}.$

Then, by the Newton-Leibniz formula and the definition of $DN$, we have

(5.13) $N_{1,k}-N_{2,k}=\int_{0}^{1}DN\left(\omega\frac{\lambda
v_{1}+(1-\lambda)v_{2}}{v_{k}}\right)\frac{v_{1}-v_{2}}{\lambda
v_{1}+(1-\lambda)v_{2}}d\lambda$

and consequently by (5.10),

(5.14)
$|N_{1,k}-N_{2,k}|\lesssim\|DN\|_{L^{\infty}}\frac{|v_{1}-v_{2}|}{\min(v_{1},v_{2})}\lesssim
u^{2}\|DN\|_{L^{\infty}}.$

Similarly,

(5.15)
$|N_{3,k}-N_{4,k}|\lesssim\|DN\|_{L^{\infty}}\frac{|v_{3}-v_{4}|}{\min(v_{3},v_{4})}\lesssim
u\|DN\|_{L^{\infty}}$

and

(5.16) $|v_{2}^{2\beta+3/2}-v_{1}^{2\beta+3/2}|\lesssim
u^{2},\qquad|v_{4}^{2\beta+3/2}-v_{3}^{2\beta+3/2}|\lesssim u^{2\beta+5/2}.$

Putting all these together, we have

(5.17)
$\begin{split}\left|\overline{\mathcal{C}}(N)(\omega)\right|&\lesssim\int_{0}^{1}u^{-2\beta-1}(1+u^{2\beta-1/2})\\\
&\qquad\qquad\left[(u^{2}+u^{2\beta+5/2})\|N\|_{L^{\infty}}^{3}+(u^{2}+u^{2\beta+5/2})\|N\|_{L^{\infty}}^{2}\|DN\|_{L^{\infty}}\right]du\\\
&\lesssim\int_{0}^{1}u^{-2\beta-1}(1+u^{2\beta-1/2})(u^{2}+u^{2\beta+5/2})du\|N\|_{L^{\infty}}^{2}(\|N\|_{L^{\infty}}+\|DN\|_{L^{\infty}}),\end{split}$

and the right-hand side integral converges if and only if $\beta\in(-1,1)$, as
the integrand is $u^{-2\beta+1}+u^{3/2}+u^{1/2}+u^{2\beta+1}$. Therefore, we
have proved the following:

######  Lemma 5.2.

We have

(5.18) $\frac{d}{dt}\|N\|_{L^{\infty}}\leq
C\|N\|_{L^{\infty}}^{2}(\|N\|_{L^{\infty}}+\|DN\|_{L^{\infty}}).$

###  5.3. A priori estimates: $\dot{W}^{1,p}\cap W^{1,\infty}$ estimate and
regularity gain

In this section, we will obtain a local-in-time $L^{2p}$ estimate for $DN$
which is uniform in $p$ for every positive integer $p\geq p_{0}$, thereby
giving an $L^{\infty}$ estimate for $DN$.

Splitting the integral Let $p$ be a positive integer. We have

(5.19)
$\begin{split}\|DN\|_{L^{2p}}^{2p-1}\frac{d}{dt}\|DN\|_{L^{2p}}=\int_{0}^{\infty}(DN)^{2p-1}(\omega)D(\overline{C}(N))(\omega)d\omega=\sum_{i=1}^{6}I_{i},\end{split}$

where

(5.20)
$\begin{split}I_{1}&=\int_{0}^{\infty}\int_{0}^{1}(DN)^{2p-1}(\omega)W(u)\sum_{k=1}^{4}(-1)^{k}v_{k}^{2\beta-1/2}(v_{2}^{2\beta+3/2}-v_{1}^{2\beta+3/2})D(N_{2,k}N_{3,k}N_{4,k})\,du\,d\omega,\\\
I_{4}&=\int_{0}^{\infty}\int_{0}^{1}(DN)^{2p-1}(\omega)W(u)\sum_{k=1}^{4}(-1)^{k}v_{k}^{2\beta-1/2}(v_{4}^{2\beta+3/2}-v_{3}^{2\beta+3/2})D(N_{1,k}N_{2,k}N_{4,k})\,du\,d\omega,\end{split}$
(5.21)
$\begin{split}I_{2}&=\int_{0}^{\infty}\int_{0}^{1}(DN)^{2p-1}(\omega)W(u)\sum_{k=1}^{4}(-1)^{k}v_{k}^{2\beta-1/2}v_{2}^{2\beta+3/2}(DN_{1,k}-DN_{2,k})N_{3,k}N_{4,k}\,du\,d\omega,\\\
I_{5}&=\int_{0}^{\infty}\int_{0}^{1}(DN)^{2p-1}(\omega)W(u)\sum_{k=1}^{4}(-1)^{k}v_{k}^{2\beta-1/2}v_{4}^{2\beta+3/2}(DN_{3,k}-DN_{4,k})N_{1,k}N_{2,k}\,du\,d\omega,\end{split}$

and

(5.22)
$\begin{split}I_{3}&=\int_{0}^{\infty}\int_{0}^{1}(DN)^{2p-1}(\omega)W(u)\sum_{k=1}^{4}(-1)^{k}v_{k}^{2\beta-1/2}v_{2}^{2\beta+3/2}(N_{1,k}-N_{2,k})D(N_{3,k}N_{4,k})\,du\,d\omega,\\\
I_{6}&=\int_{0}^{\infty}\int_{0}^{1}(DN)^{2p-1}(\omega)W(u)\sum_{k=1}^{4}(-1)^{k}v_{k}^{2\beta-1/2}v_{4}^{2\beta+3/2}(N_{3,k}-N_{4,k})D(N_{1,k}N_{2,k})\,du\,d\omega.\end{split}$

The morale of the splitting is the following: since we aim not to lose
derivatives, we take a closer look at the terms involving cancellation of
$DN$s, that is, $I_{2},I_{5}$. For the other terms, $I_{1},I_{3},I_{4}$, and
$I_{6}$, the usual cancellation between $N$s can be used. For $I_{2}$ and
$I_{5}$, it turns out that the most dangerous terms are those involving
$DN_{1,1}-DN_{2,1}$ and $DN_{1,2}-DN_{2,2}$ for $I_{2}$, and
$DN_{3,3}-DN_{4,3}$ and $DN_{3,4}-DN_{4,4}$ for $I_{5}$: the terms involving
$DN_{1,3}-DN_{2,3}$, $DN_{1,4}-DN_{2,4}$ for $I_{2}$ and its counterpart in
$I_{5}$ can be treated using integration by parts. To elaborate this, we see
that

$DN_{1,3}-DN_{2,3}=\int_{0}^{1}(DN)^{\prime}\left(\frac{\lambda
v_{1}+(1-\lambda)v_{2}}{v_{3}}\right)d\lambda\frac{v_{1}-v_{2}}{v_{3}}.$

However, we see that

$(DN)^{\prime}\left(\frac{\lambda
v_{1}+(1-\lambda)v_{2}}{v_{3}}\right)=\left(\frac{d}{du}(DN)\right)\left(\frac{\lambda
v_{1}+(1-\lambda)v_{2}}{v_{3}}\right)\left[\frac{d}{du}\left(\frac{\lambda
v_{1}+(1-\lambda)v_{2}}{v_{3}}\right)\right]^{-1},$

and the term $\frac{v_{1}-v_{2}}{v_{3}}\left[\frac{d}{du}\left(\frac{\lambda
v_{1}+(1-\lambda)v_{2}}{v_{3}}\right)\right]^{-1}\sim u^{3}$. Thus,
integration by parts with respect to $u$ can resolve the loss of derivatives
issue, while we can extract enough cancellation to keep the integral finite.

The remaining terms have good signs, which gives us an analog of parabolicity.

