&
+\left. \vphantom{\int_0^1}p_-(k) W_0\left(-y+\bar{\omega}'(k) t,-k\right)1_{[0,\bar {\omega}'(k)t]}(y)\right\}.\end{aligned}$$ Consider a semigroup of bounded operators on $L^\infty({{\mathbb R}}\times {{\mathbb T}}_*)$ defined by $$\label{010304s}
{\frak W}^{\rm un}_t(W_0)\left(y,k\right)
:=  
W^{\rm un}(t,y,k),$$ with $W_0\in L^\infty({{\mathbb R}}\times
{{\mathbb T}}_*)$, $t\ge0$ and $(y,k)\in {{\mathbb R}}_*\times {{\mathbb T}}_*$. From formula one can conclude that $\left({\frak
    W}^{\rm un}_t\right)_{t\ge0}$ forms a semigroup of contractions on both $L^1({{\mathbb R}}\times{{\mathbb T}})$ and $L^\infty({{\mathbb R}}\times{{\mathbb T}})$. Thus, by interpolation, formula defines a semigorup of contractions on any $L^p({{\mathbb R}}\times{{\mathbb T}})$, $1\le p\le+\infty$.

Note that if $W_0$ is continuous in ${{\mathbb R}}_*\times {{\mathbb T}}_*$, then $ W^{\rm un}(t,y,k)$ satisfies the interface conditions  and , with $T=0$ for all $t>0$. Therefore, $( {\frak W}^{\rm un}_t)_{t\ge0}$ is a semigroup on ${\cal C}_0$ with $W^{\rm un}(t,y,k)$ (cf ) satisfying the first equation of , the interface condition - and the initial condition $$\lim_{t\to0+} W^{\rm un}(t,y,k)=W_0(y,k),\quad (y,k)\in{{\mathbb R}}_*\times{{\mathbb T}}_*.$$

### Kinetic equation - classical solution {#sec2.6.3}

\[df013001-19\] We say that a function $W:\bar{{\mathbb R}}_+\times {{\mathbb R}}\times {{\mathbb T}}_*\to {{\mathbb R}}$ is a classical solution to equation with the interface conditions , at $y=0$, if it is bounded and continuous on ${{\mathbb R}}_+\times{{\mathbb R}}_*\times {{\mathbb T}}_*$, and the following conditions hold:

-   the restrictions of $W$ to ${{\mathbb R}}_+\times{{\mathbb R}}_\iota\times 