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

We show that any continuous semi-group on $L^1$ which is (i) $L^1-$contractive, (ii) satisfies the conservation law $\partial_t \rho+\partial_x(H(x,\rho))=0$ in $\mathbb{R}_+\times (\mathbb{R}\backslash\{0\})$ (for a space discontinuous flux $H(x,p)= H^l(p) {\bf 1}_{x<0}+ H^r(p) {\bf 1}_{x>0}$), and (iii) satisfies natural continuity and scaling properties, is necessarily given by a germ condition at the junction: $\rho(t,0)\in \mathcal G$ 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.