Estimate of $I_{1}$ and $I_{4}$ By Hölder’s inequality, (5.12), (5.16) and
Minkowski’s integral inequality,

(5.23)
$\begin{split}|I_{1}|&\lesssim\|DN\|_{L^{2p}}^{2p-1}\left[\int_{0}^{\infty}\left|\int_{0}^{1}W(u)\sum_{k=1}^{4}(-1)^{k}v_{k}^{2\beta-1/2}(v_{2}^{2\beta+\frac{3}{2}}-v_{1}^{2\beta+\frac{3}{2}})D(N_{2,k}N_{3,k}N_{4,k})\,du\right|^{2p}d\omega\right]^{1/2p}\\\
&\lesssim\|DN\|_{L^{2p}}^{2p-1}\sum_{i=2}^{4}\sum_{k=1}^{4}\int_{0}^{1}u^{-2\beta+1}v_{k}^{2\beta-1/2}\|DN_{i,k}\|_{L^{2p}}\|N\|_{L^{\infty}}^{2}\,du\end{split}$

where $\|DN_{i,k}\|_{L^{2p}}$ is an integral with respect to $\omega$, with
$u$ as a parameter, and the implicit constant is uniform in $p$. By change of
variables, we have

(5.24) $\|DN_{i,k}\|_{L^{2p}}=\|DN\|_{L^{2p}}(v_{k,i})^{1/2p}.$

Therefore, we have

(5.25)
$|I_{1}|\lesssim\|DN\|_{L^{2p}}^{2p}\|N\|_{L^{\infty}}^{2}\sum_{i=2}^{4}\sum_{k=1}^{4}\int_{0}^{1}u^{-2\beta+1}v_{k}^{2\beta-1/2}(v_{k,i})^{1/2p}\,du.$

Next, note that $v_{k,i}\lesssim u^{-1}$ for any $k,i$, and thus
$(v_{k,i})^{1/2p}\lesssim(u^{-1})^{1/2p}\leq(u^{-1})^{1/2p_{0}}$. Thus, if
$p_{0}$ satisfies $-2\beta+1-1/2p_{0}>-1$, that is,

(5.26) $p_{0}>\frac{1}{4(1-\beta)},$

then

(5.27)
$|I_{1}|\lesssim\|DN\|_{L^{2p}}^{2p}\|N\|_{L^{\infty}}^{2}\int_{0}^{1}u^{-2\beta+1-\frac{1}{2p_{0}}}(1+u^{2\beta-1/2})du\leq
C(p_{0},\beta)\|DN\|_{L^{2p}}^{2p}\|N\|_{L^{\infty}}^{2},$

where $C(p_{0},\beta)$ depends on $p_{0}$ and $\beta$ but not on $p$ itself.
$I_{4}$ is estimated in the exactly same manner: the only difference comes
from the equivalent $|v_{4}^{2\beta+3/2}-v_{3}^{2\beta+3/2}|\sim
u^{2\beta+5/2}$ instead of $u^{2}$ (see (5.16)), which leads to

(5.28)
$\begin{split}|I_{4}|&\lesssim\|DN\|_{L^{2p}}^{2p}\|N\|_{L^{\infty}}^{2}\sum_{i=1}^{3}\sum_{k=1}^{4}\int_{0}^{1}u^{3/2}v_{k}^{2\beta-1/2}(v_{k,i})^{1/2p}\,du\\\
&\lesssim\|DN\|_{L^{2p}}^{2p}\|N\|_{L^{\infty}}^{2}\int_{0}^{1}u^{3/2-\frac{1}{2p_{0}}}(1+u^{2\beta-1/2})\,du\leq
C(p_{0},\beta)\|DN\|_{L^{2p}}^{2p}\|N\|_{L^{\infty}}^{2}\end{split}$

provided that $2\beta+1-\frac{1}{2p_{0}}>-1$, that is,

(5.29) $p_{0}>\frac{1}{4(1+\beta)}.$

Estimate of $I_{3}$ and $I_{6}$. The idea is the same as for $I_{1}$ and
$I_{4}$: for $I_{3}$, we use (5.14) and for $I_{6}$ we use (5.15), together
with (5.10) to obtain

(5.30)
$\begin{split}|I_{3}|&\lesssim\|DN\|_{L^{2p}}^{2p}\|N\|_{L^{\infty}}\|DN\|_{L^{\infty}}\int_{0}^{1}u^{-2\beta-1-\frac{1}{2p_{0}}}u^{2}(1+u^{2\beta-1/2})du,\\\
|I_{6}|&\lesssim\|DN\|_{L^{2p}}^{2p}\|N\|_{L^{\infty}}\|DN\|_{L^{\infty}}\int_{0}^{1}u^{-2\beta-1-\frac{1}{2p_{0}}}u^{2\beta+3/2}u(1+u^{2\beta-1/2})du\end{split}$

which is precisely the integrals we had to bound when estimating $I_{1}$ and
$I_{4}$. Therefore,

(5.31)
$|I_{3}|+|I_{6}|\lesssim\|DN\|_{L^{2p}}^{2p}\|N\|_{L^{\infty}}\|DN\|_{L^{\infty}}$

if $p_{0}$ satisfies (5.26) and (5.29).

Further splitting of the main terms, $I_{2}$ and $I_{5}$. We can write

(5.32) $\begin{split}I_{2}&=I_{2,1}+I_{2,2},\qquad I_{5}=I_{5,1}+I_{5,2},\\\
I_{2,1}&=\int_{0}^{\infty}\int_{0}^{1}(DN)^{2p-1}(\omega)W(u)\sum_{k=1}^{2}(-1)^{k}v_{k}^{2\beta-1/2}v_{2}^{2\beta+3/2}(DN_{1,k}-DN_{2,k})N_{3,k}N_{4,k}\,du\,d\omega,\\\
I_{2,2}&=\int_{0}^{\infty}\int_{0}^{1}(DN)^{2p-1}(\omega)W(u)\sum_{k=3}^{4}(-1)^{k}v_{k}^{2\beta-1/2}v_{2}^{2\beta+3/2}(DN_{1,k}-DN_{2,k})N_{3,k}N_{4,k}\,du\,d\omega,\\\
I_{5,1}&=\int_{0}^{\infty}\int_{0}^{1}(DN)^{2p-1}(\omega)W(u)\sum_{k=1}^{2}(-1)^{k}v_{k}^{2\beta-1/2}v_{4}^{2\beta+3/2}(DN_{3,k}-DN_{4,k})N_{1,k}N_{2,k}\,du\,d\omega,\\\
I_{5,2}&=\int_{0}^{\infty}\int_{0}^{1}(DN)^{2p-1}(\omega)W(u)\sum_{k=3}^{4}(-1)^{k}v_{k}^{2\beta-1/2}v_{4}^{2\beta+3/2}(DN_{3,k}-DN_{4,k})N_{1,k}N_{2,k}\,du\,d\omega.\end{split}$

Estimate of $I_{2,2}$ and $I_{5,1}$. It is achieved through integration by
parts. First, we see that if $u>0$ is sufficiently away from $0$, then an
estimate is easily obtained. Indeed, for $\eta\in(0,1)$ to be chosen later,
and for $u\in[\eta,1]$,

(5.33)
$\left|W(u)\sum_{k=3}^{4}(-1)^{k}v_{k}^{2\beta-1/2}v_{2}^{2\beta+3/2}(DN_{1,k}-DN_{2,k})N_{3,k}N_{4,k}\right|\leq
C(\eta)\|N\|_{L^{\infty}}^{2}(|DN_{1,k}|+|DN_{2,k}|),$

with a similar bound for $I_{5,1}$. Applying the previous argument (Hölder’s
inequality, Minkowski’s integral inequality, and change of variables for
$DN_{1,k},DN_{2,k}$) gives that the portion of $I_{2,2}$ and $I_{5,1}$
corresponding to $u\in[\eta,1]$ is bounded by

(5.34) $C(\eta,\beta,p_{0})\|N\|_{L^{\infty}}^{2}\|DN\|_{L^{2p}}^{2p}.$

Again, the coefficients may depend on $\eta,\beta,p_{0}$, but they are uniform
in $p$. Next, we consider the domain $u\in[0,\eta]$. By the Newton-Leibniz
formula,

(5.35)
$\begin{split}DN_{1,k}-DN_{2,k}&=\int_{0}^{1}\frac{\omega(v_{1}-v_{2})}{v_{k}}(DN)^{\prime}\left(\frac{\omega}{v_{k}}(\lambda
v_{1}+(1-\lambda)v_{2}\right)d\lambda\\\
&=\frac{v_{1}-v_{2}}{v_{k}}\int_{0}^{1}\frac{\frac{d}{du}\left[DN\left(\frac{\omega}{v_{k}}(\lambda
v_{1}+(1-\lambda)v_{2}\right)\right]}{\frac{d}{du}\left[\frac{\lambda
v_{1}+(1-\lambda)v_{2}}{v_{k}}\right]}d\lambda,\end{split}$

so that, after integration by parts in $u$, the portion of $I_{2,2}$
corresponding to $u\in[0,\eta]$ can be written as

