In this paper, we give a unified treatment of the local well-posedness for the wave kinetic equation in almost critical weighted \(L^r\) spaces with \(2 \leq r \leq \infty.\) The proof
builds on ideas from our earlier works [1], [2]. Our approach is based solely on kinetic tools,
with no appeal to Fourier theory.
In this paper we study the 3D wave kinetic equation (WKE) with Laplacian dispersion relation \[\label{KWE}\begin{cases}
\partial_t f = \mathcal{C}[f]\\
f(t=0)=f_0
\end{cases}\tag{1}\] where \(f:[0,T] \times \mathbb{R}^3 \rightarrow \mathbb{R}\), \(T>0\) and \(f_0:{\mathbb{R}}^3\to{\mathbb{R}}\).
The collisional operator \(\mathcal{C}\) is defined as \[\begin{align} \label{collision}
\begin{aligned} \mathcal{C}[f] & := \int_{\mathbb{R}^{9}} \delta (\Sigma) \, \delta(\Omega) \, f f_1 f_2 f_3 \, \big(\frac{1}{f} + \frac{1}{f_1} - \frac{1}{f_2} - \frac{1}{f_3} \big) \,dk_1 dk_2 dk_3, \\ \Sigma&:=k+k_1 - k_2 - k_3, \\ \Omega
&:= \vert k \vert^2 + \vert k_1 \vert^2 - \vert k_2 \vert^2 - \vert k_3 \vert^2 .
\end{aligned}
\end{align}\tag{2}\] This describes the statistical properties of a system governed by the cubic NLS equation, and is the most canonical model in the field of weak turbulence. Here the unknown \(f\)
corresponds to the two-point correlation function of the system. More generally, the goal of this theory is to describe the out-of-equilibrium dynamics of interacting waves. It originated in 1929 with the work of R. Peierls [3] and in 1962 with K. Hasselman [4], [5]
independently. It has proved very versatile and has been applied to many physical systems. For a comprehensive list of concrete examples, we refer to the textbook of S. Nazarenko [6].
This theory has also been studied mathematically, with the question of derivation of kinetic equations from microscopic models as the main focus thus far. After works by several groups [7]–[10], the optimal result was obtained by Z. Hani and Y. Deng [11].
They derived the wave kinetic equation from the microscopic system on the optimal time scale, called the kinetic time.
We note that these results are all conditional on the wave kinetic equation having a good well-posedness theory. Indeed only smooth enough solutions to the kinetic equation can be derived from the microscopic system. The next natural step is thus to
develop the local well-posedness theory of the WKE, which is the subject of the present paper. We note that weak solutions were constructed by M. Escobedo and J. Velázquez in [12]. We will focus on strong solutions here, since they correspond to configurations that have been shown to be derivable from many-body systems.
The well-posedness theory of strong solutions for wave kinetic equations is still in its infancy. A particularly relevant functional analytic setting to consider are scale-critical weighted Lebesgue spaces. Indeed, they contain Rayleigh-Jeans spectra
given by \(f(k) = \frac{1}{\mu + \vert k \vert^2},\) where \(\mu\) is a free parameter. These special solutions correspond to thermodynamic equilibrium and are thus the analog of Maxwellians
for the Boltzmann equation. As such, they are important to understand the long-time behavior of 1 . We note here that kinetic equations have other physically relevant solutions, e.g. the celebrated Kolmogorov-Zakharov spectra. The
full dynamics of the equation is therefore expected to be quite complicated and different from that of the Boltzmann equation. Local results point in this direction as well: C. Collot, H. Dietert and P. Germain [13] studied the static problem near one Kolmogorov-Zakharov solution in the isotropic case, and prove that it is stable. However, some Rayleigh-Jeans
solutions have been shown to be unstable by M. Escobedo and A. Menegaki [14].
Restricting our attention to local well-posedness, the state of the art result was obtained by P. Germain, A. Ionescu and M.-B. Tran [15] who prove almost
critical local well-posedness in \(L^2\) and \(L^\infty.\) The \(L^\infty\) bound is proved by direct estimation, while for \(L^2\) the authors rely on Radon transform type techniques.
In the present paper we improve on this result in two respects: first, we extend it to all almost critical weighted \(L^r\) for \(2 \leq r \leq \infty\). Second, we give a unified
treatment of all the cases, and do not rely on multilinear interpolation to obtain the result. Indeed, our method of proof is quite different from [15], and is
inspired by our recent works on the wave kinetic and Boltzmann equations [1], [2]. We rely solely
on classical kinetic tools, such as Bobylev variables and angular averaging estimates. In particular, we make no appeal to Fourier theory.
Theorem 1. Let \(r\geq 2\) and \(0<\delta<1/r\) if \(r<\infty\), \(\delta > 0\) if \(r = \infty.\) Let \(f_0 \in {\langle}k {\rangle}^{-2 + \frac{3}{r} -\delta} L^r.\) Then there exists \(T=T(\|{\langle}k{\rangle}^{2 - \frac{3}{r} + \delta}
f_0\|_{L^r})>0\) such that the initial value problem 1 has a unique strong solution \(f(t) \in \mathcal{C} \big( [0,T] ; {\langle}k {\rangle}^{-2 + \frac{3}{r} - \delta} L^r \big),\) that is
\[\begin{align} \forall t \in [0,T], \quad f(t) = f_0 + \int_0^t \mathcal{C}[f](s)\, ds .
\end{align}\]
Moreover the solution depends continuously on the initial data: let \(f_0, g_0 \in {\langle}k {\rangle}^{-2 + \frac{3}{r} -\delta} L^r\), \(T_1,T_2>0\) be the times of existence
obtained above and \(f,g\) be the corresponding solutions of 1 in \([0,T_1]\) and \([0,T_2]\) respectively. Then, there holds the
estimate \[\begin{align}
\label{continuity32wrt32data32estimate} \sup_{t\in[0,T_{min}]}\big \Vert {\langle}k {\rangle}^{2 - \frac{3}{r} + \delta} \big( f(t) - g(t) \big) \big \Vert_{L^r} \leq 2\big \Vert {\langle}k {\rangle}^{2 - \frac{3}{r} + \delta} \big(f_0 - g_0 \big) \big
\Vert_{L^r},
\end{align}\tag{3}\] where \(T_{min}:=\min\{T_1,T_2\}\).
Finally, the flow preserves positivity, in the sense that if \(f_0 \geq 0,\) then for all \(t \in [0,T], f(t) \geq 0,\) where \(T\) is the time of
existence corresponding to \(f_0\).
Remark 2. The case \(\delta = 0\) corresponds to the critical exponent.
Remark 3. In the special cases \(r = 2, r = \infty,\) we recover the results of [15]. Of course, it is possible to
obtain the intermediate cases from their bounds by multilinear interpolation.
Remark 4. As will be clear in the proof, all the cases \(2 \leq r \leq \infty\) are treated in a unified way.
Remark 5. The proof yields a more quantitative estimate for the lifespan of the solution. More precisely, it shows that \(T \gtrsim \|{\langle}k{\rangle}^{2 - \frac{3}{r} + \delta} f_0\|_{L^r}^{-2}\).
For Laplacian dispersion relation, there is a particularly simple parametrization of the collisional kernel as a hard-sphere quantum Boltzmann-type operator. In other words, the wave interaction of the modes \(k,k_1,k_2,k_3\) can be identified with the elastic collision of two particles with pre-collisional velocities \(k,k_1\) and post-collisional velocities \(k_2,k_3\). At the level of the corresponding particle interaction, the resonant conditions \(\Sigma=0\) and \(\Omega=0\) correspond to the conservation of
momentum and energy respectively.
More precisely, consider \(F:{\mathbb{R}}^{12}\to{\mathbb{R}}\) continuously differentiable and compactly supported, and denote \[I_F(k):=\int_{{\mathbb{R}}^{9}}\delta(\Sigma)\delta(\Omega)F(k,k_1,k_2,k_3)\,dk_1\,dk_2\,dk_3.\]
Recalling \(\Sigma=k+k_1-k_2-k_3\), \(\Omega=|k|^2+|k_1|^2-|k_2|^2-|k_3|^2\), and using Fubini’s theorem and the co-area formula, we can write \[\begin{align}
I_F(k)&=\int_{{\mathbb{R}}^3}\left(\int_{{\mathbb{R}}^3}\delta(\Omega_{k,k_1}(k_2))F(k,k_1,k_2,k+k_1-k_2)\,dk_2\right)\,dk_1\\
&=\int_{{\mathbb{R}}^3}\int_{\Omega_{k,k_1}=0}\frac{1}{|\nabla_{k_2}\Omega_{k,k_1}(k_2)|}F(k,k_1,k_2,k+k_1-k_2)\,d\sigma_{k,k_1}(k_2)\,dk_1,
\end{align}\] where \[\begin{align}
\Omega_{k,k_1}(k_2)&=|k_2|^2+|k+k_1-k_2|^2-|k|^2-|k_1|^2=2\left(|k_2-K|^2-\frac{|w|^2}{4}\right),\\
K&:=\frac{k+k_1}{2},\quad w:=k-k_1,
\end{align}\] and \(\,d\sigma_{k,k_1}\) denotes the surface measure on the surface \(\Omega_{k,k_1}=0\). As a result, the surface \(\Omega_{k,k_1}=0\)
can be parametrized by \[k_2=K-\frac{|w|}{2}\sigma,\quad \sigma\in{\mathbb{S}}^2,\quad d\sigma_{k,k_1}(k_2)=\frac{|w|^2}{4}\,d\sigma.\] Finally on the surface \(\Omega_{k,k_1}=0\), we
readily compute \[\begin{align}
|\nabla_{k_2}\Omega_{k,k_1}(k_2)|=4|k_2-K|=2|w|.
\end{align}\] Combining these computations, we conclude that
\[\label{IF32parametrized}
I_F(k)=\frac{1}{8}\int_{{\mathbb{R}}^3\times{\mathbb{S}}^2}|w| F(k,k_1,k^*,k_1^*)\,d\sigma\,dk_1,\tag{4}\] where \[\label{collisional32law}
\begin{cases}
k^*=K-\frac{|w|}{2}\sigma\\
k_1^*=K+\frac{|w|}{2}\sigma
\end{cases} \quad ,
\quad K=\frac{k+k_1}{2},\quad w=k-k_1.\tag{5}\] Due to the linear growth \(|w|\) in the integrand, equation 4 essentially shows that integration over the resonant manifolds
is equivalent to a collisional integral of interacting hard-spheres. The resulting linear growth \(|w|\) is called resonant cross-section.
This idea was first implemented in the study of wave turbulence with Laplacian dispersion relation by the first author of this paper [16], where global existence,
uniqueness and stability of mild solutions to the space inhomogeneous WKE were proved for sufficiently small exponentially decaying initial data. It was later extended in [17] to small polynomially decaying initial data and to the corresponding hierarchy of equations by the first author, J.K. Miller, N. Pavlović and M. Tasković. In [1], we improved this result to translation invariant spaces in the spatial variable by introducing collisional averaging estimates combined with dispersive properties of free transport. We also obtained a
full description of the asymptotic behavior of the solutions, showing that they scatter. In [18] we used similar collisional averaging estimates to show
strong local well-posedness of MMT-type kinetic equations below a sharp ill-posedness threshold. Above said threshold the collisional averaging effect is absent and we showed that, coincidentally, the equations are ill-posed. Thus these collisional
averaging estimates characterize completely the well-posedness of these equations. These techniques are quite versatile, and can be applied to other models. We refer for example to the work of N. Pavlović, M. Tasković and L. Velasco [19] who constructed mild solutions that scatter for the six-wave WKE with exponentially decaying initial data.
Using 4 , the collisional operator can be written in gain and loss form as follows \[\label{gain-loss}
\mathcal{C}[f]=\mathcal{Q}^+[f]-\mathcal{Q}^-[f],\tag{6}\] where \[\begin{align}
\mathcal{Q^+}[f](k)&=\frac{1}{8}\int_{{\mathbb{R}}^3\times{\mathbb{S}}^2}|w|f(k^*)f(k_1^*)\big(f(k)+f(k_1)\big)\,d\sigma\,d k \color{black} _1\tag{7},\\
\mathcal{Q}^-[f](k)&=f(k)\mathcal{R}[f](k),\tag{8}\\
\mathcal{R}[f](k)&= \frac{1}{8}\int_{{\mathbb{R}}^3\times{\mathbb{S}}^2}|w|f(k_1)\big(f(k^*)+f(k_1^*)\big)\,d\sigma\,dk_1.
\end{align}\] The operator \(\mathcal{R}[f]\) is called collision frequency.
Now, we use a more refined decomposition which takes advantage of the symmetry of the collisional operators. This idea was introduced in [1] and then was
also used in [18]. For this, let us introduce the standard cut-off function \(\chi:=\mathbb{1}_{(0,+\infty)}\). Noticing
that the substitution \(\sigma\to -\sigma\) just exchanges \(k^*\) with \(k_1^*\), we obtain, denoting \(w:= k-k_1\) and
using the standard notation \(\widehat{w} = \frac{w}{\vert w \vert}\)\[\begin{align}
\mathcal{Q}^+[f](k)&= \frac{1}{8}\int_{{\mathbb{R}}^3\times{\mathbb{S}}^2}|w|f(k^*)f(k_1^*)\big(f(k)+f(k_1)\big)\chi(\widehat{w}\cdot\sigma)\,d\sigma\,d k \color{black} _1 \\
&+ \frac{1}{8}\int_{{\mathbb{R}}^3\times{\mathbb{S}}^2}|w|f(k^*)f(k_1^*)\big(f(k)+f(k_1)\big)\chi(-\widehat{w}\cdot\sigma)\,d\sigma\,d k \color{black} _1\\
&=\frac{1}{4}\int_{{\mathbb{R}}^3\times{\mathbb{S}}^2}|w|f(k^*)f(k_1^*)\big(f(k)+f(k_1)\big)\chi(\widehat{w}\cdot\sigma)\,d\sigma\,d k \color{black} _1,
\end{align}\] where for the last line we used the substitution \(\sigma\to-\sigma\) in the second part of the sum.
