A Note on $L^1-$contractive property of the solutions of the scalar conservation laws through the method by Lax-Ole\u{\i}nik

TL;DR

This paper investigates the $L^1$-contractive property of solutions to scalar conservation laws using the Lax-Olee8nik method, covering various initial data spaces and flux conditions without relying on Kruzkov's results.

Contribution

It establishes the $L^1$-contractive property for scalar conservation laws under broader conditions, including convex fluxes with super-linear growth and semi-super-linear fluxes, without Kruzkov's framework.

Findings

Proves $L^1$-contractivity for convex flux with bounded initial data.

Extends $L^1$-contractivity to initial data in $L^1$ with super-linear flux growth.

Demonstrates $L^1$-contractivity for semi-super-linear fluxes and $L^1$ initial data.

Abstract

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.