| Russian Mathematical Surveys |
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This article is cited in 2 scientific papers (total in 2 papers) Brief communications On a property of discrete models of the wave kinetic equation A. V. Bobylevab a Keldysh Institute of Applied Mathematics of Russian Academy of Sciences, Moscow b Peoples' Friendship University of Russia (RUDN University)
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     The wave kinetic equation is connected with the theory of weak turbulence for the nonlinear Schrodinger equation (see, for instance, [1], [2], and the references there). It is convenient to consider this equation (without discussing its physical meaning) as a particular case of the Boltzmann-type kinetic equation for the distribution function (that is, the probability density in $\mathbb{R}^d $) $f(v,t)$, where the variables $v \in \mathbb{R}^d $, $d \geqslant 2$, and $t \in \mathbb{R}_+$ are interpreted for clarity as the velocity of particles and time, respectively. In the general case the Boltzmann-type equation has the following form, where the argument $t$ of the function $f(v,t)$ is omitted [3]:
$$
\begin{equation}
\begin{aligned} \, \frac{df}{dt}&=\int_{\mathbb{R}^d\times\mathbb{R}^d\times\mathbb{R}^d} dv_2\, dv_3\, dv_4\, \delta[v+v_2-v_3-v_4]\, \delta[|v|^2+|v_2|^2-|v_3|^2-|v_4|^2] \\ &\qquad\times F(f_1,f_2; f_3,f_4),\qquad f_1=f(v),\quad f_i=f(v_i),\ \ i=2,3,4. \end{aligned}
\end{equation}
\tag{1}
$$
It is usually assumed that $F(x_1,x_2;x_3,x_4)=F(x_2,x_1;x_3,x_4)=-F(x_3,x_4;x_1,x_2)$. The case $F=F_{\rm B}$ corresponds to the classical Boltzmann equation, whereas the case $F=F_{\rm w}$ corresponds to the wave kinetic equation, where
$$
\begin{equation}
\begin{aligned} \, F_{\rm B}(x_1,x_2;x_3,x_4)&=x_3 x_4-x_1 x_2, \\ F_{\rm w}(x_1,x_2 ; x_3,x_4)&=x_3 x_4(x_1+x_2)-x_1 x_2(x_3+x_4). \end{aligned}
\end{equation}
\tag{2}
$$
If we substitute $F=F_{\rm B}$ into (1), then we obtain the usual Boltzmann equation for hard spheres in the case when $d=3$ or for Maxwell molecules in the case when $d=2$ (see [3]). Therefore, it is natural to try to apply some known methods of classical kinetic theory to equation (1) for $F=F_{\rm w}$. In particular, we can introduce discrete models of this equation by using an analogy with discrete models of Boltzmann’s equation, which were considered in the vast literature (see, for instance, [4], [5], and the references there). To this end we begin by defining a finite set of ‘velocities’ $ V \subset \mathbb{R}^d$ and replacing the function $f(v,t)$ by a vector $f(t) \in \mathbb{R}^n$:
$$
\begin{equation}
V=\{v_1,\dots,v_n\},\quad f(t)=\{f_1(t),\dots,f_n(t)\},\qquad n \geqslant 4.
\end{equation}
\tag{3}
$$
It is implicitly assumed here that $f_i(t)$ approximates, as $n \to \infty$ the function $f(v,t)$ at the point $v=v_i \in \mathbb{R}^d$, $i=1,\dots,n$. Equation (1) is replaced by the set of $n$ ordinary differential equations
$$
\begin{equation}
\frac{df_i}{dt}=\sum_{j,k,l=1}^{n}\Gamma_{ij}^{kl} F_{\rm w}(f_i,f_j;f_k,f_l),\qquad \Gamma_{ij}^{kl}=\Gamma_{ji}^{kl}=\Gamma_{kl}^{ij},
\end{equation}
\tag{4}
$$
where (for fixed $V$) the constants $\Gamma^{kl}_{ij} \geqslant 0$ depend only on $|v_i-v_j|=|v_k-v_l|$ for all integer indices $1 \leqslant i,j,k,l \leqslant n$. The strict inequality $\Gamma^{kl}_{ij} > 0$ is possible only if $v_i+v_j=v_k+v_l$ and $|v_i|^2+|v_j|^2=|v_k|^2+|v_l|^2$. Some restrictions on the set $V$ and coefficients of (4) are needed in order to guarantee that the model (3), (4) has the main properties of the original equation (1).