Similarly, we obtain \[\begin{align}
\mathcal{R}[f](k)&= \frac{1}{8}\int_{{\mathbb{R}}^3\times{\mathbb{S}}^2}|w|f(k_1)\big(f(k^*)+f(k_1^*)\big)\chi(\widehat{w}\cdot\sigma)\,d\sigma\,dk_1\\
& +\frac{1}{8}\int_{{\mathbb{R}}^3\times{\mathbb{S}}^2}|w|f(k_1)\big(f(k^*)+f(k_1^*)\big)\chi(-\widehat{w}\cdot\sigma)\,d\sigma\,dk_1\\
&=\frac{1}{4}\int_{{\mathbb{R}}^3\times{\mathbb{S}}^2}|w|f(k_1)\big(f(k^*)+f(k_1^*)\big)\chi(\widehat{w}\cdot\sigma)\,d\sigma\,dk_1.
\end{align}\] We conclude that we can write \[\begin{align}
\tag{9} \mathcal{Q}^+[f] &:= \mathcal{G}_0[f,f,f]+\mathcal{G}_1[f,f,f], \\ \tag{10} \mathcal{Q}^-[f] &:= \mathcal{L}_0[f,f,f] + \mathcal{L}_1[f,f,f],\\ \tag{11}\mathcal{R}[f]&=\mathcal{R}_0[f,f]+\mathcal{R}_1[f,f],
\end{align}\] where \(\mathcal{G}_0,\mathcal{G}_1,\mathcal{L}_0,\mathcal{L}_1,\mathcal{R}_0,\mathcal{R}_1\) are the cubic/quadratic restrictions of the trilinear/bilinear operators \[\begin{align}
\tag{12} \mathcal{G}_0[f,g,h](k)&=\frac{1}{4}f(k)\int_{{\mathbb{R}}^3\times{\mathbb{S}}^2}|w| g(k^*)h(k_1^*)\chi(\widehat{w}\cdot\sigma)\,d\sigma\,dk_1,\\
\tag{13} \mathcal{G}_1[f,g,h](k)&=\frac{1}{4}\int_{{\mathbb{R}}^3\times{\mathbb{S}}^2}|w| f(k_1)g(k^*)h(k_1^*)\chi(\widehat{w}\cdot\sigma)\,d\sigma\,dk_1,\\
\tag{14} \mathcal{L}_0[f,g,h](k)&=f(k) \mathcal{R}_0[g,h](k),\\
\tag{15} \mathcal{L}_1[f,g,h](k)&=f(k) \mathcal{R}_1[g,h](k),\\
\tag{16} \mathcal{R}_0[g,h](k)&=\frac{1}{4}\int_{{\mathbb{R}}^3\times{\mathbb{S}}^2}|w| g(k_1)h(k^*)\chi(\widehat{w}\cdot\sigma)\,d\sigma\,dk_1, \\
\tag{17} \mathcal{R}_1[g,h](k)&=\frac{1}{4}\int_{{\mathbb{R}}^3\times{\mathbb{S}}^2}|w| g(k_1)h(k_1^*)\chi(\widehat{w}\cdot\sigma)\,d\sigma\,dk_1.
\end{align}\]
These multilinear operators are the fundamental objects we study to prove our main result.
The paper is organized as follows: in Section 2 we record the technical tools used in the rest of the paper, namely Bobylev variables, basic geometric identities and key collisional averaging estimates. Then we use this
toolbox to prove moment preserving trilinear estimates for the gain operators (Section 3) and the loss operators (Section 4). Finally we use these bounds to prove our main result Theorem 1 in Section 5.
Throughout the paper, we will use the following notation:
We will write \(A \lesssim B\) to mean that there exists a numerical constant \(C>0\) such that \(A \leq C B.\) We will write \(A \approx B\) if \(A \lesssim B\) and \(B \lesssim A.\)
\(\chi:=\mathbb{1}_{(0,+\infty)}\) will denote the standard cut-off function.
Given \(s\in{\mathbb{R}}\) we will denote \(f_s(k):={\langle}k{\rangle}^s f(k)\).
For any \(w \in \mathbb{R}^3\setminus\{0\}\) we denote \(\widehat{w} = \frac{w}{\vert w \vert}.\)
Finally, given \(k,k_1\in{\mathbb{R}}^3\) and \(\sigma\in{\mathbb{S}}^2\), we will denote \[\begin{align} &k^*=K-\frac{|w|}{2}\sigma,\quad
k_1^*=K+\frac{|w|}{2}\sigma,\\ &K=\frac{k+k_1}{2},\quad w=k-k_1,\quad E=|k|^2+|k_1|^2.
\end{align}\]
In this section, we start by recalling basic facts about Bobylev variables. Then we record some basic geometric identities that will be used routinely in the proof. Finally, we prove collisional averaging estimates that will be crucial to offset the
linear growth of the cross section in later sections.
First, we introduce the so-called Bobylev variables [20], [21], a classical kinetic theory tool that we will
rely on to prove crucial collisional averaging estimates. The Bobylev variables appear naturally in the gain operator above and thus will often be used to provide an alternative parametrization of the collisional operator. Here, we outline their most
important properties.
Given \(\sigma\in{\mathbb{S}}^2\), we define the Bobylev variables as the maps \(R_\sigma^+,R_\sigma^-:{\mathbb{R}}^3\to{\mathbb{R}}^3\) defined by
\[\begin{align}
R_\sigma^+(y)=\frac{y}{2}+\frac{|y|}{2}\sigma,\quad R_\sigma^-(y)=\frac{y}{2}-\frac{|y|}{2}\sigma.\label{R32def}
\end{align}\tag{18}\] The following identities are easily checked: \[\begin{align}
R_\sigma^+(y)+R_\sigma^-(y)&=y,\tag{19}\\
R_\sigma^+(y)\cdot R_\sigma^-(y)&=0,\tag{20}\\
|R_\sigma^+(y)|^2+|R_\sigma^-(y)|^2&=|y|^2.\tag{21}
\end{align}\] Furthermore, recalling that \(w =
k-k_1\), we obtain \[\begin{align}
\label{R4332R-32v}
R_\sigma^+(w)=k-k^*=k_1^*-k_1,\quad R_\sigma^-(w)=k-k_1^*=k^*-k_1.
\end{align}\tag{22}\]
The next result shows that the Bobylev variables are diffeomorphisms when restricted appropriately. It also provides useful identities relating the magnitudes and angles between these various variables. The proof of this result is contained in our
previous papers [1], [2].
Proposition 6. Let \(\sigma\in{\mathbb{S}}^{2}\) and \(\epsilon \in\{+,-\}\). Then the map \[R_\sigma^{\epsilon}:\lbrace
y\in{\mathbb{R}}^3: y \cdot \sigma \neq - \epsilon \vert y\vert \rbrace\to \lbrace \nu\in{\mathbb{R}}^3: \epsilon \,(\nu \cdot \sigma) > 0 \rbrace,\] is a diffeomorphism with inverse \[\begin{align}
\label{inverse32function} y=\big(R_{\sigma}^{\epsilon}\big)^{-1}(\nu) = 2 \nu -\frac{\vert \nu \vert}{ (\widehat{\nu}\cdot \sigma )} \sigma,
\end{align}\qquad{(1)}\] and Jacobian \[\label{Jacobian} \text{Jac}\,(R_\sigma^{\epsilon})^{-1}(\nu)=\frac{4 }{(\widehat{\nu}\cdot\sigma)^2}.\qquad{(2)}\] Moreover, for any \(u\in{\mathbb{R}}^3\) with \(y\cdot\sigma\neq -\epsilon|y|\), we have \[\begin{align}
|R_\sigma^\epsilon(y)\cdot\sigma|&=\epsilon\,(R_\sigma^\epsilon(y)\cdot\sigma),\label{sign}\\
|y|&=\frac{|R_{\sigma}^{\epsilon}(y)|}{|\widehat{R}_{\sigma}^{\epsilon}(y)\cdot\sigma|}\label{magnitude},\\
\widehat{y} \cdot \sigma &= \epsilon \left(2 |\widehat{R}_{\sigma}^{\epsilon}(y)\cdot \sigma|^2 -1\right). \label{angle}
\end{align}\] {#eq: sublabel=eq:sign,eq:magnitude,eq:angle} Finally, for \(y\in {\mathbb{R}}^3\) with \(y\cdot\sigma\neq\pm |y|\), we have \[\label{R4332R-32relation} |\widehat{R}_\sigma^+(y)|^2+|\widehat{R}_\sigma^-(y)|^2=1.\qquad{(3)}\]
The most pathological interactions occur when the frequencies of the interacting waves are on different scales and cannot pointwise offset the resonant cross-section. The next lemma provides an elementary, yet very useful lower bound for relating a
small frequency with a larger one, up to a singularity on the scattering direction \(\sigma.\)
Lemma 1. Let \(k,k_1\in {\mathbb{R}}^3\) and \(\sigma\in{\mathbb{S}}^2\). Then, the following point-wise estimates hold \[\begin{align} \tag{23}
{\langle}k^* {\rangle}, {\langle}k_1^* {\rangle}&>\left(\frac{1+E}{2}\right)^{1/2} \big(1 - \lambda_E\vert \widehat{K} \cdot \sigma \vert \big)^{1/2},\quad \lambda_E=\frac{E}{1+E},\\
\tag{24}{\langle}k^*{\rangle}&\geq {\langle}k_1^*{\rangle}(1-\lambda_{k_1^*}|\widehat{k_1^*}\cdot\sigma|)^{1/2},\quad \lambda_{k_1^*}=\frac{|k_1^*|^2}{{\langle}k_1^*{\rangle}^2},\\
\tag{25}{\langle}k_1^*{\rangle}&\geq {\langle}k^*{\rangle}(1-\lambda_{k^*}|\widehat{k^*}\cdot\sigma|)^{1/2},\quad \lambda_{k^*}=\frac{|k^*|^2}{{\langle}k^*{\rangle}^2},\\
{\langle}k_1{\rangle}&> \frac{{\langle}k{\rangle}}{3}\left(1-\lambda_{k}|\widehat{(k-2R_\sigma^\epsilon(w)}\cdot\sigma|\right)^{1/2},\quad \lambda_{k}=\frac{|k|^2}{{\langle}k{\rangle}^2},\quad\epsilon\in\{+,-\},\tag{26}\\
{{\langle}k{\rangle}} &> {\frac{{\langle}k_1{\rangle}}{3}\left(1-\lambda_{k_1}|\widehat{(k_1 + 2R_\sigma^\epsilon(w))}\cdot\sigma|\right)^{1/2},\quad \lambda_{k_1}=\frac{|k_1|^2}{{\langle}k_1{\rangle}^2}, \quad \epsilon\in\{+,-\}}\tag{27}.
\end{align}\]
Proof. To prove 23 , we rely on the elementary inequality \[\begin{align}
\label{energy32inequality}
|w||K|&\leq \frac{|w|^2}{4}+|K|^2=\frac{E}{2}.
\end{align}\tag{28}\] Then, by 5 we have \[\begin{align}
|k^*|^2&=\frac{E}{2}-|w||K|(\widehat{K}\cdot\sigma) \geq \frac{E}{2}-|w||K||\widehat{K}\cdot\sigma| \geq \frac{E}{2}(1-|\widehat{K}\cdot\sigma|),
\end{align}\] which in turn implies \[{\langle}k^*{\rangle}^2=1+|k^*|^2\geq 1+\frac{E}{2}(1-|\widehat{K}\cdot\sigma|)> \frac{1+E}{2}\left(1-\lambda_E|\widehat{K}\cdot\sigma)|\right).\] The bound for \(k_1^*\) follows identically using the expression \(|k_1^*|^2=\frac{E}{2}+|w||K|(\widehat{K}\cdot\sigma)\) instead.
To prove 24 , we use 5 to write \(k_1^*-k^*=|w|\sigma\). Then we have \[|k^*|^2=\Big|k_1^*-|w|\sigma\Big|^2\geq (|k_1^*|^2+|w|^2)(1-|\widehat{k_1^*}\cdot\sigma|),\] which yields \[{\langle}k^*{\rangle}^2=1+|k^*|^2\geq
1+|k_1^*|^2(1-\widehat{k_1^*}\cdot\sigma)={\langle}k_1^*{\rangle}^2\left(1-\lambda_{k_1^*}|\widehat{k_1^*}\cdot\sigma|\right).\] Bound 25 follows identically using the expression \(k_1^*=k^*+|w|\sigma\) instead.
