$ without boundary does not affect to the conclusions; for the construction of the Friedrich’s mollifier on $S$ we refer to [@fukuoka]). The above references are, however, valid only for problems in which the dimension of the kernel ${\rm Ker}(A_\nu)$, given below, is constant on $\Gamma$ (i.e. $A_\nu$ has constant multiplicity), and thus we are not able to directly apply them. Some results for variable multiplicity can be found e.g. in [@rauch94], [@nishitani96], [@nishitani98] and [@takayama02] together with their references. The latter researches require the an additional assumption which concerns the “transition with a non-zero derivative” on the smooth $(6-2)$-dimensional manifold $\Gamma_0$. For simplicity we deal only with the case where $S_0=S_0(x)$ is independent of $E$ and use formulations of [@nishitani96]. More general formulations can be based on [@nishitani98] and related references but we omit them here.

We make the change of variables and of the unknown function \[evo2\] (x,,E):=(x,,E\_m-E) and denote $$\tilde f(x,\omega,E)=f(x,\omega,E_{\rm m}-E).$$ Making the changes, we find that the problem $(I+\tilde P_0)\phi=f$ is equivalent to $\phi$ satisfying the equation \[evo3\] [E]{}+[1]{}\_x+[1]{}(C S\_0+1)=[1]{}f=:[**f**]{}, on $G\times S\times I$, along with satisfying the following inflow boundary and initial value conditions \[evo4\] \_[|\_-]{}=0, (,,0)=0. Define $$Q(x,\omega,E,D)\Phi:={1\over{ S_0}}\omega\cdot\nabla_x\Phi
+{1\over{ S_0}}(C S_0+1)\Phi$$ and the boundary matrix $$A_{\nu}(z)={1\over{ S_0}}\omega\cdot\nu(y),\quad z=(y,\omega,E)\in\Gamma.$$

We assume that $\partial G=\{x\in{\mathbb{R}}^3\ |\ f(x)=0\}$ is a level set of class $C^\infty$, i.e. $f\in C^\infty({\mathbb{R}}^3)$ and $\nabla_x f(x)\neq 0$ whenever $f(x)=0$, and we also assume that $S_0\in C^\infty(G)$. The outward unit normal vector field $\nu(y)$ on $\partial G$ is given by $$\nu(y)={{(\nabla_x f)(y)}\over{{\left\Vert (\nabla_x f)(y)\right\Vert}}}.$$ The additional assumption is as follows. Suppose that there exist 