A Note on $L^1-$contractive property of the solutions of the scalar conservation laws through the method by Lax-Oleĭnik

In this note, we study the $L^1-$contractive property of the solutions the scalar conservation laws, got by the method of Lax-{O}leĭnik. First, it is proved when f is merely convex and the initial data is in $L^{\mathrm{\infty }}(\mathbb{R})$. 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.