To prove 26 , let us denote \(\nu=R_\sigma^\epsilon(w)\). We use ?? to write \[k_1=k-w=\left(k-2\nu\right)+\frac{|\nu|}{(\widehat{\nu}\cdot\sigma)}\sigma,\] which yields \[\begin{align}
|k_1|^2&\geq \left(|k-2\nu|^2+\frac{|\nu|^2}{|\widehat{\nu}\cdot\sigma|^2}\right)\left(1-|\widehat{(k-2\nu)}\cdot\sigma|\right)\notag\\
&\geq \left(|k-2\nu|^2+|\nu|^2\right)\left(1-|\widehat{(k-2\nu)}\cdot\sigma|\right).
\end{align}\] Now, we notice that \(|k-2\nu|^2+|\nu|^2\geq \frac{|k|^2}{9}\). Indeed, if \(|\nu|\geq |k|/3\) the claim is immediate, while if \(|\nu|<|k|/3\), the triangle inequality implies \(|k-2\nu|\geq |k|-2|\nu|> |k|/3\). Combining these facts, we obtain \[{\langle}k_1{\rangle}^2=1+|k_1|^2\geq
1+\frac{|k|^2}{9}\left(1-|\widehat{(k-2\nu)\cdot\sigma|}\right)>\frac{{\langle}k{\rangle}^2}{9}\left(1-\lambda_k|\widehat{(k-2\nu)}\cdot\sigma|\right),\quad\lambda_k=\frac{|k|^2}{{\langle}k{\rangle}^2}.\] Bound 27 follows identically using the expression \(k=w+k_1\) instead. The proof is complete. ◻
We now present the averaging estimates that are necessary to offset the linear growth of the resonant cross-section. They are largely inspired by our previous work on the wave kinetic and Boltzmann equations [1], [2].
Lemma 2. Let \(l>2\). Then for any \(k,k_1\in{\mathbb{R}}^3\), the following averaging estimates hold \[\begin{align}
\tag{29}
\int_{{\mathbb{S}}^2}\frac{1}{{\langle}k^*{\rangle}^l}\,d\sigma=\int_{{\mathbb{S}}^2}\frac{1}{{\langle}k_1^*{\rangle}^l}\,d\sigma&\lesssim \frac{1}{(l-2)(1+E)}, \\
\int_{{\mathbb{S}}^2}\frac{1}{{\langle}k^*{\rangle}^l{\langle}k_1^*{\rangle}^l}\,d\sigma&\lesssim \frac{1}{(l-2)(1+E)^{1+l/2}}.\tag{30}
\end{align}\]
Proof. Fix \(k,k_1\in{\mathbb{R}}^3\). Note that both statements are trivial if \(E \leq 1.\) Thus we may assume that \(E>1\), which implies
that \(\lambda_E=\frac{E}{1+E}\in(1/2,1)\).
We first prove 29 . By symmetry (changing \(\sigma\mapsto -\sigma\)), it is evident that \(\int_{{\mathbb{S}}^2}\frac{1}{{\langle}k^*{\rangle}^l}\,d\sigma=\int_{{\mathbb{S}}^2}\frac{1}{{\langle}k_1^*{\rangle}^l}\,d\sigma\), so it suffices to show the estimate for \(k^*\). Using 23 and integrating in spherical coordinates, we obtain \[\begin{align}
\int_{{\mathbb{S}}^2}\frac{1}{{\langle}k^*{\rangle}^l}\,d\sigma&\lesssim \frac{1}{(1+E)^{l/2}}\int_{{\mathbb{S}}^2}\frac{1}{(1-\lambda_E|\widehat{K}\cdot\sigma|)^{l/2}}\,d\sigma \approx \frac{1}{(1+E)^{l/2}}\int_{0}^{2 \pi}\frac{1}{(1-\lambda_E|\cos
\theta|)^{l/2}} \sin \theta \,d\theta \color{black} \\
& \approx \frac{1}{(1+E)^{l/2}}\int_0^1\frac{1}{(1-\lambda_E x)^{l/2}}\,d x \color{black} =\frac{1}{\lambda_E(1+E)^{1/2}}\int_0^{\lambda_E}\frac{1}{(1-y)^{l/2}}\,dy\approx \frac{(1-\lambda_E)^{1-l/2}}{(l-2)\lambda_E(1+E)^{l/2}} \\
& \approx\frac{1}{(l-2)(1+E)},
\end{align}\] where we used the facts \(\lambda_E\in(1/2,1)\) and \(1-\lambda_E=\frac{1}{1+E}\). Estimate 29 is proved.
Now, to prove 30 , the resonant condition \(|k^*|^2+|k_1^*|^2=E\) implies that either \(|k^*|^2>E/2\) or \(|k_1^*|^2>E/2\). Thus, using 29 , we obtain \[\begin{align}
\int_{{\mathbb{S}}^2}\frac{1}{{\langle}k^*{\rangle}^l{\langle}k_1^*{\rangle}^l}\,d\sigma&=\int_{{\mathbb{S}}^2}\frac{1}{{\langle}k^*{\rangle}^l{\langle}k_1^*{\rangle}^l}\mathbb{1}_{|k^*|^2>E/2}\,d\sigma+\int_{{\mathbb{S}}^2}\frac{1}{{\langle}k^*{\rangle}^l{\langle}k_1^*{\rangle}^l}\mathbb{1}_{|k_1^*|^2>E/2}\,d\sigma
\\
&\lesssim \frac{1}{(1+E)^{l/2}}\left(\int_{{\mathbb{S}}^2}\frac{1}{{\langle}k_1^*{\rangle}^l}\,d\sigma+\int_{{\mathbb{S}}^2}\frac{1}{{\langle}k^*{\rangle}^l}\,d\sigma\right)\lesssim\frac{1}{(l-2)(1+E)^{1+l/2}}.
\end{align}\] ◻
Lemma 3. Let \(1\leq p<\infty\). Then, the following estimates hold \[\begin{align}
\sup_{k\in{\mathbb{R}}^3}\int_{{\mathbb{R}}^3\times{\mathbb{S}}^2}\frac{|h(k^*)|^p\chi(\widehat{w}\cdot\sigma)}{|\widehat{R}_\sigma^-(w)\cdot\sigma|^{2\alpha}}\,d\sigma\,dk_1&\lesssim \|h\|_{L^p}^p,\quad \alpha<1\tag{31}, \\
\sup_{k\in{\mathbb{R}}^3}\int_{{\mathbb{R}}^3\times{\mathbb{S}}^2}|\widehat{R}_\sigma^-(w)\cdot\sigma|^{2\alpha}|h(k_1^*)|^p\,d\sigma\,dk_1&\lesssim \|h\|_{L^p}^p,\quad \alpha>1/2\tag{32}.\\
\sup_{k\in{\mathbb{R}}^3}\int_{{\mathbb{R}}^3\times{\mathbb{S}}^2}\frac{|h(k_1)|^p}{(1-|\widehat{K}\cdot\sigma|)^\alpha}\,d\sigma\,dk_1&\lesssim \|h\|_{L^p}^p,\quad \alpha<1\tag{33}.
\end{align}\]
Proof. For the first two bounds, we rely on the identities \[k^*=k-R_\sigma^+(w),\quad k_1^*=k-R_{\sigma}^-(w),\quad \widehat{w}\cdot\sigma=2|\widehat{R}^+_\sigma(w)\cdot\sigma|^2-1,\quad
|\widehat{R}_\sigma^+(w)|^2+|\widehat{R}_\sigma^-(w)|^2=1,\] straightforwardly deduced from 22 , ?? and ?? .
We first prove 31 . Fix \(k\in{\mathbb{R}}^3\). Using the substitution \(y:=w=k-k_1\), we write \[\begin{align}
\int_{{\mathbb{R}}^3\times{\mathbb{S}}^2}\frac{|h(k^*)|^p\chi(\widehat{w}\cdot\sigma)}{|\widehat{R}_\sigma^-(w)\cdot\sigma|^{2\alpha}}\,d\sigma\,dk_1&=\int_{{\mathbb{R}}^3\times{\mathbb{S}}^2}\frac{|h(k-R_\sigma^+(w))|^p}{(1-|\widehat{R}_\sigma^+(w)|^2)^\alpha}\chi(2|\widehat{R}_\sigma^+(w)\cdot\sigma|^2-1)\,d\sigma\,dk_1\\
&=\int_{{\mathbb{R}}^3\times{\mathbb{S}}^2}\frac{|h(k-R_\sigma^+(y))|^p}{(1-|\widehat{R}_\sigma^+(y)\cdot\sigma|^2)^\alpha}\chi(2|\widehat{R}_\sigma^+(y)\cdot\sigma|^2-1)\,d\sigma\,dy.
\end{align}\] Now, using Proposition 6 to substitute \(\nu:=R_\sigma^+(y)\), we obtain \[\begin{align}
\int_{{\mathbb{R}}^3\times{\mathbb{S}}^2}|h(k^*)|^p\chi(\widehat{w}\cdot\sigma)\,d\sigma\,dk_1&=\int_{{\mathbb{R}}^3}
|h(k-\nu)|^p\left(\int_{{\mathbb{S}}^2}\frac{\chi(2|\widehat{\nu}\cdot\sigma|^2-1)}{|\widehat{\nu}\cdot\sigma|^2(1-|\widehat{\nu}\cdot\sigma|^2)^\alpha}\,d\sigma\right)\,d\nu\\
&\approx\int_{{\mathbb{R}}^3}|h(k-\nu)|^p\left(\int_0^1 \frac{\chi(2x^2-1)}{x^2(1-x^2)^{\alpha}}\,dx\right)\,d\nu\\
&\lesssim \int_{{\mathbb{R}}^3}|h(k-\nu)|^p\left(\int_0^1 \frac{1}{(1-x^2)^{\alpha}}\,dx\right)\,d\nu\\
& \leq \int_{{\mathbb{R}}^3}|h(k-\nu)|^p\left(\int_0^1 \frac{1}{(1-x)^{\alpha}}\,dx\right)\,d\nu\\
&\approx \int_{{\mathbb{R}}^3}|h(k-\nu)|^p\,d\nu=\|h\|_{L^p}^p,
\end{align}\] where we used the basic fact \(\frac{\chi(2x^2-1)}{x^2}\lesssim 1\), \(x\neq 0\) and the fact that \(\alpha<1\) for the convergence
of the integral in \(x\). Since \(k\) is arbitrary, estimate 31 follows.
We now prove 32 . Fix \(k\in{\mathbb{R}}^3\). Using the substitution \(y:=w=k-k_1\), we obtain \[\begin{align}
\int_{{\mathbb{R}}^3\times{\mathbb{S}}^2}|\widehat{R}_\sigma^-(w)\cdot\sigma|^{2\alpha}|h(k_1^*)|^p\,d\sigma\,dk_1&=\int_{{\mathbb{R}}^3\times{\mathbb{S}}^2}|\widehat{R}_\sigma^-(w)\cdot\sigma|^{2\alpha}|h(k-R_\sigma(w))|^p\,d\sigma\,dk_1\\
&=\int_{{\mathbb{R}}^3\times{\mathbb{S}}^2}|\widehat{R}_\sigma^-(y)\cdot\sigma|^{2\alpha}|h(k-R_\sigma(y))|^p\,d\sigma\,dy.
\end{align}\] Now, using Proposition 6 to substitute \(\nu:=R_\sigma^-(y)\), we obtain \[\begin{align}
\int_{{\mathbb{R}}^3\times{\mathbb{S}}^2}|\widehat{R}_\sigma^-(w)\cdot\sigma|^{2\alpha}|h(k_1^*)|^p\,d\sigma\,dk_1&=\int_{{\mathbb{R}}^3}|h(k-\nu)|^p\left(\int_{{\mathbb{S}}^2}|\widehat{\nu}\cdot\sigma|^{2(\alpha-1)}\,d\sigma\right)\,d\nu \\
&\approx \int_{{\mathbb{R}}^3}|h(k-\nu)|^p\left(\int_0^1 x^{2(\alpha-1)}\,dx\right)\,d\nu\\ &\approx \int_{{\mathbb{R}}^3}|h(k-\nu)|^p\,d\nu=\|h\|_{L^p}^p,
\end{align}\] where we used the fact \(\alpha>1/2\) for the convergence of the integral in \(x\). Since \(k\) is arbitrary, estimate 32 follows.
Finally, to prove 33 , we fix \(k\in{\mathbb{R}}^3\). Then we integrate in spherical coordinates using the identity \[\begin{align}
\label{integr-sph-identity} \int_{{\mathbb{S}}^2} F(k \cdot \sigma) \, d\sigma \approx \int_{0}^\pi F(\cos \theta) \, \sin \theta \, d\theta \approx \int_{-1}^1 F(x) \, dx
\end{align}\tag{34}\] to obtain \[\begin{align} \int_{{\mathbb{R}}^3\times{\mathbb{S}}^2} \frac{|h(k_1)|^p}{(1-|\widehat{K}\cdot\sigma|)^\alpha}\,d\sigma\,k_1&=
\int_{{\mathbb{R}}^3}|h(k_1)|^p\int_{{\mathbb{S}}^2}\frac{1}{(1-|\widehat{K}\cdot\sigma|)^\alpha}\,d\sigma\,dk_1\\ &\approx\int_{{\mathbb{R}}^3}|h(k_1)|^p\int_{0}^1\frac{1}{(1-x)^\alpha}\,dx\,dk_1\\ &\approx \|h \|_{L^p}^p,
\end{align}\] where we used the fact that \(\alpha<1\) for the convergence of the integral in \(x\). Since \(k\) is arbitrary, estimate 33 follows. ◻
Finally, we record a very useful change of variables property, which is the analog of the involutionary pre-post collisional velocities change of variables in the context of the Boltzmann equation. For a proof of this classical result, see e.g. Lemma
2.6 in [2].
