Particle Approximation of the BGK Equation

Abstract

In this paper we prove the convergence of a suitable particle system towards the BGK model. More precisely, we consider an interacting stochastic particle system in which each particle can instantaneously thermalize locally. We show that, under a suitable scaling limit, propagation of chaos does hold and the one-particle distribution function converges to the solution of the BGK equation.

Proof of Proposition 2.2

We first observe that the smeared fields, defined in Eqs. (2.9), (2.10), and (2.11), coincide with the usual hydrodynamical fields associated to the smeared distribution function $g^\varphi (x,v) := \int \!\mathrm {d}y\, \varphi (x-y) g(y,v)$, i.e.,

$$\begin{aligned} \varrho _g^\varphi = \varrho _{g^\varphi }\,, \quad \varrho _g^\varphi u_g^\varphi = \varrho _{g^\varphi } u_{g^\varphi }\,, \quad \varrho _g^\varphi T_g^\varphi = \varrho _{g^\varphi } T_{g^\varphi }\,. \end{aligned}$$

Therefore, according to [7, Proposition 2.1], we find the following pointwise estimates for $\rho _g^\varphi$, $u_g^\varphi$, and $T_g^\varphi$:

$$\begin{aligned}&\text {(i) } \frac{\rho ^\varphi }{ (T^\varphi )^{d/2}} \le C N_0( g^\varphi )\,; \\&\text {(ii) } \rho ^\varphi ( T^\varphi +(u^\varphi )^2)^{ (q-d)/2} \le C_q N_q (g^\varphi )\text { either for } q> d+2 \text { or for }0\le q <d \,; \\&\text {(iii) } \frac{\rho ^\varphi |u^\varphi |^{d+q}}{[ T^\varphi +(u^\varphi )^2) T^\varphi ]^{d/2} } \le C_q N_q (g^\varphi )\text { for }q>1 \,. \end{aligned}$$

In the above, $C,C_q$ are constants independent of $\varphi$ and

$$\begin{aligned} N_q(f) = \sup _v |v|^q f(v)\,, \quad q\ge 0 \end{aligned}$$

for a given positive function f.

As a consequence, following [7], we infer that

$$\begin{aligned} \sup _v |v|^q M_g^{\varphi } (x,v) \le C_q N_q( g^\varphi ), \end{aligned}$$

and hence, writing the equation for g in mild form and recallig Eq. (2.17), we obtain

$$\begin{aligned} {\mathcal N}_q(g(t)) \le {\mathcal N}_q(f_0)+C_q\int _0^t\!\mathrm {d}s\, {\mathcal N}_q( g^\varphi (s))\,. \end{aligned}$$

The a priori bound ${\mathcal N}_q(g(t)) \le {\mathcal N}_q(f_0) \exp (C_q t)$ follows by the obvious inequality $N_q( g^\varphi (s)) \le N_q( g (s))$ and the Grönwall’s lemma.

Provided with this estimate, by arguing exactly as in the proof of [7, Theorem 3.1], we construct the solution g(t) by establishing the Lipschitz continuity of the operator $g \rightarrow \rho _g^\varphi M_g^{\varphi } -g$ in $L^1( (1+v^2) \mathrm {d}x \mathrm {d}v)$ and using the standard iteration scheme. Moreover, exactly as in [7], the bounds (2.18), (2.19), and (2.20) follow from this construction and the previous a priori estimates.