---
author:
- 'Michael Blaser$^{*}$, Tristan Rivière[^1]'
title: A Minimality property for entropic solutions to scalar conservation laws in $1+1$ dimensions
---

Introduction
============

We consider solutions to the following equation $$\label{scl}
\left .
\begin{array}{rclcl}
 \partial_{t}u+\partial_{x}f(u)& =&0&\mbox{in}&\mathbb{R}_+\times\mathbb{R}\,, \\ 
u(x,0) & =&u_{0}(x)\,,
\end{array}
\right \}$$ where the flux $f$ is strictly convex ( $f''\ge c>0$ )and the initial date $u_0\in L^{\infty}$. It is well known, that, even for smooth initial data, the classical solution can cease to exist in finite time, due to the possible formation of shocks (see Chapter 4.2 in [@Da]). Therefore one has to consider weak solutions of (\[scl\]), i.e. solutions, which satisfy (\[scl\]) in the distributional sense. However it turned out, that, for a given initial data, the space of weak solutions is huge (see Chapter 4.4 in [@Da]). Therefore additional conditions have to be imposed to single out the physical relevant weak solutions in some models.

In 1957 Oleinik proved in [@Ol] uniqueness of bounded weak solutions, which satisfy almost everywhere her ’*E-condition*’ $$\label{e_condition}
 u(y,t)-u(x,t)\leq \frac{y-x}{ct}\,,\quad\mbox{for}\quad x<y\,,t>0\,,$$ where $c=\inf f''$. A immediate consequence of this condition (\[e\_condition\]) is a spectacular regularization phenomena. Oleinik proved, that for bounded measurable initial data, the weak solution satisfying almost everywhere 