Lemma 4. The map \(T: (k,k_1,\sigma)\in{\mathbb{R}}^3\times{\mathbb{R}}^3\times{\mathbb{S}}^2\to (k^*,k_1^*,\eta)\in{\mathbb{R}}^3\times{\mathbb{R}}^3\times{\mathbb{S}}^2\) where \[\begin{cases} k^*=K-\frac{|w|}{2}\sigma\\ k_1^*=K+\frac{|w|}{2}\sigma\\ \eta=-\widehat{w} \end{cases},\quad K=\frac{k+k_1}{2},\quad w=k-k_1,\] is an involution of \({\mathbb{R}}^3\times{\mathbb{R}}^3\times{\mathbb{S}}^2\). Moreover, for any non-negative and continuously differentiable function \(F:{\mathbb{R}}^3\times{\mathbb{R}}^3\times [-1,1]\to
{\mathbb{R}}_+\), the following change of variables formula holds: \[\label{change32of32variables32formula}
\int_{{\mathbb{R}}^6\times{\mathbb{S}}^2}F(k^*,k_1^*,\widehat{w}\cdot\sigma)\,d\sigma\,dk_1\,dk=\int_{{\mathbb{R}}^6\times{\mathbb{S}}^2} F(k,k_1,\widehat{w}\cdot\sigma)\,d\sigma\,dk_1\,dk.\tag{35}\]
In this section we prove our main gain operator estimates. We stress that they are moment preserving, which will allow us to run a fixed point argument to prove the main result. At the technical level, the averaging in the angular variable quantified in
Lemma 2 offsets the growth of the resonant cross-section.
We prove the following estimate for \(\mathcal{G}_0\):
Proposition 7. Let \(r\geq 2\) and \(l= 2-\frac{3}{r}+\delta\), where \(0<\delta<1.\) Then the following estimate holds \[\label{estimate32for32G0}
\|{\langle}k{\rangle}^{l} \mathcal{G}_0[f,g,h]\|_{L^r}\lesssim \|{\langle}k{\rangle}^{l} f\|_{L^r} \|{\langle}k{\rangle}^l g\|_{L^r}\|{\langle}k{\rangle}^l h\|_{L^r} .\qquad{(4)}\]
Proof. We treat the cases \(2\leq r<\infty\) and \(r=\infty\) separately.
Case \(r<\infty\). By the resonant condition \(|k^*|^2+|k_1^*|^2=E\), we have either \(|k^*|^2\geq E/2\) or \(|k_1^*|^2\geq E/2\), where \(E=|k|^2+|k_1|^2\). Ignoring the measure zero set where \(|k^*|^2=|k_1^*|^2=E/2\), we obtain
\[\label{factorization32of32G0}
\mathcal{G}_0[f,g,h]=\frac{1}{4}f\left(\mathcal{Q}_0[g,h]+\mathcal{Q}_1[g,h]\right),\tag{36}\] where \[\begin{align}
\mathcal{Q}_0[g,h]&=\int_{{\mathbb{R}}^3\times{\mathbb{S}}^2}|w| |g(k^*)|\,|h(k_1^*)|\chi(\widehat{w}\cdot\sigma)\mathbb{1}_{|k^*|>E/2}\,d\sigma\,dk_1,\\
\mathcal{Q}_1[g,h]&=\int_{{\mathbb{R}}^3\times{\mathbb{S}}^2}|w| |g(k^*)|\,|h(k_1^*)|\chi(\widehat{w}\cdot\sigma)\mathbb{1}_{|k_1^*|>E/2}\,d\sigma\,dk_1.
\end{align}\] We clearly have \[\label{reduction32to32Linf32G0}
\|{\langle}k{\rangle}^l \mathcal{G}_0[f,g,h]\|_{L^r}\lesssim \|f_l\|_{L^r}\left(\|\mathcal{Q}_0[g,h]\|_{L^\infty}+\|\mathcal{Q}_1[g,h]\|_{L^\infty}\right),\tag{37}\] so it suffices to estimate \(\|\mathcal{Q}_0[g,h]\|_{L^\infty},\,\|\mathcal{Q}_1[g,h]\|_{L^\infty}\).
Define \(l_0=\frac{1-\delta}{2}\), \(l_1=\frac{1+\delta}{2}\). Clearly \(l_0+l_1=1\). Moreover, we note that for any function \(\psi\), Hölder’s inequality implies \[\label{embedding32L232to32Lr}
\|\psi_{l_0}\|_{L^2}\leq\|\psi_{1/2}\|_{L^2}\leq\|\psi_{l_1}\|_{L^2}\lesssim
\|{\langle}k{\rangle}^{\frac{1+\delta}{2}+3(\frac{1}{2}-\frac{1}{r})+\frac{\delta}{2}}\psi\|_{L^r}=\|{\langle}k{\rangle}^{2-\frac{3}{r}+\delta}\psi\|_{L^r}=\|\psi_{l}\|_{L^r}.\tag{38}\]
Fix \(k\in{\mathbb{R}}^3\). We use 25 and the inequality \(|w|\leq |k|+|k_1|\lesssim E^{1/2}\) to bound \[\begin{align}
\frac{|w|}{{\langle}k^*{\rangle}^{l_1}{\langle}k_1^*{\rangle}^{l_0}}\mathbb{1}_{|k^*|^2>E/2}\lesssim \frac{E^{1/2}}{{\langle}k^*{\rangle}(1-|\widehat{k^*}\cdot\sigma|)^{l_0/2}}\mathbb{1}_{|k^*|^2>E/2}\lesssim
\frac{1}{(1-|\widehat{k^*}\cdot\sigma|)^{l_0/2}}.
\end{align}\] Let \(\alpha=1/2+\delta/4\), and using the above bound followed by the Cauchy-Schwarz inequality, we obtain \[\begin{align}
|\mathcal{Q}_0[g,h](k)|&\leq \int_{{\mathbb{R}}^3\times{\mathbb{S}}^2}\frac{|g_{l_1}(k^*)||h_{l_0}(k_1^*)|\chi(\widehat{w}\cdot\sigma)}{(1-|\widehat{k^*}\cdot\sigma|)^{l_0/2}}\,d\sigma\,dk_1\\
&=\int_{{\mathbb{R}}^3\times{\mathbb{S}}^2}\frac{|g_{l_1}(k^*)|\chi(\widehat{w}\cdot\sigma)}{|\widehat{R}_\sigma^-(w)\cdot\sigma|^{\alpha}(1-|\widehat{k^*}\cdot\sigma|)^{l_0/2}}\,\left(|\widehat{R}_\sigma^-(w)\cdot\sigma|^{\alpha}|h_{l_0}(k_1^*)|\right)\,d\sigma\,dk_1\\
&\leq
\left(\int_{{\mathbb{R}}^3\times{\mathbb{S}}^2}\frac{|g_{l_1}(k^*)|^2\chi(\widehat{w}\cdot\sigma)}{|\widehat{R}_\sigma^-(w)\cdot\sigma|^{2\alpha}(1-|\widehat{k^*}\cdot\sigma|)^{l_0}}\,d\sigma\,dk_1\right)^{1/2}\left(\int_{{\mathbb{R}}^3\times{\mathbb{S}}^2}|\widehat{R}_\sigma^-(w)\cdot\sigma|^{2\alpha}|h_{l_0}(k_1^*)|^2\,d\sigma\,dk_1\right)^{1/2}\\
&\lesssim A^{1/2}\|h_{l_0}\|_{L^2},
\end{align}\] where \[\begin{align} A & :=
\int_{{\mathbb{R}}^3\times{\mathbb{S}}^2}\frac{|g_{l_1}(k^*)|^2\chi(\widehat{w}\cdot\sigma)}{|\widehat{R}_\sigma^-(w)\cdot\sigma|^{2\alpha}(1-|\widehat{k^*}\cdot\sigma|)^{l_0}}\,d\sigma\,dk_1.
\end{align}\] For the last inequality, we used 32 since \(\alpha>1/2\).
It remains to estimate the integral \(A\). For this, we use 22 , ?? ?? and the substitution \(y:=w=k-k_1\), to write \[\begin{align}
A&=\int_{{\mathbb{R}}^3\times{\mathbb{S}}^2}\frac{|g_{l_1}(k-R_\sigma^+(w))|^2\chi(2|\widehat{R}_\sigma^+(w)\cdot\sigma|^2-1)}{(1-|\widehat{R}_\sigma^+(w)\cdot\sigma|^2)^\alpha\left(1-|(\widehat{k-R_\sigma^+(w)})\cdot\sigma|\right)^{l_0}}\,d\sigma\,dk_1\\
&=\int_{{\mathbb{R}}^3\times{\mathbb{S}}^2}\frac{|g_{l_1}(k-R_\sigma^+(y))|^2\chi(2|\widehat{R}_\sigma^+(y)\cdot\sigma|^2-1)}{(1-|\widehat{R}_\sigma^+(y)\cdot\sigma|^2)^\alpha\left(1-|(\widehat{k-R_\sigma^+(y)})\cdot\sigma|\right)^{l_0}}\,d\sigma\,dy.
\end{align}\] Now, using Proposition 6 to substitute \(\nu:=R_\sigma^+(y)\), we obtain \[\begin{align}
A&= \int_{{\mathbb{R}}^3} |g_{l_1}(k-\nu)|^2\int_{{\mathbb{S}}^2}\frac{\chi(2|\widehat{\nu}\cdot\sigma)|^2-1)}{|\widehat{\nu}\cdot\sigma|^2(1-|\widehat{\nu}\cdot\sigma|^2)^\alpha\left(1-|\widehat{(k-\nu)}\cdot\sigma|\right)^{l_0}}\,d\sigma\,d\nu\\
&\lesssim \int_{{\mathbb{R}}^3} |g_{l_1}(k-\nu)|^2\int_{{\mathbb{S}}^2}\frac{1}{(1-|\widehat{\nu}\cdot\sigma|^2)^\alpha\left(1-|\widehat{(k-\nu)}\cdot\sigma|\right)^{l_0}}\,d\sigma\,d\nu\\
&\leq \int_{{\mathbb{R}}^3}|g_{l_1}(k-\nu)|^2\int_{{\mathbb{S}}^2}\frac{1}{(1-|\widehat{\nu}\cdot\sigma|^2)^{\alpha+l_0}}+\frac{1}{\left(1-|\widehat{(k-\nu)}\cdot\sigma|\right)^{\alpha+l_0}}\,d\sigma\,d\nu,
\end{align}\] where for the first bound we used the fact \(\frac{\chi(2x^2-1)}{x^2}\lesssim 1\), \(x\neq 0\) and for the second bound we used the standard convexity inequality
\[\label{standart32singularity32ineq}
\frac{1}{|a|^\lambda|b|^\mu}\leq\frac{1}{|a|^{\lambda+\mu}}+\frac{1}{|b|^{\lambda+\mu}},\quad a,b\neq 0,\quad \lambda,\mu>0.\tag{39}\]
Now integrating in spherical coordinates and using the fact that \(\alpha+l_0=1-\delta/4\), we obtain \[\begin{align}
&\int_{{\mathbb{S}}^2}\frac{1}{(1-|\widehat{\nu}\cdot\sigma|^2)^{\alpha+l_0}}\,d\sigma\approx \int_0^1 \frac{1}{(1-x^2)^{1-\delta/4}}\,dx\leq \int_0^1\frac{1}{(1-x)^{1-\delta/4}}\,dx\approx 1, \\
&\int_{{\mathbb{S}}^2} \frac{1}{\left(1-|\widehat{(k-\nu)}\cdot\sigma|\right)^{\alpha+l_0}}\,d\sigma\approx \int_0^1\frac{1}{(1-x)^{1-\delta/4}}\,dx\approx 1.
\end{align}\] Hence \[A\lesssim \int_{{\mathbb{R}}^3}|g_{l_1}(k-\nu)|^2\,d\nu=\|g_{l_1}\|_{L^2}^2.\] Since \(k\) is arbitrary, we conclude
\[\label{Q032Linf32bound}
\|\mathcal{Q}_0[g,h]\|_{L^\infty}\lesssim \|g_{l_1}\|_{L^2}\|h_{l_0}\|_{L^2}\lesssim \|g_l\|_{L^r}\|h_l\|_{L^r},\tag{40}\] where for the last inequality we use 38 .
Fix \(k\in{\mathbb{R}}^3\). We use the inequality \({\langle}k^*{\rangle}\geq {\langle}k_1^*{\rangle}(1-|\widehat{k_1^*}\cdot\sigma|)\) (which follows by 24 ), the fact \(l_0+l_1=1\), and the inequality \(|w|\leq |k|+|k_1|\lesssim E^{1/2}\), to bound \[\begin{align}
\frac{|w|}{{\langle}k^*{\rangle}^{l_0}{\langle}k_1^*{\rangle}^{l_1}}\mathbb{1}_{|k_1^*|^2>E/2}\lesssim \frac{E^{1/2}}{{\langle}k_1^*{\rangle}(1-|\widehat{k_1^*}\cdot\sigma|)^{l_0/2}}\mathbb{1}_{|k_1^*|^2>E/2}\lesssim
\frac{1}{(1-|\widehat{k_1^*}\cdot\sigma|)^{l_0/2}}.
