A note on contractive semi-groups on a 1:1 junction for scalar conservation laws and Hamilton-Jacobi equations

Abstract

We show that any continuous semi-group on $L^1$ which is (i) $L^1-$contractive, (ii) satisfies the conservation law ${\partial }_t\ddot{I}+{\partial }_x(H(x,\ddot{I}))=0$ in ${\mathbb{R}}_{+}\times (\mathbb{R}\backslash \{0\})$ (for a space discontinuous flux $H(x,p)=H^l(p)1_{x<0}+H^r(p)1_{x>0}$), and (iii) satisfies natural continuity and scaling properties, is necessarily given by a germ condition at the junction: $\ddot{I}(t,0)\in \mathcal{G}$ a.e., where $\mathcal{G}$ is a maximal, $L^1-$dissipative and complete germ. In a symmetric way, we prove that any continuous semi-group on $L^{\infty }$ which is (i) $L^{\infty }-$contractive, (ii) satisfies with the Hamilton-Jacobi equation ${\partial }_{t}u+H(x,{\partial }_{x}u)=0$ in ${\mathbb{R}}_{+}\times (\mathbb{R}\backslash \{0\})$ (for a space discontinuous Hamiltonian $H$ as above), and (iii) satisfies natural continuity and scaling properties, is necessarily given by a flux limited solution of the Hamilton-Jacobi equation.