(\[e\_condition\]) becomes immediately locally BV in space and locally in space-time in the complement of the initial line .

A more powerful approach was given by Kruzhkov in [@Kr], where he replaces condition (\[e\_condition\]) by a family of integral inequalities. This approach covers also cases, where $f$ is non-convex and the space dimension is bigger than one. However in the case of convex fluxes one can show, that his *entropy condition* is equivalent to Oleinik’s *E-condition* (see Chapter 8.5 in [@Da]). More precisely for $u_0\in L^{\infty}$ he proved existence and uniqueness of weak solutions satisfying the *entropy condition*: He considers the family of convex entropy flux pairs $(\eta_a, \xi_a)_{a\in\mathbb{R}}$, where $$\label{kruzhkov_family}
 \eta_{a}(u)=(u-a)^+\quad\mbox{and}\quad \xi_a(u)=\operatorname{sign}(u-a)^+(f(u)-f(a))\,,$$ and $w^+$ stands for $\max\{w, 0\}$. Then an entropy solution is a bounded function $u$, which satisfies (\[scl\]) in the sense of distributions and $$\label{e_inequality}
\partial_t\eta_a(u)+\partial_x\xi_a(u)\leq0\,.$$ Equivalently one can replace the one parameter family $(\eta_a, \xi_a)_{a\in\mathbb{R}}$ and assume, that (\[e\_inequality\]) is fulfilled for all convex $\eta$ with corresponding entropy flux $\xi$, which is defined by $\xi=\int \eta'f'$. As a consequence of this one can show, if the initial data $u_0$ is in BV, that $u$ is in BV for all later times.

Let $a\wedge b$ denote $\min\{a,b\}$. 