\end{align}\] Writing \(\alpha=1-\delta/4\), and using the bound above followed by the Cauchy-Schwarz inequality, we obtain \[\begin{align}
|\mathcal{Q}_1[g,h](k)|&\leq \int_{{\mathbb{R}}^3\times{\mathbb{S}}^2}\frac{|g_{l_0}(k^*)||h_{l_1}(k_1^*)|\chi(\widehat{w}\cdot\sigma)}{(1-|\widehat{k_1^*}\cdot\sigma|)^{l_0/2}}\,\,d\sigma\,dk_1\\
&=\int_{{\mathbb{R}}^3\times{\mathbb{S}}^2}\frac{|g_{l_0}(k^*)|\chi(\widehat{w}\cdot\sigma)}{|\widehat{R}_\sigma^-(w)\cdot\sigma|^\alpha}\frac{|\widehat{R}_\sigma^-(w)\cdot\sigma|^\alpha|h_{l_1}(k_1^*)|}{(1-|\widehat{k_1^*}\cdot\sigma|)^{l_0/2}}\,\,d\sigma\,dk_1\\
&\leq
\left(\int_{{\mathbb{R}}^3\times{\mathbb{S}}^2}\frac{|g_{l_0}(k^*)|^2\chi(\widehat{w}\cdot\sigma)}{|\widehat{R}_\sigma^-(w)\cdot\sigma|^{2\alpha}}\,d\sigma\,dk_1\right)^{1/2}\left(\int_{{\mathbb{R}}^3\times{\mathbb{S}}^2}\frac{|\widehat{R}_\sigma^-(w)\cdot\sigma|^{2\alpha}|h_{l_1}(k_1^*)|^2}{(1-|\widehat{k_1^*}\cdot\sigma|)^{l_0}}\,d\sigma\,dk_1\right)^{1/2}\\
&\lesssim \|g_{l_0}\|_{L^2}\,B^{1/2},
\end{align}\] where \[\begin{align} B:= \int_{{\mathbb{R}}^3\times{\mathbb{S}}^2}\frac{|\widehat{R}_\sigma^-(w)\cdot\sigma|^{2\alpha}|h_{l_1}(k_1^*)|^2}{(1-|\widehat{k_1^*}\cdot\sigma|)^{l_0}}\,d\sigma\,dk_1.
\end{align}\] For the last inequality, we used 31 since \(\alpha<1\).
It remains to estimate the integral \(B\). To do so, we first use 22 and the substitution \(y:=w=k-k_1\) to write \[\begin{align}
B&=\int_{{\mathbb{R}}^3\times{\mathbb{S}}^2}\frac{|\widehat{R}_\sigma^-(w)\cdot\sigma|^{2\alpha}|h_{l_1}(k-R_\sigma^-(w))|^2}{\left(1-|(\widehat{k-R_\sigma^-(w)})\cdot\sigma|\right)^{l_0}}\,d\sigma\,dk_1\\
&=\int_{{\mathbb{R}}^3\times{\mathbb{S}}^2}\frac{|\widehat{R}_\sigma^-(y)\cdot\sigma|^{2\alpha}|h_{l_1}(k-R_\sigma^-(y))|^2}{\left(1-|(\widehat{k-R_\sigma^-(y)})\cdot\sigma|\right)^{l_0}}\,d\sigma\,dy.
\end{align}\] Now, using Proposition 6 to substitute \(\nu:=R_\sigma^-(y)\), we obtain \[\begin{align}
B&= \int_{{\mathbb{R}}^3} |h_{l_1}(k-\nu)|^2\int_{{\mathbb{S}}^2}\frac{1}{|\widehat{\nu}\cdot\sigma|^{2(1-\alpha)}\left(1-|\widehat{(k-\nu)}\cdot\sigma|\right)^{l_0}}\,d\sigma\,d\nu\\
&\leq \int_{{\mathbb{R}}^3}|h_{l_1}(k-\nu)|^2\int_{{\mathbb{S}}^2}\frac{1}{|\widehat{\nu}\cdot\sigma|^{2(1-\alpha)+l_0}}+\frac{1}{\left(1-|\widehat{(k-\nu)}\cdot\sigma|\right)^{2(1-\alpha)+l_0}}\,d\sigma\,d\nu,
\end{align}\] where for the last step we use inequality 39 . Now integrating in spherical coordinates and using the fact that \(2(1-\alpha)+l_0=1/2\), we obtain \[\begin{align}
&\int_{{\mathbb{S}}^2}\frac{1}{|\widehat{\nu}\cdot\sigma|^{2(1-\alpha)+l_0}}\,d\sigma\approx \int_0^1 \frac{1}{x^{1/2}}\,dx\approx 1, \\
&\int_{{\mathbb{S}}^2} \frac{1}{\left(1-|\widehat{(k-\nu)}\cdot\sigma|\right)^{2(1-\alpha)+l_0}}\,d\sigma\approx \int_0^1\frac{1}{(1-x)^{1/2}}\,dx\approx 1.
\end{align}\] Hence \[B\lesssim \int_{{\mathbb{R}}^3}|h_{l_1}(k-\nu)|^2\,d\nu=\|h_{l_1}\|_{L^2}^2.\] Since \(k\) is arbitrary, we conclude
\[\label{Q132Linf32bound}
\|\mathcal{Q}_1[g,h]\|_{L^\infty}\lesssim \|g_{l_0}\|_{L^2}\|h_{l_1}\|_{L^2}\lesssim \|g_l\|_{L^r}\|h_l\|_{L^r},\tag{41}\] where for the last inequality we use 38 .
Combining 40 , 41 with 37 , estimate ?? for \(2\leq r<\infty\) follows.
Case \(r=\infty\). Let \(k\in{\mathbb{R}}^3\). Since \(r=\infty\), we have \(l=2+\delta\). Using 30 and the inequality \(|w|\leq |k|+|k_1|\lesssim E^{1/2}\), we obtain \[\begin{align}
|{\langle}k{\rangle}^l\mathcal{G}_0[f,g,h](k)|&\leq \|f_l\|_{L^\infty}\|g_l\|_{L^\infty}\|h_l\|_{L^\infty}\int_{{\mathbb{R}}^3}|w|\int_{{\mathbb{S}}^2}\frac{1}{{\langle}k^*{\rangle}^l{\langle}k_1^*{\rangle}^l}\,d\sigma\,dk_1\\
&\lesssim \|f_l\|_{L^\infty}\|g_l\|_{L^\infty}\|h_l\|_{L^\infty}\int_{{\mathbb{R}}^3}\frac{|w|}{E^{1+l/2}}\,dk_1\\
&\lesssim \|f_l\|_{L^\infty}\|g_l\|_{L^\infty}\|h_l\|_{L^\infty}\int_{{\mathbb{R}}^3}\frac{1}{{\langle}k_1{\rangle}^{3+\delta}}\,dk_1\\
&\lesssim \|f_l\|_{L^\infty}\|g_l\|_{L^\infty}\|h_l\|_{L^\infty}.
\end{align}\] Since \(k\) is arbitrary estimate ?? follows. ◻
We prove the following estimate for \(\mathcal{G}_1:\)
Proposition 8. Let \(r\geq 2\) and \(l= 2-\frac{3}{r}+\delta\), where \(0<\delta<1/r\) if \(r <
\infty,\) and \(\delta>0\) if \(r = \infty.\) Then the following estimate holds \[\label{estimate32on32G1}
\|{\langle}k{\rangle}^l \mathcal{G}_1[f,g,h]\|_{L^r}\lesssim \|{\langle}k{\rangle}^l f\|_{L^r} \|{\langle}k{\rangle}^l g\|_{L^r}\|{\langle}k{\rangle}^l h\|_{L^r}.\qquad{(5)}\]
Proof. We treat again the cases \(2\leq r<\infty\) and \(r=\infty\) separately.
Case \(r<\infty\). Since \(r<\infty\) and \(\delta<1/r\), we have \(l=2-\frac{3}{r}+\delta<2\). We also have \[\label{r39l602}
r'l=r'\left(2-\frac{3}{r}+\delta\right)=\frac{r}{r-1} \left(2-\frac{3}{r}+\delta\right)=\frac{2r-3+\delta r}{r-1}=2-\frac{1-\delta r}{r-1}<2.\tag{42}\]
We use the inequality \[\begin{align}
\label{lower-bound-product}
{\langle}k^*{\rangle}{\langle}k_1^*{\rangle}\gtrsim (1+E)(1-|\widehat{K}\cdot\sigma|)^{1/2}.
\end{align}\tag{43}\]
Indeed, by conservation of energy \(\vert k_1^* \vert^2 + \vert k^* \vert ^2 = E,\) thus \(\vert k_1^* \vert^2 \gtrsim E\) or \(\vert k^* \vert^2 \gtrsim
E.\) As a result \(\max \big \lbrace \langle k_1^* \rangle, \langle k^* \rangle \big \rbrace \gtrsim (1+E)^{1/2}.\) Moreover, by 23 , we have \(\min \big \lbrace \langle k_1^* \rangle, \langle k^* \rangle \big \rbrace \gtrsim (1+E)^{1/2} \big(1 - \vert \widehat{K} \cdot \sigma \vert \big)^{1/2}\) and 43 follows.
Applying 43 then yields \[\begin{align}
\frac{{\langle}k{\rangle}^l |w|}{{\langle}k^*{\rangle}^{l} {\langle}k_1^*{\rangle}^{l}}\lesssim \frac{1}{(1+E)^{\frac{l-1}{2}}(1-|\widehat{K}\cdot\sigma|)^{l/2}} \lesssim \frac{1}{{\langle}k_1{\rangle}^{l-1}(1-|\widehat{K}\cdot\sigma|)^{l/2}}.
\end{align}\] Using the above bound, we obtain \[\begin{align}
{\langle}k{\rangle}^l |\mathcal{G}_{1}[f,g,h]|&\lesssim \int_{{\mathbb{R}}^3\times{\mathbb{S}}^2}\frac{1}{(1-|\widehat{K}\cdot\sigma|)^{l/2}}|f_{1-l}(k_1)g_{l}(k^*)h_l(k_1^*)|\chi(\widehat{w}\cdot\sigma)\,d\sigma\,dk_1\notag\\
&\leq \left(\int_{{\mathbb{R}}^3\times{\mathbb{S}}^2} \frac{1}{(1-|\widehat{K}\cdot\sigma|)^{r'l/2}}|f_{1-l}(k_1)|^{r'}\,d\sigma\,dk_1\right)^{1/r'}\notag\\
&\times \left(\int_{{\mathbb{R}}^3\times{\mathbb{S}}^2}|g_l(k^*)h_l(k_1^*)|^{r}\,d\sigma\,dk_1\right)^{1/r}\notag\\
&\lesssim \|f_{1-l}\|_{L^{r'}} \left(\int_{{\mathbb{R}}^3\times{\mathbb{S}}^2}|g_l(k^*)h_l(k_1^*)|^{r}\,d\sigma\,dk_1\right)^{1/r},\notag\\
&\lesssim \| f_l\|_{L^r} \left(\int_{{\mathbb{R}}^3\times{\mathbb{S}}^2}|g_l(k^*)h_l(k_1^*)|^{r}\,d\sigma\,dk_1\right)^{1/r}\label{pointwise32bound32on32G1},
\end{align}\tag{44}\] where for the second to last line we used the fact \(r'l<2\) and 33 , and for the last line we used \[\|f_{1-l}\|_{L^{r'}}=\|{\langle}k{\rangle}^{1-l} f\|_{L^{r'}}\lesssim \|{\langle}k{\rangle}^{1-l +3\left(1-\frac{2}{r}\right)+\delta}f\|_{L^r}=\|{\langle}k{\rangle}^{2-\frac{3}{r}}f\|_{L^r}\lesssim \| f_l\|_{L^r},\]
which follows by Hölder’s inequality.
Raising 44 to the \(r\)-th power, integrating, and using 35 , we obtain \[\begin{align}
\|{\langle}k{\rangle}^l \mathcal{G}_1[f,g,h]\|_{L^r}^r&\lesssim \|f_l\|_{L^r}^r\int_{{\mathbb{R}}^6\times{\mathbb{S}}^2}|g_l(k^*) h_l(k_1^*)|^r\,d\sigma\,dk_1\,dk\\
&=\|f_l\|_{L^r}^r\int_{{\mathbb{R}}^6}|g_l(k) h_l(k_1)|^r\,\,dk_1\,dk\\
&= \|f_l\|_{L^r}^r\|g_l\|_{L^r}^r\|h_l\|_{L^r}^r,
\end{align}\] and ?? follows.
Case \(r=\infty\). Let \(k\in{\mathbb{R}}^3\). Using 30 and the inequality \(|w|\lesssim
E^{1/2}\), we bound
In this section, we prove moment preserving trilinear estimates for the loss operators. Contrary to the Boltzmann equation, such a moment gain is present for the loss. As a result, the local well-posedness result is fully perturbative, in contrast to
the Boltzmann equation.