(5.36)
$\begin{split}I_{2,2}&=\int_{0}^{\infty}d\omega(DN)^{2p-1}(\omega)\sum_{k=3}^{4}\int_{0}^{1}d\lambda\frac{W(u)(-1)^{k}v_{k}^{2\beta-1/2}v_{2}^{2\beta+3/2}}{\frac{d}{du}\left[\frac{\lambda
v_{1}+(1-\lambda)v_{2}}{v_{k}}\right]}\frac{v_{1}-v_{2}}{v_{k}}N_{3,k}N_{4,k}\\\
&\qquad\left.\times\left[DN\left(\frac{\omega}{v_{k}}(\lambda
v_{1}+(1-\lambda)v_{2}\right)\right]\right|_{u=0}^{u=\eta}\\\
&-\int_{0}^{\infty}d\omega(DN)^{2p-1}(\omega)\sum_{k=3}^{4}\int_{0}^{1}d\lambda\int_{0}^{\eta}\frac{d}{du}\left[\frac{W(u)(-1)^{k}v_{k}^{2\beta-1/2}v_{2}^{2\beta+3/2}}{\frac{d}{du}\left[\frac{\lambda
v_{1}+(1-\lambda)v_{2}}{v_{k}}\right]}\frac{v_{1}-v_{2}}{v_{k}}N_{3,k}N_{4,k}\right]\\\
&\qquad\times\left[DN\left(\frac{\omega}{v_{k}}(\lambda
v_{1}+(1-\lambda)v_{2}\right)\right]du.\end{split}$

We first investigate the term $\frac{d}{du}\left[\frac{\lambda
v_{1}+(1-\lambda)v_{2}}{v_{k}}\right]$. Since $\frac{\lambda
v_{1}+(1-\lambda)v_{2}}{v_{k}}$, $k=3,4$ can be written as

(5.37) $u^{-1}(\lambda A_{1}(u)+(1-\lambda)A_{2}(u))=:u^{-1}A_{\lambda}(u),$

where $A_{1},A_{2}$ are analytic functions in $[0,1]$, satisfying

(5.38)
$A_{i}(u)-uA_{i}^{\prime}(u),A_{i}(u)-uA_{i}^{\prime}(u)+\frac{1}{2}u^{2}A_{i}^{\prime\prime}(u)\in\left(\frac{1}{2},2\right),u\in[0,\eta],i=1,2,$

(here we choose $\eta>0$ sufficiently small), therefore the same bounds hold
for $A_{\lambda}$ for all $\lambda\in[0,1]$ and $u\in[0,\eta]$. It is clear
that $\eta>0$ is universal and independent of $\lambda$ and $p$. Now we see
that $\frac{v_{k}}{\lambda v_{1}+(1-\lambda)v_{2}}\lesssim u$ uniformly in
$\lambda$, and

(5.39) $\begin{split}\frac{d}{du}\left[\frac{\lambda
v_{1}+(1-\lambda)v_{2}}{v_{k}}\right]&=-u^{-2}(A_{\lambda}(u)-uA_{\lambda}^{\prime}(u)),\\\
\frac{d^{2}}{du^{2}}\left[\frac{\lambda
v_{1}+(1-\lambda)v_{2}}{v_{k}}\right]&=2u^{-3}\left(A_{\lambda}(u)-uA_{\lambda}^{\prime}(u)+\frac{1}{2}u^{2}A_{\lambda}^{\prime\prime}(u)\right),\end{split}$

and therefore

(5.40) $\left|\frac{d}{du}\left[\frac{\lambda
v_{1}+(1-\lambda)v_{2}}{v_{k}}\right]\right|\sim
u^{-2},\quad\left|\frac{d^{2}}{du^{2}}\left[\frac{\lambda
v_{1}+(1-\lambda)v_{2}}{v_{k}}\right]\right|\sim u^{-3}.$

Next, we investigate the size of the other terms: for $k=3,4$

(5.41)
$W(u)(-1)^{k}v_{k}^{2\beta-3/2}v_{2}^{2\beta+3/2}(v_{1}-v_{2})=u^{-2\beta-1}1^{2\beta+3/2}u^{2}u^{2\beta-3/2}C_{k}(u)=u^{-1/2}C_{k}(u),$

where $C_{k}(u)$ is an analytic function which are bounded away from 0 in
$u\in[0,\eta]$ (we can adjust $\eta>0$ to satisfy this). Therefore, we have,
for $k=3,4$,

(5.42)
$\begin{split}&\left|W(u)(-1)^{k}v_{k}^{2\beta-3/2}v_{2}^{2\beta+3/2}(v_{1}-v_{2})\right|\lesssim
u^{-1/2},\\\
&\left|\frac{d}{du}W(u)(-1)^{k}v_{k}^{2\beta-3/2}v_{2}^{2\beta+3/2}(v_{1}-v_{2})\right|\lesssim
u^{-3/2}.\end{split}$

Again, since $C_{k}$, $k=3,4$, is independent of $p$, the estimates are
uniform in $p$. Now we estimate the first term of (5.36). Assuming that
$N,DN\in L^{\infty}$, since

$\left|\sum_{k=3}^{4}\frac{W(u)(-1)^{k}v_{k}^{2\beta-1/2}v_{2}^{2\beta+3/2}}{\frac{d}{du}\left[\frac{\lambda
v_{1}+(1-\lambda)v_{2}}{v_{k}}\right]}\frac{v_{1}-v_{2}}{v_{k}}\right|\lesssim
u^{3/2},$

the contribution for $u=0$ vanishes. The contribution for $u=\eta$ can be
estimated in the usual way: Hölder inequality, Minkowski integral inequality
(this time we switch $d\lambda$ and $d\omega$), and change of variables. Then
we have the bound

(5.43) $C(\eta,\beta)\|DN\|_{L^{2p}}^{2p}\|N\|_{L^{\infty}}^{2}$

for the first term of (5.36). For the second term of (5.36), we see that, if
$k=3,4$,

(5.44)
$\begin{split}&\left|\left[\frac{W(u)(-1)^{k}v_{k}^{2\beta-1/2}v_{2}^{2\beta+3/2}}{\frac{d}{du}\left[\frac{\lambda
v_{1}+(1-\lambda)v_{2}}{v_{k}}\right]}\frac{v_{1}-v_{2}}{v_{k}}\right]\right|\lesssim
u^{3/2},\\\
&\left|\frac{d}{du}\left[\frac{W(u)(-1)^{k}v_{k}^{2\beta-1/2}v_{2}^{2\beta+3/2}}{\frac{d}{du}\left[\frac{\lambda
v_{1}+(1-\lambda)v_{2}}{v_{k}}\right]}\frac{v_{1}-v_{2}}{v_{k}}\right]\right|\lesssim
u^{1/2},\\\ \end{split}$

and

(5.45)
$\frac{d}{du}N_{i,k}=DN_{i,k}\frac{\frac{d}{du}(v_{i}/v_{k})}{v_{i}/v_{k}}.$

If $i,k\in\\{3,4\\}$, we immediately see that

(5.46)
$\left|\frac{\frac{d}{du}(v_{i}/v_{k})}{v_{i}/v_{k}}\right|=\left|\frac{d}{du}\log(v_{i}/v_{k})\right|\lesssim
1,$

Putting these estimates together, and using Hölder inequality, Minkowski
integral inequality, and change of variables, the second term of (5.36) can be
bounded by

(5.47)
$C(\beta)\|DN\|_{L^{2p}}^{2p}\sum_{k=3,4}\int_{0}^{\eta}du\int_{0}^{1}d\lambda\|N\|_{L^{\infty}}(\|N\|_{L^{\infty}}+\|DN\|_{L^{\infty}})u^{1/2}\left(\frac{v_{k}}{\lambda
v_{1}+(1-\lambda)v_{2}}\right)^{\frac{1}{2p}}.$

Applying the same argument as before, if $p_{0}$ satisfies (5.26), it is
bounded by

(5.48)
$C(\beta,\eta,p_{0})\|DN\|_{L^{2p}}^{2p}\|N\|_{L^{\infty}}(\|N\|_{L^{\infty}}+\|DN\|_{L^{\infty}})$

and again the coefficient is uniform in $p$.

