satisfies $\lambda^d(L=0)>0$ and $\lambda^d(L=\infty)=0$ (for $d=1$ see Fig. \[fig:muL\] below). Moreover $c_d(L=0)=C_d(L=0)=1$ (see , below).

![These two functions illustrate the time dependent decay estimate from . The values of $C_d,\,\lambda^d$ correspond to the 1D case with $L=2\pi$, and we chose ${\mathcal{E}}^d(\tilde f^I)=15$. We also show the two time scales of the BGK equation: $t_{{\rm init}}$ marks the intersection point of the two blue curves and it corresponds to the generic transport time. $t_2:=t_{{\rm init}}+\frac2\lambda$ marks the intersection point of the exponential curve with the value $2/e$, and $t_2-t_{{\rm init}}$ corresponds to the relaxation time scale. For larger values of $L$, $t_{{\rm init}}$ will be much larger. []{data-label="fig:H-decay"}](remark12.eps)

To prove local asymptotic stability for the nonlinear BGK equation in 3D, we make use of another set of norms: For $\gamma \geq 0$, let $H^\gamma(\tilde{\mathbf{T}}^3)$ be the Sobolev space consisting of the completion of smooth functions $\varphi$ on $\tilde{\mathbf{T}}^3$ in the Hilbertian norm $$\|\varphi\|_{H^\gamma}^2 := \sum_{k\in {\mathbb{Z}}^3} (1+|k|^2)^{\gamma}|\varphi_k|^2\ ,$$ where $\varphi_k$ ($k\in{\mathbb{Z}}^3$) is the $k$th Fourier coefficient of $\varphi$. Let $\mathcal{H}_\gamma$ denote the Hilbert space $H^\gamma(\tilde{\mathbf{T}}^3)\otimes L^2({\mathbb{R}}^3;M_1^{-1})$. Then the inner product in $\mathcal{H}_\gamma$ is given by $$\langle f,g\rangle_{\mathcal{H}_\gamma} = \int_{\tilde{\mathbf{T}}^3\times{\mathbb{R}}^3} \overline{f}(x,v) \left[\left(1 - \Delta_x\right)^{\gamma} g(x,v) \right]M_1^{-1}(v) {\,\operatorname{d}\!\tilde x} {\,\operatorname{d}\!v} \ .$$

\[BGK-decay\] 