We prove the following estimate for \(\mathcal{L}_0\):
Proposition 9. Let \(r\geq 2\) and \(l=2-\frac{3}{r}+\delta\), where \(0<\delta<1\). Then, the following estimate holds \[\begin{align}
\|{\langle}k{\rangle}^l \mathcal{L}_0[f,g,h]\|_{L^r}&\lesssim \|{\langle}k{\rangle}^{l} f\|_{L^r}\|{\langle}k{\rangle}^l g\|_{L^r}\|{\langle}k{\rangle}^l h\|_{L^r}\label{estimate32for32L950}.
\end{align}\qquad{(6)}\]
Proof. We treat the cases \(r<\infty=\) and \(r=\infty\) separately.
Case \(r<\infty\). We clearly have \(\|{\langle}k{\rangle}^l \mathcal{L}_0[f,g,h]\|_{L^r}\leq \|f_l\|_{L^r}\|\mathcal{R}_0[g,h]\|_{L^\infty}\). So it suffices to show
that \[\label{sufficient32condition32L0}
\|\mathcal{R}_0[g,h]\|_{L^\infty}\lesssim \|g_l\|_{L^r}\|h_l\|_{L^r}.\tag{45}\] We decompose \[\mathcal{R}_0[g,h]= \mathcal{R}_0^0[g,h] +\mathcal{R}_0^1[g,h],\] where \[\begin{align} \mathcal{R}_0^0[g,h]&=\frac{1}{4}\int_{{\mathbb{R}}^3\times{\mathbb{S}}^2}|w||g(k_1) h(k^*)|\mathbb{1}_{|k_1|<|k|}\chi(\widehat{w}\cdot\sigma)\,d\sigma\,dk_1,\\
\mathcal{R}_0^1[g,h]&=\frac{1}{4}\int_{{\mathbb{R}}^3\times{\mathbb{S}}^2}|w||g(k_1)h(k^*)|\mathbb{1}_{|k_1|\geq |k|}\chi(\widehat{w}\cdot\sigma)\,d\sigma\,dk_1.
\end{align}\]
Fix \(k\in{\mathbb{R}}^3\). We note that in the corresponding domain of integration, the triangle inequality implies \(|w|\leq |k|+|k_1|\leq 2|k_1|\). Then, estimate 23 and the Cauchy-Schwarz inequality imply \[\begin{align}
\mathcal{R}_0^1[g,h](k)&\lesssim \int_{{\mathbb{R}}^3\times{\mathbb{S}}^2}|k_1||g(k_1)||h(k^*)|\chi(\widehat{w}\cdot\sigma)\,d\sigma\,dk_1\\
&\lesssim \int_{{\mathbb{R}}^3\times{\mathbb{S}}^2}\frac{|g_{1/2}(k_1)|}{(1-|\widehat{K}\cdot\sigma|)^{1/4}} |h_{1/2}(k_1^*)|\chi(\widehat{w}\cdot\sigma))\,d\sigma\,dk_1\\
&\leq I_1^{1/2}I_2^{1/2},
\end{align}\] where \[\begin{align}
I_1&=\int_{{\mathbb{R}}^3\times{\mathbb{S}}^2} \frac{|g_{1/2}(k_1)|^2}{(1-|\widehat{K}\cdot\sigma)^{1/2}}\,d\sigma\,dk_1,\\
I_2&=\int_{{\mathbb{R}}^3\times{\mathbb{S}}^2} |h_{l_0}(k_1^*)|^2\chi(\widehat{w}\cdot\sigma)\,d\sigma\,dk_1.
\end{align}\]
Since \(k\) is arbitrary, we conclude \[\label{bound32on32R01}
\|\mathcal{R}_0^1[g,h]\|_{L^\infty}\lesssim \|g_{1/2}\|_{L^2}\|h_{1/2}\|_{L^2}\lesssim \|g_l\|_{L^r}\|h_l\|_{L^r},\tag{46}\] where for the last inequality we use 38 .
Fix \(k\in{\mathbb{R}}^3\). Using the triangle inequality \(|w|\leq |k|+|k_1|\leq 2|k|\), estimates 26 , 23 and the Cauchy-Schwarz inequality, we obtain \[\begin{align} |\mathcal{R}_0^0[g,h](k)|&\lesssim
\int_{{\mathbb{R}}^3\times{\mathbb{S}}^2}|k||g(k_1)||h(k^*)|\chi(\widehat{w}\cdot\sigma)\,d\sigma\,dk_1\\ &\lesssim
\int_{{\mathbb{R}}^3\times{\mathbb{S}}^2}\frac{|g_{1/2}(k_1)|}{(1-|\widehat{K}\cdot\sigma|)^{1/4}}\frac{|h_{1/2}(k^*)|}{\left(1-|\widehat{(k-2R_\sigma^-(w))}\cdot\sigma|\right)^{1/4}}\chi(\widehat{w}\cdot\sigma)\,d\sigma\,dk_1\\ &\leq
J_1^{1/2}J_2^{1/2},
\end{align}\] where \[\begin{align}
J_1&=\int_{{\mathbb{R}}^3\times{\mathbb{S}}^2}\frac{|g_{1/2}(k_1)|^2}{(1-|\widehat{K}\cdot\sigma|)^{1/2}}\,d\sigma\,dk_1,\\
J_2&=\int_{{\mathbb{R}}^3\times{\mathbb{S}}^2}\frac{|h_{1/2}(k^*)|^2\chi(\widehat{w}\cdot\sigma))}{\left(1-|\widehat{(k-2R_\sigma^+(w))}\cdot\sigma|\right)^{1/2}}\,d\sigma\,dk_1.
\end{align}\] By 33 , we have \(J_1\lesssim \|g_{1/2}\|_{L^2}^2\). For \(J_2\), we use 22 , ?? and the
substitution \(y:=w=k-k_1\) to write \[J_2=\int_{{\mathbb{R}}^3\times{\mathbb{S}}^2}\frac{|h_{1/2}(k-R_\sigma^+(y))|^2\chi(2|\widehat{R}_\sigma^+(y)\cdot\sigma|^2-1)}{\left(1-|\widehat{(k-2R_\sigma^+(y))}\cdot\sigma|\right)^{1/2}}\,d\sigma\,dy.\]
Using Proposition 6 to substitute \(\nu:=R_\sigma^+(y)\), and the basic inequality \(\frac{\chi(2z^2-1)}{z^2}\lesssim 1\), \(z\neq 0\), we obtain \[\begin{align}
J_2&=\int_{{\mathbb{R}}^3}|h_{1/2}(k-\nu)|^2\int_{{\mathbb{S}}^2}\frac{\chi(|2\widehat{\nu}\cdot\sigma|^2-1)}{|\widehat{\nu}\cdot\sigma|^2\left(1-|\widehat{(k-\nu)}\cdot\sigma|)^{1/2}\right)}\,d\sigma\,d\nu \\
&\approx \int_{{\mathbb{R}}^3}|h_{1/2}(k-\nu)|^2\int_{{\mathbb{S}}^2}\frac{1}{\left(1-|\widehat{(k-\nu)}\cdot\sigma|\right)^{1/2}}\,d\sigma\,d\nu \\
&\approx \int_{{\mathbb{R}}^3}|h_{1/2}(k-\nu)|^2\int_0^1\frac{1}{(1-x)^{1/2}}\,dx\,d\nu\\
&\approx \|h_{1/2}\|_{L^2}^2.
\end{align}\]
Since \(k\) is arbitrary, we conclude \[\label{bound32on32R00}
\|\mathcal{R}_0^0[g,h]\|_{L^\infty}\lesssim \|g_{1/2}\|_{L^2}\|h_{1/2}\|_{L^2}\lesssim \|g_l\|_{L^r}\|h_l\|_{L^r},\tag{47}\] where for the last inequality we use 38 .
Case \(r=\infty\). Fix \(k\in{\mathbb{R}}^3\). Using 29 and the inequality \(|w|\lesssim
E^{1/2}\), we bound \[\begin{align}
|{\langle}k{\rangle}^l f(k)\,\mathcal{R}_0[g,h](k)|&\leq \|f_l\|_{L^\infty}\|g_{l}\|_{L^\infty}\|h_{l}\|_{L^\infty}\int_{{\mathbb{R}}^3}\frac{|w|}{{\langle}k_1{\rangle}^{l}}\int_{{\mathbb{S}}^2}\frac{1}{{\langle}k^*{\rangle}^{l}}\,d\sigma\,dk_1\\
&\lesssim \|f_l\|_{L^\infty}\|g_{l}\|_{L^\infty}\|h_{l}\|_{L^\infty}\int_{{\mathbb{R}}^3}\frac{|w|}{{\langle}k_1{\rangle}^{l}(1+E)}\,dk_1\\
&\lesssim \|f_l\|_{L^\infty}\|g_{l}\|_{L^\infty}\|h_{l}\|_{L^\infty}\int_{{\mathbb{R}}^3}\frac{1}{{\langle}k_1{\rangle}^{3+\delta}}\,dk_1\\
&\lesssim \|f_l\|_{L^\infty}\|g_{l}\|_{L^\infty}\|h_{l}\|_{L^\infty}.
\end{align}\] Since \(k\) is arbitrary, estimate ?? follows. ◻
We prove the following estimate for \(\mathcal{L}_1\):
Proposition 10. Let \(r\geq 2\) and \(l=2-\frac{3}{r}+\delta\), where \(0<\delta<1\). Then, the following estimate holds \[\begin{align}
\|{\langle}k{\rangle}^l \mathcal{L}_1[f,g,h]\|_{L^r}&\lesssim \|{\langle}k{\rangle}^{l} f\|_{L^r}\|{\langle}k{\rangle}^l g\|_{L^r}\|{\langle}k{\rangle}^l h\|_{L^r}\label{estimate32for32L951}.
\end{align}\qquad{(7)}\]
Proof. We treat the cases \(r<\infty=\) and \(r=\infty\) separately again.
Case \(r<\infty\). We clearly have \(\|{\langle}k{\rangle}^l\mathcal{L}_1[f,g,h]\|_{L^r}\leq \|f_l\|_{L^r}\|\mathcal{R}_1[g,h]\|_{L^\infty}\). So it suffices to show
that \[\label{sufficient32condition32L1}
\|\mathcal{R}_1[g,h]\|_{L^\infty}\lesssim \|g_l\|_{L^r}\|h_l\|_{L^r}.\tag{48}\] We decompose \[\mathcal{R}_1[g,h]= \mathcal{R}_1^0[g,h] +\mathcal{R}_1^1[g,h],\] where \[\begin{align} \mathcal{R}_1^0[g,h]&=\frac{1}{4}\int_{{\mathbb{R}}^3\times{\mathbb{S}}^2}|w||g(k_1) h(k_1^*)|\mathbb{1}_{|k_1|<|w|/2}\chi(\widehat{w}\cdot\sigma)\,d\sigma\,dk_1,\\
\mathcal{R}_1^1[g,h]&=\frac{1}{4}\int_{{\mathbb{R}}^3\times{\mathbb{S}}^2}|w||g(k_1)h(k_1^*)|\mathbb{1}_{|k_1|\geq |w|/2}\chi(\widehat{w}\cdot\sigma)\,d\sigma\,dk_1.
\end{align}\]
Let \(l_0=\frac{1-\delta}{2}\), \(l_1=\frac{1+\delta}{2}\).
Let \(\alpha=1/2+\delta/4,\) and fix \(k\in{\mathbb{R}}^3\). We use estimate 23 and the Cauchy-Schwarz inequality to write \[\begin{align}
\mathcal{R}_1^1[g,h](k)&\lesssim \int_{{\mathbb{R}}^3\times{\mathbb{S}}^2}|k_1||g(k_1)||h(k_1^*)|\chi(\widehat{w}\cdot\sigma)\,d\sigma\,dk_1\\
&= \int_{{\mathbb{R}}^3\times{\mathbb{S}}^2} |g_{l_1}(k_1)|\frac{|k_1|^{l_0}|h_{l_0}(k_1^*)|}{{\langle}k_1^*{\rangle}^{l_0}}\chi(\widehat{w}\cdot\sigma)\,d\sigma\,dk_1\\
&\lesssim \int_{{\mathbb{R}}^3\times{\mathbb{S}}^2}\frac{|g_{l_1}(k_1)|\chi(\widehat{w}\cdot\sigma)}{|\widehat{R}_\sigma^-(w)\cdot\sigma|^\alpha(1-|\widehat{K}\cdot\sigma|)^{l_0/2}}
\left(|\widehat{R}_\sigma^-(w)\cdot\sigma|^\alpha|h_{l_0}(k_1^*)|\right)\,d\sigma\,dk_1\\
&\leq I_1^{1/2}I_2^{1/2},
\end{align}\] where \[\begin{align}
I_1&=\int_{{\mathbb{R}}^3\times{\mathbb{S}}^2} \frac{|g_{l_1}(k_1)|^2\chi(\widehat{w}\cdot\sigma)}{|\widehat{R}_\sigma^-(w)\cdot\sigma|^{2\alpha}(1-|\widehat{K}\cdot\sigma|)^{l_0}}\,d\sigma\,dk_1,\\
I_2&=\int_{{\mathbb{R}}^3\times{\mathbb{S}}^2} |\widehat{R}_\sigma^-(w)\cdot\sigma|^{2\alpha}|h_{l_0}(k_1^*)|^2\,d\sigma\,dk_1.