Definition. The model (3), (4) is called normal if it satisfies the following conditions on the set $V$: (a) all of its $n$ elements $v_i \in \mathbb{R}^d$ are pairwise different and do not lie in a subspace of dimension $d' \leqslant d-1$ or on a sphere in $\mathbb{R}^d$; (b) the set $V$ does not have isolated points, that is, for any index $1 \leqslant i \leqslant n$ in (4) there exist $1 \leqslant j,k,l \leqslant n$ such that that $\Gamma^{kl}_{ij} > 0$; (c) if the functional equation $h(v_i)+h(v_j)=h(v_k)+h(v_l)$ is fulfilled for all indices $(i,j;k,l)$ for which $\Gamma^{kl}_{ij}>0$, then there exist constants $\alpha,\gamma \in \mathbb{R}$ and $\beta \in \mathbb{R}^d$ such that $h(v)=\alpha+\beta \cdot v+\gamma|v|^2$. Methods of construction of normal models are well developed (see [4] and the references there), and therefore we consider only such models below. It is easy to show that $n \geqslant 6$ for them. We turn to the main result of the paper. We consider the Cauchy problem for equations (4) with initial conditions
$$
\begin{equation}
f\big|_{t=0}=f^{(0)}=\{f_1^{(0)},\dots,f_n^{(0)}\},\qquad f_i^{(0)} >0, \ \ i=1,\dots,n, \quad \sum_{i=1}^{n}f_i^{(0)}v_i=0.
\end{equation}
\tag{5}
$$
The equality for the sum in (5) simplifies the problem slightly, but this does not lead to a loss of generality.
Theorem. Let the discrete model (3), (4) of the wave kinetic equation (1), where $ F=F_w $, be normal and, in addition, let the set $ V $ be such that $ v_1=0 $ and if $v_i \in V$, then $(-v_i) \in V $ for all $ i=1,\dots,n $. Then the Cauchy problem for equations (4) with initial data satisfying conditions (5) has a unique solution $ f(t)=\{f_1(t),\dots,f_n(t)\}$ for all $ t >0 $. Moreover, for all $1 \leqslant i \leqslant n$ (i) $ 0< f^{(0)}_{i}\exp(-c\rho_{0}^{2}t) \leqslant f_i(t)<\rho_0$, $\rho_0 =\sum_{i=1}^{n} f^{(0)}_{i}$, where the constant $c > 0$ does not depend on $f^{(0)}$; (ii) $\lim_{t \to \infty}f_i(t)=a(1+b|v_i|^2)^{-1}$ and $a \sum_{i=1}^{n}(1+b |v_i|^2)^{-1}=\rho_0$, where the number $ b >-M^{-1}$, $M=\max\{|v_i|^2, \ 1 \leqslant i \leqslant n\}$, is the maximum real root of the equation It is possible to show that the function $T_0(b)$ defined by (6) decreases monotonically on the interval $[-M^{-1},\infty)$ from its maximum value $T_0(-M^{-1})=M$ to zero as $b \to \infty$. Therefore, the root $b(T_0)$ mentioned in part (ii) of the theorem is unique. The proof of the theorem is based on the fact that equations (4) admit a monotone non-increasing functional (Lyapunov function) similarly to Boltzmann’s equation [3]. Solutions of equations (4) converge, as $t \to \infty$, to the equilibrium solution, which plays the same role as the Maxwellian distribution plays for discrete models of Boltzmann’s equation. In contrast to the case of Boltzmann’s equation this property of discrete models cannot be transferred to the case of equation (1) with $F=F_{\rm w}$ (see (2)) because of the divergence of integrals. Nevertheless this property can be useful for the study of the asymptotic behaviour of solutions to (1) for $F=F_{\rm w}$, since this equation can be approximated by discrete models, as $n \to \infty$ (the proof in [6] can easily be generalized to the wave kinetic equation for $d=3$). 
Citation in format AMSBIB
\Bibitem{Bob23} |
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