the definition of the solution does not depend on the choice of function $\chi$ satisfying .* ]{}

\[rm021406-19\] [*Suppose that $W_0\in {\cal C}_T$. Then, $W(t,y,k)$ given by is the classical solution of with the interface conditions and , in the sense of Definition $\ref{df013001-19}$.*]{}

Asymptotics of the Wigner functions - the statement of the main result
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Thanks to we conclude that $$\label{030406-19}
\sup_{{\epsilon}\in(0,1]}\|  W_{{\epsilon}}\|_{L^\infty([0,\tau];{\cal
    A}')}<+\infty,\quad \mbox{for any }\tau>0.$$ Therefore $\left(W_{{\epsilon}}(\cdot)\right)$ is sequentially $\star$-weakly compact in $L^\infty_{\rm loc}([0,+\infty),{\cal A}')$, i.e. from any sequence ${\epsilon}_n\to0$ we can choose a subsequence, that we still denote by the same symbol, for which $\left( W_{{\epsilon}_n}(\cdot)\right)$ $\star$-weakly converges in $\left(L^1([0,t],{\cal
    A})\right)'$ for any $t>0$. In our main result we identify the limit as the $L^2$ solution of the kinetic equation with the interface conditions and , in the sense of Definition $\ref{df011406-19}$.

\[main-thm\] Suppose that there exist $C,\kappa>0$ such that $$\label{011812aa}
|\widehat W_{\epsilon}(0,\eta,k)|+|\widehat Y_{\epsilon}(0,\eta,k)|\le 
C\varphi(\eta),\quad (\eta,k)\in{{\mathbb T}}_{{\epsilon}}\times {{\mathbb T}}, \,{\epsilon}\in(0,1],$$ where $$\label{011812c}
\varphi(\eta):=\frac{1}{(1+\eta^2)^{3/2+\kappa}},$$ and $$ W_{{\epsilon}}(0)\mathop{\stackrel{{\rm \tiny w}^\star}{\longrightarrow}}\limits_{\tiny{{\epsilon}\to0+}} W_0\quad\mbox{ in
}{\cal A}'.$$ Then, $W_0\in L^2({{\mathbb R}}\times{{\mathbb T}})$ and for any $G\in
L^1_{\rm loc}\left([0,+\infty);{\cal A}\right)$ we have $$\label{061406-19}
\lim_{{\epsilon}\to0+}\int_0^{\tau}\langle W_{{\epsilon}}(t), G(t)\rangle dt=\int_0^{\tau}\langle W(t), G(t)\rangle dt,\quad\tau>0.$$ Here $W(t)$ is the $L^2$ solution of the kinetic equation with 