\end{align}\]
Since \(\alpha>1/2\), estimate 32 implies \(I_2\lesssim \|h_{l_0}\|_{L^2}^2\)
To estimate \(I_1\), we use ?? followed by 39 to write \[\begin{align}
I_1&=\int_{{\mathbb{R}}^3}|g_{l_1}(k_1)|^2\int_{{\mathbb{S}}^2}\frac{\chi(\widehat{w}\cdot\sigma)}{(1-\widehat{w}\cdot\sigma)^\alpha (1-|\widehat{K}\cdot\sigma|)^{l_0}}\,d\sigma\,dk_1\\
&\leq\int_{{\mathbb{R}}^3}|g_{l_1}(k_1)|^2\int_{{\mathbb{S}}^2}\frac{\chi(\widehat{w}\cdot\sigma)}{(1-\widehat{w}\cdot\sigma)^{\alpha+l_0}}+\frac{1}{(1-|\widehat{K}\cdot\sigma|)^{\alpha+l_0}}\,d\sigma\,dk_1.
\end{align}\] Integrating in spherical coordinates, we obtain \[\begin{align}
\int_{{\mathbb{S}}^2} \frac{\chi(\widehat{w}\cdot\sigma)}{(1-\widehat{w}\cdot\sigma)^{\alpha+l_0}}\,d\sigma&\approx\int_{0}^1\frac{1}{(1-x)^{\alpha+l_0}}\,dx\lesssim 1,\\
\int_{{\mathbb{S}}^2}\frac{1}{(1-|\widehat{K}\cdot\sigma|)^{\alpha+l_0}}\,d\sigma &\approx \int_0^1 \frac{1}{(1-x)^{\alpha+l_0}}\,dx\lesssim 1,
\end{align}\] where we used the fact that \(\alpha+l_0=1-\delta/4\) for the convergence of the integrals in \(x\). It follows that \(I_1\lesssim
\|g_{l_1}\|_{L^2}^2\).
Since \(k\) is arbitrary, we conclude \[\label{bound32on32R11}
\|\mathcal{R}_1^1[g,h]\|_{L^\infty}\lesssim \|g_{l_1}\|_{L^2}\|h_{l_0}\|_{L^2}\lesssim \|g_l\|_{L^r}\|h_l\|_{L^r},\tag{49}\] where for the last inequality we use 38 .
Define \(\alpha=1-\delta/4\) and fix \(k\in{\mathbb{R}}^3\). Since in the corresponding domain of integration \(\widehat{w}\cdot\sigma>0\) and \(|k_1|<|w|/2\), 22 , the triangle inequality, and ?? imply \[\begin{align}
|k_1^*|&=|k_1-R_\sigma^+(w)|\geq |R_\sigma^+(w)|-|k_1|\geq |w||\widehat{R}_\sigma^+(w)\cdot\sigma|-\frac{|w|}{2}\notag\\
&=|w|\sqrt{\frac{1+\widehat{w}\cdot\sigma}{2}}-\frac{|w|}{2}>\frac{\sqrt{2}-1}{2}|w|.
\end{align}\] In addition to that, by the triangle inequality we have \(|k|=|u+k_1|\geq |w|-|k_1|\geq |w|/2\). Using 26 for \(\epsilon=-\) as
well, we obtain \[\frac{|w|}{{\langle}k_1{\rangle}^{l_0}{\langle}k_1^*{\rangle}^{l_1}}\mathbb{1}_{|k_1|<|w|/2}\chi(\widehat{w}\cdot\sigma)\lesssim
\frac{1}{\left(1-|\widehat{(k-2R_\sigma^-(w))}\cdot\sigma|\right)^{l_0/2}}.\] Now, the above estimate followed the Cauchy-Schwarz inequality imply \[\begin{align}
|\mathcal{R}_1^0[g,h](k)|&\lesssim \int_{{\mathbb{R}}^3\times{\mathbb{S}}^2}\frac{|g_{l_0}(k_1)||h_{l_1}(k_1^*)|\chi(\widehat{w}\cdot\sigma)}{\left(1-|\widehat{(k-2R_\sigma^-(w))}\cdot\sigma|\right)^{l_0/2}}\,d\sigma\,dk_1\\
&=\int_{{\mathbb{R}}^3\times{\mathbb{S}}^2}\frac{|g_{l_0}(k_1)|\chi(\widehat{w}\cdot\sigma)}{|\widehat{R}_\sigma^-(w)\cdot\sigma|^\alpha}
\frac{|\widehat{R}_\sigma^-(w)\cdot\sigma|^\alpha|h_{l_1}(k^*)|}{\left(1-|\widehat{(k-2R_\sigma^-(w))}\cdot\sigma|\right)^{l_0/2}}\,d\sigma\,dk_1\\
&\leq J_1^{1/2}J_2^{1/2},
\end{align}\] where \[\begin{align}
J_1&=\int_{{\mathbb{R}}^3\times{\mathbb{S}}^2} \frac{|g_{l_0}(k_1)|^2\chi(\widehat{w}\cdot\sigma)}{|\widehat{R}_\sigma^-(w)\cdot\sigma|^{2\alpha}} \,d\sigma\,dk_1, \\
J_2&= \int_{{\mathbb{R}}^3\times{\mathbb{S}}^2} \frac{|\widehat{R}_\sigma^-(w)\cdot\sigma|^{2\alpha}|h_{l_1}(k^*)|^2}{\left(1-|\widehat{(k-2R_\sigma^-(w))}\cdot\sigma|\right)^{l_0}}\,d\sigma\,dk_1.
\end{align}\]
For \(J_1\), we use ?? to write \[J_1\approx
\int_{{\mathbb{R}}^3}|g_{l_0}(k_1)|^2\int_{{\mathbb{S}}^2}\frac{\chi(\widehat{w}\cdot\sigma)}{(1-\widehat{w}\cdot\sigma)^\alpha}\,d\sigma\,dk_1\approx \int_{{\mathbb{R}}^3}|g_{l_0}(k_1)|^2\int_{0}^1\frac{1}{(1-x)^\alpha}\,dx\,dk_1 \approx
\|g_{l_0}\|_{L^2}^2,\] where we used the fact that \(\alpha<1\) for the convergence of the integral in \(x\).
For \(J_2\), we first use 22 and the substitution \(y:=w=k-k_1\) to obtain \[J_2=\int_{{\mathbb{R}}^3\times{\mathbb{S}}^2}
\frac{|\widehat{R}_\sigma^-(y)\cdot\sigma|^{2\alpha}|h_{l_1}(k-R_\sigma^-(y))|^2}{\left(1-|\widehat{(k-2R_\sigma^-(y))}\cdot\sigma|\right)^{l_0}}\,d\sigma\,dy.\] Then we use Proposition 6 to substitute \(\nu:=R_\sigma^-(y)\) and obtain \[\begin{align}
J_2&\approx \int_{{\mathbb{R}}^3}|h_{l_1}(k-\nu)|^2\int_{{\mathbb{S}}^2}\frac{1}{|\widehat{\nu}\cdot\sigma|^{2(1-\alpha)}\left(1-|\widehat{(k-\nu)}\cdot\sigma|\right)^{l_0}}\,d\sigma\,d\nu\\
&\leq \int_{{\mathbb{R}}^3}|h_{l_1}(k-\nu)|^2\int_{{\mathbb{S}}^2}\frac{1}{|\widehat{\nu}\cdot\sigma|^{2(1-\alpha)+l_0}}+\frac{1}{\left(1-|\widehat{(k-\nu)}\cdot\sigma|\right)^{2(1-\alpha)+l_0}}\,d\sigma\,d\nu,
\end{align}\] where for the last step, we used 39 . Now, integrating in spherical coordinates, we obtain \[\begin{align}
&\int_{{\mathbb{S}}^2}\frac{1}{|\widehat{\nu}\cdot\sigma|^{2(1-\alpha)+l}}\,d\sigma\approx\int_0^1\frac{1}{x^{2(1-\alpha)+l_0}}\,dx\lesssim 1,\\
&\int_{{\mathbb{S}}^2}\frac{1}{\left(1-|\widehat{(k-\nu)}\cdot\sigma|\right)^{2(1-\alpha)+l_0}}\,d\sigma\approx \int_0^1\frac{1}{(1-x)^{2(1-\alpha)+l_0}}\,dx\lesssim 1,
\end{align}\] where we used the fact that \(2(1-\alpha)+l_0=1/2\) for the convergence of the integral in \(x\). It follows that \(J_2\lesssim
\|h_{l_1}\|_{L^2}^2\).
Since \(k\) is arbitrary, we conclude \[\label{bound32on32R10}
\|\mathcal{R}_1^0[g,h]\|_{L^\infty}\lesssim \|g_{l_0}\|_{L^2}\|h_{l_1}\|_{L^2}\lesssim \|g_l\|_{L^r}\|h_l\|_{L^r},\tag{50}\] where for the last inequality we use 38 .
Combining 49 , 50 , we obtain 48 , and ?? follows.
Case \(r=\infty\). The proof is identical to the corresponding case of Proposition 9. ◻
In this section, we use the estimates from the previous Sections 3 and 4 to prove our main result Theorem 1. We
fix \(2 \leq r \leq \infty\) and denote \(l = 2 - \frac{3}{r} + \delta\), where \(0 < \delta < 1/r\) if \(r<\infty\), and \(\delta>0\) if \(r=\infty\).
We first note the following auxiliary estimate: for \(\phi,\psi\in {\langle}v{\rangle}^{-l}L^r\) and \(\mathcal{T}\in\{\mathcal{G}_0,\mathcal{G}_1,\mathcal{L}_0,\mathcal{L}_1\}\),
trilinearity, the triangle inequality and ?? , ?? , ?? , ?? imply that for some numerical constant \(C\) we have
\[\label{aux32bound}
\begin{align}
\|&{\langle}k{\rangle}^l\left(\mathcal{T}[\phi,\phi,\phi]-\mathcal{T}[\psi,\psi,\psi]\right)\|_{L^r}\\
&\leq \|{\langle}k{\rangle}^l\mathcal{T}[\phi-\psi,\phi,\phi]\|_{L^r}+\|{\langle}k{\rangle}^l\mathcal{T}[\psi,\phi-\psi,\phi]\|_{L^r} + \|{\langle}k{\rangle}^l\mathcal{T}[\psi,\psi,\phi-\psi]\|_{L^r}\\
&\leq C\left(\|{\langle}k{\rangle}^l\phi\|_{L^r}^2+\|{\langle}k{\rangle}^l\phi\|_{L^r}\|{\langle}k{\rangle}^l\psi\|_{L^r}+\|{\langle}k{\rangle}^l\psi\|_{L^r}^2\right)\|{\langle}k{\rangle}^l(\phi-\psi)\|_{L^r}.
\end{align}\tag{51}\] In particular, for \(\psi=0\) we have \[\label{aux-zero} \|{\langle}k{\rangle}^l\mathcal{T}[\phi,\phi,\phi]\|_{L^r}\leq
C\|{\langle}k{\rangle}^l\phi\|_{L^r}^3.\tag{52}\]
Fix \(f_0 \in {\langle}k {\rangle}^{-l} L^r\), and let \(R := \Vert {\langle}k {\rangle}^{l} f_0 \Vert_{L^{r}},\)\(T:= \frac{1}{96 C R^2}\) where \(C\) was defined in 51 . Define \[\begin{align}
\boldsymbol{B}(2R) := \bigg \lbrace f \in \mathcal{C} \big( [0,T]; {\langle}k {\rangle}^{-l} L^{r} \big) \,:\, \sup_{t \in [0,T]} \Vert {\langle}k {\rangle}^{l} f(t) \Vert_{L^{r}} \leq 2R \bigg \rbrace.
\end{align}\]
Consider the map \[\begin{align} \Phi_{f_0}: \begin{cases} \boldsymbol{B}(2R) & \longrightarrow \, \mathcal{C}\big([0,T];{\langle}k{\rangle}^{-l}L^r\big) \\ f & \longmapsto \, f_0 + \displaystyle \int_0^t
\mathcal{C}[f](s) \, ds \end{cases}
\end{align}\] We prove that \(\Phi_{f_0}\) is a contraction on \(\boldsymbol{B}(2R).\) Stability. Let \(f \in \boldsymbol{B}(2R).\) By the triangle inequality and 52 , we obtain \[\begin{align} \Vert {\langle}k {\rangle}^{l} \Phi_{f_0} (f)(t)
\Vert_{L^r} &\leq \|{\langle}k{\rangle}^l f_0\|_{L^r}+\sum_{\mathcal{T}\in\{\mathcal{G}_0,\mathcal{G}_1,\mathcal{L}_0,\mathcal{L}_1\}}\int_0^t\|{\langle}k{\rangle}^l\mathcal{T}[f,f,f](s)\|_{L^r}\,ds\\ &\leq R + 32 C T R^3 < 2R.
\end{align}\] Therefore \(\Phi_{f_0}:\boldsymbol{B}(2R)\to\boldsymbol{B}(2R)\). Contraction. Let \(f,g\in\boldsymbol{B}(2R)\). Then, by the triangle inequality and 51 , we obtain
\[\begin{align} \big \Vert {\langle}k {\rangle}^{l} \big( \Phi_{f_0} (f) - \Phi_{f_0}(g) \big) \big \Vert_{L^r}&\leq
\sum_{\mathcal{T}\in\{\mathcal{G}_0,\mathcal{G}_1,\mathcal{L}_0,\mathcal{L}_1\}}\int_0^t\big\|{\langle}k{\rangle}^l\big(\mathcal{T}[f,f,f](s)-\mathcal{T}[g,g,g](s)\big)\big\|_{L^r}\,ds\\
&\leq 48 C R^2 T \big \Vert {\langle}k {\rangle}^{l} \big( f-g \big) \big \Vert_{L^r} = \frac{1}{2} \big \Vert {\langle}k {\rangle}^{l} \big( f-g \big) \big \Vert_{L^r}.