The term $I_{5,1}$ is estimated in a similar manner: again we split the domain
into the regions $u\in[0,\eta]$ and $u\in[\eta,1]$, and the region
$u\in[\eta,1]$ is treated in exactly the same manner. Instead of (5.46), we
have the following: if $i,k\in\\{1,2\\}$,

(5.49) $\left|\frac{\frac{d}{du}(v_{i}/v_{k})}{v_{i}/v_{k}}\right|\lesssim u.$

For the region $u\in[0,\eta]$, we note that $\frac{\lambda
v_{3}+(1-\lambda)v_{4}}{v_{k}}$, $k=1,2$ can be written as

(5.50) $u(\lambda B_{1}(u)+(1-\lambda)B_{2}(u))=:uB_{\lambda}(u),$

and the counterpart of (5.38) for $B_{i}$s hold for sufficiently small $\eta$
as well. Thus, we have

(5.51) $\left|\frac{\lambda v_{3}+(1-\lambda)v_{4}}{v_{k}}\right|\sim
u,\quad\left|\frac{d}{du}\left[\frac{\lambda
v_{3}+(1-\lambda)v_{4}}{v_{k}}\right]\right|\sim
1,\quad\left|\frac{d^{2}}{du^{2}}\left[\frac{\lambda
v_{3}+(1-\lambda)v_{4}}{v_{k}}\right]\right|\lesssim 1$

uniformly in $\lambda$. Also, for $k=1,2$

(5.52)
$W(u)(-1)^{k}v_{k}^{2\beta-3/2}v_{4}^{2\beta+3/2}(v_{3}-v_{4})=u^{-2\beta-1}u^{2\beta+3/2}u^{2}(1^{2\beta-3/2}C_{k}(u))=u^{5/2}C_{k}(u)$

for an analytic function $C_{k}(u)$ which is bounded away from $0$ in
$u\in[0,\eta]$, and we have, if $k=1,2$,

(5.53)
$\begin{split}&\left|\left[\frac{W(u)(-1)^{k}v_{k}^{2\beta-1/2}v_{4}^{2\beta+3/2}}{\frac{d}{du}\left[\frac{\lambda
v_{3}+(1-\lambda)v_{4}}{v_{k}}\right]}\frac{v_{3}-v_{4}}{v_{k}}\right]\right|\lesssim
u^{5/2},\\\
&\left|\frac{d}{du}\left[\frac{W(u)(-1)^{k}v_{k}^{2\beta-1/2}v_{4}^{2\beta+3/2}}{\frac{d}{du}\left[\frac{\lambda
v_{3}+(1-\lambda)v_{4}}{v_{k}}\right]}\frac{v_{3}-v_{4}}{v_{k}}\right]\right|\lesssim
u^{3/2}.\end{split}$

Then, the same arguments as for $I_{2,2}$ gives the same bound as in (5.48).

Estimate of $I_{2,1}$ and $I_{5,2}$. It turns out that these terms involve
higher derivatives in an essential way, but they have a good sign, which can
be interpreted as parabolicity. We show the estimates for $I_{5,2}$, the case
$I_{2,1}$ being essentially identical. We write

(5.54)
$\begin{split}I_{5,2}&=\int_{0}^{\infty}\int_{0}^{1}(DN)^{2p-1}(\omega)W(u)v_{4}^{2\beta-1/2}v_{4}^{2\beta+3/2}(DN_{3,4}-DN_{4,4})N_{1,4}N_{2,4}\,du\,d\omega\\\
&-\int_{0}^{\infty}\int_{0}^{1}(DN)^{2p-1}(\omega)W(u)v_{3}^{2\beta-1/2}v_{4}^{2\beta+3/2}(DN_{3,3}-DN_{4,3})N_{1,3}N_{2,3}\,du\,d\omega.\end{split}$

We apply the change of variables

(5.55) $\overline{\omega}=\omega\frac{v_{4}}{v_{3}},$

to the second integral, which leads to

(5.56)
$\begin{split}&I_{5,2}=\int_{0}^{\infty}\int_{0}^{1}(DN)^{2p-1}(\omega)W(u)v_{4}^{2\beta-1/2}v_{4}^{2\beta+3/2}(DN_{3,4}-DN_{4,4})N_{1,4}N_{2,4}\,du\,d\omega\\\
&\qquad\qquad-\int_{0}^{\infty}\int_{0}^{1}(DN_{3,4})^{2p-1}W(u)v_{3}^{2\beta-1/2}v_{4}^{2\beta+3/2}(DN_{3,4}-DN_{4,4})N_{1,4}N_{2,4}\,du\frac{v_{3}}{v_{4}}\,d\omega\\\
&=\int_{0}^{\infty}\int_{0}^{1}\left[(DN_{4,4})^{2p-1}-(DN_{3,4})^{2p-1}\right]W(u)v_{4}^{2\beta-1/2}v_{4}^{2\beta+3/2}(DN_{3,4}-DN_{4,4})N_{1,4}N_{2,4}\,du\,d\omega\\\
&-\int_{0}^{\infty}\int_{0}^{1}(DN_{3,4})^{2p-1}W(u)(v_{3}^{2\beta+1/2}v_{4}^{2\beta+1/2}-v_{4}^{2\beta+1/2}v_{4}^{2\beta+1/2})(DN_{3,4}-DN_{4,4})N_{1,4}N_{2,4}\,du\,d\omega\\\
&=-\int_{0}^{\infty}\int_{0}^{1}\left[(DN_{4,4})^{2p-1}-(DN_{3,4})^{2p-1}\right]\left[DN_{4,4}-DN_{3,4}\right]W(u)v_{4}^{4\beta+1}N_{1,4}N_{2,4}\,du\,d\omega\\\
&-\int_{0}^{\infty}\int_{0}^{1}(DN(\omega))^{2p-1}(DN_{3,3}-DN_{4,3})N_{1,3}N_{2,3}W(u)v_{4}^{2\beta+1/2}(v_{3}^{2\beta+1/2}-v_{4}^{2\beta+1/2})\,du\,d\omega,\end{split}$

where we changed the integration variable once more in the last line above.

The first term is negative, due to nonnegativity of $N$ and monotonicity of
$x\rightarrow x^{2p-1}$. The second term is controlled in the usual way:
Holder, Minkowski integral inequality, and change of variables. The
computation is essentially same as $I_{4}$. To summarize, we have the
following:

######  Lemma 5.3.

For every positive integer $p>p_{0}$, we have

(5.57)
$\begin{split}&\frac{d}{dt}\|DN\|_{L^{2p}}+\|DN\|_{L^{2p}}^{-(2p-1)}\int_{0}^{\infty}\int_{0}^{1}\left[(DN_{4,4})^{2p-1}-(DN_{3,4})^{2p-1}\right]\\\
&\qquad\qquad\qquad\qquad\qquad\left[DN_{4,4}-DN_{3,4}\right]W(u)v_{4}^{2\beta-1/2}v_{4}^{2\beta+3/2}N_{1,4}N_{2,4}\,du\,d\omega\\\
&+\|DN\|_{L^{2p}}^{-(2p-1)}\int_{0}^{\infty}\int_{0}^{1}\left[(DN_{2,2})^{2p-1}-(DN_{1,2})^{2p-1}\right]\\\
&\qquad\qquad\qquad\qquad\qquad\left[DN_{2,2}-DN_{1,2}\right]W(u)v_{2}^{2\beta-1/2}v_{2}^{2\beta+3/2}N_{3,2}N_{4,2}\,du\,d\omega\\\
&\leq\mathbf{C}(\beta,p_{0})\|DN\|_{L^{2p}}\|N\|_{L^{\infty}}(\|N\|_{L^{\infty}}+\|DN\|_{L^{\infty}}).\end{split}$

Here, $\mathbf{C}(\beta,p_{0})$ depends only on $\beta$ and $p_{0}$ and in
particular uniform in $p$.

