$\mathbf {L}^1$ stability of semigroups with respect to their generators

This note is concerned with the $\mathbf {L}^1$–theory for the system $\partial _t u = \operatorname {div}_x A(u) + B \cdot \Delta u + C(u)$ in several space dimensions. First, an existence result is proved for data in $\mathbf {L}^1\cap \mathbf {L}^\infty \cap \mathbf {BV}$. Then, the $\mathbf {L}^1$–Lipschitz dependence of the solutions with respect to the natural norms of $A$, $B$ and $C$ is achieved. As a corollary, the vanishing viscosity limit for conservation laws in 1D recently obtained in a work by Bianchini and Bressan is slightly extended.