# Chapter 11: Optimal local well-posedness theory for the kinetic wave equation.

This post is based on my paper

We investigate the local well-posedness theory for the space-homogeneous 4-wave kinetic equation

$\partial_t f(t,p) = \mathcal{Q}[f](t,p), \mbox{ on } \mathbb{R}_+\times\mathbb{R}^3, (1)$

$f(0,p) = f_0(p) \mbox{ on } \mathbb{R}^3.$

The trilinear operator $\mathcal{Q}$ is given by

$\mathcal{Q}[f](p) = \iiint_{\mathbb{R}^{3\times3}}\delta(p+p_1-p_2-p_3)\delta(\omega + \omega_1 -\omega_2 - \omega_3) [f_2 f_3 (f_1+f) -f f_1 (f_2 + f_3)]dp_1dp_2dp_3,$

where we denoted

$\omega= \omega(p), \omega_i = \omega(p_i), f = f(p), f_i = f(p_i).$

In the above, $p \mapsto \omega(p)$ is the dispersive relation of the underlying dispersive problem.

The question of the local existence and uniqueness of solutions to (1) was first studied in Escobedo-Velazquez’15, where the dispersive relation is of classical type $\omega(p)=|p|^2$, and the solution $f$ is radial (velocity-isotropic). Abusing notations by denoting $p$ for $|p|$ and $f(p)$ for $f(|p|)$, the equation (1) reduces to a one-dimensional Boltzmann equation

$\partial_t f=\int_{\mathbb{R}_+^2} \frac{p_2 p_3 \min\{p,p_1,p_2,p_3\}}{p} [f_2 f_3(f+f_1)- f f_1(f_2+f_3)] dp_3dp_4, (2)$

where $p_1^2 = p_2^2+ p_3^2 - p^2$.

It is proved in~Escobedo-Velazquez’15 that the above equation admits global, measure valued, weak solutions. This functional framework allows in particular for condensation, namely the development of a point mass at the origin. It is furthermore showed that condensation can occur, and that, as $t \to \infty$, most of the energy is transferred to high frequencies.

The reduction to the radial model (2) is restricted to the case $\omega(p) = |p|^2$. It is therefore the goal of our paper to construct a local existence and uniqueness theory, which does not rely on the various forms of the dispersion laws and is valid without the assumption that the solutions are radial.

In the theory of the classical Boltzmann equation, the conservation laws $p+p_1=p_2+p_3, |p|^2+|p_1|^2=|p_2|^2+|p_3|^2$ play a very important role. Since this implies that $p$, $p_1,$ $p_2,$ $p_3$ are on the sphere centered at $\frac{p+p_1}{2}$ with radius $\frac{|p-p_1|}{2}$, the Boltzmann collision operators can be considered as integrals on spheres and the Carleman representation can be used. This is not the case for more general dispersion relations, for which the resonant manifolds do not admit such simple parameterizations.

In this work we provide an optimal local existence and uniqueness theory for general dispersion relations of the form below and the radial assumption is removed:

$\omega(p)=\Omega(|p|)$

and satisfies:

(i) $\Omega(0) = 0$ (this is simply a convenient normalization).

(ii) $\Omega\in C^{1}(\mathbb{R}_+)$ and $\Omega(x)\ge 0$ for all $x$ in $\mathbb{R}_+$.

(iii) There exists a constant $c_1>0$ such that $\Omega'(x)\ge c_1x$, for all $x$ in $\mathbb{R}_+$.

(iv) There exists a constant $c_2>0$ such that $\Omega(x) \le \frac{1}{2} \Omega(c_2x)$, for all $x$ in $\mathbb{R}_+$.