\, \Longleftrightarrow  \, 
\forall n\in \mathbb{N},\, f\in C^n(\Omega), 
\; \sup _{\Omega}\, \sum _{k=0}^{n} \, \bigl| \, D^kf\, \bigr|<\infty$$ where $\Omega \subset \mathbb{R}^d$ and $D^k$ denote the all of $k$-th order derivatives.

Preliminaries
=============

In this section, we shall arrange the several lemmas concerning with the basic properties of the rarefaction wave and the viscous contact wave for accomplishing the proof of the main theorem. Since the rarefaction wave $u^r$ is not smooth enough, we need some smooth approximated one as in the previous works in [@hashimoto-matsumura], [@liu-matsumura-nishihara], [@matsu-nishi1], [@matsumura-yoshida]. We start with the well-known arguments on $u^r$ and the method of constructing its smooth approximation. We first consider the rarefaction wave solution $w^r$ to the Riemann problem for the non-viscous Burgers equation $$\label{riemann-burgers}
  \left\{\begin{array}{l}
  \partial _t w + 
  \displaystyle{ \partial _x \left( \, \frac{1}{2} \, w^2 \right) } = 0 
  \, \, \; \; \qquad \quad \qquad ( t > 0,\,x\in \mathbb{R}),\\[7pt]
  w(0,x) = w_0 ^{\rm{R}} ( \, x\: ;\: w_- ,w_+):= \left\{\begin{array}{ll}
                                                  w_+ & \, \: \; \quad (x>0),\\[5pt]
                                                  w_- & \, \: \; \quad (x<0),
                                                  \end{array}
                                                  \right.
  \end{array}
  \right.$$ where $w_\pm \in \mathbb{R} \: (w_-<w_+)$ are the prescribed far field states. The unique global weak solution $w=w^r\left( \, 