Combining with (5.18), for every $p>p_{0}$ we have

(5.58)
$\frac{d}{dt}\left(\|N\|_{L^{\infty}}+\|DN\|_{L^{2p}}\right)\leq\mathbf{C}(\beta,p_{0})(\|N\|_{L^{\infty}}+\|DN\|_{L^{\infty}})\|N\|_{L^{\infty}}\left(\|N\|_{L^{\infty}}+\|DN\|_{L^{2p}}\right),$

and thus

(5.59) $\begin{split}&\|N(t)\|_{L^{\infty}}+\|DN(t)\|_{L^{2p}}\\\
&\qquad\leq\left(\|N(0)\|_{L^{\infty}}+\|DN(0)\|_{L^{2p}}\right)\exp\left(\int_{0}^{t}\mathbf{C}(\beta,p_{0})(\|N(s)\|_{L^{\infty}}+\|DN(s)\|_{L^{\infty}})\|N(s)\|_{L^{\infty}}ds\right)\end{split}$

and by taking the limit $p\rightarrow\infty$,

(5.60)
$\begin{split}\|N(t)\|_{L^{\infty}}+\|DN(t)\|_{L^{\infty}}&\leq\left(\|N_{0}\|_{L^{\infty}}+\|DN_{0}\|_{L^{\infty}}\right)\\\
&\times\exp\left(\int_{0}^{t}\mathbf{C}(\beta,p_{0})(\|N(s)\|_{L^{\infty}}+\|DN(s)\|_{L^{\infty}})\|N(s)\|_{L^{\infty}}ds\right)\end{split}$

and therefore we can close the a priori estimate. Also, if we define $F(t)$ by

(5.61)
$F(t)=\left(\|N_{0}\|_{L^{\infty}}+\|DN_{0}\|_{L^{\infty}}\right)\exp\left(\int_{0}^{t}\mathbf{C}(\beta,p_{0})F(s)^{2}ds\right)$

then $F(t)$ is a supersolution to
$\|N(t)\|_{L^{\infty}}+\|DN(t)\|_{L^{\infty}}$, and satisfies

(5.62) $F^{\prime}(t)=\mathbf{C}(\beta,p_{0})F(t)^{3}$

whose solution exists for a positive time $T=T(\beta,p_{0},N_{0})$:

(5.63)
$\sup_{t\in(0,T)}\|DN(t)\|_{L^{\infty}}\leq\mathbf{C}(\beta,p_{0},N_{0}),$

and for every positive integer $p\geq p_{0}$

(5.64)
$\begin{split}&\int_{0}^{T}\int_{0}^{\infty}\int_{0}^{1}\left[(DN_{4,4})^{2p-1}-(DN_{3,4})^{2p-1}\right]\left[DN_{4,4}-DN_{3,4}\right]W(u)v_{4}^{2\beta-1/2}v_{4}^{2\beta+3/2}N_{1,4}N_{2,4}dud\omega
dt\\\
&+\int_{0}^{T}\int_{0}^{\infty}\int_{0}^{1}\left[(DN_{2,2})^{2p-1}-(DN_{1,2})^{2p-1}\right]\left[DN_{2,2}-DN_{1,2}\right]W(u)v_{2}^{2\beta-1/2}v_{2}^{2\beta+3/2}N_{3,2}N_{4,2}dud\omega
dt\\\ &\leq C(N_{0},p,\beta,p_{0})<\infty.\end{split}$

The positivity and the following change of variable simplifies (5.64): we set

$v=\frac{u^{2}}{1+u+u^{2}}.$

Then $v$ is an increasing function for $u\in[0,1]$, $v\in[0,1/3]$,
$DN_{1,2}=DN(\omega(1-v))$, $du=\frac{(1+u+u^{2})^{2}}{u(2+u)}dv$, and
$u^{2}\sim v$. Thus, we have

(5.65)
$\begin{split}\int_{0}^{T}\int_{0}^{\infty}\int_{0}^{1}&|DN(\omega)-DN(\omega(1+u))|^{2}u^{2\beta}\\\
\times&\left[\int_{0}^{1}|\lambda
DN(\omega)+(1-\lambda)DN(\omega(1+u))|^{2(p-1)}d\lambda\right]dud\omega dt\\\
+\int_{0}^{T}\int_{0}^{\infty}\int_{0}^{1/3}&|DN(\omega)-DN(\omega(1-u))|^{2}u^{-(\beta+1)}\\\
\times&\left[\int_{0}^{1}|\lambda
DN(\omega)+(1-\lambda)DN(\omega(1+u))|^{2(p-1)}d\lambda\right]dud\omega dt\\\
&\leq C(N_{0},p,\beta,p_{0})<\infty,\text{ for every positive integer }p\geq
p_{0}.\end{split}$

In particular, for $\beta\in(-3/4,3/4)$, by choosing $p=1$, we obtain

(5.66)
$\begin{split}&\int_{0}^{T}\int_{0}^{\infty}\int_{-1/3}^{1}|DN(\omega)-DN(\omega(1+u))|^{2}\left(\mathbf{1}_{[-1/3,0]}(u)|u|^{-(\beta+1)}+\mathbf{1}_{[0,1]}(u)|u|^{2\beta}\right)dud\omega
dt\\\ &\qquad\leq C(N_{0},\beta,p_{0})<\infty.\end{split}$

Finally, letting $\omega^{\prime}=\omega(1+u)$, and symmetrizing the norm of
(5.66) in $\omega$ and $\omega^{\prime}$, it is equivalent to (1.5) and we
obtain (1.4).

To summarize, we have the following a priori estimate:

######  Lemma 5.4.

There exists a time $T=T(N_{0},p_{0},\beta)$ and constant
$\mathbf{C}(\beta,p_{0},N_{0})$ such that

(5.67) $\|N\|_{L^{\infty}(0,T;X)}\leq\mathbf{C}(\beta,p_{0},N_{0}).$

Moreover, we have (5.65) and for $\beta\in(-3/4,3/4)$, we have (1.4).

###  5.4. Local well-posedness

Let $p_{0}>\max\left(\frac{1}{4(1-\beta)},\frac{1}{4(1+\beta)}\right)$ and
recall that

(5.68) $X=\\{N\in
L^{\infty}((0,\infty))|\|DN\|_{L^{2p_{0}}((0,\infty))}+\|DN\|_{L^{\infty}((0,\infty))}<\infty\\}$

with the natural norm

(5.69)
$\|N\|_{X}=\|N\|_{L^{\infty}((0,\infty))}+\|DN\|_{L^{2p_{0}}((0,\infty))}+\|DN\|_{L^{\infty}((0,\infty))}.$

The approximation scheme. For a $N_{0}\in X,N_{0}\geq 0$, we consider the
following approximate problem:

(5.70)
$\begin{split}\partial_{t}N^{\varepsilon}(t,\omega)&=\overline{C}_{\varepsilon}(N^{\varepsilon})(\omega):=\int_{\varepsilon}^{1}\frac{\left(\prod_{i=1}^{4}v_{i}\right)^{-\beta-\frac{1}{2}}}{1+u+u^{2}}\sum_{j,k=1}^{4}(-1)^{j+k}v_{j}^{2\beta+\frac{3}{2}}v_{k}^{2\beta-\frac{1}{2}}\frac{\prod_{i=1}^{4}N_{i,k}^{\varepsilon}}{N_{j,k}^{\varepsilon}}du,\\\
N^{\varepsilon}(0,\omega)&=N_{0}(\omega).\end{split}$

The problem (5.70) admits a unique solution $N^{\varepsilon}\in
L^{\infty}(0,T_{\varepsilon};X)$ for some time $T_{\varepsilon}>0$: this can
be regarded as an ODE with values in $X$: we obviously have

(5.71)
$\begin{split}\|\overline{C}_{\varepsilon}(N^{\varepsilon})\|_{L^{\infty}((0,\infty))}&\leq
C_{\varepsilon}\|N^{\varepsilon}\|_{L^{\infty}((0,\infty))}^{3},\\\
\|D\overline{C}_{\varepsilon}(N^{\varepsilon})\|_{L^{\infty}((0,\infty))}&\leq
C_{\varepsilon}\|DN^{\varepsilon}\|_{L^{\infty}((0,\infty))}\|N^{\varepsilon}\|_{L^{\infty}((0,\infty))}^{2},\end{split}$

and

(5.72)
$\begin{split}\|D\overline{C}_{\varepsilon}(N^{\varepsilon})\|_{L^{2p_{0}}((0,\infty))}&\leq
C_{\varepsilon}\sum_{j,k=1}^{4}\left\|\int_{\varepsilon}^{1}D\left(\prod_{i=1,i\neq
j}^{4}N_{i,k}^{\varepsilon}\right)du\right\|_{L^{2p_{0}}((0,\infty))}\\\ &\leq
C_{\varepsilon,p_{0}}\|DN^{\varepsilon}\|_{L^{2p_{0}}((0,\infty))}\|N^{\varepsilon}\|_{L^{\infty}((0,\infty))}^{2}.\end{split}$

To sum up, we have $\|\partial_{t}N^{\varepsilon}\|_{X}\leq
C_{\varepsilon,p_{0}}\|N^{\varepsilon}\|_{X}^{3}$, so we have local well-
posedness for $N^{\varepsilon}$ in $X$.

