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 ${\mathrm{\partial }}_t\rho +{\mathrm{\partial }}_x(H(x,\rho ))=0$ in ${\mathbb{R}}_{+}\times (\mathbb{R}\mathrm{\backslash }\{0\})$ (for a space discontinuous flux $H(x,p)=H^l(p){\mathbf{1}}_{x<0}+H^r(p){\mathbf{1}}_{x>0}$), 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 $L^{\mathrm{\infty }}$ which is (i) $L^{\mathrm{\infty }}-$contractive, (ii) satisfies with the Hamilton-Jacobi equation ${\mathrm{\partial }}_{t}u+H(x,{\mathrm{\partial }}_{x}u)=0$ in ${\mathbb{R}}_{+}\times (\mathbb{R}\mathrm{\backslash }\{0\})$ (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.