𝐋¹ stability of semigroups with respect to their generators

Abstract

This note is concerned with the L 1 \mathbf {L}^1 –theory for the system ∂ t u = div x ⁡ A ( u ) + B ⋅ Δ u + C ( u ) \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 L 1 ∩ L ∞ ∩ B V \mathbf {L}^1\cap \mathbf {L}^\infty \cap \mathbf {BV} . Then, the L 1 \mathbf {L}^1 –Lipschitz dependence of the solutions with respect to the natural norms of A A , B B and C 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.