Uniform bounds in $\epsilon$. The key observation is that all the estimates in
Subsection 5.1, Subsection 5.2, and Subsection 5.3 carry over in the exact
same manner: we just replace $u\in[0,1]$ by $u\in[\varepsilon,1]$. The only
nontrivial part is the treatment of $I_{2,2}$, $I_{5,1}$, where we have
employed integration by parts, so the contribution of $u=\varepsilon$ might
not be zero. However, the contribution for $u=\varepsilon$ is smaller than the
contribution for $u=\eta$: in the case of $I_{2,2}$, the corresponding term is
of the form $O(u^{3/2})N^{2}DN$. In particular, this implies that all
$N^{\varepsilon}$ can be extended to the time $T=T(N_{0})>0$ which is uniform
in $\varepsilon$, and moreover we have uniform estimates

(5.73)
$\sup_{\varepsilon\in(0,1)}\|N^{\varepsilon}\|_{L^{\infty}(0,T;X)}\lesssim 1.$

In addition to that, we also have that for every $p>p_{0}$,

(5.74)
$\begin{split}&\sup_{\varepsilon\in(0,1)}\int_{0}^{T}\int_{0}^{\infty}\int_{\varepsilon}^{1}\left[(DN^{\varepsilon}_{4,4})^{2p-1}-(DN^{\varepsilon}_{3,4})^{2p-1}\right]\\\
&\qquad\qquad\qquad\qquad\qquad\left[DN^{\varepsilon}_{4,4}-DN^{\varepsilon}_{3,4}\right]W(u)v_{4}^{2\beta-1/2}v_{4}^{2\beta+3/2}N^{\varepsilon}_{1,4}N^{\varepsilon}_{2,4}\,du\,d\omega\,dt\\\
&\qquad+\int_{0}^{T}\int_{0}^{\infty}\int_{\varepsilon}^{1}\left[(DN^{\varepsilon}_{2,2})^{2p-1}-(DN^{\varepsilon}_{1,2})^{2p-1}\right]\\\
&\qquad\qquad\qquad\qquad\qquad\left[DN^{\varepsilon}_{2,2}-DN^{\varepsilon}_{1,2}\right]W(u)v_{2}^{2\beta-1/2}v_{2}^{2\beta+3/2}N^{\varepsilon}_{3,2}N^{\varepsilon}_{4,2}\,du\,d\omega\,dt\\\
&\leq C_{p,p_{0},\beta}.\end{split}$

Convergence of a subsequence in the uniform topology. To apply a compactness
argument: we first rely on the classical Arzela-Ascoli theorem. Fix $M>0$. We
see that the collection of functions

(5.75) $\\{N^{\varepsilon}(t,e^{s}),\,0\leq t\leq T,\,-M\leq s\leq M\\}$

are uniformly bounded (by
$\|N^{\varepsilon}\|_{L^{\infty}(0,T;L^{\infty}((0,\infty))}$) and uniformly
equicontinuous: we have

(5.76)
$\begin{split}|N^{\varepsilon}(t,e^{s_{2}})-N^{\varepsilon}(t,e^{s_{1}})|&=\left|\int_{e^{s_{1}}}^{e^{s_{2}}}DN^{\varepsilon}(\omega)\frac{d\omega}{\omega}\right|=\left|\int_{s_{1}}^{s_{2}}DN^{\varepsilon}(e^{s})ds\right|\\\
&\leq|s_{2}-s_{1}|\|DN^{\varepsilon}\|_{L^{\infty}(0,T;L^{\infty}((0,\infty))},\\\
|N^{\varepsilon}(t_{2},e^{s})-N^{\varepsilon}(t_{1},e^{s})|&=\left|\int_{t_{1}}^{t_{2}}\overline{C}_{\varepsilon}(N^{\varepsilon}(s))ds\right|\\\
&\lesssim\|N^{\varepsilon}\|_{L^{\infty}(0,T;L^{\infty})}^{2}(\|N^{\varepsilon}\|_{L^{\infty}(0,T;L^{\infty})}+\|DN^{\varepsilon}\|_{L^{\infty}(0,T;L^{\infty})})(t_{2}-t_{1})\end{split}$

and thanks to the uniform estimates we obtain the desired conclusion.
Therefore, by the Arzela-Ascoli theorem, we obtain a subsequence
$\\{N^{\varepsilon_{k,M}}\\}_{k}$ which converges to a function $N$ uniformly
in $[0,T]\times[e^{-M},e^{M}]$, and by a diagonalization argument, we can
extract a subsequence $N^{\varepsilon_{k}}$ which converges to $N$ uniformly
on compact subsets of $[0,T]\times(0,\infty)$. In the remainder, we slightly
abuse the notation and still denote this subsequence by $N^{\varepsilon}$.
This also shows that $N\in L^{\infty}((0,T;L^{\infty}(0,\infty))$. Also, this
implies that $DN^{\varepsilon}$ converges to $DN$ in the sense of distribution
in $[0,T]\times(0,\infty)$, and Banach-Alaoglu theorem applied to
$L^{\infty}(0,T;L^{\infty}(0,\infty))=L^{1}(0,T;L^{1}(0,\infty))^{*}$ and
$L^{\infty}(0,T;L^{2p_{0}}(0,\infty))=L^{1}(0,T;L^{2p_{0}^{*}}(0,\infty))^{*}$
implies that up to subsequence $DN^{\varepsilon}$ is weak-$*$ convergent, and
the limit should be therefore $DN$. Therefore, $N\in L^{\infty}(0,T;X)$. This
also implies that $\overline{\mathcal{C}}(N)\in
L^{\infty}(0,T;L^{\infty}((0,\infty)))$ and moreover, the integrand of
$\overline{\mathcal{C}}(N)$ is in $L^{1}([0,T]\times[0,1];dtdu)$ for almost
every $\omega$, following the argument in (5.17).

In the remainder of the argument, we shall abuse notations by denoting
$N^{\epsilon}$ instead of $N^{\epsilon_{k}}$ the converging subsequence.

Solution in the mild sense. The next step is to show that $N$ is a mild
solution, using the Vitali convergence theorem. Writing down (5.70) in Duhamel
form, we have

(5.77)
$N^{\varepsilon}(t,\omega)-N^{\varepsilon}(0,\omega)=\int_{0}^{t}\overline{C}_{\varepsilon}(N^{\varepsilon}(s))(\omega)ds,$

and for any compact subset $[0,T]\times C\subset[0,T]\times(0,\infty)$ the
left-hand side uniformly converges to $N(t,\omega)-N(0,\omega)$. For the
right-hand side, we claim that for every $\omega\in(0,\infty)$,
$\overline{C}_{\varepsilon}(N^{\varepsilon}(s))(\omega)$ converges to
$\overline{C}(N(s))(\omega)$ in $L^{1}([0,t])$.

