Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2411.12326 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866912126372675584 |
|---|---|
| author | Cardaliaguet, P |
| author_facet | Cardaliaguet, P |
| contents | We show that any continuous semi-group on $L^1$ which is (i) $L^1-$contractive, (ii) satisfies the conservation law $\partial_t ρ+\partial_x(H(x,ρ))=0$ 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: $ρ(t,0)\in \mathcal G$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_12326 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A note on contractive semi-groups on a 1:1 junction for scalar conservation laws and Hamilton-Jacobi equations Cardaliaguet, P Analysis of PDEs We show that any continuous semi-group on $L^1$ which is (i) $L^1-$contractive, (ii) satisfies the conservation law $\partial_t ρ+\partial_x(H(x,ρ))=0$ 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: $ρ(t,0)\in \mathcal G$ 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. |
| title | A note on contractive semi-groups on a 1:1 junction for scalar conservation laws and Hamilton-Jacobi equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2411.12326 |