\end{align}\] Existence and uniqueness of a solution to 1 in \([0,T]\) follows by the contraction mapping principle. Continuous dependence on the initial data. Let \(f_0, g_0 \in{\langle}k{\rangle}^{-l}L^r\). Denote \(R_1:=\|{\langle}k{\rangle}^l f_0\|_{L^r}\), \(R_2:=\|{\langle}k{\rangle}^{l}g_0\|_{L^r}\) and let \(T_1=\frac{1}{96CR_1^2}\), \(T_2=\frac{1}{96CR_2^2}\) be the corresponding times of existence. Let \(f\in\boldsymbol{B}(2R_1)\) and \(g\in\boldsymbol{B}(2R_2)\) be the solutions corresponding to \(f_0\) and \(g_0\) respectively,
and denote \(T_{min}:=\min\{T_1,T_2\}=\frac{1}{96C\max\{R_1^2,R_2^2\}}\).
Then, for all \(t\in[0,T]\), the definition of a solution, the triangle inequality, and 51 imply \[\begin{align}
&\big \Vert {\langle}k {\rangle}^{l} \big( f(t) - g(t) \big) \big \Vert_{L^r} \\
&\leq \big \Vert {\langle}k {\rangle}^{l} \big( f_0 - g_0 \big)\|_{L^r}+
\sum_{\mathcal{T}\in\{\mathcal{G}_0,\mathcal{G}_1,\mathcal{L}_0,\mathcal{L}_1\}}\int_0^t\big\|{\langle}k{\rangle}^l\big(\mathcal{T}[f,f,f](s)-\mathcal{T}[g,g,g](s)\big)\big\|_{L^r}\,ds\\
&\leq \big \Vert {\langle}k {\rangle}^{l} \big( f_0 - g_0 \big)\|_{L^r}+48C\big(\max\{R_1,R_2\}\big)^2T_{min} \sup_{t\in[0,T_{min}]}\|{\langle}k{\rangle}^l \big(f(t)-g(t)\big)\|_{L^r}\\
&=\big \Vert {\langle}k {\rangle}^{l} \big( f_0 - g_0 \big)\|_{L^r}+\frac{1}{2} \sup_{t\in[0,T_{min}]}\|{\langle}k{\rangle}^l \big(f(t)-g(t)\big)\|_{L^r}.
\end{align}\] Taking supremum on the left hand side of the above estimate, 3 follows.
We now prove the last part of the statement in Theorem 1, namely that the flow preserves positivity. Again denote \(R=\|{\langle}k{\rangle}^l
f_0\|_{L^r}\) and \(T=\frac{1}{96CR^2}\).
We rely on the classical idea of the Kaniel-Shinbrot iteration [22], [23], which we used in
our previous work [1] to prove positivity as well. We sketch the approach for the convenience of the reader. Definition of lower and upper solutions. Let \(u_0, l_0:[0,\infty)\times{\mathbb{R}}^3\to{\mathbb{R}}\) with \(0\leq l_0\leq u_0\). Recalling 9 and 11 , define the coupled initial value problems \[\label{lower32IVP32n}
\begin{cases}
\partial_t l_n+ l_n \mathcal{R}[u_{n-1}] &=\mathcal{Q}^{+}[l_{n-1}]\\
l_n(0)=f_0
\end{cases},\quad n\in\mathbb{N},\tag{53}\] and \[\label{upper32IVP32n}
\begin{cases}
\partial_t u_n+ u_n \mathcal{R}[l_{n-1}]&=\mathcal{Q}^{+}[u_{n-1}]\\
u_n(0)=f_0
\end{cases},\quad n\in \mathbb{N}.\tag{54}\] After integration, we find the integro-differential equations \[\label{solution32lower32IVP32n}
l_n(t)= \exp \left( -\int_0^t \mathcal{R}[u_{n-1}](s)\,ds \right) f_0 +\int_0^t \mathcal{Q}^{+}[l_{n-1}](s)\exp\left(-\int_s^t \mathcal{R}[u_{n-1}](\tau)\,d\tau \right)\,ds,\tag{55}\]\[\label{solution32upper32IVP32n}
u_n(t)=\exp\left( -\int_0^t \mathcal{R}[l_{n-1}](s)\,ds\right) f_0 +\int_0^t \mathcal{Q}^{+}[u_{n-1}](s)\exp\left(-\int_s^t \mathcal{R}[l_{n-1}](\tau)\,d\tau \right)\,ds.\tag{56}\]
Initialization. We define \(l_0 = 0\) and \(u_0 = \widetilde{f}\), where \(\widetilde{f}\) is the strong solution of the gain only initial value
problem (recalling the notation 9 ) \[\begin{align} \label{IVP32Q43} \begin{cases} \partial_t f = \mathcal{Q}^+[f], \\ f(0) = f_0. \end{cases}
\end{align}\tag{57}\] One can show that this problem is well-posed in \([0,T]\), and that the solution \(\widetilde{f}\) satisfies
\[\label{gain32solution32estimate}
\sup_{t\in[0,T]}\|{\langle}k{\rangle}^l \widetilde{f}(t)\|_{L^r}\leq 2R.\tag{58}\] The proof is identical to the proof in Subsection 5.1 (just ignore the loss terms), so we omit the details. Moreover \(\widetilde{f} \geq 0\) by the positivity of successive Picard iterates. Monotonicity.
It is standard to prove that the sequences \((l_n)_{n=1}^\infty,\,(u_n)_{n=1}^\infty\) are nested, that is for every \(n \in \mathbb{N},\)\[\label{qojlnieg}
0= l_0\leq l_1\leq\dots\leq l_{n-1}\leq l_n\leq u_n\leq u_{n-1}\leq\dots \leq u_1= u_0.\tag{59}\] Indeed, this follows directly by induction, combined with the integral representations 5556 , as well as the monotonicity of the operators \(\mathcal{R}\) and \(\mathcal{Q}^{+}.\) Convergence. For fixed \(t\geq 0\), [nesting32condition] implies that the sequence \((l_n(t))_{n=0}^\infty\) is increasing and upper bounded, therefore it converges \(l_n(t)\nearrow l(t)\). Similarly, the sequence \((u_n(t))_{n=0}^\infty\) is
decreasing and lower bounded, so \(u_n(t)\searrow u(t)\). Clearly, by [nesting32condition] we have
\[\label{comparison32u4432l}
0\leq l(t)\leq u(t)\leq u_0.\tag{60}\]
Next we integrate 5354 in time and let \(n\to\infty.\) By the dominated convergence theorem, we obtain \[\begin{align}
l(t)+ \int_0^t \big(l\mathcal{R}[u]\big)(s) \,ds&=f_0+\int_0^t \mathcal{Q}^{+}[l](s) \,ds,\tag{61}\\
u(t)+ \int_0^t \big( u\mathcal{R}[l] \big)(s)\,ds &=f_0+\int_0^t \mathcal{Q}^{+}[u](s)\,ds.\tag{62}
\end{align}\] Subtracting 62 from 61 and using the fact that
\(u\geq l\), we find that \(w := u-l\geq 0\) satisfies \[\begin{align} w(t)&=\int_0^t \left(
\mathcal{Q}^{+}[u]-\mathcal{Q}^{+}[l] \right)(s)\,ds-\int_{0}^t (u\mathcal{R}[l]-l\mathcal{R}[u])(s)\,ds\notag\\ &\leq \int_0^t\left(\mathcal{Q}^{+}[u]-\mathcal{Q}^{+}[l]\right)(s)\,ds,\label{equation32satisfied32by32w}
\end{align}\tag{63}\] where for the last inequality, we used \[\begin{align}
u\mathcal{R}[l]-l\mathcal{R}[u]&=w\mathcal{R}[l]+l(\mathcal{R}[u]-\mathcal{R}[l]) \geq 0
\end{align}\] since used \(w\geq 0.\) We now use 51 followed by 60 , the fact that \(u_0=\widetilde{f}\) and 58 , to bound the right hand side of 63 as follows: \[\begin{align}
\Vert {\langle}k {\rangle}^{l} w \Vert_{L^r} &\leq \sum_{\mathcal{T}\in\{\mathcal{G}_0,\mathcal{G}_1\}}\int_0^t \big\|{\langle}k{\rangle}^l\big(\mathcal{T}(u,u,u)(s)-\mathcal{T}(l,l,l)(s)\big)\big\|_{L^r}\,ds\\
&\leq 24CR^2T \|{\langle}k{\rangle}^l w\|_{L^r} = \frac{1}{4} \Vert {\langle}k {\rangle}^{l} w \Vert_{L^r} .
\end{align}\] We conclude that \(w=0\) so \(u=l\).
Defining \(f:=l=u\) and using 61 or 62 , we obtain that \(f\) is a solution to the initial value problem 1 . Moreover \(f\geq 0\) since \(l\geq 0\) by 60 . By uniqueness, the positivity of the solution follows.
I. Ampatzoglou, T. Léger, Global existence of strong solutions to the Inhomogeneous Kinetic Wave equation, Comm. Math. Phys. 407, 144 (2026).
[2]
I. Ampatzoglou, T. Léger, Moment-preserving Young’s inequality for the hard potential Boltzmann gain operator, arXiv:2505.09554 (2025).
[3]
R. Peierls, Zur kinetischen theorie der Wärmeleitung in kristallen, Annalen der Physik 395(1929) 1055-1101.
[4]
K. Hasselmann, On the non-linear energy transfer in a gravity-wave spectrum. I. General theory, J. Fluid Mech. 12(1962), 481–500.
[5]
K. Hasselmann, On the non-linear energy transfer in a gravity wave spectrum. II. Conservation theorems; wave-particle analogy; irreversibility, J. Fluid Mech.
15(1963), 273–281.
[6]
S. Nazarenko, Wave turbulence, Lecture Notes in Physics, 825. Springer, Heidelberg, 2011.
[7]
T. Buckmaster, P. Germain, Z. Hani, J. Shatah, Onset of the wave turbulence description of the longtime behavior of the nonlinear Schrödinger equation, Invent. Math.
225(2021), no. 3, 787–855.
[8]
C. Collot, P. Germain, On the derivation of the homogeneous wave kinetic equation, to appear in Comm. Pure Appl. Math., arXiv:1912.10368.
[9]
C. Collot, P. Germain, Derivation of the homogeneous wave kinetic equation: longer time scales, arXiv:2007.03508.
[10]
Y. Deng, Z. Hani, On the derivation of the wave kinetic equation for NLS, Forum Math. Pi 9(2021), Paper No. e6, 37 pp.
[11]
Y. Deng, Z. Hani, Full derivation of the wave kinetic equation, Invent. Math. 233(2023), no. 2, 543–724.
[12]
M. Escobedo, J.J.L. Velázquez, On the theory of weak turbulence for the nonlinear Schrödinger equation, Mem. Amer. Math. Soc. 238(2015), no. 1124, v+107
pp.
[13]
C. Collot, H. Dietert, P. Germain, Stability and cascades for the Kolmogorov-Zakharov spectrum of wave turbulence, Arch. Ration. Mech. Anal. 248 (2024), no. 1,
no. 7, 31 pp.
[14]
M. Escobedo, A. Menegaki, Instability of singular equilibria of a wave kinetic equation, arXiv:2406.05280.
[15]
P. Germain, A. D. Ionescu, M.-B. Tran, Optimal local well-posedness theory for the wave kinetic equation, J. Funct. Anal. 279(2020), no. 4,
108570, 28 pp.
[16]
I. Ampatzoglou, Global well-posedness of the inhomogeneous wave kinetic equation near vacuum, Kinet. Relat. Models 17(2024), no. 6, 838-854.
[17]
I. Ampatzoglou, J.K. Miller, N. Pavlović, M. Tasković, Inhomogeneous wave kinetic equation and its hierarchy in polynomially weighted \(L^\infty\)
spaces, Comm. Partial Differential Equations 50(2025), no. 6, 723-765.
[18]
I. Ampatzoglou, T. Léger, On the ill-posedness of wave kinetic equations, Nonlinearity 38 115004 (2025).
[19]
N. Pavlović, M. Tasković, L. Velasco, Inhomogeneous six-wave kinetic equation in exponentially weighted \(L^\infty\) spaces, arXiv:2501.10565
(2024).
[20]
A.V. Bobylev, Fourier transform method in the theory of the Boltzmann equation for Maxwellian molecules, Dokl. Akad. Nauk USSR 225(1975), 1041–
1044.
[21]
A.V. Bobylev, The theory of the nonlinear spatially uniform Boltzmann equation for Maxwell molecules, Sov. Sci. Rev. C. Math. Phys. 7(1988),
111–233.
[22]
S. Kaniel, M. Shinbrot, The Boltzmann Equation: Uniqueness and Local Existence, Comm. Math. Phys. 58, 65-84 (1978).
[23]
R. Illner, M. Shinbrot, The Boltzmann Equation: Global Existence for a Rare Gas in an Infinite Vacuum, Commun. Math. Phys. 95, 217-226 (1984).