To show this, we first define

(5.78)
$\begin{split}f^{\varepsilon}(u,s)&:=\mathbf{1}_{[\varepsilon,1]}(u)\frac{\left(\prod_{i=1}^{4}v_{i}\right)^{-\beta-\frac{1}{2}}}{1+u+u^{2}}\sum_{j,k=1}^{4}(-1)^{j+k}v_{j}^{2\beta+\frac{3}{2}}v_{k}^{2\beta-\frac{1}{2}}\frac{\prod_{i=1}^{4}N_{i,k}^{\varepsilon}(s)}{N_{j,k}^{\varepsilon}(s)},\\\
f(u,s)&=\frac{\left(\prod_{i=1}^{4}v_{i}\right)^{-\beta-\frac{1}{2}}}{1+u+u^{2}}\sum_{j,k=1}^{4}(-1)^{j+k}v_{j}^{2\beta+\frac{3}{2}}v_{k}^{2\beta-\frac{1}{2}}\frac{\prod_{i=1}^{4}N_{i,k}(s)}{N_{j,k}(s)},\end{split}$

with the measure space $([0,1]\times[0,t],du\,ds)$. We first show that
$f^{\varepsilon}$ converges to $f$ in measure. Let $\zeta>0$ be an arbitrary
small number. Then there exists a number $L(\zeta),U(\zeta)>0$ such that if
$u\in[\zeta,1]$, then $\omega v_{i,k}\in[L(\zeta),U(\zeta)]$ for all
$\omega\in C$ and $i,k=1,2,3,4.$ Then for sufficiently small $\varepsilon>0$,
due to uniform convergence of $N^{\varepsilon}$ on compact subsets,
$f^{\varepsilon}$ converges uniformly to $f$ in
$(u,s)\in[\zeta,1]\times[0,t]$. Therefore, for sufficiently small
$\varepsilon>0$, depending on $\zeta$,

(5.79)
$\\{(u,s)\in[0,1]\times[0,t]||f(u,s)-f^{\varepsilon}(u,s)|>\zeta\\}\subset[0,\zeta]\times[0,t],$

which demonstrates that for each $\omega$ $f^{\varepsilon}$ converges to $f$
in measure in $(u,s)\in[0,1]\times[0,t]$. Next, we show that
$\\{f^{\varepsilon}\\}$ is uniformly integrable. By the same calculation as
that of (5.17) we see that

(5.80) $\begin{split}|f^{\varepsilon}(u,s)|&\leq
u^{-2\beta-1}(1+u^{2\beta-1/2})(u^{2}+u^{2\beta+5/2})\|N^{\varepsilon}\|_{L^{\infty}(0,T;L^{\infty}((0,\infty)))}^{2}\\\
&\qquad\qquad(\|N^{\varepsilon}\|_{L^{\infty}(0,T;L^{\infty}((0,\infty)))}+\|DN^{\varepsilon}\|_{L^{\infty}(0,T;L^{\infty}((0,\infty)))}),\end{split}$

and by the uniform estimates we have

(5.81) $|f^{\varepsilon}(u,s)|\leq
Cu^{-2\beta-1}(1+u^{2\beta-1/2})(u^{2}+u^{2\beta+5/2}),$

and the right-hand side is integrable for $\beta\in(-1,1)$. Thus, the uniform
integrability condition is satisfied

(5.82)
$\lim_{C\rightarrow\infty}\sup_{\varepsilon\in[0,1]}\int_{|f^{\varepsilon}>C|}|f^{\varepsilon}|duds=0.$

Therefore, the Vitali convergence theorem gives $L^{1}$ convergence, namely
for every $\omega>0$,

(5.83)
$\overline{C}_{\varepsilon}(N^{\varepsilon}(s))(\omega)\rightarrow\overline{C}(N(s))(\omega)\text{
in }L^{1}((0,t),ds).$

In conclusion, for each $\omega>0$, $N$ satisfies

(5.84) $N(t,\omega)=N(0,\omega)+\int_{0}^{t}\overline{C}(N(s))(\omega)ds$

so $N$ is a mild solution.

Solution in the strong sense. We prove that the solution is in fact strong. We
have already seen that for every $\omega$, the integrand of
$\overline{\mathcal{C}}(N)$ is integrable in $[0,T]\times[0,1]$. In
particular, by Fubini’s theorem, for each $\omega\in(0,\infty)$, and for
almost $t\in[0,T]$, $\overline{\mathcal{C}}(N(t))(\omega)$ is well-defined and
is an integrable (and therefore measurable) function of $t\in[0,T]$.
Therefore, by Lebesgue differentiation theorem,
$\partial_{t}N(t,\omega)=\lim_{h\rightarrow
0}\frac{1}{h}\int_{t}^{t+h}\overline{C}(N(s))(\omega)ds$ exists and equals
$\overline{C}(N(t))(\omega)$ for almost every $t\in[0,T]$.

Uniqueness of the solution. Finally, we conclude that the solution to the
Cauchy problem is unique, by establishing a stability estimate. Suppose that
$\beta,p_{0}$ satisfies (5.5), $N,M\in L^{\infty}(0,T;X)$ be two strong
solutions of (5.2) with initial data $N_{0},M_{0}\in X$ respectively, and
$N_{0}-M_{0}\in L^{2p}((0,\infty))$ for some positive integer $p\geq p_{0}$.
If we let $\delta N=N-M$, we have

(5.85) $\begin{split}\partial_{t}\delta
N&=\int_{0}^{1}W(u)\sum_{k=1}^{4}(-1)^{k}v_{k}^{2\beta-\frac{1}{2}}\times\\\
&\left[(v_{2}^{2\beta+3/2}-v_{1}^{2\beta+3/2})\left((\delta
N)_{2,k}N_{3,k}N_{4,k}+M_{2,k}(\delta N)_{3,k}N_{4,k}+M_{2,k}M_{3,k}(\delta
N)_{4,k}\right)\right.\\\ &+v_{2}^{2\beta+3/2}((\delta N)_{1,k}-(\delta
N)_{2,k})N_{3,k}N_{4,k})\\\ &+v_{2}^{2\beta+3/2}(M_{1,k}-M_{2,k})\left((\delta
N)_{3,k}N_{4,k}+M_{3,k}(\delta N)_{4,k}\right)\\\
&+(v_{4}^{2\beta+3/2}-v_{3}^{2\beta+3/2})\left((\delta
N)_{1,k}N_{2,k}N_{4,k}+M_{1,k}(\delta N)_{2,k}N_{4,k}+M_{1,k}M_{2,k}(\delta
N)_{4,k}\right)\\\ &+v_{4}^{2\beta+3/2}((\delta N)_{3,k}-(\delta
N)_{4,k})N_{1,k}N_{2,k}\\\
&\left.+v_{4}^{2\beta+3/2}(M_{3,k}-M_{4,k})\left((\delta
N)_{1,k}N_{2,k}+M_{1,k}(\delta N)_{2,k}\right)\right]du\\\
&=:J_{1}+J_{2}+J_{3}+J_{4}+J_{5}+J_{6}.\end{split}$

Then we multiply $(\delta N)^{2p-1}$ to (5.85) and integrate over $\omega$ to
measure $\frac{1}{2p}\frac{d}{dt}\int_{0}^{\infty}(\delta
N)^{2p}(\omega)d\omega$. The calculation proceeds exactly same as in Section
5.3: for $\ell=1,3,4,6$,

(5.86) $\int_{0}^{\infty}(\delta N)^{2p-1}J_{\ell}d\omega\leq\|\delta
N\|_{L^{2p}(0,\infty)}^{2p-1}\|J_{\ell}\|_{L^{2p}(0,\infty)}$

and from (5.10), (5.12) and (5.16) and by Minkowski integral inequality, we
have

(5.87) $\begin{split}&\int_{0}^{\infty}(\delta
N)^{2p-1}(J_{1}+J_{4})d\omega\\\ &\lesssim\|\delta
N\|_{L^{2p}}^{2p}(\|N\|_{L^{\infty}}+\|M\|_{L^{\infty}})^{2}\times\\\
&\int_{0}^{1}u^{-2\beta-1}\sum_{k=1}^{4}v_{k}^{2\beta-1/2}\left[u^{2}(v_{k,2}^{1/2p}+v_{k,3}^{1/2p}+v_{k,4}^{1/2p})+u^{2\beta+5/2}(v_{k,1}^{1/2p}+v_{k,2}^{1/2p}+v_{k,4}^{1/2p})\right]du\\\
&\lesssim\|\delta
N\|_{L^{2p}}^{2p}(\|N\|_{L^{\infty}}+\|M\|_{L^{\infty}})^{2}\int_{0}^{1}(u^{-2\beta+1}+u^{3/2})(1+u^{-1/2p})+(u^{1/2}+u^{2\beta+1})(1+u^{1/2p})du\\\
&\lesssim\|\delta
N\|_{L^{2p}}^{2p}(\|N\|_{L^{\infty}}+\|M\|_{L^{\infty}})^{2}\end{split}$

by the conditions on $p,p_{0},\beta$. Similarly, we have

(5.88) $\begin{split}&\int_{0}^{\infty}(\delta
N)^{2p-1}(J_{3}+J_{6})d\omega\lesssim\|\delta
N\|_{L^{2p}}^{2p}(\|N\|_{L^{\infty}}+\|M\|_{L^{\infty}})\|DM\|_{L^{\infty}}\times\\\
&\int_{0}^{1}u^{1-2\beta-1/2p}+u^{3/2}+u^{1/2}+u^{1+2\beta+1/2p}du\lesssim\|\delta
N\|_{L^{2p}}^{2p}(\|N\|_{L^{\infty}}+\|M\|_{L^{\infty}})\|DM\|_{L^{\infty}}.\end{split}$

