  In this note, we study the $L^1-$contractive property of the solutions the
scalar conservation laws, got by the method of Lax-{O}le\u{\i}nik. First, it is
proved when f is merely convex and the initial data is in
$L^{\infty}(\mathbb{R})$. And then, it is shown for the case when the initial
data is in $L^1(\mathbb{R})$ with the convex flux having super-linear growth.
Finally, the $L^1-$contractive property is shown for the scalar conservation
laws with the initial data in $L^1(\mathbb{R})$ and the flux is
"semi-super-linear". This entire note does not assume any results mentioned
through the approach by Kruzkov.
