Let $u_0 \in BV(\mathbb R)$ and $f:\mathbb R \to \mathbb R$ be Lipschitz. Consider the Cauchy problem $$ \begin{align*} u_t +( v(x)f(u))_x&=0\\ u(0,\cdot) &= u_0 \end{align*} $$
What is the minimal assumption that we need to put on $v$ to prove $u(t,\cdot) \in BV(\mathbb R)$?
From a formal computation, I think we need $v' \in BV(\mathbb R)$, but my intuition would have been the weaker condition $v' \in L^\infty(\mathbb R)$.