The estimates on $I_{2}$ and $I_{5}$ in Section 5.3, mutatis mutandis, give
the estimate on $J_{2}$ and $J_{5}$: we just replace $DN$ by $\delta N$ and we
omit the details. To summarize, we have

(5.89) $\frac{d}{dt}\|\delta N\|_{L^{2p}}\lesssim\|\delta
N\|_{L^{2p}}(\|N\|_{L^{\infty}(0,T;X)}+\|M\|_{L^{\infty}(0,T;X)})^{2},$

which gives the uniqueness.

## References

  * [1] R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu, and T. Yang.  Regularizing effect and local existence for the non-cutoff boltzmann equation.  _A_ rchive for Rational Mechanics and Analysis, 198(1):39–123, 2010\.
  * [2] R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu, and T. Yang.  Smoothing effect of weak solutions for the spatially homogeneous Boltzmann equation without angular cutoff.  _K_ yoto Journal of Mathematics, 52(3):433 – 463, 2012.
  * [3] T. Buckmaster, P. Germain, Z. Hani, and J. Shatah.  Onset of the wave turbulence description of the longtime behavior of the nonlinear schrödinger equation.  _I_ nventiones mathematicae, 225:787–855, 2021.
  * [4] O. Bühler and R. Du.  The impact of frequency bandwidth on a one-dimensional model for dispersive wave turbulence.  _J_. Nonlinear Sci.
  * [5] D. Cai, A. J. Majda, D. W. McLaughlin, and E. G. Tabak.  Spectral bifurcations in dispersive wave turbulence.  _P_ roceedings of the National Academy of Sciences, 96(25):14216–14221, 1999.
  * [6] D. Cai, A. J. Majda, D. W. McLaughlin, and E. G. Tabak.  Dispersive wave turbulence in one dimension.  _P_ hysica D: Nonlinear Phenomena, 152:551–572, 2001.
  * [7] S. Chibbaro, F. De Lillo, and M. Onorato.  Weak versus strong wave turbulence in the majda-mclaughlin-tabak model.  _P_ hysical Review Fluids, 2(5):052603, 2017.
  * [8] C. Collot, H. Dietert, and P. Germain.  Stability and cascades for the kolmogorov-zakharov spectrum of wave turbulence.  _a_ rXiv preprint arXiv:2208.00947, 2022.
  * [9] C. Collot and P. Germain.  On the derivation of the homogeneous kinetic wave equation.  _a_ rXiv preprint arXiv:1912.10368, 2019.
  * [10] C. Collot and P. Germain.  Derivation of the homogeneous kinetic wave equation: longer time scales.  _a_ rXiv preprint arXiv:2007.03508, 2020.
  * [11] Y. Deng and Z. Hani.  On the derivation of the wave kinetic equation for nls.  In _F_ orum of Mathematics, Pi, volume 9, page e6. Cambridge University Press, 2021.
  * [12] Y. Deng and Z. Hani.  Propagation of chaos and the higher order statistics in the wave kinetic theory.  _a_ rXiv preprint arXiv:2110.04565, 2021.
  * [13] Y. Deng and Z. Hani.  Derivation of the wave kinetic equation: full range of scaling laws.  _a_ rXiv preprint arXiv:2301.07063, 2023.
  * [14] Y. Deng and Z. Hani.  Full derivation of the wave kinetic equation.  _I_ nventiones mathematicae, pages 1–182, 2023.
  * [15] E. Di Nezza, G. Palatucci, and E. Valdinoci.  Hitchhiker?s guide to the fractional sobolev spaces.  _B_ ulletin des Sciences Mathématiques, 136(5):521–573, 2012.
  * [16] M. Escobedo, S. Mischler, and J. J. Velazquez.  On the fundamental solution of a linearized uehling–uhlenbeck equation.  _A_ rchive for rational mechanics and analysis, 186:309–349, 2007\.
  * [17] M. Escobedo, S. Mischler, and J. J. Velazquez.  Singular solutions for the uehling–uhlenbeck equation.  _P_ roceedings of the Royal Society of Edinburgh Section A: Mathematics, 138(1):67–107, 2008.
  * [18] M. Escobedo and J. J. Velázquez.  _O_ n the theory of weak turbulence for the nonlinear Schrodinger equation.  American Mathematical Soc., 2015.
  * [19] S. Galtier.  _P_ hysics of Wave Turbulence.  Cambridge University Press, 2022.
  * [20] P. Germain, A. D. Ionescu, and M.-B. Tran.  Optimal local well-posedness theory for the kinetic wave equation.  _J_ ournal of Functional Analysis, 279(4):108570, 2020.
  * [21] P. Gressman and R. Strain.  Global classical solutions of the boltzmann equation without angular cut-off.  _J_ ournal of the American Mathematical Society, 24(3):771–847, mar 2011.
  * [22] A. Hannani, M. Rosenzweig, G. Staffilani, and M.-B. Tran.  On the wave turbulence theory for a stochastic kdv type equation–generalization for the inhomogeneous kinetic limit.  _a_ rXiv preprint arXiv:2210.17445, 2022.
  * [23] J. W. Jang and R. M. Strain.  Asymptotic stability of the relativistic boltzmann equation without angular cut-off.  _A_ nnals of PDE, 8(2):20, 2022.
  * [24] A. Majda, D. McLaughlin, and E. Tabak.  A one-dimensional model for dispersive wave turbulence.  _J_ ournal of Nonlinear Science, 7(1):9–44, 1997.
  * [25] S. Nazarenko.  _W_ ave turbulence, volume 825.  Springer Science & Business Media, 2011.
  * [26] T. T. Nguyen and M.-B. Tran.  On the kinetic equation in zakharov’s wave turbulence theory for capillary waves.  _S_ IAM Journal on Mathematical Analysis, 50(2):2020–2047, 2018.
  * [27] A. Pushkarev and V. E. Zakharov.  Quasibreathers in the mmt model.  _P_ hysica D: Nonlinear Phenomena, 248:55–61, 2013.
  * [28] B. Rumpf and L. Biven.  Weak turbulence and collapses in the majda–mclaughlin–tabak equation: fluxes in wavenumber and in amplitude space.  _P_ hysica D: Nonlinear Phenomena, 204(3-4):188–203, 2005.
  * [29] A. Soffer and M.-B. Tran.  On the energy cascade of 3-wave kinetic equations: beyond kolmogorov–zakharov solutions.  _C_ ommunications in Mathematical Physics, 376(3):2229–2276, 2020\.
  * [30] G. Staffilani and M.-B. Tran.  On the wave turbulence theory for a stochastic kdv type equation.  _a_ rXiv preprint arXiv:2106.09819, 2021.
  * [31] V. Zakharov, F. Dias, and A. Pushkarev.  One-dimensional wave turbulence.  _P_ hysics Reports, 398(1):1–65, 2004.
  * [32] V. Zakharov, P. Guyenne, A. Pushkarev, and F. Dias.  Wave turbulence in one-dimensional models.  _P_ hysica D: Nonlinear Phenomena, 152:573–619, 2001.
  * [33] V. E. Zakharov, V. S. L’vov, and G. Falkovich.  _K_ olmogorov spectra of turbulence I: Wave turbulence.  Springer Science & Business Media, 2012.